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Exploratory use of periodic pumping tests for hydraulic characterization of faults

Exploratory use of periodic pumping tests for hydraulic characterization of faults Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Geophysical Journal International Geophys. J. Int. (2018) 212, 543–565 doi: 10.1093/gji/ggx390 Advance Access publication 2017 September 16 GJI Seismology Exploratory use of periodic pumping tests for hydraulic characterization of faults Yan Cheng and Joerg Renner Institute for Geology, Mineralogy, and Geophysics, Ruhr-Universitat Bochum, D-44780 Bochum, Germany. E-mail: yan.cheng@rub.de Accepted 2017 September 15. Received 2017 September 8; in original form 2017 April 5 SUMMARY Periodic pumping tests were conducted using a double-packer probe placed at four different depth levels in borehole GDP-1 at Grimselpass, Central Swiss Alps, penetrating a hydrother- mally active fault. The tests had the general objective to explore the potential of periodic testing for hydraulic characterization of faults, representing inherently complex heterogeneous hy- draulic features that pose problems for conventional approaches. Site selection reflects the specific question regarding the value of this test type for quality control of hydraulic stimu- lations of potential geothermal reservoirs. The performed evaluation of amplitude ratio and phase shift between pressure and flow rate in the pumping interval employed analytical so- lutions for various flow regimes. In addition to the previously presented 1-D and radial-flow models, we extended the one for radial flow in a system of concentric shells with varying hydraulic properties and newly developed one for bilinear flow. In addition to these injectivity analyses, we pursued a vertical-interference analysis resting on observed amplitude ratio and phase shift between the periodic pressure signals above or below packers and in the interval by numerical modeling of the non-radial-flow situation. When relying on the same model the order of magnitude of transmissivity values derived from the analyses of periodic tests agrees with that gained from conventional hydraulic tests. The field campaign confirmed several advantages of the periodic testing, for example, reduced constraints on testing time relative to conventional tests since a periodic signal can easily be separated from changing background pressure by detrending and Fourier transformation. The discrepancies between aspects of the results from the periodic tests and the predictions of the considered simplified models indicate a hydraulically complex subsurface at the drill site that exhibits also hydromechanical features in accord with structural information gained from logging. The exploratory modeling of verti- cal injectivity shows its potential for analysing hydraulic anisotropy. Yet, more comprehensive modeling will be required to take full advantage of all the pressure records typically acquired when using a double-packer probe for periodic tests. Key words: Fracture and flow; Hydrothermal systems; Europe; Fourier analysis; Numerical modelling; Mechanics, theory, and modelling. constraining effective hydraulic properties (e.g. Rasmussen et al. 1 INTRODUCTION 2003; Renner & Messar 2006), diagnosing the flow regime (e.g. Analyses of pressure and/or flow transients associated with pump- Hollaender et al. 2002) and providing valuable information on sub- ing operations in wells, that is, injection or production, constitute surface heterogeneity (Renner & Messar 2006; Ahn & Horne 2010; the primary means for hydraulic characterization of the subsurface Fokker et al. 2012;Cardiff et al. 2013). The periodic signals can be (e.g. Fetter 2001). Effective hydraulic properties are determined re- easily separated from noise (Renner & Messar 2006; Bakhos et al. lying on models considering specific flow geometries and boundary 2014) or background variations associated with long-term operation conditions. Specific aspects of observations may guide the selec- (e.g. Guiltinan & Becker 2015). In addition, numerical modeling of tion of one out of a number of alternative models (e.g. Matthews & periodic tests in the frequency domain is much faster than that of Russell 1967; Bourdet et al. 1989). typical transients in the time domain (Cardiff et al. 2013; Fokker Periodic pumping tests involve consecutive periods of injection et al. 2013). and/or production resulting in alternating flow and pressure and As is the case for conventional testing, effective (or equivalent) promise several practical and analytical benefits. They may be seen hydraulic parameters may be derived from periodic pressure and as either an alternative or a complement to conventional testing for flow-rate observations by relying on a simple diffusion equation for The Authors 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 544 Y. Cheng and J. Renner a homogeneous and isotropic medium in the first step of analyses. Yet, their variation with pumping period can be attributed to— among other possible reasons—spatial heterogeneity (e.g. Renner & Messar 2006; Rabinovich et al. 2015). Renner & Messar (2006) performed comprehensive field tests and found that diffusivity val- ues estimated from the analysis of flow-rate and pressure responses at the pumping well increased with increasing period. In contrast, the diffusivity gained from analysis of pressure responses at the pump- ing and a monitoring well decreased with increasing period, as also found in field tests by Becker & Guiltinan (2010), Rabinovich et al. (2015) and Guiltinan & Becker (2015). The estimated diffusivity values of the two analyses agree at a critical period corresponding to a penetration depth on the order of the well distances. A similar phenomenon was observed in oscillatory hydraulic tests on rock samples (Song & Renner 2007). Fokker et al. (2013) successfully modeled the field observations using a heterogeneous subsurface suggesting that the disparity of hydraulic properties below the crit- ical period reflects effective hydraulic properties of different parts of the subsurface probed depending on period. The work described here aimed at assessing the use of periodic testing for in situ hydraulic characterization of fault structures on Figure 1. Sketch of the borehole site. The trajectory of borehole GDP-1 the scale of metres to decametres which is of considerable inter- is indicated by the orange line. The depth of the Transitgas AG Tunnel (trajectory in light blue) is 200 m; the zone of inflow is indicated in dark est in fundamental science and industrial applications. Previous blue. Estimated surface traces of shear zones and faults are represented by fundamental research covered the relation between fault zone ar- grey and dashed black lines, respectively. GBFZ: Grimsel Breccia Fault chitecture as constrained by structure investigations (e.g. Wibber- zone. GSZ 2: Grimsel Shear Zone. SBSZ: Southern Boundary Shear Zone. ley & Shimamoto 2003), hydraulic properties (Caine et al. 1996; Lopez ´ & Smith 1996) in cases derived from laboratory experi- 2.1 Pumping tests ments (e.g. Mitchell & Faulkner 2009), mineralization (Sheldon & Micklethwaite 2007) and earthquake-source mechanics (Sibson 2.1.1 Location et al. 1975; Bizzarri 2012). From a perspective of industrial ap- plications, the hydraulic characteristics of faults play an important The close to E–W striking Grimsel Breccia Fault zone (GBFZ, role for assessing the performance of a reservoir in general (e.g. Fig. 1) crops out at Grimselpass, Central Swiss Alps, with an al- Fisher & Knipe 2001) but also for stimulations. The term hydraulic titude of 2160 m above sea level. To the north and the south, two stimulation refers to an operation in wells initially developed by shear zones subparallel to GBFZ, namely Grimsel Shear Zone (GSZ the hydrocarbon industry (Economides & Nolte 2000; Shaoul et al. 2) and the Southern Boundary Shear Zone, crop out at distances 2007) but now also used for developing deep geothermal reservoirs of about 80 and 500 m, respectively (Fig. 1). The fault zone is (Baria et al. 1999; Schindler et al. 2008; Genter et al. 2010; Houwers hydrothermally active as evidenced by a 250 m wide zone of hy- et al. 2015; Garcia et al. 2016). drothermal discharge in the Transitgas AG Tunnel that runs in NS We present and analyse results from periodic pumping tests in direction at 200 m depth. The origin of the warm fluid is esti- a well penetrating a strike-slip fault considered an analogue for a mated at 4 ± 1 km depth with a reservoir temperature of at least heat exchanger as needed for economic recovery of petrothermal ◦ 100 C(Belgrano et al. 2016). Borehole GDP-1 (coordinates of energy. Our work addresses the need to develop testing protocols 669 464/157 025) with a radius of 0.048 m was drilled about 100 m and evaluation methods suitable for heterogeneous reservoirs, such west of the freshwater reservoir Totensee with a deviation of 24 as faulted rocks. Equivalent hydraulic parameters are estimated by from the vertical and an azimuth of 164 to intersect the GBFZ ap- comparing the relation between flow rate and pressure with a range proximately normal to its strike. The Transitgas AG tunnel passes of previously presented and newly derived analytical solutions. The by the borehole at about 180 m distance to the west. The deviated pressure record in the borehole section separated by packers is borehole was drilled to a final depth of 125.3 m using a thixotropic compared to that above or below the probe and the potential of polymer fluid. The true vertical depth is about 114.5 m. The bore- the found amplitude ratios and phase shifts for the derivation of hole is open except for the casing stabilizing the first 2.1 m at the hydraulic parameters is explored relying on a numerical model. top of the well (Fig. 2). 2 EXPERIMENTAL DETAILS AND DATA ANALYSIS 2.1.2 Equipment and procedures We use the term injectivity analysis when analysis involves flow The test system (Fig. 2) consisted of a reservoir filled with wa- rate and pressure in the actively tested section of the pumping well, ter from Totensee, a rotary pump and several magnetic-inductive −1 addressed as interval. Advanced double-packer tests provide the flow meters covering a range from 0.35 to 400 L min (accuracy pressure responses above and below the interval, too. We refer to the better than 0.5 per cent of full scale) at the surface. A variable- comparison of these various pressures measured in a single well as length double-packer probe with downhole pressure gauges (full vertical-interference test following the convention of addressing the scale 2 MPa, accuracy 0.1 per cent of full scale) was lowered into analyses of pressure responses in monitoring wells as interference theborehole. Thepackerswith1mlengthwereinflated fromtheir tests (Matthews & Russell 1967). initial diameter of 72 mm using nitrogen with pressures of up to Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 545 (e.g. constant head, constant flow, slug, pulse and recovery tests) were performed at all intervals, while periodic pumping tests were conducted at all depth sections but i4. Initial selection of oscillation period and flow rate was based on forward calculations assuming radial flow in a homogeneous medium with diffusivity values considered typical for a fractured formation. The programme was then adjusted on the basis of the first results such that ultimately tests were performed at periods τ between 60 and 1080 s (Table 2). A single period consisted of two injection phases, each with a duration of half of the period, with −1 nominal flow rates that differed by about 3 L min . By varying the −1 lower rate in a period between 0 and 7 L min , we also modified the average pressure at which periodic testing took place (Fig. 3). Ac- cording to the simple scaling relation z ∼ Dτ, the chosen periods correspond to penetration depths z of several decimetres to tens of metres for assumed values of hydraulic diffusivityD between 1E–5 2 −1 and 1E–2 m s . Thus, neither Totensee nor the tunnel are expected to constitute relevant boundary conditions for the exerted flow. 2.1.3 Undisturbed hydraulic heads We constrained the natural hydraulic heads from the pressure levels above, in, and below the interval asymptotically approached with time after packer inflation (Fig. 4). The recorded pressures suggest that the unperturbed water level of about 32 m below the surface, corresponding to about 700 L of water in the well, is actually the consequence of a dynamic situation with a prominent outflow at the borehole bottom, that is, at a true vertical depth of more than 110 m, and an inflow between about 73 and 110 m. Relative to the Figure 2. Sketch of the measurement setup. The flow rate recorded by the unperturbed pressure the pumping operations employed overpres- flow meter at the surface is denoted as Q . Pressure recorded above the surface sures of at most 300 kPa (Fig. 4). Pumping pressures reached the upper packer, in the interval and below the lower packer are represented as level corresponding to a water column filling the entire borehole p , p and p , respectively. abo int bel only for the interval i5. 2.5 MPa. Data were digitally recorded with a resolution of 16 bit 2.2 Fourier analysis and a time step of 1 or 2 s. Four depth levels were selected for the testing relying on the We performed two basic steps of data processing before Fourier preliminary results of borehole logging and inspection of drilled transformation. Recorded time-series were cut to lengths corre- cores (Table 1 and Appendix A). Tests with an interval length of sponding to integer multiples of the pumping period and pressure 9.4 m at all four depth levels (i1–i4) were complemented by one records were automatically detrended to avoid artefacts due to dif- test with a short interval of 1.7 m (i5) as part of i2. The latter ferences between pressure levels at start and end of a pumping se- isolated the prominent fault at a depth of ∼105 m with an orien- quence (Fig. 5). Transformation yield amplitude and phase spectra tation of 150/85 estimated from an optical log and core analyses from which we determined amplitude ratios and phase shifts be- (Appendix A). The orientation of this fracture is actually also repre- tween corrected flow rate (see Section 2.3.1) and interval pressure sentative of the rather steeply dipping fractures striking subparallel but also between interval pressure and pressure below and above to GBFZ that dominate in interval i1 and all of i2 (Table 1). Struc- the interval for the imposed period (see Table 2). The ratio between tural characterization of the main fault (i3) was not possible owing the amplitudes of flow rate and interval pressure corresponds to to extensive borehole breakout and core loss. The shallowest inter- the injectivity index determined from conventional tests (e.g. Lyons val at ∼45 m (i4) exhibits a slightly higher fracture density than i2 2010). and i3 mainly resulting from two perpendicular sets of E–W strik- To constrain the uncertainty of phase shift and amplitude ratio, ing fractures, of which one is steeply dipping. Conventional tests we performed a sliding-window analysis, that is, a window with a Table 1. Specifics of the test intervals. Int. Packer position Depth at centre of probe Fracture density Comment Lower Upper −1 (m) (m) (m) (1 m ) i1 121.8 112.4 117.1 7.6 Several small breccia i2 111.4 102.0 106.7 5.5 Prominent, isolated fault i3 86.6 77.2 81.9 – Main fault with breccia i4 49.6 40.2 44.9 8.8 Several fault strands i5 105.8 104.1 105.0 6.4 Prominent, isolated fault Severe breakouts and core loss, neither logging (see Appendix A) nor core analysis revealed structural information. Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 546 Y. Cheng and J. Renner Figure 3. Pressures (left y-axis) and flow rate (green, right y-axis) recorded during the periodic testing at a depth of about 105 m (i5) for oscillation periods of (a) 60 s (i5–1, i5–2, see Table 2) and (b) 180 and 540 s (i5–3, i5–4). length of three nominal periods was successively moved over the entire data set using a step size of 10–20 per cent of a period. For each window position, phase shift and amplitude ratio was determined and the standard deviation of all values is here reported as uncertainty. 2.3 Injectivity analysis 2.3.1 Correction for the storage capacity of the tubing Injectivity analysis requires the true flow rate into the rock that differs from the flow rate measured at the surface owing to the storage of fluid in the hydraulic system, composed of the tubing and the borehole section enclosed by the packers. For a closed system entirely filled with fluid, its capacity to store fluid stems from the finite fluid compressibility, the deformability of the tubing and of the pressurized rock section in the interval, and changes in packer seating due to changes in interval pressure. For an open system, one has to additionally account for a geometrical storage capacity that Table 2. Results of the Fourier analyses of recorded pressures and flow rate. Amplitude ratios and phase shifts are only reported when the amplitude spectrum confirmed that the imposed frequency is the dominant frequency in the signal. The results for tests that included sequences of zero-flow rate and therefore were manually shifted along the time axis are indicated by italic numbers. Int. Period Nominal flow rate Injectivity analysis Vertical-interference analysis Q –p Q –p p –p p –p cor int surface int bel int abo int Phase shift Amplitude ratio Amplitude ratio Phase shift Amplitude ratio Phase shift Amplitude ratio −1 3 −1 −1 3 −1 −1 (s) (L min ) (cycles) (m s Pa )(m s Pa ) (cycles) (–) (cycles) (–) i1 180 3/0 3.65 ± 0.03E−01 4.54 ± 0.26E−10 2.05 ± 0.02E−09 4.43 ± 0.31E−01 6.90 ± 0.59E−03 6.14 ± 0.35E−01 1.40 ± 0.23E−03 1.08 ± 0.05E−01 5.21 ± 0.12E−10 i2–1 180 10/7 9.89 ± 0.22E−02 4.06 ± 0.09E−09 5.23 ± 0.08E−09 7.71 ± 0.25E−01 2.01 ± 0.30E−01 – – i2–2 180 3/0 1.70 ± 0.03E−01 3.76 ± 0.05E−09 5.26 ± 0.01E−09 8.74 ± 0.05E−01 3.44 ± 0.09E−01 – – i3–1 180 7/4 2.03 ± 0.05E−01 2.86 ± 0.30E−09 4.23 ± 0.09E−09 8.62 ± 0.03E−01 1.22 ± 0.04E−02 – – i3–2 180 3/0 3.24 ± 0.05E−01 1.06 ± 0.03E−09 2.67 ± 0.02E−09 8.91 ± 0.03E−01 1.51 ± 0.03E−02 – – 2.65 ± 0.05E−01 9.88 ± 0.31E−10 i3–3 480 3/0 1.19 ± 0.02E−01 9.64 ± 0.10E−10 1.47 ± 0.02E−09 8.97 ± 0.04E−01 2.74 ± 0.07E−02 – – i3–5 1080 7/4 2.39 ± 0.08E−02 6.19 ± 0.20E−10 7.19 ± 0.23E−10 8.80 ± 0.10E−01 3.85 ± 0.17E−02 – – i5–1 60 3/0 3.80 ± 0.08E−01 5.19 ± 0.13E−09 9.43 ± 0.11E−09 9.46 ± 0.06E−01 1.53 ± 0.05E−01 – – 1.40 ± 0.01E−01 4.90 ± 0.03E−09 i5–2 60 10/7 1.02 ± 0.03E−01 6.60 ± 0.13E−09 1.04 ± 0.01E−08 9.16 ± 0.17E−01 2.24 ± 0.20E−01 – – i5–3 180 10/7 7.40 ± 0.17E−02 4.65 ± 0.05E−09 5.63 ± 0.04E−09 8.94 ± 0.28E−01 2.85 ± 0.46E−01 7.04 ± 0.77E−01 7.60 ± 3.40E−03 i5–4 540 10/7 8.40 ± 0.11E−02 3.21 ± 0.03E−09 3.50 ± 0.04E−09 2.62 ± 1.67E−02 2.70 ± 0.15E−01 7.15 ± 0.51E−01 3.50 ± 0.59E−03 Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 547 Figure 4. Equilibrium pressure of intervals (p ) and below probe (p ) int bel compared to pumping pressure in the interval and two hydrostatic water pressures for water levels at the surface (0 m) or at 32 m depth in the well. specifies how much fluid has to be added to/removed from the water column for a unit change in its height and thus in pressure it exerts. Figure 5. Comparison of (a) recorded and (b) detrended pressure during The used tubing with an inner diameter of 23.7 mm exhibits the periodic testing at a depth of about 105 m (i5–3). For detrending, the a storage capacity dominated by its geometrical component of average of the upper and lower envelopes is subtracted from the signal. −8 3 −1 S = A /ρg = 4.8 × 10 m Pa where A , ρ and g de- tube tube tube 2 −3 note cross-section of the tubes (m ), fluid density (kg m )and −2 gravitational acceleration (m s ), respectively. The contribution from the compressibility of water is at least three orders of mag- nitude smaller, as is the contribution from the deformation of the steel tubes with a thickness of almost 5 mm. We constrained the combined contribution of rock deformation and changes in packer seating to interval-storage capacity by a rapid pulse test for the long interval of 9.4 m length (corresponding to an interval vol- ume of about 60 L). The valve of the probe was briefly opened to pressurize the interval by the water column in the tubing. The change Figure 6. Interval pressure (blue, left y-axis) and flow rate (orange, right y-axis) recorded during a periodic pumping test with a period of 180 s at in volume was determined from the change in water-column height a depth of 115 m (i1, see Tables 1 and 2) exhibiting a delay between start in the tubing and the change in pressure was recorded by the interval (stop) of pumping, indicated by the steep rise (fall) in flow rate, and increase sensor. The ratio between these changes yields an interval-storage −10 3 −1 in pressure, indicated by the kinks. capacity of S (9.4m) = 6.0 × 10 m Pa , that is, almost two int orders of magnitude smaller than the geometrical storage capacity 2.3.2 Flow delay of the tubing (or 1.3 per cent). The shorter interval length of 1.7 m (corresponding to an interval volume of about 10 L) was not explic- All tests with a water level in the tube below surface level are poten- itly tested, but the reduced length should lead to a further reduction tially affected by the conventional approach of remotely measuring of the storage capacity roughly proportional to interval length, that flow rate at the surface and the associated time delay between flow −10 3 −1 is, S (1.7m)  1.1 × 10 m Pa . Thus, we neglected all con- int in the gauge and actual addition of fluid to the water column load- tributions to storage capacity but the geometrical one of the tubing ing the interval. Yet, when correcting flow records after eq. (1), when determining true flow rates into or out of the rock forma- tests employing a ‘zero’-flow rate, which we address as zero-flow −1 tion from the flow rate recorded at the surface, Q (L min ), surface rate tests below, yield suspicious delays of up to 10 s between flow according to rate at the surface and pressure in the interval exceeding (Fig. 6). These large delays presumably result from the actual time of flow Q = Q − S p˙ . (1) cor surface tube int in the partly empty tube. These data cannot be evaluated in a stan- The time derivative of the interval pressure, p˙ , was calculated dard way. We pursue two different approaches to derive information int using a Fourier transformation of the recorded pressure. on hydraulic properties from these tests. First, we determine the Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 548 Y. Cheng and J. Renner Shell model. Analytical solutions for the injectivity analysis con- sidering a shell model, known as multiregion composite model in reservoir engineering (Ambastha & Ramey 1992; Acosta & Ambastha 1994), are given in Appendix C. These solutions com- plement the work by Ahn & Horne (2010), who reported a semi- analytical solution for interference analysis of a multicomposite radial ring system. For the analysis of the field data, we focus on a single cylindrical shell concentrically surrounding the borehole and exhibiting hydraulic properties that differ from the medium farther away from the well (comparable to the situation envisioned for the so-called skin effect, Matthews & Russell 1967). The prominent feature of this model is a perturbation of the transition from zero phase shift to a phase shift of 1/4 in comparison to radial flow in a ho- mogeneous medium. The perturbations occur for values of ∼0.01–1 of the dimensionless parameter r ω/D ,where D is the hy- i shell shell draulic diffusivity of the shell, r the radius of the injection well and ω the angular frequency. This dimensionless parameter gives the Figure 7. Effect of interval length (as given in labels; radial flow corre- inverse of the ratio between the hydraulic penetration depth into the sponds to an infinitely long interval) on phase shift between flow rate and shell material and borehole radius. Prominent maxima, successions interval pressure according to the mirror-symmetric model described in Ap- of maxima and minima, or a focusing of the transition occur de- pendix B. Curves were calculated after eq. (B17) with the summation up to pending on the ratios between the hydraulic parameters representing n = 100. the shell and the surrounding medium (see e.g. Appendix C2). amplitude ratio of uncorrected flow rate, that is, surface-flow rate and interval pressure (Table 2) for all tests and compare their re- Bilinear-flow model. Bilinear flow occurs when fluid is drained lation to pumping characteristics with that of amplitude ratios of from or injected into a permeable matrix through an enclosed frac- the tests for which flow-rate correction was possible. Secondly, we ture of finite conductivity intersecting a well along its axis. A com- test whether manual shifting of flow records guided by the promi- bination of two approximately linear flow regimes may result, one nent kinks in pressure records provides reasonable estimates for the in the matrix with flow essentially perpendicular to the fracture and phase shift between flow rate and pressure and thus opens the way the other in the fracture itself associated with the non-negligible for an inversion towards hydraulic parameters. pressure drop or increase in it (Cinco-Ley et al. 1978; Cinco-Ley & Samaniego-V 1981; Ortiz et al. 2013). In Appendix D, we present an analytical solution of the coupled diffusion equations for bilinear 2.3.3 Flow regimes considered for estimation flow in a homogeneous medium subjected to periodic pumping. For of hydraulic properties infinite fracture length, the phase shift is bounded by asymptotes 2 −1 Equivalent or effective hydraulic properties, diffusivity D (m s ), to 1/16 and 1/8 of a cycle for large and small periods, respectively, 2 −1 transmissivity T (m s ) and storativity S (−), are typically es- with a smooth transition in-between (Fig. D2 in Appendix D). The timated relying on analytical solutions of the pressure-diffusion asymptotic value for long periods is consistent with the value de- equation for specific flow regimes, that is, the dimensional and di- rivedbyHollaender et al. (2002). In case of fractures with finite rectional characteristics of the flow pattern (e.g. Fetter 2001). Here, length, only the asymptote of 1/8 of a cycle for small periods holds, too, as expected because the pressure perturbation does not reach we considered: (i) 1-D flow, (ii) radial flow, (iii) radial flow in con- centric cylindrical shells and (iv) bilinear flow. Analytical solutions the fracture tip. For a constant-pressure boundary at the fracture tip, of the 1-D-flow and radial-flow models were previously derived for phase shift monotonically decreases to 0 with increasing period. various boundary conditions (Black & Kipp 1981; Rasmussen et al. The relation between phase shift and period is not monotonic for a 2003; Renner & Messar 2006). Below, we investigate the effect of no-flow boundary, but minima (and possibly also maxima exceed- finite interval length on flow in homogeneous isotropic medium and ing 1/8) occur at intermediate periods before 1/8 is asymptotically present solutions for the other scenarios. approached for very large periods when the pressure in the fracture is homogeneous and the total response is dominated by linear flow into the surrounding medium. Effect of interval length. The use of a double-packer probe poses a significant difference to previously performed periodic pumping tests mandating to investigate the consequences of a restricted ‘ac- 2.3.4 Flow-regime diagnosis from spectral analyses tive’ length of the formation. For this purpose, we considered a radially infinite, mirror-symmetric model vertically bounded by a For conventional tests, flow regimes can be inferred from diagnos- no-flow condition (see Appendix B). To allow for analytical treat- tic log–log plots of pressure or pressure derivative versus time (e.g. ment, we assumed pressure to evolve linearly along the packers and Renard et al. 2009). Analogously, slopes of amplitude spectra of to be constant above the upper (below the lower) packer. non-harmonic periodic tests allow for a diagnosis of flow regimes According to the model, the effects of interval length on the (Hollaender et al. 2002). In either type of diagram, slopes of 1/4, phase shift between flow rate and interval pressure are stronger the 1/2, 0 and 1 are indicative of bilinear flow, linear flow, radial flow larger the period, that is, the penetration depth (Fig. 7). Relying and pseudo-steady-state flow, respectively. Since our excitation is on the radial-flow model for the inversion of hydraulic diffusivity not truly harmonic, spectra of flow rate and pressure contain signif- from phase shift, diffusivity is the more overestimated the shorter icant amplitudes at periods shorter than the nominally excited one, the interval and the smaller the phase shift are. Small phase shifts too. Yet, simple division of entire flow-rate and pressure spectra correspond to large periods (small frequencies) for which a double- proved to be impractical (Fig. 8). Thus, we restricted to specific packer probe acts as a point source rather than a line source. local maxima of pressure and flow rate, as did Fokker et al. (2013). Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 549 Figure 8. Spectrum of the amplitude ratio modulus for the interval at 105 m tested with a period of 540 s (i5–4). Coloured lines represent exponents of 1/4, 1/2 and 1. The restriction to local maxima of pressure and flow rate in the frequency domain (red asterisks) reduces the scatter and suggests Figure 10. Phase shift and amplitude ratio of the interference response bilinear (slope 1/4) to linear flow (1/2). between pressure recorded (a) below the lower packer and (b) above the upper packer and in the interval calculated using the model described in Fig. 9 with different diffusivities and boundary conditions as specified in Table 3 (expressed by the numbers used as labels). The dots represent individual 2 −1 calculations labeled by the used diffusivity (m s ). The lines intended to help the eye to follow the trends. The red square with error bars represents the observations at a depth of 85 m for a period of 180 s (i5–3). cally using COMSOL. We determined values of phase shift and amplitude ratio of pressures measured at different locations of a double-packer probe considering different diffusivities and combi- nations of boundary conditions (Table 3). The effect of the inves- tigated scenarios (1–8) is subordinate for the interference response between below lower packer and in the interval (Fig. 10). The results for pressure above upper packer separate into two groups according to the top boundary conditions labeled as ‘no flow’ and ‘constant pressure’. Effective diffusivity values D or D are estimated from am- δ ϕ plitude ratio or phase shift, respectively, by linearly interpolating between the modeling results. Deviation of model results from ob- servations could be explained by several aspects. The model as- Figure 9. Illustration of the model used for numerical analysis of vertical sumes homogeneity and isotropy for the subsurface which may not interference. Blue colour indicates water-filled well sections and the two be true in reality. Furthermore, the model only considers relations packers are represented in grey. The sketch is not to scale: distance of the between pressures but a more comprehensive model should also lateral boundaries from the symmetry axis is 1000r ; distance between top integrate the injectivity analysis. Finally, hydromechanical effects and bottom boundary is 240 m, that is, the bottom boundary is more than were observed but not considered in the current model. 100 m away from the bottom of the borehole. 2.4 Vertical-interference analysis 3 RESULTS When placing a double-packer probe in an open borehole, the pump- 3.1 Injectivity analysis ing operations in the interval may also cause detectable pressure variations above or below the probe due to probe-parallel flow Amplitude ratios of data sets resulting from performing the flow- through the formation bearing additional information on its hy- rate correction after eq. (1) are systematically smaller than those draulic properties. To take advantage of these pressure records, we of uncorrected data (Fig. 11), but the relative relations among the analysed an axisymmetric model (Fig. 9) and treated it numeri- ratios for the different test intervals as well as their trends with the Table 3. Combinations of boundary conditions considered for the performed eight calculations employing the model sketched in Fig. 9. Boundary conditions 1 2 3 4 5 6 7 8 Top boundary No flow No flow No flow No flow Constant pressure Constant pressure Constant pressure Constant pressure Bottom boundary No flow Constant pressure Constant pressure No flow No flow Constant pressure Constant pressure No flow Lateral boundary Constant pressure Constant pressure No flow No flow Constant pressure Constant pressure No flow No flow Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 550 Y. Cheng and J. Renner systematically with increasing pumping period for i5 as well as for i3. 3.1.2 Effective hydraulic parameters Effective transmissivity is the least variable of all hydraulic parame- ters when evaluating the results of the Fourier analyses of corrected flow rate and interval pressure relying on the analytical solution for radial flow in a homogeneous medium (Table 4). It varies by about −6 2 −1 half an order of magnitude from 3 to 9 × 10 m s . Apart from one outlier (i3–5), the values for hydraulic diffusivity and storativ- −5 2 −1 ity stay within about one order of magnitude (4–40 × 10 m s −3 and 2–20 × 10 , respectively). The outlier may to some extent be explained by the finite length of the interval. The modeling of the length effect (see Appendix B) predicts the largest error for test i3– 5, with diffusivity potentially overestimated by about one order of 2 −1 magnitude (0.37 m s from conventional radial-flow model versus Figure 11. Injectivity quantified by the amplitude ratio of flow rate and 2 −1 0.04 m s from mirror-symmetric double-packer model). For the injection pressure as dependent on mean interval pressure. Amplitude ratios other tests, yielding phase shifts of at least 0.074 cycles, the error were calculated for all tests before (labeled uncorrected and represented by due to neglecting the finite interval length is negligible (Fig. 7). solid circles) and after (labeled corrected and represented by open circles) applying the correction to flow rate for the storage of the tubing. The colour Phase-shift values are particularly diagnostic for hydraulic coding represents the five investigated test intervals (see Table 1). Data boundaries. For example, phase shifts larger than 1/8 unequivo- points with uncorrected flow rate are also labeled by the imposed oscillation cally require bounded reservoirs (Fig. 12). The phase-shift value period in seconds. Vertical error bars indicate the uncertainty in amplitude for the test at a depth of 85 m and a period of 180 s (i3–2) remains ratio gained from the sliding-window fast Fourier transform (FFT) analyses. larger than the upper bound of 1/4 of all considered bounded models Horizontal error bars represent the range of pressures at which the tests were even after manual shifting. Only models with a no-flow boundary conducted. can explain the phase shift of the test i3–1 at 85 m and a period of 180 s, the largest phase shift observed for non-zero-flow rate tests parameters characterizing a test, mean pressure and period, remain (Fig. 12). However, the increase of amplitude ratios with phase shift qualitatively similar. We therefore use the larger set of uncorrected between i3–1 and the other test at the same depth for a period of data for the subsequent analysis and refer to this approach as sim- 1080 s (i3–5) cannot be explained by any of the no-flow models. plified injectivity analysis. Independent of interval length the phase-shift values for a depth of ∼105 m (i2, i5) are all smaller than 1/8 of a cycle, excluding the 1-D-flow model with a no-flow boundary or no boundary. The 3.1.1 Simplified injectivity analysis non-monotonically varying amplitude ratios for i5 also exclude a Amplitude ratios of uncorrected flow rates and interval pressures, constant pressure boundary or no boundary for all the flow types. here treated as tentative measures of injectivity, tend to increase For the interval with a depth of ∼115 m (i1), the only phase shift with increasing mean pressure but variations at a given pressure are lies between 1/16 and 1/8, which can be fit by all the models except significant (Fig. 11). A period of 180 s was chosen for more than 1-D flow with no-flow boundary or no boundary. half of the periodic pumping tests allowing us to compare the hy- At a depth of ∼105 m, three tests were performed with different draulic behaviour of the four tested intervals. The deepest interval periods using the short interval (60, 180 and 540 s correspond- with a depth of 115 m (i1) exhibits the lowest injectivity despite its ing to i5–2, i5–3, and i5–4 in Table 2). Since they all included association with the largest absolute mean pressure. The shallowest only non-zero flow rates their results are not affected by the prob- interval at 85 m depth (i3) shows a strong positive correlation be- lems associated with the finite distance between the location of tween injectivity and mean pressure (also found for a period of 60 s the flow meter and the top of the water column loading the inter- applied in i5) indicating that the dominant hydraulic conduits are val. The non-monotonic succession of phase-shift values, with the perceptible to hydromechanical effects. For a depth of 105 m (i2, lowest occurring for the intermediate period, can be modeled by a i5), we neither find a pronounced dependence of amplitude ratio shell with a thickness of about 6–15 times the borehole radius of −3 on mean pressure nor on interval length. Amplitude ratios decrease ∼0.05 m, and with a diffusivity between just below 10 to above Table 4. Hydraulic properties derived from the results of the Fourier analyses of flow rate and interval pressure relying on the analytical solution for radial flow in a homogeneous medium. Int. Period Nominal flow rate Depth Diffusivity Transmissivity Storativity −1 2 −1 2 −1 (s) (L min ) (m) (m s)(m s)(–) i1 180 3/0 117.1 1.3 ± 0.9E−05 2.7 ± 0.8E−07 2.7 ± 1.3E−02 i2–1 180 10/7 106.7 3.9 ± 0.9E−05 3.5 ± 0.3E−06 9.4 ± 1.5E−02 i3–3 480 3/0 81.9 5.0 ± 3.1E−07 1.8 ± 0.5E−07 4.4 ± 1.8E−01 i3–5 1080 7/4 81.9 3.7 ± 1.3E−01 5.1 ± 0.2E−06 1.5 ± 0.5E−05 i5–2 60 10/7 105.0 8.4 ± 3.4E−05 4.9 ± 0.8E−06 6.4 ± 1.5E−02 i5–3 180 10/7 105.0 3.9 ± 0.3E−04 9.0 ± 0.2E−06 2.3 ± 0.2E−02 i5–4 540 10/7 105.0 5.7 ± 0.6E−05 4.8 ± 0.1E−06 8.5 ± 0.6E−02 Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 551 Figure 12. Amplitude ratio versus phase shift for the injectivity analysis. Red, blue and black lines represent the solutions for radial flow, 1-D flow and bilinear flow, respectively. Solid lines are for infinite reservoirs and dashed and dashed–dotted lines represent constant pressure (pb) and no-flow (fb) boundary at distances of 20r , respectively, for radial and bilinear flows. Changing the assumed hydraulic parameters shifts the theoretical curves vertical without distortion. Filled and open symbols represent non-zero-flow-rate sequences and zero-flow-rate sequences for which manual shift was applied, respectively (see Table 2). Errors for the data points do not exceed symbol size. Two periods are also labeled for the convenience of discussion (see Table 4). Table 5. Parameters of the shell model (see Appendix C) derived flow is that the pressure at the fracture tip has risen to a substantial from the phase-shift values observed for the tests i5–3 and i5–4 (see fraction of the associated pressure increase in the well at the end of also Fig. 13). bilinear flow and the flow field transforms towards formation linear flow. The transition from pseudo-steady-state flow to bilinear flow is k /k D /D r /r Diffusivity (D ) 1 2 1 2 1 i 1 2 −1 consistent with a succession of the domination of wellbore storage (–) (–)(–)(m s ) at early time by bilinear flow (finite conductivity fracture). 6.7 8 15 8.1E−04 7.1 10 15 7.4E−04 6.3 10 10 7.1E−03 5.9 9 6.1 2.3E−02 3.3 Vertical-interference analysis 5.6 9 6.1 2.1E−02 The periodic excitations in the injection intervals lead to detectable periodic responses in the pressures recorded below the lower and −2 2 −1 10 m s (Table 5) exceeding that for the enclosing medium by above the upper packer (Table 2). The vertical-interference analy- a factor between about 5 and 10 (Fig. 13). sis based on these pressure records has the advantage that all tests can be analysed even those involving the problematic zero-flow-rate sequences. Yet, the magnitude of interference response in a single 3.2 Flow regime diagnosis from spectral analyses borehole critically depends on the storage capacity of the borehole Flow-regime identification from full spectra proved difficult (Fig. 8), section in which it is determined. Above the upper packer we had not the least because the spectra of flow rate and pressure contain lit- a free, unconstrained water column corresponding to a ‘large’ stor- tle significant contributions for high frequencies. We thus restricted age capacity determined by the cross-section of the borehole. The to the first two odd multiples of the test frequency that correspond borehole section below the lower packer, in contrast, constitutes an to pronounced local maxima in the spectra. As the analysis is sen- enclosed fluid volume with a ‘small’ storage capacity determined sitive to the timing of flow-rate changes (Hollaender et al. 2002), by the fluid compressibility and the deformability of packer and we only analyse non-zero-flow-rate sequences. The results for over- borehole wall. In agreement with these qualitative storage consid- tones from the various tests in a specific interval agree closely with erations, the interference response is much larger below the probe those for nominally excited periods (Fig. 14). The slope for the than above the probe where it is in fact detectable only in some interval at 105 m (i5) increases from less than 1/4 and approaches cases (Table 2). All amplitude ratios derived from pressures below 1/2 as period increases. In contrast, the slope for interval i3 de- the lower packer, but that for the deepest test interval (i1), follow creases from around 1 to a value just above 1/4. At face value, these the expected inverse correlation with the length of the enclosed results indicate a transition from bilinear to linear flow for i5 and borehole section (Fig. 15): the larger the enclosed volume the larger a transition from pseudo-steady-state flow to bilinear flow for i3. the storage capacity and the less sensitive the section, that is, the The physical interpretation for a transition from bilinear to linear amplitude ratio between interference response and excitation Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 552 Y. Cheng and J. Renner Figure 14. Amplitude ratio in frequency domain for the deepest interval (i5) represented by asterisks and the interval at depth 81.9 m (i3) represented by circles at the main period of the pumping tests and the first overtone, that is, a third of the imposed period. The dotted lines represent exponents of 1/4, 1/2 and 1. Figure 13. Examples phase-shift–frequency relations for the shell model calculated using eq. (C2) in relation to observations for the periodic tests at 105 m for a period of 180 s (i5–3) and for a period of 540 s (i5–4). The labels give the ratios between the diffusivity of the shell and the surrounding medium and the lateral extension of the shell in multiples of the borehole radius. decreases with the length of the enclosed section. Relative to the general trend, the response below the probe observed for the test in the deepest interval is weak demonstrating exceptional hydraulic properties near the borehole bottom. Including phase shift in the comparison, observations for the interference with the borehole section above the interval are only in Figure 15. Correlation between interference responses, quantified by the agreement with model results for a ‘no-flow’ condition (Fig. 10). amplitude ratio of pressure oscillations, and the length of the enclosed The effective diffusivities estimated from phase shift (denoted D section below the lower packer of the probe. We present the amplitude ratio in Table 6) tend to be larger than the ones estimated from amplitude between the pressure above the interval and in the interval in this graph, too, though it should not be controlled by the length of the enclosed section ratio (D ) for i2 and i5, the long and short intervals at 105 m. below the probe. The joint presentation of results is chosen to highlight the The reverse seems to hold for the shallowest interval i3. Effective difference in sensitivity of vertical response between the enclosed section diffusivity values using p –p data or consistently on the order of a bel int below the probe and the unconstrained water column above it. 2 −1 few m s for the long and the short interval (i2 and i5) at the same depth, except for two relatively small values (i2–1, i5–4). Those derived from the conventional methods and periodic radial-flow estimated by p –p analysis are smaller than those estimated by abo int analysis are of similar order of magnitude (Table 4). Flow regimes p –p for the short interval i5 without considering the extremely bel int were constrained using diagnostic log–log plots of pressure and small D value for i5–4. A decreasing trend of diffusivity with pressure derivative versus elapsed time. Intervals i1 und i3 indicate increasing period is indicated for i5. bilinear flow and interval i2 and i5 display features of linear flow. 3.4 Conventional methods 4 DISCUSSION Conventional tests yield the smallest transmissivity and storativity The variable differences between equilibrium pressure for a cer- for the shallowest interval (i4, Table 7). Disregarding the excep- tain depth interval and a hydrostatic gradient indicate heterogeneity tionally low value found for interval i4, effective transmissivity is of the hydraulic system. The fractured aquifer is dominated by the least variable hydraulic property, consistent with the observa- steep conduits that appear to have limited interconnectivity despite tions from periodic radial-flow analysis. The hydraulic parameters their small lateral distance (at most a few metres in the case of the Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 553 Table 6. Effective diffusivity D estimated by amplitude ratio and D by phase shift of vertical-interference analysis. δ ϕ Top and bottom boundaries are set as no flow. 2 −1 2 −1 Int. Period Nominal flow rate D (m s ) D (m s ) δ ϕ −1 (s) (L min ) p –p p –p p –p p –p bel int abo int bel int abo int i1 180 3/0 5E−35E−33E−34E−3 i2–1 180 10/7 1 – 3E−2– i2–2 180 3/0 4–2– i3–1 180 7/4 2E−2–3E−1– i3–2 180 3/0 3E−2–4E−1– i3–3 480 3/0 3E−2–3E−1– i3–5 1080 7/4 2E−2–1E−1– i5–1 60 3/0 4–8– i5–2 60 10/7 7–6– i5–3 180 10/7 5 9E−233E−2 i5–4 540 10/7 3 1E−23E−47E−3 Table 7. Hydraulic properties derived from conventional tests. 2 −1 2 −1 a Int. Test phase Diffusivity (m s ) Transmissivity (m s ) Storativity (–) Remarks i1 Shut-in <1.14E−04 <1.6E−06 1.4E−02 RSLA i2 Constant flow rate <1.39E−04 <4.3E−06 3.1E−02 SLA i3 Shut-in <2.29E−04 <1.9E−06 8.3E−03 RSLA i4 Shut-in <1.45E−03 <2.9E−08 2.0E−05 RSLA i5 Shut-in <3.18E−05 <2.7E−06 8.5E−02 RSLA SLA: ‘Straight Line Analysis’ after Jacob & Lohman (1952). RSLA: ‘Recovery Straight Line Analysis’ after Agarwal (1980). borehole’s bottom section). The diffusivity variations with pumping for the shallowest of the tested intervals (i3), diffusivity values periods suggest spatial heterogeneity or complexity of the conduits derived from periodic testing deviate significantly from the one in a specific interval, too, that is, simple radial flow is a poor ap- gained from conventional testing. The lower diffusivity value results proximation of the flow excited during the tests. The prominent from a test sequence with zero flow rate (i3–3) and thus has a role of discrete hydraulic conduits is documented by the similarity limited reliability. The higher diffusivity value corresponds to test of injectivity values for the two tests at a borehole depth of about i3–5 giving a phase shift so low that it falls within the range for 105 m (i2 and i5) despite their differences in interval length. In the which the finite length of the probe cannot be neglected, but the following, we discuss to what extent the employed methods yield flow field has a significant axial component (Fig. 7). The estimate comparable values for hydraulic parameters, comment on the con- based on the mirror-symmetric double-packer model is more than straints for the pressure dependence of the hydraulic response, and one order of magnitude smaller than the value assuming simple finally speculate on the flow-regime model that is most consistent radial flow. Furthermore, the analyses of periodic tests demonstrates with the entirety of observations in a specific interval. a dependence of the effective hydraulic parameters on period and mean pressure. Thus, differences between the methods likely reflect the simplifications inherent in the radial-flow model that are not justified for the investigated structures necessitating to investigate 4.1 Comparison of methods more complex flow models. Periodic pumping tests can be easily implemented using field equip- Vertical-interference analysis is a valuable by-product of peri- ment for conventional testing. The excitation signals do not have to odic testing when packer-separated intervals are investigated and be perfectly harmonic but Fourier analysis of non-harmonic signals pressure is monitored at several depth levels. We refer to its re- sults as effective axial diffusivity to distinguish them from results might actually provide information regarding periods shorter than of the injectivity analysis and conventional methods, addressed as the imposed main period (spectral analyses, e.g. Hollaender et al. effective radial diffusivity. The attributes ‘axial’ and ‘radial’ are 2002; Fokker et al. 2013). The periodic excitation can be applied chosen from the perspective of the pumping well, addressing flow even when the hydraulic pressure is not equilibrated, in fact it can dominantly subparallel and normal to well axis, respectively. Ef- be superposed to any transient, be it associated with a terminated or fective axial diffusivities are here found to be generally larger than ongoing pumping operation. The superposition is valid as long as effective radial diffusivities (Fig. 16). The axial diffusivities scatter linearity can be assumed, that is, the diffusion equation in its basic significantly for the intervals i2 and i5 at similar mean depths but form holds. If it does not hold, the results of conventional meth- with different probe lengths. The scatter is larger for the shorter ods become questionable, too. As a consequence of the freedom to interval i5 tested for a larger spread in periods and apart from the start periodic testing at any time, operational time is reduced and result for the longest period of 540 s, communication to the deeper to a larger degree plannable than for conventional tests that involve borehole section is characterized by larger diffusivity values than long waiting times for equilibration of apriori unknown duration. to the shallower sections. The vertical response to shallower depth During the current test campaign, a total of 11 hr was spent on sections is actually lost with the increase in probe length suggest- conventional tests and 7 hr for periodic tests. ing that faults outside the short interval but inside the long interval Our results show that the analyses of periodic pumping provide are hydraulically connected to the prominent fault isolated by the effective hydraulic parameters comparable to those gained from short interval. The decrease in diffusivity with increasing period or conventional methods when assuming radial flow (Fig. 16). Only Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 554 Y. Cheng and J. Renner geological situation, an increase in injectivity with mean pressure likely results from the pressure-induced increase in effective hy- draulic aperture of the joints and faults intersecting the borehole. The more pronounced pressure dependence of the short interval i5 compared to the long interval i2 supports the notion that the pressure dependence reflects hydromechanical behaviour of discrete faults. Observations for i3 (∼85 m depth) with a relatively large period of 180 s suggest that the hydromechanical effects are not restricted to the very intersection between faults and the well. The significance of hydromechanical effects is also evidenced by the pronounced reverse pressure response observed below the lower packer when injecting in i1 (not documented here). 4.3 Constraints on flow regime Details of our observations are obviously at conflict with simple radial flow. From periodic tests, flow regimes can be constrained by (i) investigating spectra of non-harmonic signals (Fig. 14)orby (ii) comparing the variation patterns of amplitude ratio and phase shift with theoretical flow models (Fig. 12). The few constraints we derived from the analysis of spectra yield flow regimes identical to those indicated by the conventional methods. The periodic injectiv- ity analysis tends to indicate more complex flow regimes than the conventional methods (Table 8) which we interpret as evidence for the superior resolution of the periodic approach. Effective injectivity values determined relying on simple radial flow (Fig. 11) systematically decrease with increasing pumping pe- Figure 16. (a) Radial diffusivity determined from periodic injectivity anal- riod for intervals i3 and i5 (factor of 3–5 for less than one order ysis and conventional methods. (b) Axial diffusivity determined by vertical- of magnitude in period). The conventional scaling relation for dif- interference analysis. Labels refer to the effective diffusivity values esti- fusion processes (e.g. Weir 1999) suggests that the covered range mated from amplitude ratio (Ddel: D ) and phase shift (Dphi: D )using δ ϕ in period corresponds to a change of half an order of magnitude in pressure records above (pabo), in (pint), and below (pbel) the interval. nominal penetration depth. Thus, the period dependence of injec- tivity indicates a change in hydraulic characteristics with distance penetration depth (Table 6) also indicates that the axial connectivity from the borehole. When addressing the period dependence by the is a local phenomenon. shell model for the deepest interval (i5), we find a decrease in effective diffusivity and permeability by about one order of mag- nitude beyond a zone of about several decimetres to one metre 4.2 Pressure dependence of hydraulic properties (Fig. 13). This change in hydraulic characteristics could be related The tested intervals exhibit a general trend of increasing injectivity to drilling-associated damage of the borehole wall or remnants of with increasing pressure, with the prominent exception of the deep- the thixotropic polymer fluid used during drilling. The flushing of est interval (i1) that also has the lowest injectivity. In the current the well during logging preceding the hydraulic testing may have Table 8. Constraints on flow regimes gained from injectivity analysis, spectral analysis, or conventional methods: possible (o), excluded (x), no constraint (–) for radial (R), bilinear (BL) and 1-D (1-D) flow. Periodic pumping method Conventional methods Injectivity analysis Spectral analysis (see Fig. 14) No flow Constant pressure No boundary No boundary No boundary i1 R o o o – x 1-D x o x – x BL o o o – o i2 R o o o – x 1-D x o x – o BL x x x – x i3 R x x x x x 1-D x x x x x BL x x x o o i5 R o x x x x 1-D x x x o o BL o x x o x Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 555 well resulted in a cleansing of only the hydraulic conduits nearest approach, addressed as vertical-interference analysis. We demon- to the borehole. strated that the analyses of the periodic pumping tests yield results For the short interval i5, the conventional method and the spectral comparable to conventional methods when relying on the same analysis (Fig. 14) suggest a linear flow regime while injectivity model. The field campaign revealed, however, several methodolog- analysis indicates radial or bilinear flow with a no-flow boundary. ical advantages of the periodic testing, for example, its plannable For the longer interval at the same depth (i2), the injectivity analysis testing time and the ease in separating the response to the actual suggests radial flow with any boundary condition or linear flow pumping operation from noise and transients gained from data pro- with constant pressure boundary. Assuming the two intervals to cessing in frequency space. Our analytical and numerical model- be governed by the same boundary condition, the only regime in ing of periodic testing extends the suite of evaluation tools and common by i5 and i2 is radial flow with a no-flow boundary. specifically allowed us to decipher more details of the encountered For the deepest interval (i1), the injectivity is the lowest of all complex flow geometry and further conduit characteristics than the intervals though it was tested at the highest mean injection pres- conventional methods. sure. The low equilibrium pressure below i1 (Fig. 4) suggests that The two approaches, the previously employed injectivity anal- the section below i1 is disconnected from the ones above. In addi- ysis and the vertical-interference analysis, are complementary by tion, higher axial diffusivity than radial diffusivity (Fig. 16) implies providing information on the hydraulic properties in different di- axially oriented or subvertical upflow in the sections above i1. rections relative to the well axis. The variations in effective ‘axial’ and ‘radial’ diffusivity with pumping periods and mean pumping pressure indicate significant spatial heterogeneity of conduits that 4.4 Synopsis also exhibit hydromechanical effects for the investigated site. While we showed the potential of the vertical-interference analysis, clearly The periodic testing revealed that the flow geometry in the pene- more work is needed in the future regarding the modeling of records trated subsurface is quite variable on the metre (lateral) to decametre available when using double- or multipacker probes. (axial) scale and that the conduit system exhibits hydromechanical effects, two characteristics that appear quite reasonable for a strike- slip fault in crystalline rock. Yet, at this point, it is difficult to distin- ACKNOWLEDGEMENTS guish between a situation characterized by a lateral no-flow bound- ary, possibly related to relics of the polymer drilling fluid with high The financial support by BfE, Switzerland is gratefully acknowl- viscosity or drilling-associated near-welbore damage, and a truly edged as is the coordinator, Marco Herwegh, for involving us in anisotropic hydraulic system with the high axial diffusivity due to project NFP70-P1. Field tests were performed together with Sacha subvertically oriented fluid pathways. The latter notion is not only Reinhardt and Markus Bosshard (SolExperts), and Daniel Eggli supported by the logging observations (Appendix A) that document (Uni Bern) who is particularly thanked for generously sharing the a dominance of steeply dipping fractures intersecting the well, but results of the structural core and log analyses. The presented treat- is also consistent with the conduit geometry deduced by Belgrano ment of the shell model builds on an unpublished analyses of Eugen et al. (2016), who suggested that the fault zone is controlled by Petkau. We are grateful to the three anonymous reviewers for their localized subvertically oriented ‘pipe’-like upflow zones. The verti- valuable comments. cal variability observed for the undisturbed hydraulic heads and the low hydraulic transmissivity of the deepest test interval compared to the major fault zone support this suggestion derived solely from REFERENCES structural investigations. 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Int., 202, 1613–1626. https://pangea.stanford.edu/ERE/db/WGC/papers/WGC/2015/12072.pdf, Weir, G.J., 1999. Single-phase flow regimes in a discrete fracture model, last accessed 6 October 2017. Water Resour. Res., 35, 65–73. Jacob, C.E. & Lohman, S.W., 1952. Nonsteady flow to a well of constant Wibberley, C.A.J. & Shimamoto, T., 2003. Internal structure and perme- drawdown in an extensive aquifer, Trans. Am. Geophys. Union, 33, 559– ability of major strike-slip fault zones: the Median Tectonic Line in Mie 569. Prefecture, Southwest Japan, J. Struct. Geol., 25, 59–78. Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 557 APPENDIX A: STRUCTURAL INFORMATION FOR INTERVALS Figure A1. Images of the borehole wall for the five intervals as gained from logging with an optical camera. For i2 (i5), a section of about 1 m length is also represented by a photograph of the recovered cores. APPENDIX B: MIRROR-SYMMETRIC DOUBLE-PACKER MODEL For an isotropic and homogenous subsurface, the governing diffusion equation reads in cylindrical coordinates: 2 2 ∂ p(r, z, t) 1 ∂p(r, z, t) ∂ p(r, z, t) 1 ∂p(r, z, t) + + = (B1) 2 2 ∂r r ∂r ∂z D ∂t where r is radial distance from the centre of the borehole and z denotes the vertical direction (upright positive). Employing separation of variables, that is, p(r, z, t) = P(r, z)(t) = P(r, z)exp(i ωt), the spatial variation of pressure obeys 2 2 ∂ P(r, z) 1 ∂ P(r, z) ∂ P(r, z) i ω + + = P(r, z). (B2) 2 2 ∂r r ∂r ∂z D We consider a double-packer probe enclosing an interval with a total height of 2h in a well with radius r . Instead of modeling two packers, 0 i we assume mirror symmetry to the horizontal plane bisecting the interval (Fig. B1) and thus we can restrict to treating the upper half of the Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 558 Y. Cheng and J. Renner Figure B1. Model geometry for the analytical solution of a double-packer test. The model has mirror symmetry with respect to the horizontal plane that bisects the interval. model space. The locations of the bottom and top of the packer are z = h and z = h , respectively, and the aquifer extends vertically to z = b 0 p (Fig. B1). We adapt boundary conditions from Rehbinder (1996) and Mathias & Butler (2007): ∂ P(r, z) BC1: = 0 z = b no-flow condition for vertical bounds of model ∂z BC2: P(r, z) = 0 r →∞ radially infinite reservoir T ∂ P(r, z) BC3: Q = 4πr q(r , z)dz with q(r , z) =− | 0 ≤ z ≤ h integrating flux q over the injection interval has to i i i 0 2ρgh ∂r r 0 i yield the constant flow rate Q BC4: P(r , z) = p h ≤ z ≤ b average pressure above packer (neglecting effect of fluid column) i 1 p BC5: P(r , z) = p 0 ≤ z ≤ h average pressure in injection interval (neglecting effect of fluid column) i 2 0 h − z P(r , z) = (p − p ) + p h ≤ z BC6: ≤ h linear pressure change along packer section i 2 1 1 0 p h − h p 0 where the pressures in the well above and in the interval are denoted p and p , respectively. 1 2 After Fourier cosine transforming of z,thatis, 2 nπ P (r) = P(r, z)cos(a z)dz, a = (B3) n n n b b and accounting for BC1, eq. (B2) becomes ∂ P (r) 1 ∂ P (r) i ω n n + − a + P (r) = 0 (B4) ∂r r ∂r D with a general solution in the form of i ω P (r) = A K (η r) + B I (η r),η = a + (B5) n n 0 n n 0 n n where K and I denote modified Bessel functions of the first and second kind of zero order. Since I →∞ when r →∞ (BC2), we have to 0 0 0 require B = 0, that is, P (r) = A K (η r). (B6) n n 0 n The backtransform reads A K (η r) 0 0 0 P (r, z) = + A K (η r)cos (a z). (B7) n 0 n n n=1 Application of BC4–BC6 gives p 0 ≤ z ≤ h ⎪ 2 0 A K (η r ) 0 0 0 i h −z + A K (η r )cos (a z) = (p − p )( ) + p h ≤ z ≤ h (B8) n 0 n i n 2 1 1 0 p h −h p 0 2 ⎪ n=1 ⎩ p h ≤ z ≤ b 1 p Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 559 which upon Fourier-cosine transformation of z yields b b A K (η r ) 0 0 0 i cos (a z) dz + A K (η r ) cos (a z) cos (a z) dz m n 0 n i n m 0 0 n=1 =0 h h b 0 p h − z = p cos (a z) dz + (p − p ) + p cos (a z) dz + p cos (a z) dz. (B9) 2 m 2 1 1 m 1 m h − h p 0 0 h h 0 p For a − a = 0, the second integrand on the left-hand side can be modified as n m cos [(a − a )z] + cos [(a + a )z] n m n m cos (a z) cos (a z) = n m and thus z z cos [(a − a )z] + cos [(a + a )z] sin (n − n)π sin (n + n)π n m n m b b dz = +  = 0, 2 2(a − a ) 2(a + a ) n m n m that is, all elements of the sum vanish but the one for which a − a = 0. In this case, the left-hand side of eq. (B9) can be algebraically n m manipulated as A K (η r ) cos (a z) dz = A K (η r ) . (B10) m 0 m i m m 0 m i After integration, the right-hand side sums up to ( ) cos a h ( ) cos a h cos a h m p cos a h m p m 0 m 0 p   − p   − p   −   . (B11) 2 2 1 2 2 2 2 h − h a h − h a h − h a h − h a p 0 p 0 p 0 p 0 m m m m With both, left- and right-hand sides, eqs (B10) and (B11), respectively, we can solve for the unknown coefficients cos (a h ) cos a h cos (a h ) cos a h m 0 m p m 0 m p p   −   − p   − 2 1 2 2 2 2 h − h a h − h a h − h a h − h a p 0 p 0 p 0 p 0 m m m m A = . (B12) K (η r ) 0 m i To derive A , we apply Darcy’s law to eq. (B7) and obtain the flow rate according to (BC3), that is, T A η K (η r ) sin (a h ) 0 0 1 0 i n 0 Q = 2πr h + A η K (η r ) . (B13) i 0 n n 1 n i ρgh 2 a 0 n n=1 and thus Qρg η K (η r ) sin (a h ) n 1 n i n 0 A = − 2 A . (B14) 0 n πr T η K (η r ) η K (η r ) a h i 0 1 0 i 0 1 0 i n 0 n=1 Finally, the pressure function eq. (B7) reads ∞ ∞ Qρg K (η r) K (η r) sin (a h ) 0 0 0 0 n 0 P (r, z) = − A η K (η r ) + A K (η r)cos (a z) (B15) n n 1 n i n 0 n n 2πr T η K (η r ) η K (η r ) a h i 0 1 0 i 0 1 0 i n 0 n=1 n=1 2(p − p )[cos(a h ) − cos(a h )] 2 1 n 0 n p where A = . bK (η r )(h − h )a 0 n i p 0 For the injection interval, with P(r , z < h ) = p , we now find a relation between the ratios of flow rate and interval pressure on the one i 0 2 hand and between interval pressure and pressure outside the interval on the other hand: 1 ∞ 1 − cos (a h ) − cos a h Q 2πr T η K (η r ) n 0 n p η K (η r ) η K (η r )K (η r ) i 0 1 0 i p n 1 n i 0 0 n i 1 0 i = + 2   sin (a h ) − cos (a z) . (B16) n 0 n p ρg K (η r ) K (η r )a h a K (η r ) b h − h 2 0 0 i p 0 0 n i 0 n 0 0 i n=1 It can be seen that solution consists of two parts: standard radial flow for infinite line source and an additional term depending on packer positioning. When the pressure outside of the interval remains unaltered by the pumping operation, that is, p = p = const., we can—without 1 0 loss of generality—chose the reference pressure to p = 0. In this case, eq. (B16) simplifies to cos (a h ) − cos a h Q 2πr T η K (η r ) 2 η K (η r ) η K (η r )K (η r ) i 0 1 0 i n 0 n p n 1 n i 0 0 n i 1 0 i = +   sin (a h ) − cos (a z) , (B17) n 0 n p ρg K (η r ) K (η r )a h a K (η r ) b h − h 2 0 0 i p 0 0 n i 0 n 0 0 i n=1 an analytical expression for the ratio of flow rate and interval pressure. Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 560 Y. Cheng and J. Renner Figure C1. Geometry of radial flow from a well into N concentric shells and the surrounding medium. APPENDIX C: CONCENTRIC CYLINDRICAL SHELLS SURROUNDING THE BOREHOLE C1 Analytical solution We consider radial flow from a borehole into and beyond a single concentric cylindrical shell located between r , the radius of the injection well and r (Fig. C1). The diffusion equation can be written as: ∂ p(r, t) 1 ∂p(r, t) 1 ∂p(r, t) + = . (C1) ∂r r ∂r D ∂t Here, p is fluid pressure, t is time, r is radial distance and D is hydraulic diffusivity. The hydraulic properties of the shell (subscript 1) and the surrounding medium (subscript 2) are described by permeability k = 1,2 μT /(ρgh) and diffusivity D = k /(s μ) = T /S where μ denotes the fluid viscosity, T , S and s transmissivity, storativity 1,2 1,2 1,2 1,2 1,2 1,2 1,2 1,2 1,2 and specific storage capacity of the two media, respectively, g gravitational acceleration and h the length of the injection interval. Solving the diffusion equation for periodic pressure variations in the well by separation of variables (compare Renner Messar 2006) and account for boundary conditions p (r ≥ r , t) = 0 r →∞ radially infinite reservoir 2 1 p (r, t) = p (r, t) r = r continuity of pressure at outer shell radius 1 2 1 ∂ p (r, t) ∂ p (r, t) 1 2 continuity of flow rate at outer shell radius T = T r = r 1 2 1 ∂r ∂r yields amplitude ratio and phase shift between flow rate and interval pressure of T η (−m K (η r ) + I (η r )) 1 1 1 1 1 i 1 1 i δ = 2πr , (C2) qp i ρg m K (η r ) + I (η r ) 1 0 1 i 0 1 i and η (−m K (η r ) + I (η r )) 1 1 1 1 i 1 1 i ϕ = arg , (C3) qp m K (η r ) + I (η r ) 1 0 1 i 0 1 i respectively. The constant m is related to the ratios of permeability (or equivalently transmissivity) and diffusivity according to I (η r ) K (η r ) + I (η r ) K (η r ) 1 1 1 0 2 1 0 1 1 1 2 1 m =  , (C4) K (η r ) K (η r ) − K (η r ) K (η r ) 1 1 1 0 2 1 0 1 1 1 2 1 where I and I represent the modified Bessel functions of the first kind of zero and first order and K and K the modified Bessel functions 0 1 0 1 of the second kind of zero and first order. The arguments of these Bessel functions contain the complex parameters η = i ω/D that 1,2 1,2 depend on angular frequency ω and diffusivity of the two media. The analytical solution for N concentric cylindrical shells (Fig. C1) is given as ( ( ) ( )) T η −c K η r + I η r 1 1 1 1 1 i 1 1 i δ = 2πr , (C5) qp i ρg c K (η r ) + I (η r ) 1 0 1 i 0 1 i and η (−c K (η r ) + I (η r )) 1 1 1 1 i 1 1 i ϕ = arg , (C6) qp ( ) ( ) c K η r + I η r 1 0 1 i 0 1 i Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 561 Figure C2. Results of the shell model given with phase shift as a function of dimensionless frequency (a) for various radius ratios and indicated fixed permeability (transmissivity) and diffusivity ratios, (b) for various diffusivity ratios and indicated fixed permeability (transmissivity) and radius ratios, (c) for various permeability (transmissivity) ratios when storativity is covaried such that the diffusivity of shell and medium are identical and (d) for various identical diffusivity and permeability (transmissivity) ratios that correspond to an identical storativity for shell and medium at indicated fixed radius ratio. For identical permeability (transmissivity) of shell and medium, the diffusivity ratio is related to an inverse ratio of specific storage capacity (storativity), that is, D /D = s /s = S /S . 1 2 2 1 2 1 and j = 1, . . . N−1. For N = 1, c = m but for N > 1, to obtain c one has to successively determine c , c ... and c from 1 1 1 2 3 N k N N +1 K (η r )I (η r ) + I (η r )K (η r ) 1 N +1 N 0 N N 1 N N 0 N +1 N N +1 c =− k D N +1 N K (η r )K (η r ) − K (η r )K (η r ) 1 N +1 N 0 N N 1 N N 0 N +1 N k D N N +1 k D j +1 j I (η r ) c K (η r ) − I (η r ) + I (η r ) c K (η r ) + I (η r ) 0 j j j +1 1 j +1 j 1 j +1 j 1 j j j +1 0 j +1 j 0 j +1 j k D j j +1 c =− k D j +1 j K (η r ) c K (η r ) − I (η r ) − K (η r ) c K (η r ) + I (η r ) 0 j j j +1 1 j +1 j 1 j +1 j 1 j j j +1 0 j +1 j 0 j +1 j k D j j +1 C2 Parameter study For the parameter study, we restrict to the simple case of a single shell. We investigated the sensitivity of phase shift to the various model parameters by systematic parameter variation. Phase shift is reported as a function of r ω/D where D = D denotes the i shell shell 1 hydraulic diffusivity of the shell (Fig. C1). This expression can be understood either as a dimensionless frequency or an inverse dimensionless penetration depth given in multiples of the borehole radius. A sigmoidal shape of the relation between phase shift and dimensionless frequency is characteristic for a homogeneous medium. Small phase shifts occur at low frequencies and vice versa with a rather steep switch between the two limits around a dimensionless frequency of 1. This shape is significantly perturbed when ratios between the parameters of shell and Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 562 Y. Cheng and J. Renner medium between one-tenth and a factor of 10 are allowed. The high-frequency limit of 1/8 for phase shift is not strictly valid for the compound model but values as high as 1/5 are found. The phase shift for a shell with a low storage property compared to the medium exhibits a pronounced maximum that forms a shoulder on the steep increase characteristic for the homogeneous medium (Fig. C2a). The location of the maximum moves towards lower frequencies with increasing radius of the shell. A variation of the ratio of storage parameters for shell and medium for a fixed shell radius results in qualitatively similar phase-shift curves as the variation in shell radius (Fig. C2b). Yet, the maximum in phase transforms into a minimum preceded by a new maximum at lower frequency when the storage capacity in the shell switches from lower to higher than the storage capacity in the medium. When permeability (transmissivity) and specific storage capacity (storativity) are modified in the same way, that is, diffusivity is kept constant, the transition from the low- to the high-frequency regime is significantly perturbed. Low permeability and storativity in the shell relative to the surrounding medium leads to a transition that resembles a Heavyside function (Fig. C2c). High permeability and storativity in the shell gives rise to a pronounced maximum in phase shift at a dimensionless frequency of about 0.1 that reaches values as large as 1/5, that is, well above the limit for a homogeneous medium of 1/8. Covarying peremability and diffusivity and keeping the storage parameter uniform for shell and medium has qualitatively similar effects (Fig. C2d). APPENDIX D: BILINEAR FLOW MODEL The model consists of a slit with distinct hydraulic properties embedded in a homogeneous matrix and assumes that the direct volume flow from the source into the matrix along the x-direction is negligible (Fig. D1), but the fracture boundary is considered a harmonic source for flow into the matrix. The governing equations in the matrix and the fracture correspond to diffusion equations: ∂ p (x , y, t) = D ∇ p (x , y, t) (D1) m m ∂t ∂ p (x , t) ∂ p (x , t) 2D ∂ p (x , y, t) f f m m = D + (D2) ∂t ∂x δ ∂y |y|=δ/2 where p (x, y, t) is the pressure in the matrix, p (x, t) the pressure in the fracture, δ the width of the fracture and D and D the diffusivity of m f m f matrix and fracture, respectively. 2 2 As the direct volume flow from the source into the matrix along the x-direction is negligible, ∂ p (x, y, t)/∂x ≈ 0, so that eq. (D1) reduces to ∂ p (x , y, t) ∂ p (x , y, t) m m = D . (D3) ∂t ∂ y The pressure in the matrix, that is, the solution of eq. (D1), is given by the standard solution for diffusion from a harmonic source into a homogeneous semi-infinite medium (Carslaw & Jaeger 1986): (|y| − δ/2) exp − 4D (t − λ) | | m y − δ/2 p (x , y, t) = √ p (x,λ) dλ (D4) m f 3/2 2 π D (t − λ) m 0 Figure D1. Model for bilinear flow: the fracture boundary is a source for flow into the matrix in y-direction; no direct volume flow from the source into the matrix along the x-direction. Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 563 Figure D2. Phase shift for (a) constant-pressure boundary and (b) no-flow boundary at the tip of a fracture with finite length L and width δ = 0.096 m. (I) 2 −1 2 −1 2 −1 2 −1 2 −1 2 −1 D = 1m s , D = 10 m s and L = 20δ, (II) D = 1m s , D = 100 m s and L = 20δ, (III) D = 1m s , D = 100 m s and L = 100δ and m f m f m f 2 −1 2 −1 (IV) D = 1m s , D = 10 m s and L = 4δ. The relation for an infinite fracture is shown for comparison in (a). m f whose insertion into eq. (D2) yields ⎡ ⎤ (|y| − δ/2) exp − ⎢ ⎥ 4D (t − λ) ⎢ m ⎥ ∂ p (x , t) 2D ∂ |y| − δ/2 ∂ p (x , t) f m f ⎢ ⎥ D + √ p (x,λ) dλ = . (D5) f f ⎢ ⎥ 2 3/2 ∂x δ ∂y 2 π D ( ) ∂t 0 t − λ ⎣ m ⎦ |y|−δ/2=0 Differentiation of the second term by part restricting to the section y ≥ 0 because of the symmetry of the problem gives ⎡ % ⎤ (|y| − δ 2) exp − √ ⎢ ⎥ 2 t 4D (t − λ) ⎢ ⎥ (y − δ 2) m ∂ p (x , t) D ∂ p (x , t) f m f ⎢ ⎥ D + √ p (x,λ) 1 + dλ = . (D6) f f ⎢ 3/2 ⎥ ∂x πδ 2D (t − λ) ∂t (t − λ) 0 m ⎣ ⎦ y=δ 2 Applying y = δ/2weget 2 t ∂ p (x , t) D p (x,λ) ∂ p (x , t) f m f f D + √ dλ = (D7) ∂x πδ ∂t (t − λ) Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 564 Y. Cheng and J. Renner and thus for harmonic pressure variations in the fracture ∂ p (x) D A exp(i ωλ) f m D A exp(i ωt) + √ p (x) dλ = i ωA exp(i ωt)p (x). (D8) f f f ∂x πδ 2 (t − λ) The integral solution is A exp(i ωλ) dλ = γ (−0.5, i ωt) exp(i ωt) = [ (−0.5) − (−0.5, i ωt)] i ω exp(i ωt)(D9) (t − λ) where γ (−0.5, i ωt), (−0.5) and (−0.5, i ωt) represent the lower incomplete gamma function, the gamma function and the upper incomplete gamma function, respectively. For times longer than 1.25 times the period, the deviation of γ (−0.5, i ωt) from (−0.5) is less than 1 per cent and beyond 2.2 times the period the deviation falls below 0.5 per cent. Thus, we can approximate the integral as √ √ exp (i ωλ) dλ (−0.5) i ω exp (i ωt) =−2 iπω exp (i ωt) (D10) 3/2 (t − λ) to arrive at ∂ p (x) 2 D − iD ω p (x) = i ω p (x) . (D11) f m f f ∂x δ when canceling the harmonic time dependence. Eq. (D11) is a homogeneous second-order differential equation with the solution 1/2 & ' p (x) = c exp − √ 2 iπω+i ω x + dH (L) (D12) δD π subject to the boundary conditions p (x)→ 0as x →∞ (infinite fracture), or p (x) = p as x = L (finite fracture with constant-pressure f f 0 boundary), or ∂ p (x)/∂x = 0as x = L (finite fracture with no-flow boundary) where c is a constant, H(L) is a parameter defined by 0 L →∞ H (L) = (D13) 1 L finite and 1/2 & ' ⎪ D p − c exp − √ 2 iπω + i ω L for constant-pressure boundary ⎪ 0 δD π d =   (D14) √ √ 1/2 1/2 ⎪ & ' & ' √ √ ⎪ D D m m 2 iπω + i ω cx exp − 2 iπω + i ω L + const. for no-flow boundary √ √ δD π δD π f f To simplify the calculation, we assume the constant appearing for the no-flow boundary to equal zero without loss of generality because it represents a constant shift in pressure that can be accommodated by scale shifting. Using Darcy’s law, we get the flow rate as T ∂ p (x , t) Q (x , t) = 2πr hq (x , t) =−2πr f i i ρg ∂x 1/2 & ' TcA D = 2πr √ 2 iπω + i ω ρg δD π √   √ 1/2 1/2 & ' & ' √ √ D D m m · exp − √ 2 iπω+i ω x − H exp − √ 2 iπω + i ω L exp (i ωt) (D15) bc δD π δD π f f where T is transmissivity, ρ is fluid density and h is the height of the injection zone. We introduce a parameter reflecting the boundary condition at the fracture tip as 0 constant pressure boundary H = . (D16) bc 1 no flow boundary Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 565 Finally, the complex injectivity results to 1/2 & ' ( ) Q x , t Tc D f m = 2πr √ 2 iπω + i ω p (x , t) ρg δD π f f √ √ 1/2 1/2 & ' & ' √ √ D D m m exp − √ 2 iπω + i ω x − H exp − √ 2 iπω + i ω L bc δD π δD π f f ·   . (D17) 1/2 & ' c exp − √ 2 iπω + i ω x + dH (L) δD π Specifically, for an infinite fracture, we find √ √ 1/2 Q (x = 0, t) T 2 i π D ω + i ω D f m m = 2πr √ (D18) p (x = 0, t) ρg πδD f f H(L)=0 with amplitude ratio and phase shift of √ √ 1/2 T 2 π D ω + ω D m m δ = 2πr (D19) qp i ρg δD and ϕ = arg 2 iπω + i ω , (D20) qp respectively, at the injection point x = 0(Fig. D2a). For a fracture with finite length and a constant-pressure boundary at its tip and assuming p = 0, we find Q (x = 0, t) p (x = 0, t) Q (x = 0, t) f f H(L)=0 =   (D21) 1/2 & ' p (x = 0, t) 1 − exp − √ 2 iπω + i ω L δD π at x = 0(Fig. D2a). Analogously, we find 1/2 & ' Q (x = 0, t) Q (x = 0, t) D f f m = 1 − exp − √ 2 iπω + i ω L (D22) p (x = 0, t) p (x = 0, t) δD π f f f H(L)=0 at x = 0(Fig. D2b). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Geophysical Journal International Oxford University Press

Exploratory use of periodic pumping tests for hydraulic characterization of faults

Geophysical Journal International , Volume 212 (1) – Jan 1, 2018

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Copyright © 2022 The Royal Astronomical Society
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1365-246X
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10.1093/gji/ggx390
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Abstract

Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Geophysical Journal International Geophys. J. Int. (2018) 212, 543–565 doi: 10.1093/gji/ggx390 Advance Access publication 2017 September 16 GJI Seismology Exploratory use of periodic pumping tests for hydraulic characterization of faults Yan Cheng and Joerg Renner Institute for Geology, Mineralogy, and Geophysics, Ruhr-Universitat Bochum, D-44780 Bochum, Germany. E-mail: yan.cheng@rub.de Accepted 2017 September 15. Received 2017 September 8; in original form 2017 April 5 SUMMARY Periodic pumping tests were conducted using a double-packer probe placed at four different depth levels in borehole GDP-1 at Grimselpass, Central Swiss Alps, penetrating a hydrother- mally active fault. The tests had the general objective to explore the potential of periodic testing for hydraulic characterization of faults, representing inherently complex heterogeneous hy- draulic features that pose problems for conventional approaches. Site selection reflects the specific question regarding the value of this test type for quality control of hydraulic stimu- lations of potential geothermal reservoirs. The performed evaluation of amplitude ratio and phase shift between pressure and flow rate in the pumping interval employed analytical so- lutions for various flow regimes. In addition to the previously presented 1-D and radial-flow models, we extended the one for radial flow in a system of concentric shells with varying hydraulic properties and newly developed one for bilinear flow. In addition to these injectivity analyses, we pursued a vertical-interference analysis resting on observed amplitude ratio and phase shift between the periodic pressure signals above or below packers and in the interval by numerical modeling of the non-radial-flow situation. When relying on the same model the order of magnitude of transmissivity values derived from the analyses of periodic tests agrees with that gained from conventional hydraulic tests. The field campaign confirmed several advantages of the periodic testing, for example, reduced constraints on testing time relative to conventional tests since a periodic signal can easily be separated from changing background pressure by detrending and Fourier transformation. The discrepancies between aspects of the results from the periodic tests and the predictions of the considered simplified models indicate a hydraulically complex subsurface at the drill site that exhibits also hydromechanical features in accord with structural information gained from logging. The exploratory modeling of verti- cal injectivity shows its potential for analysing hydraulic anisotropy. Yet, more comprehensive modeling will be required to take full advantage of all the pressure records typically acquired when using a double-packer probe for periodic tests. Key words: Fracture and flow; Hydrothermal systems; Europe; Fourier analysis; Numerical modelling; Mechanics, theory, and modelling. constraining effective hydraulic properties (e.g. Rasmussen et al. 1 INTRODUCTION 2003; Renner & Messar 2006), diagnosing the flow regime (e.g. Analyses of pressure and/or flow transients associated with pump- Hollaender et al. 2002) and providing valuable information on sub- ing operations in wells, that is, injection or production, constitute surface heterogeneity (Renner & Messar 2006; Ahn & Horne 2010; the primary means for hydraulic characterization of the subsurface Fokker et al. 2012;Cardiff et al. 2013). The periodic signals can be (e.g. Fetter 2001). Effective hydraulic properties are determined re- easily separated from noise (Renner & Messar 2006; Bakhos et al. lying on models considering specific flow geometries and boundary 2014) or background variations associated with long-term operation conditions. Specific aspects of observations may guide the selec- (e.g. Guiltinan & Becker 2015). In addition, numerical modeling of tion of one out of a number of alternative models (e.g. Matthews & periodic tests in the frequency domain is much faster than that of Russell 1967; Bourdet et al. 1989). typical transients in the time domain (Cardiff et al. 2013; Fokker Periodic pumping tests involve consecutive periods of injection et al. 2013). and/or production resulting in alternating flow and pressure and As is the case for conventional testing, effective (or equivalent) promise several practical and analytical benefits. They may be seen hydraulic parameters may be derived from periodic pressure and as either an alternative or a complement to conventional testing for flow-rate observations by relying on a simple diffusion equation for The Authors 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 544 Y. Cheng and J. Renner a homogeneous and isotropic medium in the first step of analyses. Yet, their variation with pumping period can be attributed to— among other possible reasons—spatial heterogeneity (e.g. Renner & Messar 2006; Rabinovich et al. 2015). Renner & Messar (2006) performed comprehensive field tests and found that diffusivity val- ues estimated from the analysis of flow-rate and pressure responses at the pumping well increased with increasing period. In contrast, the diffusivity gained from analysis of pressure responses at the pump- ing and a monitoring well decreased with increasing period, as also found in field tests by Becker & Guiltinan (2010), Rabinovich et al. (2015) and Guiltinan & Becker (2015). The estimated diffusivity values of the two analyses agree at a critical period corresponding to a penetration depth on the order of the well distances. A similar phenomenon was observed in oscillatory hydraulic tests on rock samples (Song & Renner 2007). Fokker et al. (2013) successfully modeled the field observations using a heterogeneous subsurface suggesting that the disparity of hydraulic properties below the crit- ical period reflects effective hydraulic properties of different parts of the subsurface probed depending on period. The work described here aimed at assessing the use of periodic testing for in situ hydraulic characterization of fault structures on Figure 1. Sketch of the borehole site. The trajectory of borehole GDP-1 the scale of metres to decametres which is of considerable inter- is indicated by the orange line. The depth of the Transitgas AG Tunnel (trajectory in light blue) is 200 m; the zone of inflow is indicated in dark est in fundamental science and industrial applications. Previous blue. Estimated surface traces of shear zones and faults are represented by fundamental research covered the relation between fault zone ar- grey and dashed black lines, respectively. GBFZ: Grimsel Breccia Fault chitecture as constrained by structure investigations (e.g. Wibber- zone. GSZ 2: Grimsel Shear Zone. SBSZ: Southern Boundary Shear Zone. ley & Shimamoto 2003), hydraulic properties (Caine et al. 1996; Lopez ´ & Smith 1996) in cases derived from laboratory experi- 2.1 Pumping tests ments (e.g. Mitchell & Faulkner 2009), mineralization (Sheldon & Micklethwaite 2007) and earthquake-source mechanics (Sibson 2.1.1 Location et al. 1975; Bizzarri 2012). From a perspective of industrial ap- plications, the hydraulic characteristics of faults play an important The close to E–W striking Grimsel Breccia Fault zone (GBFZ, role for assessing the performance of a reservoir in general (e.g. Fig. 1) crops out at Grimselpass, Central Swiss Alps, with an al- Fisher & Knipe 2001) but also for stimulations. The term hydraulic titude of 2160 m above sea level. To the north and the south, two stimulation refers to an operation in wells initially developed by shear zones subparallel to GBFZ, namely Grimsel Shear Zone (GSZ the hydrocarbon industry (Economides & Nolte 2000; Shaoul et al. 2) and the Southern Boundary Shear Zone, crop out at distances 2007) but now also used for developing deep geothermal reservoirs of about 80 and 500 m, respectively (Fig. 1). The fault zone is (Baria et al. 1999; Schindler et al. 2008; Genter et al. 2010; Houwers hydrothermally active as evidenced by a 250 m wide zone of hy- et al. 2015; Garcia et al. 2016). drothermal discharge in the Transitgas AG Tunnel that runs in NS We present and analyse results from periodic pumping tests in direction at 200 m depth. The origin of the warm fluid is esti- a well penetrating a strike-slip fault considered an analogue for a mated at 4 ± 1 km depth with a reservoir temperature of at least heat exchanger as needed for economic recovery of petrothermal ◦ 100 C(Belgrano et al. 2016). Borehole GDP-1 (coordinates of energy. Our work addresses the need to develop testing protocols 669 464/157 025) with a radius of 0.048 m was drilled about 100 m and evaluation methods suitable for heterogeneous reservoirs, such west of the freshwater reservoir Totensee with a deviation of 24 as faulted rocks. Equivalent hydraulic parameters are estimated by from the vertical and an azimuth of 164 to intersect the GBFZ ap- comparing the relation between flow rate and pressure with a range proximately normal to its strike. The Transitgas AG tunnel passes of previously presented and newly derived analytical solutions. The by the borehole at about 180 m distance to the west. The deviated pressure record in the borehole section separated by packers is borehole was drilled to a final depth of 125.3 m using a thixotropic compared to that above or below the probe and the potential of polymer fluid. The true vertical depth is about 114.5 m. The bore- the found amplitude ratios and phase shifts for the derivation of hole is open except for the casing stabilizing the first 2.1 m at the hydraulic parameters is explored relying on a numerical model. top of the well (Fig. 2). 2 EXPERIMENTAL DETAILS AND DATA ANALYSIS 2.1.2 Equipment and procedures We use the term injectivity analysis when analysis involves flow The test system (Fig. 2) consisted of a reservoir filled with wa- rate and pressure in the actively tested section of the pumping well, ter from Totensee, a rotary pump and several magnetic-inductive −1 addressed as interval. Advanced double-packer tests provide the flow meters covering a range from 0.35 to 400 L min (accuracy pressure responses above and below the interval, too. We refer to the better than 0.5 per cent of full scale) at the surface. A variable- comparison of these various pressures measured in a single well as length double-packer probe with downhole pressure gauges (full vertical-interference test following the convention of addressing the scale 2 MPa, accuracy 0.1 per cent of full scale) was lowered into analyses of pressure responses in monitoring wells as interference theborehole. Thepackerswith1mlengthwereinflated fromtheir tests (Matthews & Russell 1967). initial diameter of 72 mm using nitrogen with pressures of up to Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 545 (e.g. constant head, constant flow, slug, pulse and recovery tests) were performed at all intervals, while periodic pumping tests were conducted at all depth sections but i4. Initial selection of oscillation period and flow rate was based on forward calculations assuming radial flow in a homogeneous medium with diffusivity values considered typical for a fractured formation. The programme was then adjusted on the basis of the first results such that ultimately tests were performed at periods τ between 60 and 1080 s (Table 2). A single period consisted of two injection phases, each with a duration of half of the period, with −1 nominal flow rates that differed by about 3 L min . By varying the −1 lower rate in a period between 0 and 7 L min , we also modified the average pressure at which periodic testing took place (Fig. 3). Ac- cording to the simple scaling relation z ∼ Dτ, the chosen periods correspond to penetration depths z of several decimetres to tens of metres for assumed values of hydraulic diffusivityD between 1E–5 2 −1 and 1E–2 m s . Thus, neither Totensee nor the tunnel are expected to constitute relevant boundary conditions for the exerted flow. 2.1.3 Undisturbed hydraulic heads We constrained the natural hydraulic heads from the pressure levels above, in, and below the interval asymptotically approached with time after packer inflation (Fig. 4). The recorded pressures suggest that the unperturbed water level of about 32 m below the surface, corresponding to about 700 L of water in the well, is actually the consequence of a dynamic situation with a prominent outflow at the borehole bottom, that is, at a true vertical depth of more than 110 m, and an inflow between about 73 and 110 m. Relative to the Figure 2. Sketch of the measurement setup. The flow rate recorded by the unperturbed pressure the pumping operations employed overpres- flow meter at the surface is denoted as Q . Pressure recorded above the surface sures of at most 300 kPa (Fig. 4). Pumping pressures reached the upper packer, in the interval and below the lower packer are represented as level corresponding to a water column filling the entire borehole p , p and p , respectively. abo int bel only for the interval i5. 2.5 MPa. Data were digitally recorded with a resolution of 16 bit 2.2 Fourier analysis and a time step of 1 or 2 s. Four depth levels were selected for the testing relying on the We performed two basic steps of data processing before Fourier preliminary results of borehole logging and inspection of drilled transformation. Recorded time-series were cut to lengths corre- cores (Table 1 and Appendix A). Tests with an interval length of sponding to integer multiples of the pumping period and pressure 9.4 m at all four depth levels (i1–i4) were complemented by one records were automatically detrended to avoid artefacts due to dif- test with a short interval of 1.7 m (i5) as part of i2. The latter ferences between pressure levels at start and end of a pumping se- isolated the prominent fault at a depth of ∼105 m with an orien- quence (Fig. 5). Transformation yield amplitude and phase spectra tation of 150/85 estimated from an optical log and core analyses from which we determined amplitude ratios and phase shifts be- (Appendix A). The orientation of this fracture is actually also repre- tween corrected flow rate (see Section 2.3.1) and interval pressure sentative of the rather steeply dipping fractures striking subparallel but also between interval pressure and pressure below and above to GBFZ that dominate in interval i1 and all of i2 (Table 1). Struc- the interval for the imposed period (see Table 2). The ratio between tural characterization of the main fault (i3) was not possible owing the amplitudes of flow rate and interval pressure corresponds to to extensive borehole breakout and core loss. The shallowest inter- the injectivity index determined from conventional tests (e.g. Lyons val at ∼45 m (i4) exhibits a slightly higher fracture density than i2 2010). and i3 mainly resulting from two perpendicular sets of E–W strik- To constrain the uncertainty of phase shift and amplitude ratio, ing fractures, of which one is steeply dipping. Conventional tests we performed a sliding-window analysis, that is, a window with a Table 1. Specifics of the test intervals. Int. Packer position Depth at centre of probe Fracture density Comment Lower Upper −1 (m) (m) (m) (1 m ) i1 121.8 112.4 117.1 7.6 Several small breccia i2 111.4 102.0 106.7 5.5 Prominent, isolated fault i3 86.6 77.2 81.9 – Main fault with breccia i4 49.6 40.2 44.9 8.8 Several fault strands i5 105.8 104.1 105.0 6.4 Prominent, isolated fault Severe breakouts and core loss, neither logging (see Appendix A) nor core analysis revealed structural information. Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 546 Y. Cheng and J. Renner Figure 3. Pressures (left y-axis) and flow rate (green, right y-axis) recorded during the periodic testing at a depth of about 105 m (i5) for oscillation periods of (a) 60 s (i5–1, i5–2, see Table 2) and (b) 180 and 540 s (i5–3, i5–4). length of three nominal periods was successively moved over the entire data set using a step size of 10–20 per cent of a period. For each window position, phase shift and amplitude ratio was determined and the standard deviation of all values is here reported as uncertainty. 2.3 Injectivity analysis 2.3.1 Correction for the storage capacity of the tubing Injectivity analysis requires the true flow rate into the rock that differs from the flow rate measured at the surface owing to the storage of fluid in the hydraulic system, composed of the tubing and the borehole section enclosed by the packers. For a closed system entirely filled with fluid, its capacity to store fluid stems from the finite fluid compressibility, the deformability of the tubing and of the pressurized rock section in the interval, and changes in packer seating due to changes in interval pressure. For an open system, one has to additionally account for a geometrical storage capacity that Table 2. Results of the Fourier analyses of recorded pressures and flow rate. Amplitude ratios and phase shifts are only reported when the amplitude spectrum confirmed that the imposed frequency is the dominant frequency in the signal. The results for tests that included sequences of zero-flow rate and therefore were manually shifted along the time axis are indicated by italic numbers. Int. Period Nominal flow rate Injectivity analysis Vertical-interference analysis Q –p Q –p p –p p –p cor int surface int bel int abo int Phase shift Amplitude ratio Amplitude ratio Phase shift Amplitude ratio Phase shift Amplitude ratio −1 3 −1 −1 3 −1 −1 (s) (L min ) (cycles) (m s Pa )(m s Pa ) (cycles) (–) (cycles) (–) i1 180 3/0 3.65 ± 0.03E−01 4.54 ± 0.26E−10 2.05 ± 0.02E−09 4.43 ± 0.31E−01 6.90 ± 0.59E−03 6.14 ± 0.35E−01 1.40 ± 0.23E−03 1.08 ± 0.05E−01 5.21 ± 0.12E−10 i2–1 180 10/7 9.89 ± 0.22E−02 4.06 ± 0.09E−09 5.23 ± 0.08E−09 7.71 ± 0.25E−01 2.01 ± 0.30E−01 – – i2–2 180 3/0 1.70 ± 0.03E−01 3.76 ± 0.05E−09 5.26 ± 0.01E−09 8.74 ± 0.05E−01 3.44 ± 0.09E−01 – – i3–1 180 7/4 2.03 ± 0.05E−01 2.86 ± 0.30E−09 4.23 ± 0.09E−09 8.62 ± 0.03E−01 1.22 ± 0.04E−02 – – i3–2 180 3/0 3.24 ± 0.05E−01 1.06 ± 0.03E−09 2.67 ± 0.02E−09 8.91 ± 0.03E−01 1.51 ± 0.03E−02 – – 2.65 ± 0.05E−01 9.88 ± 0.31E−10 i3–3 480 3/0 1.19 ± 0.02E−01 9.64 ± 0.10E−10 1.47 ± 0.02E−09 8.97 ± 0.04E−01 2.74 ± 0.07E−02 – – i3–5 1080 7/4 2.39 ± 0.08E−02 6.19 ± 0.20E−10 7.19 ± 0.23E−10 8.80 ± 0.10E−01 3.85 ± 0.17E−02 – – i5–1 60 3/0 3.80 ± 0.08E−01 5.19 ± 0.13E−09 9.43 ± 0.11E−09 9.46 ± 0.06E−01 1.53 ± 0.05E−01 – – 1.40 ± 0.01E−01 4.90 ± 0.03E−09 i5–2 60 10/7 1.02 ± 0.03E−01 6.60 ± 0.13E−09 1.04 ± 0.01E−08 9.16 ± 0.17E−01 2.24 ± 0.20E−01 – – i5–3 180 10/7 7.40 ± 0.17E−02 4.65 ± 0.05E−09 5.63 ± 0.04E−09 8.94 ± 0.28E−01 2.85 ± 0.46E−01 7.04 ± 0.77E−01 7.60 ± 3.40E−03 i5–4 540 10/7 8.40 ± 0.11E−02 3.21 ± 0.03E−09 3.50 ± 0.04E−09 2.62 ± 1.67E−02 2.70 ± 0.15E−01 7.15 ± 0.51E−01 3.50 ± 0.59E−03 Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 547 Figure 4. Equilibrium pressure of intervals (p ) and below probe (p ) int bel compared to pumping pressure in the interval and two hydrostatic water pressures for water levels at the surface (0 m) or at 32 m depth in the well. specifies how much fluid has to be added to/removed from the water column for a unit change in its height and thus in pressure it exerts. Figure 5. Comparison of (a) recorded and (b) detrended pressure during The used tubing with an inner diameter of 23.7 mm exhibits the periodic testing at a depth of about 105 m (i5–3). For detrending, the a storage capacity dominated by its geometrical component of average of the upper and lower envelopes is subtracted from the signal. −8 3 −1 S = A /ρg = 4.8 × 10 m Pa where A , ρ and g de- tube tube tube 2 −3 note cross-section of the tubes (m ), fluid density (kg m )and −2 gravitational acceleration (m s ), respectively. The contribution from the compressibility of water is at least three orders of mag- nitude smaller, as is the contribution from the deformation of the steel tubes with a thickness of almost 5 mm. We constrained the combined contribution of rock deformation and changes in packer seating to interval-storage capacity by a rapid pulse test for the long interval of 9.4 m length (corresponding to an interval vol- ume of about 60 L). The valve of the probe was briefly opened to pressurize the interval by the water column in the tubing. The change Figure 6. Interval pressure (blue, left y-axis) and flow rate (orange, right y-axis) recorded during a periodic pumping test with a period of 180 s at in volume was determined from the change in water-column height a depth of 115 m (i1, see Tables 1 and 2) exhibiting a delay between start in the tubing and the change in pressure was recorded by the interval (stop) of pumping, indicated by the steep rise (fall) in flow rate, and increase sensor. The ratio between these changes yields an interval-storage −10 3 −1 in pressure, indicated by the kinks. capacity of S (9.4m) = 6.0 × 10 m Pa , that is, almost two int orders of magnitude smaller than the geometrical storage capacity 2.3.2 Flow delay of the tubing (or 1.3 per cent). The shorter interval length of 1.7 m (corresponding to an interval volume of about 10 L) was not explic- All tests with a water level in the tube below surface level are poten- itly tested, but the reduced length should lead to a further reduction tially affected by the conventional approach of remotely measuring of the storage capacity roughly proportional to interval length, that flow rate at the surface and the associated time delay between flow −10 3 −1 is, S (1.7m)  1.1 × 10 m Pa . Thus, we neglected all con- int in the gauge and actual addition of fluid to the water column load- tributions to storage capacity but the geometrical one of the tubing ing the interval. Yet, when correcting flow records after eq. (1), when determining true flow rates into or out of the rock forma- tests employing a ‘zero’-flow rate, which we address as zero-flow −1 tion from the flow rate recorded at the surface, Q (L min ), surface rate tests below, yield suspicious delays of up to 10 s between flow according to rate at the surface and pressure in the interval exceeding (Fig. 6). These large delays presumably result from the actual time of flow Q = Q − S p˙ . (1) cor surface tube int in the partly empty tube. These data cannot be evaluated in a stan- The time derivative of the interval pressure, p˙ , was calculated dard way. We pursue two different approaches to derive information int using a Fourier transformation of the recorded pressure. on hydraulic properties from these tests. First, we determine the Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 548 Y. Cheng and J. Renner Shell model. Analytical solutions for the injectivity analysis con- sidering a shell model, known as multiregion composite model in reservoir engineering (Ambastha & Ramey 1992; Acosta & Ambastha 1994), are given in Appendix C. These solutions com- plement the work by Ahn & Horne (2010), who reported a semi- analytical solution for interference analysis of a multicomposite radial ring system. For the analysis of the field data, we focus on a single cylindrical shell concentrically surrounding the borehole and exhibiting hydraulic properties that differ from the medium farther away from the well (comparable to the situation envisioned for the so-called skin effect, Matthews & Russell 1967). The prominent feature of this model is a perturbation of the transition from zero phase shift to a phase shift of 1/4 in comparison to radial flow in a ho- mogeneous medium. The perturbations occur for values of ∼0.01–1 of the dimensionless parameter r ω/D ,where D is the hy- i shell shell draulic diffusivity of the shell, r the radius of the injection well and ω the angular frequency. This dimensionless parameter gives the Figure 7. Effect of interval length (as given in labels; radial flow corre- inverse of the ratio between the hydraulic penetration depth into the sponds to an infinitely long interval) on phase shift between flow rate and shell material and borehole radius. Prominent maxima, successions interval pressure according to the mirror-symmetric model described in Ap- of maxima and minima, or a focusing of the transition occur de- pendix B. Curves were calculated after eq. (B17) with the summation up to pending on the ratios between the hydraulic parameters representing n = 100. the shell and the surrounding medium (see e.g. Appendix C2). amplitude ratio of uncorrected flow rate, that is, surface-flow rate and interval pressure (Table 2) for all tests and compare their re- Bilinear-flow model. Bilinear flow occurs when fluid is drained lation to pumping characteristics with that of amplitude ratios of from or injected into a permeable matrix through an enclosed frac- the tests for which flow-rate correction was possible. Secondly, we ture of finite conductivity intersecting a well along its axis. A com- test whether manual shifting of flow records guided by the promi- bination of two approximately linear flow regimes may result, one nent kinks in pressure records provides reasonable estimates for the in the matrix with flow essentially perpendicular to the fracture and phase shift between flow rate and pressure and thus opens the way the other in the fracture itself associated with the non-negligible for an inversion towards hydraulic parameters. pressure drop or increase in it (Cinco-Ley et al. 1978; Cinco-Ley & Samaniego-V 1981; Ortiz et al. 2013). In Appendix D, we present an analytical solution of the coupled diffusion equations for bilinear 2.3.3 Flow regimes considered for estimation flow in a homogeneous medium subjected to periodic pumping. For of hydraulic properties infinite fracture length, the phase shift is bounded by asymptotes 2 −1 Equivalent or effective hydraulic properties, diffusivity D (m s ), to 1/16 and 1/8 of a cycle for large and small periods, respectively, 2 −1 transmissivity T (m s ) and storativity S (−), are typically es- with a smooth transition in-between (Fig. D2 in Appendix D). The timated relying on analytical solutions of the pressure-diffusion asymptotic value for long periods is consistent with the value de- equation for specific flow regimes, that is, the dimensional and di- rivedbyHollaender et al. (2002). In case of fractures with finite rectional characteristics of the flow pattern (e.g. Fetter 2001). Here, length, only the asymptote of 1/8 of a cycle for small periods holds, too, as expected because the pressure perturbation does not reach we considered: (i) 1-D flow, (ii) radial flow, (iii) radial flow in con- centric cylindrical shells and (iv) bilinear flow. Analytical solutions the fracture tip. For a constant-pressure boundary at the fracture tip, of the 1-D-flow and radial-flow models were previously derived for phase shift monotonically decreases to 0 with increasing period. various boundary conditions (Black & Kipp 1981; Rasmussen et al. The relation between phase shift and period is not monotonic for a 2003; Renner & Messar 2006). Below, we investigate the effect of no-flow boundary, but minima (and possibly also maxima exceed- finite interval length on flow in homogeneous isotropic medium and ing 1/8) occur at intermediate periods before 1/8 is asymptotically present solutions for the other scenarios. approached for very large periods when the pressure in the fracture is homogeneous and the total response is dominated by linear flow into the surrounding medium. Effect of interval length. The use of a double-packer probe poses a significant difference to previously performed periodic pumping tests mandating to investigate the consequences of a restricted ‘ac- 2.3.4 Flow-regime diagnosis from spectral analyses tive’ length of the formation. For this purpose, we considered a radially infinite, mirror-symmetric model vertically bounded by a For conventional tests, flow regimes can be inferred from diagnos- no-flow condition (see Appendix B). To allow for analytical treat- tic log–log plots of pressure or pressure derivative versus time (e.g. ment, we assumed pressure to evolve linearly along the packers and Renard et al. 2009). Analogously, slopes of amplitude spectra of to be constant above the upper (below the lower) packer. non-harmonic periodic tests allow for a diagnosis of flow regimes According to the model, the effects of interval length on the (Hollaender et al. 2002). In either type of diagram, slopes of 1/4, phase shift between flow rate and interval pressure are stronger the 1/2, 0 and 1 are indicative of bilinear flow, linear flow, radial flow larger the period, that is, the penetration depth (Fig. 7). Relying and pseudo-steady-state flow, respectively. Since our excitation is on the radial-flow model for the inversion of hydraulic diffusivity not truly harmonic, spectra of flow rate and pressure contain signif- from phase shift, diffusivity is the more overestimated the shorter icant amplitudes at periods shorter than the nominally excited one, the interval and the smaller the phase shift are. Small phase shifts too. Yet, simple division of entire flow-rate and pressure spectra correspond to large periods (small frequencies) for which a double- proved to be impractical (Fig. 8). Thus, we restricted to specific packer probe acts as a point source rather than a line source. local maxima of pressure and flow rate, as did Fokker et al. (2013). Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 549 Figure 8. Spectrum of the amplitude ratio modulus for the interval at 105 m tested with a period of 540 s (i5–4). Coloured lines represent exponents of 1/4, 1/2 and 1. The restriction to local maxima of pressure and flow rate in the frequency domain (red asterisks) reduces the scatter and suggests Figure 10. Phase shift and amplitude ratio of the interference response bilinear (slope 1/4) to linear flow (1/2). between pressure recorded (a) below the lower packer and (b) above the upper packer and in the interval calculated using the model described in Fig. 9 with different diffusivities and boundary conditions as specified in Table 3 (expressed by the numbers used as labels). The dots represent individual 2 −1 calculations labeled by the used diffusivity (m s ). The lines intended to help the eye to follow the trends. The red square with error bars represents the observations at a depth of 85 m for a period of 180 s (i5–3). cally using COMSOL. We determined values of phase shift and amplitude ratio of pressures measured at different locations of a double-packer probe considering different diffusivities and combi- nations of boundary conditions (Table 3). The effect of the inves- tigated scenarios (1–8) is subordinate for the interference response between below lower packer and in the interval (Fig. 10). The results for pressure above upper packer separate into two groups according to the top boundary conditions labeled as ‘no flow’ and ‘constant pressure’. Effective diffusivity values D or D are estimated from am- δ ϕ plitude ratio or phase shift, respectively, by linearly interpolating between the modeling results. Deviation of model results from ob- servations could be explained by several aspects. The model as- Figure 9. Illustration of the model used for numerical analysis of vertical sumes homogeneity and isotropy for the subsurface which may not interference. Blue colour indicates water-filled well sections and the two be true in reality. Furthermore, the model only considers relations packers are represented in grey. The sketch is not to scale: distance of the between pressures but a more comprehensive model should also lateral boundaries from the symmetry axis is 1000r ; distance between top integrate the injectivity analysis. Finally, hydromechanical effects and bottom boundary is 240 m, that is, the bottom boundary is more than were observed but not considered in the current model. 100 m away from the bottom of the borehole. 2.4 Vertical-interference analysis 3 RESULTS When placing a double-packer probe in an open borehole, the pump- 3.1 Injectivity analysis ing operations in the interval may also cause detectable pressure variations above or below the probe due to probe-parallel flow Amplitude ratios of data sets resulting from performing the flow- through the formation bearing additional information on its hy- rate correction after eq. (1) are systematically smaller than those draulic properties. To take advantage of these pressure records, we of uncorrected data (Fig. 11), but the relative relations among the analysed an axisymmetric model (Fig. 9) and treated it numeri- ratios for the different test intervals as well as their trends with the Table 3. Combinations of boundary conditions considered for the performed eight calculations employing the model sketched in Fig. 9. Boundary conditions 1 2 3 4 5 6 7 8 Top boundary No flow No flow No flow No flow Constant pressure Constant pressure Constant pressure Constant pressure Bottom boundary No flow Constant pressure Constant pressure No flow No flow Constant pressure Constant pressure No flow Lateral boundary Constant pressure Constant pressure No flow No flow Constant pressure Constant pressure No flow No flow Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 550 Y. Cheng and J. Renner systematically with increasing pumping period for i5 as well as for i3. 3.1.2 Effective hydraulic parameters Effective transmissivity is the least variable of all hydraulic parame- ters when evaluating the results of the Fourier analyses of corrected flow rate and interval pressure relying on the analytical solution for radial flow in a homogeneous medium (Table 4). It varies by about −6 2 −1 half an order of magnitude from 3 to 9 × 10 m s . Apart from one outlier (i3–5), the values for hydraulic diffusivity and storativ- −5 2 −1 ity stay within about one order of magnitude (4–40 × 10 m s −3 and 2–20 × 10 , respectively). The outlier may to some extent be explained by the finite length of the interval. The modeling of the length effect (see Appendix B) predicts the largest error for test i3– 5, with diffusivity potentially overestimated by about one order of 2 −1 magnitude (0.37 m s from conventional radial-flow model versus Figure 11. Injectivity quantified by the amplitude ratio of flow rate and 2 −1 0.04 m s from mirror-symmetric double-packer model). For the injection pressure as dependent on mean interval pressure. Amplitude ratios other tests, yielding phase shifts of at least 0.074 cycles, the error were calculated for all tests before (labeled uncorrected and represented by due to neglecting the finite interval length is negligible (Fig. 7). solid circles) and after (labeled corrected and represented by open circles) applying the correction to flow rate for the storage of the tubing. The colour Phase-shift values are particularly diagnostic for hydraulic coding represents the five investigated test intervals (see Table 1). Data boundaries. For example, phase shifts larger than 1/8 unequivo- points with uncorrected flow rate are also labeled by the imposed oscillation cally require bounded reservoirs (Fig. 12). The phase-shift value period in seconds. Vertical error bars indicate the uncertainty in amplitude for the test at a depth of 85 m and a period of 180 s (i3–2) remains ratio gained from the sliding-window fast Fourier transform (FFT) analyses. larger than the upper bound of 1/4 of all considered bounded models Horizontal error bars represent the range of pressures at which the tests were even after manual shifting. Only models with a no-flow boundary conducted. can explain the phase shift of the test i3–1 at 85 m and a period of 180 s, the largest phase shift observed for non-zero-flow rate tests parameters characterizing a test, mean pressure and period, remain (Fig. 12). However, the increase of amplitude ratios with phase shift qualitatively similar. We therefore use the larger set of uncorrected between i3–1 and the other test at the same depth for a period of data for the subsequent analysis and refer to this approach as sim- 1080 s (i3–5) cannot be explained by any of the no-flow models. plified injectivity analysis. Independent of interval length the phase-shift values for a depth of ∼105 m (i2, i5) are all smaller than 1/8 of a cycle, excluding the 1-D-flow model with a no-flow boundary or no boundary. The 3.1.1 Simplified injectivity analysis non-monotonically varying amplitude ratios for i5 also exclude a Amplitude ratios of uncorrected flow rates and interval pressures, constant pressure boundary or no boundary for all the flow types. here treated as tentative measures of injectivity, tend to increase For the interval with a depth of ∼115 m (i1), the only phase shift with increasing mean pressure but variations at a given pressure are lies between 1/16 and 1/8, which can be fit by all the models except significant (Fig. 11). A period of 180 s was chosen for more than 1-D flow with no-flow boundary or no boundary. half of the periodic pumping tests allowing us to compare the hy- At a depth of ∼105 m, three tests were performed with different draulic behaviour of the four tested intervals. The deepest interval periods using the short interval (60, 180 and 540 s correspond- with a depth of 115 m (i1) exhibits the lowest injectivity despite its ing to i5–2, i5–3, and i5–4 in Table 2). Since they all included association with the largest absolute mean pressure. The shallowest only non-zero flow rates their results are not affected by the prob- interval at 85 m depth (i3) shows a strong positive correlation be- lems associated with the finite distance between the location of tween injectivity and mean pressure (also found for a period of 60 s the flow meter and the top of the water column loading the inter- applied in i5) indicating that the dominant hydraulic conduits are val. The non-monotonic succession of phase-shift values, with the perceptible to hydromechanical effects. For a depth of 105 m (i2, lowest occurring for the intermediate period, can be modeled by a i5), we neither find a pronounced dependence of amplitude ratio shell with a thickness of about 6–15 times the borehole radius of −3 on mean pressure nor on interval length. Amplitude ratios decrease ∼0.05 m, and with a diffusivity between just below 10 to above Table 4. Hydraulic properties derived from the results of the Fourier analyses of flow rate and interval pressure relying on the analytical solution for radial flow in a homogeneous medium. Int. Period Nominal flow rate Depth Diffusivity Transmissivity Storativity −1 2 −1 2 −1 (s) (L min ) (m) (m s)(m s)(–) i1 180 3/0 117.1 1.3 ± 0.9E−05 2.7 ± 0.8E−07 2.7 ± 1.3E−02 i2–1 180 10/7 106.7 3.9 ± 0.9E−05 3.5 ± 0.3E−06 9.4 ± 1.5E−02 i3–3 480 3/0 81.9 5.0 ± 3.1E−07 1.8 ± 0.5E−07 4.4 ± 1.8E−01 i3–5 1080 7/4 81.9 3.7 ± 1.3E−01 5.1 ± 0.2E−06 1.5 ± 0.5E−05 i5–2 60 10/7 105.0 8.4 ± 3.4E−05 4.9 ± 0.8E−06 6.4 ± 1.5E−02 i5–3 180 10/7 105.0 3.9 ± 0.3E−04 9.0 ± 0.2E−06 2.3 ± 0.2E−02 i5–4 540 10/7 105.0 5.7 ± 0.6E−05 4.8 ± 0.1E−06 8.5 ± 0.6E−02 Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 551 Figure 12. Amplitude ratio versus phase shift for the injectivity analysis. Red, blue and black lines represent the solutions for radial flow, 1-D flow and bilinear flow, respectively. Solid lines are for infinite reservoirs and dashed and dashed–dotted lines represent constant pressure (pb) and no-flow (fb) boundary at distances of 20r , respectively, for radial and bilinear flows. Changing the assumed hydraulic parameters shifts the theoretical curves vertical without distortion. Filled and open symbols represent non-zero-flow-rate sequences and zero-flow-rate sequences for which manual shift was applied, respectively (see Table 2). Errors for the data points do not exceed symbol size. Two periods are also labeled for the convenience of discussion (see Table 4). Table 5. Parameters of the shell model (see Appendix C) derived flow is that the pressure at the fracture tip has risen to a substantial from the phase-shift values observed for the tests i5–3 and i5–4 (see fraction of the associated pressure increase in the well at the end of also Fig. 13). bilinear flow and the flow field transforms towards formation linear flow. The transition from pseudo-steady-state flow to bilinear flow is k /k D /D r /r Diffusivity (D ) 1 2 1 2 1 i 1 2 −1 consistent with a succession of the domination of wellbore storage (–) (–)(–)(m s ) at early time by bilinear flow (finite conductivity fracture). 6.7 8 15 8.1E−04 7.1 10 15 7.4E−04 6.3 10 10 7.1E−03 5.9 9 6.1 2.3E−02 3.3 Vertical-interference analysis 5.6 9 6.1 2.1E−02 The periodic excitations in the injection intervals lead to detectable periodic responses in the pressures recorded below the lower and −2 2 −1 10 m s (Table 5) exceeding that for the enclosing medium by above the upper packer (Table 2). The vertical-interference analy- a factor between about 5 and 10 (Fig. 13). sis based on these pressure records has the advantage that all tests can be analysed even those involving the problematic zero-flow-rate sequences. Yet, the magnitude of interference response in a single 3.2 Flow regime diagnosis from spectral analyses borehole critically depends on the storage capacity of the borehole Flow-regime identification from full spectra proved difficult (Fig. 8), section in which it is determined. Above the upper packer we had not the least because the spectra of flow rate and pressure contain lit- a free, unconstrained water column corresponding to a ‘large’ stor- tle significant contributions for high frequencies. We thus restricted age capacity determined by the cross-section of the borehole. The to the first two odd multiples of the test frequency that correspond borehole section below the lower packer, in contrast, constitutes an to pronounced local maxima in the spectra. As the analysis is sen- enclosed fluid volume with a ‘small’ storage capacity determined sitive to the timing of flow-rate changes (Hollaender et al. 2002), by the fluid compressibility and the deformability of packer and we only analyse non-zero-flow-rate sequences. The results for over- borehole wall. In agreement with these qualitative storage consid- tones from the various tests in a specific interval agree closely with erations, the interference response is much larger below the probe those for nominally excited periods (Fig. 14). The slope for the than above the probe where it is in fact detectable only in some interval at 105 m (i5) increases from less than 1/4 and approaches cases (Table 2). All amplitude ratios derived from pressures below 1/2 as period increases. In contrast, the slope for interval i3 de- the lower packer, but that for the deepest test interval (i1), follow creases from around 1 to a value just above 1/4. At face value, these the expected inverse correlation with the length of the enclosed results indicate a transition from bilinear to linear flow for i5 and borehole section (Fig. 15): the larger the enclosed volume the larger a transition from pseudo-steady-state flow to bilinear flow for i3. the storage capacity and the less sensitive the section, that is, the The physical interpretation for a transition from bilinear to linear amplitude ratio between interference response and excitation Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 552 Y. Cheng and J. Renner Figure 14. Amplitude ratio in frequency domain for the deepest interval (i5) represented by asterisks and the interval at depth 81.9 m (i3) represented by circles at the main period of the pumping tests and the first overtone, that is, a third of the imposed period. The dotted lines represent exponents of 1/4, 1/2 and 1. Figure 13. Examples phase-shift–frequency relations for the shell model calculated using eq. (C2) in relation to observations for the periodic tests at 105 m for a period of 180 s (i5–3) and for a period of 540 s (i5–4). The labels give the ratios between the diffusivity of the shell and the surrounding medium and the lateral extension of the shell in multiples of the borehole radius. decreases with the length of the enclosed section. Relative to the general trend, the response below the probe observed for the test in the deepest interval is weak demonstrating exceptional hydraulic properties near the borehole bottom. Including phase shift in the comparison, observations for the interference with the borehole section above the interval are only in Figure 15. Correlation between interference responses, quantified by the agreement with model results for a ‘no-flow’ condition (Fig. 10). amplitude ratio of pressure oscillations, and the length of the enclosed The effective diffusivities estimated from phase shift (denoted D section below the lower packer of the probe. We present the amplitude ratio in Table 6) tend to be larger than the ones estimated from amplitude between the pressure above the interval and in the interval in this graph, too, though it should not be controlled by the length of the enclosed section ratio (D ) for i2 and i5, the long and short intervals at 105 m. below the probe. The joint presentation of results is chosen to highlight the The reverse seems to hold for the shallowest interval i3. Effective difference in sensitivity of vertical response between the enclosed section diffusivity values using p –p data or consistently on the order of a bel int below the probe and the unconstrained water column above it. 2 −1 few m s for the long and the short interval (i2 and i5) at the same depth, except for two relatively small values (i2–1, i5–4). Those derived from the conventional methods and periodic radial-flow estimated by p –p analysis are smaller than those estimated by abo int analysis are of similar order of magnitude (Table 4). Flow regimes p –p for the short interval i5 without considering the extremely bel int were constrained using diagnostic log–log plots of pressure and small D value for i5–4. A decreasing trend of diffusivity with pressure derivative versus elapsed time. Intervals i1 und i3 indicate increasing period is indicated for i5. bilinear flow and interval i2 and i5 display features of linear flow. 3.4 Conventional methods 4 DISCUSSION Conventional tests yield the smallest transmissivity and storativity The variable differences between equilibrium pressure for a cer- for the shallowest interval (i4, Table 7). Disregarding the excep- tain depth interval and a hydrostatic gradient indicate heterogeneity tionally low value found for interval i4, effective transmissivity is of the hydraulic system. The fractured aquifer is dominated by the least variable hydraulic property, consistent with the observa- steep conduits that appear to have limited interconnectivity despite tions from periodic radial-flow analysis. The hydraulic parameters their small lateral distance (at most a few metres in the case of the Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 553 Table 6. Effective diffusivity D estimated by amplitude ratio and D by phase shift of vertical-interference analysis. δ ϕ Top and bottom boundaries are set as no flow. 2 −1 2 −1 Int. Period Nominal flow rate D (m s ) D (m s ) δ ϕ −1 (s) (L min ) p –p p –p p –p p –p bel int abo int bel int abo int i1 180 3/0 5E−35E−33E−34E−3 i2–1 180 10/7 1 – 3E−2– i2–2 180 3/0 4–2– i3–1 180 7/4 2E−2–3E−1– i3–2 180 3/0 3E−2–4E−1– i3–3 480 3/0 3E−2–3E−1– i3–5 1080 7/4 2E−2–1E−1– i5–1 60 3/0 4–8– i5–2 60 10/7 7–6– i5–3 180 10/7 5 9E−233E−2 i5–4 540 10/7 3 1E−23E−47E−3 Table 7. Hydraulic properties derived from conventional tests. 2 −1 2 −1 a Int. Test phase Diffusivity (m s ) Transmissivity (m s ) Storativity (–) Remarks i1 Shut-in <1.14E−04 <1.6E−06 1.4E−02 RSLA i2 Constant flow rate <1.39E−04 <4.3E−06 3.1E−02 SLA i3 Shut-in <2.29E−04 <1.9E−06 8.3E−03 RSLA i4 Shut-in <1.45E−03 <2.9E−08 2.0E−05 RSLA i5 Shut-in <3.18E−05 <2.7E−06 8.5E−02 RSLA SLA: ‘Straight Line Analysis’ after Jacob & Lohman (1952). RSLA: ‘Recovery Straight Line Analysis’ after Agarwal (1980). borehole’s bottom section). The diffusivity variations with pumping for the shallowest of the tested intervals (i3), diffusivity values periods suggest spatial heterogeneity or complexity of the conduits derived from periodic testing deviate significantly from the one in a specific interval, too, that is, simple radial flow is a poor ap- gained from conventional testing. The lower diffusivity value results proximation of the flow excited during the tests. The prominent from a test sequence with zero flow rate (i3–3) and thus has a role of discrete hydraulic conduits is documented by the similarity limited reliability. The higher diffusivity value corresponds to test of injectivity values for the two tests at a borehole depth of about i3–5 giving a phase shift so low that it falls within the range for 105 m (i2 and i5) despite their differences in interval length. In the which the finite length of the probe cannot be neglected, but the following, we discuss to what extent the employed methods yield flow field has a significant axial component (Fig. 7). The estimate comparable values for hydraulic parameters, comment on the con- based on the mirror-symmetric double-packer model is more than straints for the pressure dependence of the hydraulic response, and one order of magnitude smaller than the value assuming simple finally speculate on the flow-regime model that is most consistent radial flow. Furthermore, the analyses of periodic tests demonstrates with the entirety of observations in a specific interval. a dependence of the effective hydraulic parameters on period and mean pressure. Thus, differences between the methods likely reflect the simplifications inherent in the radial-flow model that are not justified for the investigated structures necessitating to investigate 4.1 Comparison of methods more complex flow models. Periodic pumping tests can be easily implemented using field equip- Vertical-interference analysis is a valuable by-product of peri- ment for conventional testing. The excitation signals do not have to odic testing when packer-separated intervals are investigated and be perfectly harmonic but Fourier analysis of non-harmonic signals pressure is monitored at several depth levels. We refer to its re- sults as effective axial diffusivity to distinguish them from results might actually provide information regarding periods shorter than of the injectivity analysis and conventional methods, addressed as the imposed main period (spectral analyses, e.g. Hollaender et al. effective radial diffusivity. The attributes ‘axial’ and ‘radial’ are 2002; Fokker et al. 2013). The periodic excitation can be applied chosen from the perspective of the pumping well, addressing flow even when the hydraulic pressure is not equilibrated, in fact it can dominantly subparallel and normal to well axis, respectively. Ef- be superposed to any transient, be it associated with a terminated or fective axial diffusivities are here found to be generally larger than ongoing pumping operation. The superposition is valid as long as effective radial diffusivities (Fig. 16). The axial diffusivities scatter linearity can be assumed, that is, the diffusion equation in its basic significantly for the intervals i2 and i5 at similar mean depths but form holds. If it does not hold, the results of conventional meth- with different probe lengths. The scatter is larger for the shorter ods become questionable, too. As a consequence of the freedom to interval i5 tested for a larger spread in periods and apart from the start periodic testing at any time, operational time is reduced and result for the longest period of 540 s, communication to the deeper to a larger degree plannable than for conventional tests that involve borehole section is characterized by larger diffusivity values than long waiting times for equilibration of apriori unknown duration. to the shallower sections. The vertical response to shallower depth During the current test campaign, a total of 11 hr was spent on sections is actually lost with the increase in probe length suggest- conventional tests and 7 hr for periodic tests. ing that faults outside the short interval but inside the long interval Our results show that the analyses of periodic pumping provide are hydraulically connected to the prominent fault isolated by the effective hydraulic parameters comparable to those gained from short interval. The decrease in diffusivity with increasing period or conventional methods when assuming radial flow (Fig. 16). Only Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 554 Y. Cheng and J. Renner geological situation, an increase in injectivity with mean pressure likely results from the pressure-induced increase in effective hy- draulic aperture of the joints and faults intersecting the borehole. The more pronounced pressure dependence of the short interval i5 compared to the long interval i2 supports the notion that the pressure dependence reflects hydromechanical behaviour of discrete faults. Observations for i3 (∼85 m depth) with a relatively large period of 180 s suggest that the hydromechanical effects are not restricted to the very intersection between faults and the well. The significance of hydromechanical effects is also evidenced by the pronounced reverse pressure response observed below the lower packer when injecting in i1 (not documented here). 4.3 Constraints on flow regime Details of our observations are obviously at conflict with simple radial flow. From periodic tests, flow regimes can be constrained by (i) investigating spectra of non-harmonic signals (Fig. 14)orby (ii) comparing the variation patterns of amplitude ratio and phase shift with theoretical flow models (Fig. 12). The few constraints we derived from the analysis of spectra yield flow regimes identical to those indicated by the conventional methods. The periodic injectiv- ity analysis tends to indicate more complex flow regimes than the conventional methods (Table 8) which we interpret as evidence for the superior resolution of the periodic approach. Effective injectivity values determined relying on simple radial flow (Fig. 11) systematically decrease with increasing pumping pe- Figure 16. (a) Radial diffusivity determined from periodic injectivity anal- riod for intervals i3 and i5 (factor of 3–5 for less than one order ysis and conventional methods. (b) Axial diffusivity determined by vertical- of magnitude in period). The conventional scaling relation for dif- interference analysis. Labels refer to the effective diffusivity values esti- fusion processes (e.g. Weir 1999) suggests that the covered range mated from amplitude ratio (Ddel: D ) and phase shift (Dphi: D )using δ ϕ in period corresponds to a change of half an order of magnitude in pressure records above (pabo), in (pint), and below (pbel) the interval. nominal penetration depth. Thus, the period dependence of injec- tivity indicates a change in hydraulic characteristics with distance penetration depth (Table 6) also indicates that the axial connectivity from the borehole. When addressing the period dependence by the is a local phenomenon. shell model for the deepest interval (i5), we find a decrease in effective diffusivity and permeability by about one order of mag- nitude beyond a zone of about several decimetres to one metre 4.2 Pressure dependence of hydraulic properties (Fig. 13). This change in hydraulic characteristics could be related The tested intervals exhibit a general trend of increasing injectivity to drilling-associated damage of the borehole wall or remnants of with increasing pressure, with the prominent exception of the deep- the thixotropic polymer fluid used during drilling. The flushing of est interval (i1) that also has the lowest injectivity. In the current the well during logging preceding the hydraulic testing may have Table 8. Constraints on flow regimes gained from injectivity analysis, spectral analysis, or conventional methods: possible (o), excluded (x), no constraint (–) for radial (R), bilinear (BL) and 1-D (1-D) flow. Periodic pumping method Conventional methods Injectivity analysis Spectral analysis (see Fig. 14) No flow Constant pressure No boundary No boundary No boundary i1 R o o o – x 1-D x o x – x BL o o o – o i2 R o o o – x 1-D x o x – o BL x x x – x i3 R x x x x x 1-D x x x x x BL x x x o o i5 R o x x x x 1-D x x x o o BL o x x o x Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 555 well resulted in a cleansing of only the hydraulic conduits nearest approach, addressed as vertical-interference analysis. We demon- to the borehole. strated that the analyses of the periodic pumping tests yield results For the short interval i5, the conventional method and the spectral comparable to conventional methods when relying on the same analysis (Fig. 14) suggest a linear flow regime while injectivity model. The field campaign revealed, however, several methodolog- analysis indicates radial or bilinear flow with a no-flow boundary. ical advantages of the periodic testing, for example, its plannable For the longer interval at the same depth (i2), the injectivity analysis testing time and the ease in separating the response to the actual suggests radial flow with any boundary condition or linear flow pumping operation from noise and transients gained from data pro- with constant pressure boundary. Assuming the two intervals to cessing in frequency space. Our analytical and numerical model- be governed by the same boundary condition, the only regime in ing of periodic testing extends the suite of evaluation tools and common by i5 and i2 is radial flow with a no-flow boundary. specifically allowed us to decipher more details of the encountered For the deepest interval (i1), the injectivity is the lowest of all complex flow geometry and further conduit characteristics than the intervals though it was tested at the highest mean injection pres- conventional methods. sure. The low equilibrium pressure below i1 (Fig. 4) suggests that The two approaches, the previously employed injectivity anal- the section below i1 is disconnected from the ones above. In addi- ysis and the vertical-interference analysis, are complementary by tion, higher axial diffusivity than radial diffusivity (Fig. 16) implies providing information on the hydraulic properties in different di- axially oriented or subvertical upflow in the sections above i1. rections relative to the well axis. The variations in effective ‘axial’ and ‘radial’ diffusivity with pumping periods and mean pumping pressure indicate significant spatial heterogeneity of conduits that 4.4 Synopsis also exhibit hydromechanical effects for the investigated site. While we showed the potential of the vertical-interference analysis, clearly The periodic testing revealed that the flow geometry in the pene- more work is needed in the future regarding the modeling of records trated subsurface is quite variable on the metre (lateral) to decametre available when using double- or multipacker probes. (axial) scale and that the conduit system exhibits hydromechanical effects, two characteristics that appear quite reasonable for a strike- slip fault in crystalline rock. Yet, at this point, it is difficult to distin- ACKNOWLEDGEMENTS guish between a situation characterized by a lateral no-flow bound- ary, possibly related to relics of the polymer drilling fluid with high The financial support by BfE, Switzerland is gratefully acknowl- viscosity or drilling-associated near-welbore damage, and a truly edged as is the coordinator, Marco Herwegh, for involving us in anisotropic hydraulic system with the high axial diffusivity due to project NFP70-P1. Field tests were performed together with Sacha subvertically oriented fluid pathways. The latter notion is not only Reinhardt and Markus Bosshard (SolExperts), and Daniel Eggli supported by the logging observations (Appendix A) that document (Uni Bern) who is particularly thanked for generously sharing the a dominance of steeply dipping fractures intersecting the well, but results of the structural core and log analyses. The presented treat- is also consistent with the conduit geometry deduced by Belgrano ment of the shell model builds on an unpublished analyses of Eugen et al. (2016), who suggested that the fault zone is controlled by Petkau. We are grateful to the three anonymous reviewers for their localized subvertically oriented ‘pipe’-like upflow zones. The verti- valuable comments. cal variability observed for the undisturbed hydraulic heads and the low hydraulic transmissivity of the deepest test interval compared to the major fault zone support this suggestion derived solely from REFERENCES structural investigations. 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For i2 (i5), a section of about 1 m length is also represented by a photograph of the recovered cores. APPENDIX B: MIRROR-SYMMETRIC DOUBLE-PACKER MODEL For an isotropic and homogenous subsurface, the governing diffusion equation reads in cylindrical coordinates: 2 2 ∂ p(r, z, t) 1 ∂p(r, z, t) ∂ p(r, z, t) 1 ∂p(r, z, t) + + = (B1) 2 2 ∂r r ∂r ∂z D ∂t where r is radial distance from the centre of the borehole and z denotes the vertical direction (upright positive). Employing separation of variables, that is, p(r, z, t) = P(r, z)(t) = P(r, z)exp(i ωt), the spatial variation of pressure obeys 2 2 ∂ P(r, z) 1 ∂ P(r, z) ∂ P(r, z) i ω + + = P(r, z). (B2) 2 2 ∂r r ∂r ∂z D We consider a double-packer probe enclosing an interval with a total height of 2h in a well with radius r . Instead of modeling two packers, 0 i we assume mirror symmetry to the horizontal plane bisecting the interval (Fig. B1) and thus we can restrict to treating the upper half of the Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 558 Y. Cheng and J. Renner Figure B1. Model geometry for the analytical solution of a double-packer test. The model has mirror symmetry with respect to the horizontal plane that bisects the interval. model space. The locations of the bottom and top of the packer are z = h and z = h , respectively, and the aquifer extends vertically to z = b 0 p (Fig. B1). We adapt boundary conditions from Rehbinder (1996) and Mathias & Butler (2007): ∂ P(r, z) BC1: = 0 z = b no-flow condition for vertical bounds of model ∂z BC2: P(r, z) = 0 r →∞ radially infinite reservoir T ∂ P(r, z) BC3: Q = 4πr q(r , z)dz with q(r , z) =− | 0 ≤ z ≤ h integrating flux q over the injection interval has to i i i 0 2ρgh ∂r r 0 i yield the constant flow rate Q BC4: P(r , z) = p h ≤ z ≤ b average pressure above packer (neglecting effect of fluid column) i 1 p BC5: P(r , z) = p 0 ≤ z ≤ h average pressure in injection interval (neglecting effect of fluid column) i 2 0 h − z P(r , z) = (p − p ) + p h ≤ z BC6: ≤ h linear pressure change along packer section i 2 1 1 0 p h − h p 0 where the pressures in the well above and in the interval are denoted p and p , respectively. 1 2 After Fourier cosine transforming of z,thatis, 2 nπ P (r) = P(r, z)cos(a z)dz, a = (B3) n n n b b and accounting for BC1, eq. (B2) becomes ∂ P (r) 1 ∂ P (r) i ω n n + − a + P (r) = 0 (B4) ∂r r ∂r D with a general solution in the form of i ω P (r) = A K (η r) + B I (η r),η = a + (B5) n n 0 n n 0 n n where K and I denote modified Bessel functions of the first and second kind of zero order. Since I →∞ when r →∞ (BC2), we have to 0 0 0 require B = 0, that is, P (r) = A K (η r). (B6) n n 0 n The backtransform reads A K (η r) 0 0 0 P (r, z) = + A K (η r)cos (a z). (B7) n 0 n n n=1 Application of BC4–BC6 gives p 0 ≤ z ≤ h ⎪ 2 0 A K (η r ) 0 0 0 i h −z + A K (η r )cos (a z) = (p − p )( ) + p h ≤ z ≤ h (B8) n 0 n i n 2 1 1 0 p h −h p 0 2 ⎪ n=1 ⎩ p h ≤ z ≤ b 1 p Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 559 which upon Fourier-cosine transformation of z yields b b A K (η r ) 0 0 0 i cos (a z) dz + A K (η r ) cos (a z) cos (a z) dz m n 0 n i n m 0 0 n=1 =0 h h b 0 p h − z = p cos (a z) dz + (p − p ) + p cos (a z) dz + p cos (a z) dz. (B9) 2 m 2 1 1 m 1 m h − h p 0 0 h h 0 p For a − a = 0, the second integrand on the left-hand side can be modified as n m cos [(a − a )z] + cos [(a + a )z] n m n m cos (a z) cos (a z) = n m and thus z z cos [(a − a )z] + cos [(a + a )z] sin (n − n)π sin (n + n)π n m n m b b dz = +  = 0, 2 2(a − a ) 2(a + a ) n m n m that is, all elements of the sum vanish but the one for which a − a = 0. In this case, the left-hand side of eq. (B9) can be algebraically n m manipulated as A K (η r ) cos (a z) dz = A K (η r ) . (B10) m 0 m i m m 0 m i After integration, the right-hand side sums up to ( ) cos a h ( ) cos a h cos a h m p cos a h m p m 0 m 0 p   − p   − p   −   . (B11) 2 2 1 2 2 2 2 h − h a h − h a h − h a h − h a p 0 p 0 p 0 p 0 m m m m With both, left- and right-hand sides, eqs (B10) and (B11), respectively, we can solve for the unknown coefficients cos (a h ) cos a h cos (a h ) cos a h m 0 m p m 0 m p p   −   − p   − 2 1 2 2 2 2 h − h a h − h a h − h a h − h a p 0 p 0 p 0 p 0 m m m m A = . (B12) K (η r ) 0 m i To derive A , we apply Darcy’s law to eq. (B7) and obtain the flow rate according to (BC3), that is, T A η K (η r ) sin (a h ) 0 0 1 0 i n 0 Q = 2πr h + A η K (η r ) . (B13) i 0 n n 1 n i ρgh 2 a 0 n n=1 and thus Qρg η K (η r ) sin (a h ) n 1 n i n 0 A = − 2 A . (B14) 0 n πr T η K (η r ) η K (η r ) a h i 0 1 0 i 0 1 0 i n 0 n=1 Finally, the pressure function eq. (B7) reads ∞ ∞ Qρg K (η r) K (η r) sin (a h ) 0 0 0 0 n 0 P (r, z) = − A η K (η r ) + A K (η r)cos (a z) (B15) n n 1 n i n 0 n n 2πr T η K (η r ) η K (η r ) a h i 0 1 0 i 0 1 0 i n 0 n=1 n=1 2(p − p )[cos(a h ) − cos(a h )] 2 1 n 0 n p where A = . bK (η r )(h − h )a 0 n i p 0 For the injection interval, with P(r , z < h ) = p , we now find a relation between the ratios of flow rate and interval pressure on the one i 0 2 hand and between interval pressure and pressure outside the interval on the other hand: 1 ∞ 1 − cos (a h ) − cos a h Q 2πr T η K (η r ) n 0 n p η K (η r ) η K (η r )K (η r ) i 0 1 0 i p n 1 n i 0 0 n i 1 0 i = + 2   sin (a h ) − cos (a z) . (B16) n 0 n p ρg K (η r ) K (η r )a h a K (η r ) b h − h 2 0 0 i p 0 0 n i 0 n 0 0 i n=1 It can be seen that solution consists of two parts: standard radial flow for infinite line source and an additional term depending on packer positioning. When the pressure outside of the interval remains unaltered by the pumping operation, that is, p = p = const., we can—without 1 0 loss of generality—chose the reference pressure to p = 0. In this case, eq. (B16) simplifies to cos (a h ) − cos a h Q 2πr T η K (η r ) 2 η K (η r ) η K (η r )K (η r ) i 0 1 0 i n 0 n p n 1 n i 0 0 n i 1 0 i = +   sin (a h ) − cos (a z) , (B17) n 0 n p ρg K (η r ) K (η r )a h a K (η r ) b h − h 2 0 0 i p 0 0 n i 0 n 0 0 i n=1 an analytical expression for the ratio of flow rate and interval pressure. Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 560 Y. Cheng and J. Renner Figure C1. Geometry of radial flow from a well into N concentric shells and the surrounding medium. APPENDIX C: CONCENTRIC CYLINDRICAL SHELLS SURROUNDING THE BOREHOLE C1 Analytical solution We consider radial flow from a borehole into and beyond a single concentric cylindrical shell located between r , the radius of the injection well and r (Fig. C1). The diffusion equation can be written as: ∂ p(r, t) 1 ∂p(r, t) 1 ∂p(r, t) + = . (C1) ∂r r ∂r D ∂t Here, p is fluid pressure, t is time, r is radial distance and D is hydraulic diffusivity. The hydraulic properties of the shell (subscript 1) and the surrounding medium (subscript 2) are described by permeability k = 1,2 μT /(ρgh) and diffusivity D = k /(s μ) = T /S where μ denotes the fluid viscosity, T , S and s transmissivity, storativity 1,2 1,2 1,2 1,2 1,2 1,2 1,2 1,2 1,2 and specific storage capacity of the two media, respectively, g gravitational acceleration and h the length of the injection interval. Solving the diffusion equation for periodic pressure variations in the well by separation of variables (compare Renner Messar 2006) and account for boundary conditions p (r ≥ r , t) = 0 r →∞ radially infinite reservoir 2 1 p (r, t) = p (r, t) r = r continuity of pressure at outer shell radius 1 2 1 ∂ p (r, t) ∂ p (r, t) 1 2 continuity of flow rate at outer shell radius T = T r = r 1 2 1 ∂r ∂r yields amplitude ratio and phase shift between flow rate and interval pressure of T η (−m K (η r ) + I (η r )) 1 1 1 1 1 i 1 1 i δ = 2πr , (C2) qp i ρg m K (η r ) + I (η r ) 1 0 1 i 0 1 i and η (−m K (η r ) + I (η r )) 1 1 1 1 i 1 1 i ϕ = arg , (C3) qp m K (η r ) + I (η r ) 1 0 1 i 0 1 i respectively. The constant m is related to the ratios of permeability (or equivalently transmissivity) and diffusivity according to I (η r ) K (η r ) + I (η r ) K (η r ) 1 1 1 0 2 1 0 1 1 1 2 1 m =  , (C4) K (η r ) K (η r ) − K (η r ) K (η r ) 1 1 1 0 2 1 0 1 1 1 2 1 where I and I represent the modified Bessel functions of the first kind of zero and first order and K and K the modified Bessel functions 0 1 0 1 of the second kind of zero and first order. The arguments of these Bessel functions contain the complex parameters η = i ω/D that 1,2 1,2 depend on angular frequency ω and diffusivity of the two media. The analytical solution for N concentric cylindrical shells (Fig. C1) is given as ( ( ) ( )) T η −c K η r + I η r 1 1 1 1 1 i 1 1 i δ = 2πr , (C5) qp i ρg c K (η r ) + I (η r ) 1 0 1 i 0 1 i and η (−c K (η r ) + I (η r )) 1 1 1 1 i 1 1 i ϕ = arg , (C6) qp ( ) ( ) c K η r + I η r 1 0 1 i 0 1 i Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 561 Figure C2. Results of the shell model given with phase shift as a function of dimensionless frequency (a) for various radius ratios and indicated fixed permeability (transmissivity) and diffusivity ratios, (b) for various diffusivity ratios and indicated fixed permeability (transmissivity) and radius ratios, (c) for various permeability (transmissivity) ratios when storativity is covaried such that the diffusivity of shell and medium are identical and (d) for various identical diffusivity and permeability (transmissivity) ratios that correspond to an identical storativity for shell and medium at indicated fixed radius ratio. For identical permeability (transmissivity) of shell and medium, the diffusivity ratio is related to an inverse ratio of specific storage capacity (storativity), that is, D /D = s /s = S /S . 1 2 2 1 2 1 and j = 1, . . . N−1. For N = 1, c = m but for N > 1, to obtain c one has to successively determine c , c ... and c from 1 1 1 2 3 N k N N +1 K (η r )I (η r ) + I (η r )K (η r ) 1 N +1 N 0 N N 1 N N 0 N +1 N N +1 c =− k D N +1 N K (η r )K (η r ) − K (η r )K (η r ) 1 N +1 N 0 N N 1 N N 0 N +1 N k D N N +1 k D j +1 j I (η r ) c K (η r ) − I (η r ) + I (η r ) c K (η r ) + I (η r ) 0 j j j +1 1 j +1 j 1 j +1 j 1 j j j +1 0 j +1 j 0 j +1 j k D j j +1 c =− k D j +1 j K (η r ) c K (η r ) − I (η r ) − K (η r ) c K (η r ) + I (η r ) 0 j j j +1 1 j +1 j 1 j +1 j 1 j j j +1 0 j +1 j 0 j +1 j k D j j +1 C2 Parameter study For the parameter study, we restrict to the simple case of a single shell. We investigated the sensitivity of phase shift to the various model parameters by systematic parameter variation. Phase shift is reported as a function of r ω/D where D = D denotes the i shell shell 1 hydraulic diffusivity of the shell (Fig. C1). This expression can be understood either as a dimensionless frequency or an inverse dimensionless penetration depth given in multiples of the borehole radius. A sigmoidal shape of the relation between phase shift and dimensionless frequency is characteristic for a homogeneous medium. Small phase shifts occur at low frequencies and vice versa with a rather steep switch between the two limits around a dimensionless frequency of 1. This shape is significantly perturbed when ratios between the parameters of shell and Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 562 Y. Cheng and J. Renner medium between one-tenth and a factor of 10 are allowed. The high-frequency limit of 1/8 for phase shift is not strictly valid for the compound model but values as high as 1/5 are found. The phase shift for a shell with a low storage property compared to the medium exhibits a pronounced maximum that forms a shoulder on the steep increase characteristic for the homogeneous medium (Fig. C2a). The location of the maximum moves towards lower frequencies with increasing radius of the shell. A variation of the ratio of storage parameters for shell and medium for a fixed shell radius results in qualitatively similar phase-shift curves as the variation in shell radius (Fig. C2b). Yet, the maximum in phase transforms into a minimum preceded by a new maximum at lower frequency when the storage capacity in the shell switches from lower to higher than the storage capacity in the medium. When permeability (transmissivity) and specific storage capacity (storativity) are modified in the same way, that is, diffusivity is kept constant, the transition from the low- to the high-frequency regime is significantly perturbed. Low permeability and storativity in the shell relative to the surrounding medium leads to a transition that resembles a Heavyside function (Fig. C2c). High permeability and storativity in the shell gives rise to a pronounced maximum in phase shift at a dimensionless frequency of about 0.1 that reaches values as large as 1/5, that is, well above the limit for a homogeneous medium of 1/8. Covarying peremability and diffusivity and keeping the storage parameter uniform for shell and medium has qualitatively similar effects (Fig. C2d). APPENDIX D: BILINEAR FLOW MODEL The model consists of a slit with distinct hydraulic properties embedded in a homogeneous matrix and assumes that the direct volume flow from the source into the matrix along the x-direction is negligible (Fig. D1), but the fracture boundary is considered a harmonic source for flow into the matrix. The governing equations in the matrix and the fracture correspond to diffusion equations: ∂ p (x , y, t) = D ∇ p (x , y, t) (D1) m m ∂t ∂ p (x , t) ∂ p (x , t) 2D ∂ p (x , y, t) f f m m = D + (D2) ∂t ∂x δ ∂y |y|=δ/2 where p (x, y, t) is the pressure in the matrix, p (x, t) the pressure in the fracture, δ the width of the fracture and D and D the diffusivity of m f m f matrix and fracture, respectively. 2 2 As the direct volume flow from the source into the matrix along the x-direction is negligible, ∂ p (x, y, t)/∂x ≈ 0, so that eq. (D1) reduces to ∂ p (x , y, t) ∂ p (x , y, t) m m = D . (D3) ∂t ∂ y The pressure in the matrix, that is, the solution of eq. (D1), is given by the standard solution for diffusion from a harmonic source into a homogeneous semi-infinite medium (Carslaw & Jaeger 1986): (|y| − δ/2) exp − 4D (t − λ) | | m y − δ/2 p (x , y, t) = √ p (x,λ) dλ (D4) m f 3/2 2 π D (t − λ) m 0 Figure D1. Model for bilinear flow: the fracture boundary is a source for flow into the matrix in y-direction; no direct volume flow from the source into the matrix along the x-direction. Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 563 Figure D2. Phase shift for (a) constant-pressure boundary and (b) no-flow boundary at the tip of a fracture with finite length L and width δ = 0.096 m. (I) 2 −1 2 −1 2 −1 2 −1 2 −1 2 −1 D = 1m s , D = 10 m s and L = 20δ, (II) D = 1m s , D = 100 m s and L = 20δ, (III) D = 1m s , D = 100 m s and L = 100δ and m f m f m f 2 −1 2 −1 (IV) D = 1m s , D = 10 m s and L = 4δ. The relation for an infinite fracture is shown for comparison in (a). m f whose insertion into eq. (D2) yields ⎡ ⎤ (|y| − δ/2) exp − ⎢ ⎥ 4D (t − λ) ⎢ m ⎥ ∂ p (x , t) 2D ∂ |y| − δ/2 ∂ p (x , t) f m f ⎢ ⎥ D + √ p (x,λ) dλ = . (D5) f f ⎢ ⎥ 2 3/2 ∂x δ ∂y 2 π D ( ) ∂t 0 t − λ ⎣ m ⎦ |y|−δ/2=0 Differentiation of the second term by part restricting to the section y ≥ 0 because of the symmetry of the problem gives ⎡ % ⎤ (|y| − δ 2) exp − √ ⎢ ⎥ 2 t 4D (t − λ) ⎢ ⎥ (y − δ 2) m ∂ p (x , t) D ∂ p (x , t) f m f ⎢ ⎥ D + √ p (x,λ) 1 + dλ = . (D6) f f ⎢ 3/2 ⎥ ∂x πδ 2D (t − λ) ∂t (t − λ) 0 m ⎣ ⎦ y=δ 2 Applying y = δ/2weget 2 t ∂ p (x , t) D p (x,λ) ∂ p (x , t) f m f f D + √ dλ = (D7) ∂x πδ ∂t (t − λ) Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 564 Y. Cheng and J. Renner and thus for harmonic pressure variations in the fracture ∂ p (x) D A exp(i ωλ) f m D A exp(i ωt) + √ p (x) dλ = i ωA exp(i ωt)p (x). (D8) f f f ∂x πδ 2 (t − λ) The integral solution is A exp(i ωλ) dλ = γ (−0.5, i ωt) exp(i ωt) = [ (−0.5) − (−0.5, i ωt)] i ω exp(i ωt)(D9) (t − λ) where γ (−0.5, i ωt), (−0.5) and (−0.5, i ωt) represent the lower incomplete gamma function, the gamma function and the upper incomplete gamma function, respectively. For times longer than 1.25 times the period, the deviation of γ (−0.5, i ωt) from (−0.5) is less than 1 per cent and beyond 2.2 times the period the deviation falls below 0.5 per cent. Thus, we can approximate the integral as √ √ exp (i ωλ) dλ (−0.5) i ω exp (i ωt) =−2 iπω exp (i ωt) (D10) 3/2 (t − λ) to arrive at ∂ p (x) 2 D − iD ω p (x) = i ω p (x) . (D11) f m f f ∂x δ when canceling the harmonic time dependence. Eq. (D11) is a homogeneous second-order differential equation with the solution 1/2 & ' p (x) = c exp − √ 2 iπω+i ω x + dH (L) (D12) δD π subject to the boundary conditions p (x)→ 0as x →∞ (infinite fracture), or p (x) = p as x = L (finite fracture with constant-pressure f f 0 boundary), or ∂ p (x)/∂x = 0as x = L (finite fracture with no-flow boundary) where c is a constant, H(L) is a parameter defined by 0 L →∞ H (L) = (D13) 1 L finite and 1/2 & ' ⎪ D p − c exp − √ 2 iπω + i ω L for constant-pressure boundary ⎪ 0 δD π d =   (D14) √ √ 1/2 1/2 ⎪ & ' & ' √ √ ⎪ D D m m 2 iπω + i ω cx exp − 2 iπω + i ω L + const. for no-flow boundary √ √ δD π δD π f f To simplify the calculation, we assume the constant appearing for the no-flow boundary to equal zero without loss of generality because it represents a constant shift in pressure that can be accommodated by scale shifting. Using Darcy’s law, we get the flow rate as T ∂ p (x , t) Q (x , t) = 2πr hq (x , t) =−2πr f i i ρg ∂x 1/2 & ' TcA D = 2πr √ 2 iπω + i ω ρg δD π √   √ 1/2 1/2 & ' & ' √ √ D D m m · exp − √ 2 iπω+i ω x − H exp − √ 2 iπω + i ω L exp (i ωt) (D15) bc δD π δD π f f where T is transmissivity, ρ is fluid density and h is the height of the injection zone. We introduce a parameter reflecting the boundary condition at the fracture tip as 0 constant pressure boundary H = . (D16) bc 1 no flow boundary Downloaded from https://academic.oup.com/gji/article/212/1/543/4160098 by DeepDyve user on 13 July 2022 Periodic pumping tests for hydraulic characterization of faults 565 Finally, the complex injectivity results to 1/2 & ' ( ) Q x , t Tc D f m = 2πr √ 2 iπω + i ω p (x , t) ρg δD π f f √ √ 1/2 1/2 & ' & ' √ √ D D m m exp − √ 2 iπω + i ω x − H exp − √ 2 iπω + i ω L bc δD π δD π f f ·   . (D17) 1/2 & ' c exp − √ 2 iπω + i ω x + dH (L) δD π Specifically, for an infinite fracture, we find √ √ 1/2 Q (x = 0, t) T 2 i π D ω + i ω D f m m = 2πr √ (D18) p (x = 0, t) ρg πδD f f H(L)=0 with amplitude ratio and phase shift of √ √ 1/2 T 2 π D ω + ω D m m δ = 2πr (D19) qp i ρg δD and ϕ = arg 2 iπω + i ω , (D20) qp respectively, at the injection point x = 0(Fig. D2a). For a fracture with finite length and a constant-pressure boundary at its tip and assuming p = 0, we find Q (x = 0, t) p (x = 0, t) Q (x = 0, t) f f H(L)=0 =   (D21) 1/2 & ' p (x = 0, t) 1 − exp − √ 2 iπω + i ω L δD π at x = 0(Fig. D2a). Analogously, we find 1/2 & ' Q (x = 0, t) Q (x = 0, t) D f f m = 1 − exp − √ 2 iπω + i ω L (D22) p (x = 0, t) p (x = 0, t) δD π f f f H(L)=0 at x = 0(Fig. D2b).

Journal

Geophysical Journal InternationalOxford University Press

Published: Jan 1, 2018

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