TY - JOUR
AU - Han, Tongcheng
AB - SUMMARY Understanding the electrical properties of rocks under varying pressure is important for a variety of geophysical applications. This study proposes an approach to modelling the pressure-dependent electrical properties of porous rocks based on an effective medium model. The so-named Textural model uses the aspect ratios and pressure-dependent volume fractions of the pores and the aspect ratio and electrical conductivity of the matrix grains. The pores were represented by randomly oriented stiff and compliant spheroidal shapes with constant aspect ratios, and their pressure-dependent volume fractions were inverted from the measured variation of total porosity with differential pressure using a dual porosity model. The unknown constant stiff and compliant pore aspect ratios and the aspect ratio and electrical conductivity of the matrix grains were inverted by best fitting the modelled electrical formation factor to the measured data. Application of the approach to three sandstone samples covering a broad porosity range showed that the pressure-dependent electrical properties can be satisfactorily modelled by the proposed approach. The results demonstrate that the dual porosity concept is sufficient to explain the electrical properties of porous rocks under pressure through the effective medium model scheme. Electrical properties, High-pressure behaviour, Permeability and porosity 1 INTRODUCTION Electrical survey is an important geophysical method to obtain the deep Earth’s tectonic structure (e.g. MacGergor et al.2001; Ichiki et al.2009; Key 2012), to estimate the geo-resources content residing in the mid-depth sedimentary rocks (e.g. Ellis & Singer 2007; Constable 2010) and to provide the useful geotechnical parameters in the surface sediments (e.g. Cosenza et al.2006; Long et al.2012). Knowledge about the pressure-dependent electrical properties of rocks plays a key role in better understanding and interpreting the exploration data, especially when they are associated with varying depths. It is generally observed that electrical conductivity (σ, reciprocal of electrical resistivity ρ) decreases rapidly with initial increase in differential pressure (defined as the difference between confining and pore pressure) and the decrease becomes much gentler as the differential pressure is further increased, following a relationship described by an empirical expression (e.g. Jing et al.1990, 1992; Mahmood et al.1991; Han et al.2011):   \begin{equation*} \frac{1}{{\sigma (P)}} = \rho (P) = A + KP - B\exp ( - DP), \end{equation*} (1)where P is the differential pressure and coefficients A, K, B and D are fitting parameters for a given set of measurements. The pressure-dependent conductivity behaviour is usually considered to be controlled by the closure or dilation of pores and cracks with a broad distribution of pore stiffnesses, described by the pore structure of the rock (e.g. Cheng & Toksöz 1979; Tod 2002; David & Zimmerman 2012), and pore geometry models are accordingly developed to model and explain the dependence of electrical properties on pressure. Bernabe (1991) proposed three categories of pores based on 3-D qualitative observations of the pore structure of sandstones: large spherical pores at four-grain vertices (or nodel pores), tube-like throats at three-grain edges and narrow sheet-like throats at two-grain faces. The capillary model was then mathematically applied to the different types of pores to model the strongly pressure-dependent electrical properties associated with the sheet-like throats and the approximately constant parts associated with tube-like throats and the large nodel pores. While the above approach accounting for three types of pores remains valid, it has been recognized, more recently, that the pore structure of porous rocks can be plausibly simplified to a binary distribution (Shapiro 2003; Kaselow & Shapiro 2004; Shapiro & Kaselow 2005): stiff pores, which form most of the pore space and decrease linearly with pressure, and a small amount of compliant (or soft) pores that reduce exponentially with pressure. By applying the pressure-dependent dual porosity concept to Archie equation (Archie 1942), Kaselow & Shapiro (2004) gave a mathematical model that explained well the dependence of electrical properties on pressure. Unlike the mathematical models described above, which are either fitting equations with no physical meanings for the parameters or too complicated to carry out in practice, this paper aims to present an alternative effective medium approach to model the pressure-dependent electrical properties of porous rocks. Effective medium models are widely employed for simulating electrical properties of porous rocks (e.g. Bruggeman 1935; Hanai 1960, 1961; Bussian 1983; Berryman 1995; Asami 2002; Berg 2007; Han et al.2015) based simply and explicitly on the volume fractions and electrical properties of each of the rock-forming components and the geometric details of how the phases are arranged relative to each other (Mavko et al.2009). To achieve the goal, an effective medium model that calculates rock conductivity from the properties of both pores and matrix grains is first introduced and tested. The pore geometry is then assumed to be composed of stiff and compliant pores each with a constant aspect ratio. The pressure-dependent electrical properties of porous rocks are finally modelled by the effective medium model through the pressure-dependent volume fractions of the stiff and compliant pores. A similar way has been employed by Han (2016) to model the pressure dependency of rock velocity, and this study serves as a continuation and an extension of the work of Han (2016) to the area of electrical properties of porous rocks. 2 THEORETICAL MODELS 2.1 Maxwell–Garnett model The Maxwell–Garnett effective medium model (Garnett 1904, 1906) calculates homogeneous electrical conductivity by assuming inclusions with specific shapes randomly dispersed in the host background (Fig. 1a), through the expression:   \begin{equation*} {\sigma _{\mathrm{ MG}}} = {\sigma _\mathrm{ h}} + \frac{{\frac{1}{3}\sum\limits_{i = 1}^n {{f_i}\left( {{\sigma _i} - {\sigma _\mathrm{ h}}} \right)\sum\limits_{j = x,y,z} {\frac{{{\sigma _\mathrm{ h}}}}{{{\sigma _\mathrm{ h}} + L_i^j\left( {{\sigma _i} - {\sigma _\mathrm{ h}}} \right)}}} } }}{{1 - \frac{1}{3}\sum\limits_{i = 1}^n {{f_i}\left( {{\sigma _i} - {\sigma _\mathrm{ h}}} \right)\sum\limits_{j = x,y,z} {\frac{{L_i^j}}{{{\sigma _\mathrm{ h}} + L_i^j\left( {{\sigma _i} - {\sigma _\mathrm{ h}}} \right)}}} } }}, \end{equation*} (2)where σh is the conductivity of the host background, σi and fi are the conductivity and volume fraction of the ith inclusion, respectively, and Li is the depolarization factor (e.g. Asami 2002) of the oblate inclusion (Rx = Ry ≥ Rz, where Rx, Ry and Rz are the semi-axes along the x-, y- and z-axes, respectively) with aspect ratio of αi (α = Rz/Rx):   \begin{equation*} L_i^z = \frac{1}{{1 - {\alpha _i}^2}} - \frac{{{\alpha _i}}}{{{{\left( {1 - {\alpha _i}^2} \right)}^{{3 / 2}}}}}{\cos ^{ - 1}}{\alpha _i} \end{equation*} (3)and   \begin{equation*} L_i^x = L_i^y = \frac{{1 - L_i^z}}{2}. \end{equation*} (4) Figure 1. View largeDownload slide Schematic diagrams showing the difference between the microstructures of (a) the Maxwell–Garnett model and (b) the Textural model. In the Textural model, the new background (with conductivity of σCRI) is described by the complex refractive index (CRI) mixing law while the host material (with conductivity of σh) in the Maxwell–Garnett model is now assumed as an inclusion also. σ1, σ2 and σi are the conductivity of the inclusions. Figure 1. View largeDownload slide Schematic diagrams showing the difference between the microstructures of (a) the Maxwell–Garnett model and (b) the Textural model. In the Textural model, the new background (with conductivity of σCRI) is described by the complex refractive index (CRI) mixing law while the host material (with conductivity of σh) in the Maxwell–Garnett model is now assumed as an inclusion also. σ1, σ2 and σi are the conductivity of the inclusions. 2.2 Textural model The Maxwell–Garnett model assumes the host background to be always connected while the inclusions dilute. This, however, is not the case for water saturated rocks where both water and the solid matrix are continuous, and it is therefore not possible to assign one phase as the host background. Based on experimental investigations, Seleznev et al. (2006) developed a modified version of the Maxwell–Garnett model where both pores and the matrix grains are imbedded in the background with electrical properties described by the complex refractive index (CRI) mixing law (Schön 1996). The so-called Textural model (Fig. 1b) gives   \begin{equation*} {\sigma _{T\mathrm{ ex}}} = {\sigma _{\mathrm{ CRI}}} + \frac{{\frac{1}{3}\sum\limits_{i = 1}^n {{f_i}\left( {{\sigma _i} - {\sigma _{\mathrm{ CRI}}}} \right)\sum\limits_{j = x,y,z} {\frac{{{\sigma _{\mathrm{ CRI}}}}}{{{\sigma _{\mathrm{ CRI}}} + L_i^j\left( {{\sigma _i} - {\sigma _{\mathrm{ CRI}}}} \right)}}} } }}{{1 - \frac{1}{3}\sum\limits_{i = 1}^n {{f_i}\left( {{\sigma _i} - {\sigma _{\mathrm{ CRI}}}} \right)\sum\limits_{j = x,y,z} {\frac{{L_i^j}}{{{\sigma _{\mathrm{ CRI}}} + L_i^j\left( {{\sigma _i} - {\sigma _{\mathrm{ CRI}}}} \right)}}} } }}, \end{equation*} (5)where   \begin{equation*} {\sigma _{\mathrm{ CRI}}} = {\left( {\sum\limits_{i = 1}^n {{f_i}\sqrt {{\sigma _i}} } } \right)^2}. \end{equation*} (6) Fig. 1 shows schematically the difference between the microstructures described by the Maxwell–Garnett model and the Textural model. It should be noted that the host background in the Maxwell–Garnett model is now employed as inclusion in the Textural model, and therefore the volume fractions (fi) of the inclusions in eq. (5) should sum up to 1, which differs from that in the Maxwell–Garnett model where the host material is excluded from the inclusions. To test the validity of the multicomponent Textural model, a multiple-salinity partially saturated clay-rich sandstone (sample 3279B from Clavier et al.1984) is considered. The sample with a porosity (ϕ) of 0.264 was saturated with four saline brines at various saturations as can be seen in Fig. 2, which shows the comparison of the Textural model prediction with the experimental data. In the calculation, the volume fraction of the matrix grain (fm) was determined as fm = 1 – ϕ, and the volume factions of water (fw) and gas (fg) as fw = ϕSw and fg = ϕ(1 – Sw), respectively, where Sw is the water saturation obtained from the experimental measurements. By using aspect ratios of 0.07, 0.11 and 0.14 for the water, gas and matrix grain phases, respectively and a reasonable matrix grain conductivity of σm = 0.0236 S m−1 (the sandstone sample considered was high in clay content showing concentration of clay counter-ions per unit pore volume Qv = 0.36 meq cm−3), the Textural model gives satisfactory fit to the experimental data with a squared correlation coefficient R2 = 0.9994. Figure 2. View largeDownload slide Comparison of the Textural model prediction (solid curves) with the measured conductivity (filled squares) of a clay-rich sandstone 3279B from Clavier et al. (1984). The modelled and measured results are in good agreement with squared correlation coefficient R2 = 0.9994. Figure 2. View largeDownload slide Comparison of the Textural model prediction (solid curves) with the measured conductivity (filled squares) of a clay-rich sandstone 3279B from Clavier et al. (1984). The modelled and measured results are in good agreement with squared correlation coefficient R2 = 0.9994. 2.3 Dual porosity model Having tested the validity of the Textural model, we proceed to apply the model to calculate the pressure-dependent electrical properties of porous rocks. To do this, the pore geometry of the rock is assumed to be composed of stiff and compliant pores as described by the dual porosity model of Shapiro (2003), who showed that the linear reduction of stiff porosity (ϕs) and exponential decrease of compliant porosity (ϕc) with increasing differential pressure P can be written as   \begin{equation*} {\phi _\mathrm{ s}}\left( P \right) = {\phi _{\mathrm{ s}_{ 0}}} - P\left( {{C_{\mathrm{ d}\mathrm{ s}_{ 0}}} - {C_\mathrm{ m}}} \right), \end{equation*} (7)and   \begin{equation*} {\phi _\mathrm{ c}}\left( P \right) = {\phi _{\mathrm{ c}_{ 0}}}\exp \left( { - {\theta _\mathrm{ c}}{C_{\mathrm{ d}\mathrm{ s}_{ 0}}}P} \right), \end{equation*} (8)where ϕs0 and ϕc0 are the stiff and compliant porosity at zero pressure, Cds0 is the stiff limit compressibility (reciprocal of elastic bulk modulus) of the dry rock at high enough confining stress where all compliant porosity is closed, Cm = 1/Km (where Km is the bulk modulus of the rock matrix grain and is set to be 37 GPa for the sandstones in the following analysis, see e.g. Mavko et al.2009) is the compressibility of the matrix grain and θc is the pressure sensitivity coefficient for compliant pores. The pressure-dependent total porosity of a rock is then expressed as   \begin{equation*} \phi \left( P \right) = {\phi _\mathrm{ s}}\left( P \right) + {\phi _\mathrm{ c}}\left( P \right). \end{equation*} (9) The variations of the stiff and compliant porosity with differential pressure are estimated by fitting the dual porosity model (i.e. eq. 9) to the measured total porosity as a function of pressure and the obtained pressure-dependent stiff and compliant porosity is then employed as input parameters for the Textural model to determine the pressure dependency of electrical properties of porous rocks. 3 MODELLING THE PRESSURE-DEPENDENT ELECTRICAL PROPERTIES OF POROUS ROCKS 3.1 Workflow To model the pressure-dependent electrical properties of porous rocks through effective medium model (i.e. the Textural model), dual porosity of the pore structure is assumed. It is further assumed that the stiff and compliant pores can be represented by spheroids with specific aspect ratios, and the stiff and compliant pore aspect ratios stay constant as the differential pressure changes (see Section 4 for the discussion of the effects of varying pore aspect ratios on the modelled electrical properties). The Textural model is then employed to calculate the electrical properties as a function of differential pressure through the workflow summarized below. Note that in this work the electrical properties are expressed in terms of electrical formation factor (F), a ratio of the conductivity of the water saturating the rock (σw, which is assumed to be 4.69 S m−1 for sea water) to the conductivity of the fully saturated sandstone (σr), as F = σw/σr. The variations of stiff and compliant porosity with differential pressure are inverted by fitting the dual porosity model (i.e. eq. 9) to the measured total porosity of the rock as a function of pressure using nonlinear Levenberg–Marquardt algorithm (Levenberg 1944; Marquardt 1963). The aspect ratios of the stiff (αs) and compliant pores (αc) and the aspect ratio and electrical conductivity of the matrix grain, αm and σm, respectively, are unknown parameters. They are estimated by best fitting the modelled pressure-dependent electrical formation factor to the measured values, using the Levenberg–Marquardt regression algorithm (Levenberg 1944; Marquardt 1963). Now that the aspect ratios of all the phases (i.e. the stiff and compliant pores and the matrix grain), the pressure-dependent stiff and compliant porosity and the matrix grain electrical conductivity are obtained, they are used as input parameters into the Textural model to simulate the pressure-dependent electrical formation factor of porous rocks. 3.2 Real data examples This section gives some examples of the modelled pressure-dependent electrical properties of porous sandstones using the proposed approach in comparison with their experimental measurement results. The samples consist of a high (25.57 per cent), medium (15.2 per cent) and low porosity (∼4.5 per cent) sandstone to cover a broad porosity range of typical reservoir sandstones. The approach is first applied to modelling the pressure-dependent electrical formation factor of the Castlegate-31 sandstone with high porosity of 25.57 per cent (Hausenblas 1995), for which the electrical formation factor was measured in the differential pressure range 1–70 MPa. The variations of the stiff and compliant porosity as a function of differential pressure are determined by fitting the dual porosity model (i.e. eq. 9) to the measured total porosity data (R2 = 0.9980 as shown in Fig. 3a). With the stiff and compliant pore aspect ratios determined to be 0.1 and 0.001, respectively, and the aspect ratio and electrical conductivity of the matrix grain set to be 0.2 and 0.048 S m−1, respectively (see Table 1 for the value of the parameters for all the samples), the modelled pressure-dependent electrical formation factor (Fig. 3b) gives excellent fit to the laboratory measurements (R2 = 0.9937). Figure 3. View largeDownload slide (a) Estimation of the pressure-dependent stiff and compliant porosity by fitting the dual porosity model to the measured variation of total porosity with differential pressure and (b) comparison of the measured and modelled pressure-dependent electrical formation factor for the high porosity Castlegate-31 sandstone (Hausenblas 1995). The aspect ratios for the stiff and compliant pores are determined to be 0.1 and 0.001, respectively and the matrix grain aspect ratio and conductivity are taken to be 0.2 and 0.048 S m−1, respectively to give best fit to the measured pressure-dependent formation factor with squared correlation coefficients R2 = 0.9937. Figure 3. View largeDownload slide (a) Estimation of the pressure-dependent stiff and compliant porosity by fitting the dual porosity model to the measured variation of total porosity with differential pressure and (b) comparison of the measured and modelled pressure-dependent electrical formation factor for the high porosity Castlegate-31 sandstone (Hausenblas 1995). The aspect ratios for the stiff and compliant pores are determined to be 0.1 and 0.001, respectively and the matrix grain aspect ratio and conductivity are taken to be 0.2 and 0.048 S m−1, respectively to give best fit to the measured pressure-dependent formation factor with squared correlation coefficients R2 = 0.9937. Table 1. Measured porosity and parameters employed to model the pressure-dependent electrical properties and the corresponding squared correlation coefficients between the modelled and measured formation factors for the three sandstone samples used in the study.    Castlegate-31  Kirkwood  Fahler-154  Porosity  25.57%  15.2%  ∼4.5%  αs  0.1  0.07  0.0062  αc  0.001  0.0044  0.0005  αm  0.2  0.3  0.9  σm (S m−1)  0.048  0.000062  0.000030  R2  0.9937  0.9985  0.9750     Castlegate-31  Kirkwood  Fahler-154  Porosity  25.57%  15.2%  ∼4.5%  αs  0.1  0.07  0.0062  αc  0.001  0.0044  0.0005  αm  0.2  0.3  0.9  σm (S m−1)  0.048  0.000062  0.000030  R2  0.9937  0.9985  0.9750  View Large Table 1. Measured porosity and parameters employed to model the pressure-dependent electrical properties and the corresponding squared correlation coefficients between the modelled and measured formation factors for the three sandstone samples used in the study.    Castlegate-31  Kirkwood  Fahler-154  Porosity  25.57%  15.2%  ∼4.5%  αs  0.1  0.07  0.0062  αc  0.001  0.0044  0.0005  αm  0.2  0.3  0.9  σm (S m−1)  0.048  0.000062  0.000030  R2  0.9937  0.9985  0.9750     Castlegate-31  Kirkwood  Fahler-154  Porosity  25.57%  15.2%  ∼4.5%  αs  0.1  0.07  0.0062  αc  0.001  0.0044  0.0005  αm  0.2  0.3  0.9  σm (S m−1)  0.048  0.000062  0.000030  R2  0.9937  0.9985  0.9750  View Large The obtained stiff and compliant pore aspect ratios are very consistent with the values (0.12 and 0.0012 for the stiff and compliant pores, respectively) inverted by Han (2016) from the pressure-dependent rock velocity of a sandstone sample, and are also in general agreement with the pore aspect ratios employed by Xu & White (1995) for the elastic properties of sandstones. The inverted matrix grain aspect ratio of 0.2 is in the same order of magnitude for sandstone grains determined by Friedman (2005) and Louis et al. (2005) based on fitting to experimental data and computer processing of the microstructure images, respectively. An explanation to the obtained non-insulating matrix grains is given in Section 4. The successful fit of the modelling results to the experimental data using reasonable parameters preliminarily indicates the validity of the proposed effective medium approach for modelling the pressure-dependent electrical properties by simply employing constant aspect ratios for the stiff and compliant pores and their volume variations with differential pressure. To better confirm the applicability of the approach, it is further applied to another two sandstone samples with lower porosities. Fig. 4(a) shows the comparison of the measured pressure dependency of the total porosity with the inverted results based on the dual porosity model (i.e. eq. 9) for the (15.2 per cent porosity) Kirkwood sandstone sample (Wyble 1958), with R2 = 0.9869. Using the obtained variations of the stiff and compliant porosity in combination with their constant aspect ratios, as well as the matrix grain aspect ratio and electrical conductivity as shown in Table 1, the proposed modelling approach gives successful fit to the measured pressure-dependent electrical formation factor (R2 = 0.9985) as shown in Fig. 4(b). Figure 4. View largeDownload slide (a) Estimation of the pressure-dependent stiff and compliant porosity by fitting the dual porosity model to the measured variation of total porosity with differential pressure and (b) comparison of the measured and modelled pressure-dependent electrical formation factor for the medium porosity Kirkwood sandstone (Wyble 1958). The aspect ratios for the stiff and compliant pores are determined to be 0.07 and 0.0044, respectively and the matrix grain aspect ratio and conductivity are taken to be 0.3 and 0.000062 S m−1, respectively to give best fit to the measured pressure-dependent formation factor with squared correlation coefficients R2 = 0.9985. Figure 4. View largeDownload slide (a) Estimation of the pressure-dependent stiff and compliant porosity by fitting the dual porosity model to the measured variation of total porosity with differential pressure and (b) comparison of the measured and modelled pressure-dependent electrical formation factor for the medium porosity Kirkwood sandstone (Wyble 1958). The aspect ratios for the stiff and compliant pores are determined to be 0.07 and 0.0044, respectively and the matrix grain aspect ratio and conductivity are taken to be 0.3 and 0.000062 S m−1, respectively to give best fit to the measured pressure-dependent formation factor with squared correlation coefficients R2 = 0.9985. The determined grain aspect ratio of 0.3 is higher than that for the Castlegate sandstone with higher porosity, and this is consistent with the observation (e.g. Selley 1976) that total porosity decreases with increasing grain sphericity and roundness (i.e. increasing grain aspect ratio). It is also reasonable that when the grains become more rounded the pores tend to be more ‘squeezed’ and therefore the pore aspect ratios become smaller. This explains the lower stiff pore aspect ratio obtained in the Kirkwood sandstone than that in the higher porosity Castlegate sandstone, although the compliant pore aspect ratio of the Kirkwood sample is slightly higher but in the same order of magnitude than Castlegate sandstone. The fit of the dual porosity model (eq. 9) to the measured variations of total porosity with differential pressure up to 36 MPa for the low porosity (∼4.5 per cent) Fahler-154 sandstone sample (Yale 1984) with R2 = 0.9996 is given in Fig. 5(a). By using the obtained pressure-dependent stiff and compliant porosity, the inverted aspect ratios of the stiff and compliant pores, and the estimated matrix grain aspect ratio and electrical conductivity listed in Table 1, the modelled pressure-dependent electrical formation factor gives satisfactory fit to the measured data with squared correlation coefficient R2 = 0.9750, as shown in Fig. 5(b). Figure 5. View largeDownload slide (a) Estimation of the pressure-dependent stiff and compliant porosity by fitting the dual porosity model to the measured variation of total porosity with differential pressure and (b) comparison of the measured and modelled pressure-dependent electrical formation factor for the low porosity Fahler-154 sandstone (Yale 1984). The aspect ratios for the stiff and compliant pores are determined to be 0.0062 and 0.0005, respectively and the matrix grain aspect ratio and conductivity are taken to be 0.9 and 0.000030 S m−1, respectively to give best fit to the measured pressure-dependent formation factor with squared correlation coefficients R2 = 0.9750. Figure 5. View largeDownload slide (a) Estimation of the pressure-dependent stiff and compliant porosity by fitting the dual porosity model to the measured variation of total porosity with differential pressure and (b) comparison of the measured and modelled pressure-dependent electrical formation factor for the low porosity Fahler-154 sandstone (Yale 1984). The aspect ratios for the stiff and compliant pores are determined to be 0.0062 and 0.0005, respectively and the matrix grain aspect ratio and conductivity are taken to be 0.9 and 0.000030 S m−1, respectively to give best fit to the measured pressure-dependent formation factor with squared correlation coefficients R2 = 0.9750. The stiff and compliant pore aspect ratios further decrease whereas the matrix grain aspect ratio further increases as the total porosity gets lower, and this is in general consistency with the results presented above. The excellent fit between the modelled and measured pressure-dependent electrical formation factor for the three sandstone samples covering a broad porosity range, together with the reasonable parameters employed for the calculation strongly, confirms the applicability of the proposed effective medium approach to model the pressure-dependent electrical properties of porous rocks. 4 DISCUSSION An approach has been proposed and validated to model the pressure-dependent electrical properties of porous rocks on the basis of an effective medium model making use of two constant aspect ratios for the spheroids representing the stiff and compliant pores of the dual porosity model (Shapiro 2003), their volume fraction changes with differential pressure and the matrix grain properties. The electrical model that the approach based on is the Textural model (Seleznev et al.2006). The Textural model is a modified version of the Maxwell–Garnett model (Garnett 1904, 1906) but assumes both ellipsoidal grains and pores to be dispersed in the host background described by the CRI mixing law, thus keeping the expected microstructure consistent with real rocks where both conductive pore fluids and relatively insulating grains are continuous. This is the main advantage of the Textural model over the majority of other electrical effective medium models (e.g. Bruggeman 1935; Hanai 1960, 1961; Bussian 1983; Asami 2002; Berg 2007; Han et al.2015) that imply isolated grains are imbedded into the connecting fluid background, which does not correspond to reality. However, there are many other models (e.g. Sheng 1991; Koelman & Kuijper 1997; Kazatchenko et al.2004; Aquino-Lopez et al.2011) that assume realistic rock microstructures can be potentially employed to form the basis of the current modelling approach, and a comparison of the modelling results of these models will be presented in a separate study. As mentioned in the context, the electrical properties are expressed in terms of electrical formation factor. This is because the conductivity of the fluids saturating the sandstone samples is not reported in the cited experimental studies (i.e. Wyble 1958; Yale 1984; Hausenblas 1995), and without such fluid conductivity it is impossible to calculate the bulk conductivity of the rocks. A conductivity of 4.69 S m−1 for sea water is assumed for the pore fluid in the calculation. Further investigation of the effect of the pore water conductivity on the inverted pore aspect ratios and the grain properties (Fig. 6) is as an example for the medium porosity Kirkwood sandstone sample with varying pore water conductivity from 0.01 to 10 S m−1, a reasonable water salinity that may appear in most of the underground environment. The constant aspect ratios obtained for the two types of pores and the matrix grain indicate that they are unique to the sample and not influenced by the pore water conductivity. These aspect ratios together with the stiff and compliant porosity determine the exponential dependence of the electrical properties on differential pressure. The only changing parameter with water conductivity from the inversion is the matrix grain conductivity, which is found to be in positive linear correlation with the water conductivity on a logarithmic scale. This is contradictory to concept of surface conductivity associated with clay minerals arising from the electrochemical interactions of the clay–water system (e.g. Revil 2013), which is essentially the same as increasing the bulk conductivity of the sand grain to get a new effective value (De Lima & Sharma 1992), and decreases with increasing pore water conductivity. Therefore, the inverted matrix grain conductivity is not interpreted as surface conductivity but can be regarded as a fitting parameter that contributes (in addition to the water conductivity) to the magnitude of the exponential variation of the electrical properties with differential pressure determined by the grain and pore aspect ratios and their volume fractions. Figure 6. View largeDownload slide Variations of the inverted stiff (αs) and compliant (αc) pore aspect ratios and aspect ratio (αm) and conductivity (σm) of the matrix grains with pore water conductivity. The red dot sands for the grain conductivity obtained for the water conductivity of 4.69 S m−1 employed in the context. Figure 6. View largeDownload slide Variations of the inverted stiff (αs) and compliant (αc) pore aspect ratios and aspect ratio (αm) and conductivity (σm) of the matrix grains with pore water conductivity. The red dot sands for the grain conductivity obtained for the water conductivity of 4.69 S m−1 employed in the context. The pore structure in the modelling approach is simplified to be composed of stiff and compliant pores according to the dual porosity model (Shapiro 2003). While the variations of the stiff and compliant porosity with differential pressure and their constant aspect ratios are already obtained, it is useful and interesting to investigate the contributions of the different types of pores to the bulk electrical properties of the rock sample. Fig. 7 gives an example of the modelled pressure-dependent electrical formation factor from the different pores for the Castlegate-31 sandstone (Hausenblas 1995). With the increase in differential pressure, the formation factor increases approximately linearly with the decrease in stiff porosity, and the stiff porosity dominates the formation factor at high pressure where the compliant pores have all closed. The compliant porosity, on the other hand, although contributes a small amount to the total porosity, has an effect of significantly increasing the formation factor in an exponential manner with increasing differential pressure. The effective medium modelled contributions of the stiff and compliant porosity to the pressure dependency of electrical properties of porous rocks are consistent with the results of Kaselow & Shapiro (2004) based on mathematical derivation, suggesting the simplification of the complicated pore structure to stiff and compliant pores is sufficient to explain the observed variations of electrical properties with pressure, and also confirming the feasibility of using an effective medium model to simulate the pressure-dependent electrical properties of porous rocks. Figure 7. View largeDownload slide Modelled contributions of the stiff and compliant porosity to the pressure dependency of electrical formation factor for the Castlegate-31 sandstone (Hausenblas 1995). The dashed red and blue curves are for a hypothetical rock with only stiff and compliant pores, respectively. Figure 7. View largeDownload slide Modelled contributions of the stiff and compliant porosity to the pressure dependency of electrical formation factor for the Castlegate-31 sandstone (Hausenblas 1995). The dashed red and blue curves are for a hypothetical rock with only stiff and compliant pores, respectively. The aspect ratios for the stiff and compliant pores are assumed to be constant regardless of the changing pressure. This is not the case in real rocks where in addition to the change of the pore volume, the aspect ratios of the pores also change with pressure (e.g. Toksöz et al.1976). However, as shown in Fig. 8 for the Kirkwood sandstone (Wyble 1958), the variations in the compliant pore aspect ratio only have a minor effect in affecting the formation factor, especially when the stiff pore aspect ratio is below 0.1, a range obtained for the stiff pore aspect ratio of all the three samples in the study. The change in the stiff pore aspect ratio affects the electrical properties more significantly, but it is less possible for the stiff pores to deform dramatically (e.g. Shapiro et al.2015), as least in the pressure range under consideration (Fortin et al.2007). In fact the decrease in the pore aspect ratio with increasing differential pressure (if porosity keeps unchanged) will not increase the formation factor but reduce it (as can be seen in Fig. 8), as the decreasing pore aspect ratio may enhance the connectivity between the conductive pore fluids leading to an increasing electrical conductivity but decreasing electrical formation factor. Therefore, the observed increasing formation factor with pressure is a balanced result of the decreased pore aspect ratio and volume, where the effects of the reducing pore volume dominate and the effects of changes in the pore aspect ratio are minor and thus the pore aspect ratios can be effectively considered constant. Figure 8. View largeDownload slide Modelled effects of varying stiff and compliant pore aspect ratios on the electrical formation factor for the Kirkwood sandstone (Wyble 1958). Figure 8. View largeDownload slide Modelled effects of varying stiff and compliant pore aspect ratios on the electrical formation factor for the Kirkwood sandstone (Wyble 1958). It has been shown that the proposed approach models successfully the pressure-dependent electrical properties of porous rock by assuming a binary distribution of the pore structure. On the other hand, a pore aspect ratio spectrum has been used to model the pressure dependency of elastic and electrical properties (e.g. Cheng & Toksöz 1979; David & Zimmerman 2012; Han et al.2016a). Theoretically, the more the number of pores assigned in the spectrum, the better it can model the pressure-dependent physical properties, but also the larger number of unknown parameters to be inverted for. It is therefore logical and interesting to investigate whether the pressure-dependent electrical properties can be modelled by regarding the whole pore structure as one pore type with a constant aspect ratio, because if this is possible it will make the modelling procedure much easier and simpler. Fig. 9 shows the misfit (expressed as the difference between the modelled and the measured formation factor divided by the measured values) between the measured and the modelled variations of formation factor with pressure using the approach proposed in this work (i.e. assuming the pore structure to be composed of stiff and compliant pores) in comparison with that between the measured and modelled formation factor change by assuming that all the pores have the same aspect ratio (i.e. directly from the total porosity) for the Kirkwood sandstone (Wyble 1958). The significant and systematic discrepancy between the measurement and the modelling results based on the total porosity suggests that it is practically impossible to accurately model the measured pressure-dependent electrical properties by regarding the pore structure of one type of pore with a constant aspect ratio. This, in turn, indicates that the separation of the pores to stiff and compliant pores as demonstrated in this work seems to be the minimum number in the pore spectrum required to best describe the pressure-dependent electrical properties of porous rocks. Figure 9. View largeDownload slide A comparison of the modelled and measured electrical formation factor misfit based on the pressure-dependent dual porosity and total porosity for the Kirkwood sandstone (Wyble 1958). Figure 9. View largeDownload slide A comparison of the modelled and measured electrical formation factor misfit based on the pressure-dependent dual porosity and total porosity for the Kirkwood sandstone (Wyble 1958). The electrical properties have been presented as functions of the differential pressures in the context; however, it is the porosity which is affected by the differential pressure that directly causes the changes in the electrical properties. Therefore, it is interesting to show the relationship between the electrical formation factor and the varying porosity as the differential pressure changes. As given in Fig. 10, the increasing differential pressure reduces the porosity (in the form of normalized porosity defined as porosity variations with respect to initial porosity), which in turn leads to an increase in the formation factor. The formation factor is approximately linearly correlated with porosity, this is consistent with the results obtained by Han et al. (2016b) who show that the dual porosity concept is sufficient to explain the pressure dependency of elastic, electrical and joint elastic–electrical properties of saturated porous sandstones. Figure 10. View largeDownload slide Measured (squares) and modelled (curves) variations of the formation factor with normalized porosity. The normalized porosity is the varying porosity with pressure divided by the lowest measured porosity of each sample. Figure 10. View largeDownload slide Measured (squares) and modelled (curves) variations of the formation factor with normalized porosity. The normalized porosity is the varying porosity with pressure divided by the lowest measured porosity of each sample. The stiff and compliant pore aspect ratios obtained in this work for the electrical properties are consistent with the results of Han (2016) for the elastic properties of porous rocks. This suggests the pore aspect ratios inverted from the pressure-dependent electrical properties can be used to model the pressure dependency of elastic properties (and vice versa) if the matrix grain properties are also available. This is theoretically possible because the pressure dependency of both elastic and electrical properties is controlled by the microstructure of the rock. In fact, a similar concept of dual type of pores with varying aspect ratios as a function of porosity has been employed by Kazatchenko et al. (2004) and Aquino-Lopez et al. (2011, 2015) to jointly model the elastic and electrical properties of reservoir rocks, although the lack of direct electrical measurements on the samples constrain the validity of their modelling results. Nevertheless, more work will be needed to develop rock physics models for the elastic and electrical properties based on the same microstructure (Han et al.2016a), and to acquire in the laboratory simultaneously the variations of porosity, elastic and electrical properties with differential pressure to accurately validate the models built. This will form part of the future research. 5 CONCLUSIONS An approach has been proposed to model the pressure-dependent electrical properties of porous rocks based on an effective medium model, namely the Textural model, which describes the microstructure that resembles the reality of real rocks. Using the variations of stiff and compliant porosity with differential pressure extracted from the measured pressure-dependent total porosity, the constant stiff and compliant pore aspect ratios and the aspect ratio and electrical conductivity of the matrix grains inverted by best fitting the modelled electrical formation factor to the measured data, the approach modelled satisfactorily the pressure-dependent electrical properties of three sandstone samples covering a broad porosity range. The results showed that the variations of electrical properties of porous rocks with pressure can be sufficiently modelled by the dual porosity concept through the effective medium model scheme, and the dual porosity separation of the pore structure seemed to be the simplest pore geometry required to accurately model the pressure-dependent electrical properties of porous rocks. ACKNOWLEDGEMENTS The author would like to thank Prof. L. 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TI - An effective medium approach to modelling the pressure-dependent electrical properties of porous rocks
JF - Geophysical Journal International
DO - 10.1093/gji/ggy125
DA - 2018-07-01
UR - https://www.deepdyve.com/lp/oxford-university-press/an-effective-medium-approach-to-modelling-the-pressure-dependent-uYibw9vC60
SP - 70
EP - 78
VL - 214
IS - 1
DP - DeepDyve
ER -