Porous media’s porosity value is commonly taken as a constant for a given granular texture free from any type of imposed loads. Although such definition holds for those media at hydrostatic equilibrium, it might not be hydrodynamically true for media subjected to the flow of fluids. This article casts light on an alternative vision describing porosity as a function of fluid velocity, though the media’s solid skeleton does not undergo any changes and remain essentially intact. Carefully planned laboratory experiments support such as hypothesis and may help reducing reported disagreements between observed and actual behaviors of nonlinear flow regimes. Findings indicate that the so-called Stephenson relationship that enables esti - mating actual flow velocity is a case that holds true only for the Darcian conditions. In order to investigate the relationship, an accurate permeability should be measured. An alternative relationship, therefore, has been proposed to estimate actual pore flow velocity. On the other hand, with introducing the novel concept of effective porosity, that should be determined not only based on geotechnical parameters, but also it has to be regarded as a function of the flow regime. Such a porosity may be affected by the flow regime through variations in the effective pore volume and effective shape factor. In a numerical justification of findings, it is shown that unsatisfactory results, obtained from nonlinear mathematical models of unsteady flow, may be due to unreliable porosity estimates. Keywords Porosity function · Granular porous media · Non-Darcy flow · Boundary layer Introduction saturated and unsaturated zone, and transportation of water- soluble pollutants in the soil. Hydrodynamically speaking flow of fluids through porous Unfortunately, it is recognized that (1) is valid only as media may be described by a simple rule commonly known the pressure gradient or flow velocity is very small. As the as the Darcy Law that may be rewritten as (Leps 1975; Reynolds number (Re) increases up to a critical value, the Ahmed and Sunada 1969; Haber and Mauri 1983; Ward relationship will become nonlinear. To provide a univer- 1965; Pavlovski 1940; Shokri et al. 2014): sal relation, including this nonlinear effect, Forchheimer proposed an empirical formula (Forchheimer equation) V =−K ∇h (1) (Mccorquodale et al. 1978; Leu et al. 2009; Chai et al. 2010; where the conductivity term (m/s) in this linear relation- Wahyudi et al. 2002; Shokri and Sabour 2014) ship is described by K with a negative sign indicating an occurrence of the fluid flow from a relatively high head to a −∇P = aV + bV (2) relatively low head locates, and the gradient in the relevant where P , V denote the pore pressure, flow velocity and a and driving head is ∇h . In groundwater hydrology, the knowl- b are nonlinear coefficients depending on fluid properties, edge of saturated hydraulic conductivity of porous media is the pore size, porosity and shape. necessary for modeling the water o fl w in the soil, both in the In a 2D space, one may combine continuity equation + = 0 with (1) to arrive at a so-called Laplacian x y * Morteza Shokri equation that describes the flow domain as: msh.shokri@gmail.com; m.shokri@basu.ac.ir ( + )= 0 xx yy (3) Civil Engineering Department, Bu-Ali Sina University, Kabodarahang, Hamedan, Iran Vol.:(0123456789) 1 3 92 Page 2 of 8 Applied Water Science (2018) 8:92 where is a scalar function of the head (h), and are using (8) and (10) for estimating actual velocity. He eventually y x differentiations of in the x and y directions, respectively. ended up with an alternative relationship (Parkin 1971) as: In coarse granular media, the flow velocity is much higher V = Q∕(A ⋅ n ) (11) under the same driving head compared to fine-grained soils where is a coefficient which 0.75 ≤ ≤ 1 depending on the due to the higher hydraulic conductivity that in turn makes surface geometry of the pores. Eq. (1) invalid. This nonlinear laminar flow regime persists to a Reynolds number = 150 (Burcharth and Andersen 1995). An alternative equation may be employed instead, there- Theoretical basis of the porosity function fore, enabling more reliable estimates of energy-loss through pores of such media. That equation may be rewritten as: In the past five decades, numerous studies have been focused � � on the evaluation of the effects of physical properties of the V = ∇h ∕m (4) granular porous media on the hydraulic conductivity for both linear and nonlinear flows. This is due to the fact that for the where m and are empirical values depending on the media/ Darcy flow conditions in a saturated media, one may define a fluid properties. Parkin (1971) could combine (4 ) with conti- fluid flow in terms of hydraulic conductivity as follows: nuity equation to develop a partial differential equation gov - h h h erning non-Darcian flows in porous media as well (Bazargan V =−K V =−K V =−K (12) z z y y x x z y x 2002). The Parkin equation may be written as: where V , V and V represent fluid flow velocity in the x , y x y z 2 2 2 2 ( + )( + ) +(N − 1) ( ) + 2 +( ) = 0 xx yy xx x y xy yy x y x y and z directions, respectively, K , K and K are correspond- x y z (5) ing hydraulic conductivities and h is the driving head. If the −1 where N = , and two velocity vectors V and V in a 2D x y medium is assumed to be homogenous and isotropic (i.e., Cartesian coordinates are defined by: k = k = k = k ) the transient flow equation may be simpli- x y z fied as: 2 2 (N−0.5) V = ( + ) (6) x x x y 2 2 2 ( n) h h h + + = (13) 2 2 2 x y z t 2 2 (N−0.5) V = ( + ) . y y (7) x y where n is porosity and ρ represents the fluid’s density. Under nonlinear conditions where a governing equation Actual fluid velocity V within pores of coarse soils is such as Lu and Likos (2004) should be adapted, the media clearly much higher than apparent velocity, and it is gener- properties as well as the properties of the flowing fluid should ally estimated using the Stephenson equation (Leps 1975) be considered. Using the Forchheimer’s nonlinear relationship as: i = aV + bV as the basis for his comprehensive study of V = Q∕(A ⋅ n) (8) the nonlinearity in flow through granular porous media, Ward where V is the actual velocity, Q is the discharge rate, A is a (1965) proposed following empirical equation to approximate the cross-sectional area of the specimen and n is the soil’s the medium’s property (b): porosity. A search in the literature shows that at early days (1 − n) 6.72 of evaluating the Darcy Law’s validity range using the b = ⋅ (14) 3∕2 g d n Reynolds number, the actual velocity V was calculated S e differently. For instance, Pavlovski (1940) reported a special where is a dimensionless constant reflecting the effect version of the Reynolds number reexamining validity of the of the particle shape (equal to unity for spheres), d is the Darcy Law (Lu and Likos 2004), in which the actual velocity effective particle size and g is the gravitational acceleration. was defined by: In interpreting (13) and (14), one may easily n fi d the cru - cial role of the porosity in both flow conditions (i.e., linear V = Q∕[A (0.23 + 0.75n)]. (9) and nonlinear). This is mainly due to the hydraulic conduc- Assuming a cubical unit volume of the porous media to tivity–grain size interrelation that was reviewed by Odong assess validity limitations of the Darcy Law, Bakhmeteff (2007). Within the framework of his work, Odong proposed and Scobey (1932) made use of a relationship to calculate the following general equation for comparing certain types of the actual velocity (Odong 2007) that may be rewritten as: empirical formulae in current use for estimating K: 2∕3 V = Q∕(A ⋅ n ). (10) g ⋅ K = ⋅ C ⋅ f (n) ⋅ d (15) Bazargan (2002) conducted carefully planned experiments e to compare the range of accuracies of the results of Eq. (5) 1 3 Applied Water Science (2018) 8:92 Page 3 of 8 92 where K denotes hydraulic conductivity, is dynamic vis- Kozeny–Carman equation. Other attempts were made by cosity, C the dimensionless constant related to the geometry Shepherd (1989), Alyamani and Şen (1993) and Terzaghi of the soil pores and f(n) represents porosity function. (1996). Odong’s literature survey shows different researchers The applicability of these formulae depends on the type approach in quantifying porosity function f(n) in Eq. (15) of soil for which hydraulic conductivity is to be estimated. none of which addressing the way porosity controls the flow Moreover, few formulas give reliable estimates of results through porous media. Though in an alternative conceptual because of the difficulty of including all possible variables assessment of n Vukovic and Soro (1992) made use of a in porous media. Vukovic and Soro (1992) noted that the uniformity coefficient ( U = ) to estimate porosity as: applications of different empirical formulae to the same porous medium material can yield different values of hydraulic conductivity, which may differ by a factor of 10 n = 0.255 1 + 0.83 . (16) or even 20. The objective of those researches, therefore, It seems to be confined to geotechnical applications too. is to evaluate the applicability and reliability of some of Now concerning to Vukovic and Soro (1992), it may be the commonly used empirical formulae for the determina- postulated that the governing parameter in either hydraulic tion of hydraulic conductivity of unconsolidated soil/rock porosity concept or in a broader theme, the media’s resist- materials. ance is the actual velocity that controls effective porosity The Terzaghi equation is one of the most widely values as well. In other words, resistance of a given medium accepted and used derivations of permeability as a func- to the flow of a fluid can be better described if its effec- tion of the characteristics of the soil medium. The empiri- tive porosity is defined as a function of actual flow velocity. cal formulae for the determination of hydraulic conductiv- The priority of such definition relies on the physics of the ity based on Terzaghi rewritten: boundary-layer development in the capillaries that reduces � � a free cross-sectional area of the tubes available for the fluid 2 n − 0.13 flow. In fact, internal surfaces of tiny capillaries formed by K = C d (17) interconnected pores are covered by the boundary layers, the 1 − n thickness of which is a function of flow velocity (Fig. 1). The where the C is sorting coefficient with higher the flow velocity, the thicker will be the boundary −3 −3 6.1 × 10 ≤ C ≤ 10.7 × 10 . In this study, we used an layer. This approach is not in contradiction with geotechnical −3 average value of C (C ≅ 8.4 × 10 ). Terzaghi formula is concept of the porosity as a function of grain size, uniform- t t most applicable for coarse granular media (Cheng and Chen ity coefficient or the packing characteristics of the media as 2007). far as the system has not been subjected to the flow of fluids. Numerous investigators have studied this relationship, and several formulae have resulted based on experimen- tal work. Kozeny (1927) proposed a formula which was Experimental setup and methods then modified by Carman (1937, 1956) to become the The experimental setup employed for the current investiga- tion was the same as that described by Shokri et al. (2014); thus, it is sufficient to address it briefly here. The experi - ments were conducted in a recirculating glass-sided flume to allow visual as well as electronic monitoring of o fl w pattern. The test section is in nearly one-fifth length of the setup as shown in Fig. 2. The flume is 0.6 m in depth, 0.6 m wide and 13 m long; one-fifth of which has been designated to accom- modate modeled media; an inlet valve has been controlling discharge rate that could be measured using a calibrated V-notch weir fitted at the outlet of the flume. A test run was always followed by full saturation of the tested medium to remove air bobbles’ blockage of the pores. Three temperature recording sensors were placed at the entrance, middle and the end of the flume to enable monitoring possible heat buildup due to the recirculation Fig. 1 Schematic illustration of the boundary layer in distorted capil- of the liquid. laries within two soil grain 1 3 92 Page 4 of 8 Applied Water Science (2018) 8:92 making it necessary to consider the boundary-layer disper- sion (Koch and Brady 1985). In a microscopic scale, once the boundary layer forms on the internal surface of a pore space, the free pore diameter o the penetrating flow might be smaller than the geometrical diameter as shown in Fig. 3, as suggested by Shokri et al. (2014). It may be concluded, therefore, the hydraulic poros- ity is: ∗ 2 d − 2 n = n (19) where is the boundary-layer thickness, d denotes the ini- tial pore diameter, n is the porosity that can be measured by geotechnical means and n refers the so-called hydraulic porosity. In other words, the hydraulic porosity is a function Fig. 2 Experimental setup containing a packed medium through which phreatic line can be seen of geotechnical porosity and the Reynolds number Re, or: −1 n = f n, Re . (20) Test materials and experimental result Although the effect of boundary-layer growth may have limited application in practice, it seems to have a noticeable Two different types of pre-washed coarse granular materials role in small-scale experimental setups where the effects of were prepared and coded as CM1 and CM2 having general viscosity may be overlooked. characteristics as shown in Table 1. −1 A discharge rate of 7.22 L s ranging between 0.059 and 0.350 was maintained for CGM1 series. For CGM2 series, −1 Experimental verifications a discharge rate of 13.19 L s under hydraulic gradients ranging between and 0.065–0.300 was adapted. To create To cross-check any probable effects of boundary-layer unsteady flow condition, a flap gate placed at the down- growth of the porosity concept, the non-Darcy flow regimes stream end of the flume was used that was maneuvering open ought to be passed through coarse granular materials causing and close repeatedly following a prescribed-preset period. sufficiently thick boundary layers within pore spaces of the Once flow velocity was determined, its corresponding Reyn- media. This could be achieved by the cyclic changes of tail olds numbers (Re) were calculated by: water level. Fast photographic means were used to observe VL Re = (18) where V denotes average flow velocity, L is a characteristic length (assumed to be equal to d in the present study) and represents the fluid kinematic viscosity. The values of for CGM1 , and CGM2 materials are 0.915 and 1.004 mm /s, respectively. A crucial point in describing the flow regime was to estimate the thickness of the boundary layer through pore spaces. It might exceed the surface roughness of the grains, Fig. 3 Schematic illustration showing possible effects of boundary layer on the velocity distribution Table 1 General characteristics Media (coarse Size distri- d (mm) Tempera- Porosity (n) Coefficient of Coefficient of of the tested granular media granular materi- bution (mm) ture (°C) uniformity C concavity C u c als) CGM1 2.5–28 14.20 23 0.472 1.50 1.08 CGM2 10.5–63 30.40 20 0.498 1.85 0.820 1 3 Applied Water Science (2018) 8:92 Page 5 of 8 92 Fig. 4 Variation of hydraulic gradient versus time observed in CGM1 and CGM2 materials Fig. 6 Variation of hydraulic porosity versus Reynolds number for CGM2 with d = 30.40 mm 0.400 0.350 0.300 0.250 0.200 0.150 0.100 0.050 0.000 0.0250.025 0.0260.027 0.028 0.031 0.034 0.0380.044 0.056 Velocity(m/s) Fig. 5 Variation of hydraulic porosity versus Reynolds number for i(Cal.)with nh i(Obs.) CGM1 with d = 14.20 mm Fig. 7 Comparing plots of hydraulic gradient versus velocity for variation of the hydraulic gradients which were visualized CGM1 series. Results are start of run. Curve “A” shows variations of (i) versus (V) based on common porosity concept, curve “B” shows as series of inclined piezometers being installed across the variations of (i) versus (V) based on hydraulic porosity concept media. Figure 4 shows such as piezometric variation versus time. Calculated effective porosity—i.e., hydraulic porosity— (i) versus flow velocity, i.e., data obtained from observed values versus Reynolds number for both types of media (i) in deferent types of media (designated with C curves), are shown in Figs. 5 and 6. It is seen from these figures calculated (i) using common understanding of porosity (des- that increasing the Reynolds number causes to increase the ignated with A curves) and calculated (i) using the proposed hydraulic porosity as also seen in reference (Shokri et al. hydraulic porosity concept (designated with B curves in the 2014). following plots Figs. 7, 8, 9, 10, 11, 12, 13 and 14. It is noteworthy that the calculated hydraulic gradient As shown in Figs. 7, 8, 9, 10, 11, 12, 13 and 14, with values were based on the recommendation of Shokri et al. regard to the increasing velocity as a result of increasing (2014) using Ergun’s equation as follows: turbulence, the predicted equation was near to the observed values of the hydraulic gradient with regard to Ergun’s 150 (1 − n) 1 − n 1.75 equation. More adaptation of this subject observed with i = V + V (21) 3 2 the unsteady flow, especially. The analysis indicated that n g ⋅ d g ⋅ d the predicted equation agrees with the theory and practical where n may be taken as either measured porosity or effec- procedures. Thus, with the unsteady flow, the nature of the tive porosity leading to two sets of output. boundary-layer thickness decreases, and hence, the porous To show probable effects of boundary-layer growth (due porosity with the mentioned Reynolds number (the rigid to increased velocity), it was deemed sufficient to plot our structure constant) increases. So, following the theory of three sets of data on the variations of the hydraulic gradient porous media flow regime is confirmed. 1 3 Hydraulic gradient(m/m) 92 Page 6 of 8 Applied Water Science (2018) 8:92 0.300 0.110 0.100 0.250 0.090 0.200 0.080 C 0.070 0.150 0.060 0.050 0.100 0.040 0.050 0.030 0.020 0.000 0.0270.0270.0270.027 0.028 0.029 0.029 0.0300.031 0.031 0.0250.026 0.026 0.0270.028 0.0310.033 0.0370.040 0.047 Velocity (m/s) Velocity(m/s) i(Cal.)with nh i(Obs.) i(Calc)-Ergun i(Cal.)with nh i(Obs.) Fig. 10 Comparing plots of hydraulic gradient versus velocity for Fig. 8 Comparing plots of hydraulic gradient versus velocity for CGM1 series. Result is after 1200 s of run start. Curve “A” shows CGM1 series. Result is after 480 s of run start. Curve “A” shows vari- variations of (i) versus (V) based on common porosity concept, curve ations of (i) versus (V) based on common porosity concept, curve “B” “B” shows variations of (i) versus (V) based on hydraulic porosity shows variations of (i) versus (V) based on hydraulic porosity con- concept, curve “C” shows variations of (i) versus (V) based on actual cept, curve “C” shows variations of (i) versus (V) based on actual model observations model observations 0.180 0.350 0.160 0.300 0.140 0.120 0.250 0.100 0.200 0.080 0.150 0.060 0.040 0.100 0.020 0.050 0.000 0.0260.027 0.0270.028 0.028 0.030 0.032 0.034 0.036 0.039 0.000 0.0390.040 0.041 0.0430.045 0.0500.0550.061 0.0720.090 Velocity(m/s) Velocity (m/s) i(Cal.)with nh i(Obs.) i(Calc)-Ergun i(Cal.)with nh i(Obs.) i(Calc)-Ergun Fig. 9 Comparing plots of hydraulic gradient versus velocity for Fig. 11 Comparing plots of hydraulic gradient versus velocity for CGM1 series. Result is after 840 s of run start. Curve “A” shows vari- CGM2 series. Results are start of run. Curve “A” shows variations of ations of (i) versus (V) based on common porosity concept, curve “B” (i) versus (V) based on common porosity concept. Curve “B” shows shows variations of (i) versus (V) based on hydraulic porosity con- variations of (i) versus (V) based on hydraulic porosity concept. cept, curve “C” shows variations of (i) versus (V) based on actual Curve “C” shows variations of (i) versus (V) based on actual model model observations observations The hydraulic conductivity (K) of the porous media is cal- culated using the hydraulic porosity concept, and plot varia- tion of hydraulic conductivity versus superficial velocity of current research by the author of this paper shows that it is series materials in Figs. 15 and 16 is with d = 10.44 mm, necessary to use a porosity correction coefficient, which is and d = 22 mm for CGM1, and CGM2, respectively. always smaller than one, to be porosity, in order to obtain a more logical estimate than the actual velocity. The analysis of the mathematical model under study shows that using Conclusions such true corrected speeds, the partial differential equa- tion governing the leakage current in frictional soils yields Based on the analysis and results, methods of estimating the acceptable solutions. hydraulic conductivity from empirical formulae based on In order to design granular porous media with fixed hydraulic porosity have been developed and used to over- texture as rubble-mound breakwaters, the hydraulic gra- come relevant issues and problems. dient should be evaluated reliably. For this purpose, the The determination of the actual flow velocity in fric- extended Forchheimer’s equation (EFE) has been ana- tional soils pores cannot be based on Stephens’ theory, lyzed and the equations for coefficients a, b and c have which is widely used in geotechnical engineering. The been derived. However, reported experimental results did 1 3 Hydraulic gradient(m/m) Hydraulic gradient(m/m) Hydraulic gradient(m/m) Hydraulic gradient(m/m) Applied Water Science (2018) 8:92 Page 7 of 8 92 0.300 0.160 0.140 0.250 0.120 0.200 0.100 0.150 0.080 0.060 0.100 0.040 0.050 0.020 0.000 0.000 0.0390.040 0.041 0.0430.045 0.0490.054 0.0590.067 0.079 0.0410.042 0.0430.0440.046 0.049 0.051 0.054 0.058 0.063 Velocity (m/s) Velocity (m/s) i(Cal.)with nh i(Obs.) i(Calc)-Ergun i(Cal.)with nh i(Obs.) i(Calc)-Ergun Fig. 12 Comparing plots of hydraulic gradient versus velocity for Fig. 14 Comparing plots of hydraulic gradient versus velocity for CGM2 series. Result is after 840 s of run start. Curve “A” shows vari- CGM2 series. Result is after 1200 s of run start. Curve “A” shows ations of (i) versus (V) based on common porosity concept, curve “B” variations of (i) versus (V) based on common porosity concept, curve shows variations of (i) versus (V) based on hydraulic porosity con- “B” shows variations of (i) versus (V) based on hydraulic porosity cept, curve “C” shows variations of (i) versus (V) based on actual concept, curve “C” shows variations of (i) versus (V) based on actual model observations model observations 1.420 1.400 0.250 1.380 0.200 1.360 1.340 0.150 1.320 0.100 1.300 0.050 1.280 0.027 0.027 0.027 0.027 0.028 0.029 0.029 0.030 0.0310.031 0.032 Velocity (m/s) 0.000 0.0400.041 0.0420.0430.045 0.048 0.051 0.056 0.063 0.072 Calculated K using hydraulic porosity concept Velocity (m/s) i(Cal.)with nh i(Obs.) i(Calc)-Ergun Fig. 15 Variation of hydraulic conductivity based on hydraulic poros- ity versus superficial velocity: CGM1 materials Fig. 13 Comparing plots of hydraulic gradient versus velocity for CGM2 series. Result is after 840 s of run start. Curve “A” shows vari- ations of (i) versus (V) based on common porosity concept, curve “B” shows variations of (i) versus (V) based on hydraulic porosity con- 6.000 cept, curve “C” shows variations of (i) versus (V) based on actual 5.800 model observations 5.600 5.400 5.200 not agree with that theory, and present study shows that 5.000 this contradiction stems from a misleading in evaluat- 4.800 ing hydraulic porosity due to some scale effects in the 0.041 0.042 0.043 0.044 0.046 0.049 0.051 0.054 0.058 0.063 0.065 experiments. Velocity (m/s) In this paper, emphasizing that the expansion of the calculated K using hydraulic porosity concept boundary layer changes the space available for flow, the porosity and shape of the pores are a function of the Fig. 16 Variation of hydraulic conductivity based on hydraulic poros- hydraulic gradient of flow through the porous medium. ity versus superficial velocity: CGM2 materials If this theory is valid, the assumption that the afore- mentioned coefficients are constant in the porous medium is not correct and it is necessary a full scale of future 1 3 Hydraulic gradient(m/m) Hydraulic gradient(m/m) Hydraulic conductivity (m/sec) Hydraulic conductivity (m/sec) Hydraulic gradient(m/m) 92 Page 8 of 8 Applied Water Science (2018) 8:92 Leps TM (1975) Flow through rockl fi l: in Embankment-dam engineer - experiments to predict better understanding how they ing. Textbook. Eds. RC Hirschfeld and SJ Paulos. JOHN WILEY change with the flow regime. AND SONS INC., PUB., NY, 1973, 22P. In: International Jour- nal of Rock Mechanics and Mining Sciences & Geomechanics Open Access This article is distributed under the terms of the Crea- Abstracts. Pergamon tive Commons Attribution 4.0 International License (http://creat iveco Leu J et al (2009) Velocity distribution of non-Darcy flow in a porous mmons.or g/licenses/b y/4.0/), which permits unrestricted use, distribu- medium. J Mech 25(01):49–58 tion, and reproduction in any medium, provided you give appropriate Lu N, Likos WJ (2004) Unsaturated soil mechanics. Wiley, New York credit to the original author(s) and the source, provide a link to the Mccorquodale JA, Hannoura AAA, Nasser MS (1978) Hydraulic con- Creative Commons license, and indicate if changes were made. ductivity of rockfill. J Hydraul Res 16(2):123–137 Odong J (2007) Evaluation of empirical formulae for determination of hydraulic conductivity based on grain-size analysis. 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J Fluid Mech 154:399–427 Kozeny J (1927) Über kapillare Leitung des Wassers im Boden: (Aufstieg, Versickerung und Anwendung auf die Bewässerung). Hölder-Pichler-Tempsky 1 3
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Published: Jun 1, 2018
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