Physics, Wright State University, Dayton, Ohio, USA, 3 Department of Earth & Environmental Sciences, Wright State University, Dayton, Ohio, USA

Transcription

1 PUBLICATIONS Water Resources Research RESEARCH ARTICLE Special Section: Applications of percolation theory to porous media Key Points: An adapted Poiseuille s approximation is proposed to describe flow in rough-walled pore tubes A theoretical relationship between pore length and its radius is proposed Our new approach predicted water permeability more accurately than the original Poiseuille s law Correspondence to: B. Ghanbarian, ghanbarian@austin.utexas.edu Citation: Ghanbarian, B., A. G. Hunt, and H. Daigle (2016), Fluid flow in porous media with rough pore-solid interface, Water Resour. Res., 52, , doi:. Received 15 JUL 2015 Accepted 16 FEB 2016 Accepted article online 22 FEB 2016 Published online 16 MAR 2016 Fluid flow in porous media with rough pore-solid interface Behzad Ghanbarian 1, Allen G. Hunt 2,3, and Hugh Daigle 1 1 Department of Petroleum & Geosystems Engineering, University of Texas at Austin, Austin, Texas, USA, 2 Department of Physics, Wright State University, Dayton, Ohio, USA, 3 Department of Earth & Environmental Sciences, Wright State University, Dayton, Ohio, USA Abstract Quantifying fluid flow through porous media hinges on the description of permeability, a property of considerable importance in many fields ranging from oil and gas exploration to hydrology. A common building block for modeling porous media permeability is consideration of fluid flow through tubes with circular cross section described by Poiseuille s law in which flow discharge is proportional to the fourth power of the tube s radius. In most natural porous media, pores are neither cylindrical nor smooth; they often have an irregular cross section and rough surfaces. This study presents a theoretical scaling of Poiseuille s approximation for flow in pores with irregular rough cross section quantified by a surface fractal dimension D s2. The flow rate is a function of the average pore radius to the power 2(3-D s2 ) instead of 4 in the original Poiseuille s law. Values of D s2 range from 1 to 2, hence, the power in the modified Poiseuille s approximation varies between 4 and 2, indicating that flow rate decreases as pore surface roughness (and surface fractal dimension D s2 ) increases. We also proposed pore length-radius relations for isotropic and anisotropic fractal porous media. The new theoretical derivations are compared with standard approximations and with experimental values of relative permeability. The new approach results in substantially improved prediction of relative permeability of natural porous media relative to the original Poiseuille equation. 1. Introduction 1.1. Laminar Flow Through Cylindrical Tubes: Poiseuille s Law Poiseuille s law describing viscous fluid flow in a single cylindrical tube (also known as Hagen-Poiseuille s law) has been widely applied in different fields ranging from fluid mechanics [White, 2003] to applied physics [Wong, 1999], chemical and petroleum engineering [Dullien, 1992], hydrogeology [Adler and Thovert, 1999], soil physics [Hillel, 2004], and biology and medicine [Davidovits, 2001]. Poiseuille s law for flow through a cylindrical tube is in the following form [Bryant and Blunt, 1992]: g5s pr4 ll (1) where R is the tube radius, s is a shape factor equal to one eighth for circular cross section, l is the dynamic fluid viscosity, and g the hydraulic conductance is equal to Q/Dp in which Q is the flow discharge and Dp is the pressure drop along the tube with length L. Poiseuille s law is valid for viscous, incompressible, and Newtonian fluid movement through a tube of constant circular cross section, longer than its diameter, where the flow is laminar (creeping) with negligible inertial forces. The validity of equation (1) is often a fundamental assumption in modeling single-phase and multiphase flow in porous media. Note that in order to apply equation (1) to model flow and transport in porous media, different pore length-radius relationships have been assumed in the literature. As we discuss in the following, both dependence and independence of the pore length with respect to the pore radius have been considered. VC American Geophysical Union. All Rights Reserved Deviation From Poiseuille s Law Experimental studies [e.g., Ahuja, 1975; Giordano et al., 1978; Mala and Li, 1999] have shown consistent deviations (often overestimation) from Poiseuille s law. Results show that flow in micron-scale tubes is more complicated than flow in millimeter-scale or larger tubes due to substantial pressure drop and changes in velocity profile, particularly when the tube wall is rough [Wang and Wang, 2007]. The surface roughness may influence the volume flow discharge in rough pipes mainly in two ways: (1) it leads to an irregular and GHANBARIAN ET AL. FLOW IN ROUGH-WALLED POROUS MEDIA 2045

2 Figure 1. Examples of rough fractal surfaces. Both have the same root-mean-square surface height but different surface fractal dimensions. (a) D and (b) D 5 2 [after Brown, 1989]. rough cross-sectional area that subsequently changes the power of the tube radius-discharge dependency in Poiseuille s law as we discuss in section 2, and (2) it affects the velocity profile and consequently increases the pressure drop. The latter has been investigated by Hu et al. [2003], Wang and Wang [2007], and Chen et al. [2009], who found that the roughness changes the velocity distribution for laminar flow in microchannels, which consequently results in a considerable pressure drop along the channel length. Chen et al. [2009] state, [...] the peaks and valleys of rough surface obviously perturb the local flow, and swirl patterns are observed near the channel walls. In addition, the pressure gradient at the location of vortex formation is significantly larger than that in no vortex region. This phenomenon implies that the presence of rough elements results in a counterpressure distribution near the surface and consequently leads to an increase of friction factor for laminar flow. Therefore, for constant volume flow rate, the pressure drop in a rough channel is greater than that in a smooth one. We should note that in addition to surface roughness, eletrokinetic effects, i.e., the interaction between molecules of polar liquids or electrolytic solutions and the solid wall may influence flow in microtubes [see e.g., Li et al., 2014a]. However, results of Phares and Smedley [2004] indicated that deviation from Poiseuille s law is more likely due to surface roughness than eletrokinetic effects for tubes larger than 50 lm in diameter. Brown and his coworkers [Brown and Scholz, 1985; Brown, 1987, 1989; Brown et al., 1995] implied that fluid flow in fractures with rough surfaces depends strongly on topology and surface roughness characteristics. In particular, Brown [1987] found that parameters most affecting fluid flow through rough fractures are the surface fractal dimension and the ratio of the mean distance between pairs of pore-solid interface points to the root-mean-square surface height. The former scales the roughness of the surface, and the latter controls the asperity height (roughness thickness). In Figure 1, examples of two rough fractal surfaces with the same root-mean-square surface height but different surface fractal dimensions (D s3 5 2 and 2.5) are given [after Brown, 1989]. As can be seen, the surface roughness is controlled by both surface fractal dimension and root-mean-square surface height. Brown [1987] also indicated that the actual flow rate through rough fracture surfaces was about 70 90% of that predicted by the cubic law, which is an analog of the Poiseuille equation for viscous flow through parallel smooth plates. Brown [1989] states, Deviations from the parallel plate model (cubic law) are expected, since real fracture surfaces are rough and in partial contact. Although rough-walled fractures were numerically investigated in the literature, to our knowledge, rough-walled cylindrical pore tubes, which might be more realistic representatives in some soils and rocks, have not been systematically used to study fluid flow. The generalization of the Poiseuille law to flow though rough cylindrical tubes by including effects of surface roughness offers a promise for various applications Laminar Flow Through Irregularly Shaped Tubes With Constant Cross Section The geometrical definition of void space of natural porous media, particularly a single pore, is invariably ambiguous, making a geometrical determination of pore length, radius, and volume difficult [Lindquist et al., 2000]. Although both circular and noncircular, e.g., elliptical, triangular, and rectangular cross sections have been used in the literature, natural pores of rocks and soils are often irregular in shape. The dependence of the hydraulic conductance g on the cross-sectional area of flow in tubes of irregular shape may be approximated by the concept of hydraulic radius [Carman, 1956; Dullien, 1992] g5c h R 2 h A ll (2) where the hydraulic radius R h 5 A/P is the ratio of cross-sectional area for flow A to the wetted pore perimeter P, and c h is a constant depending on the shape of the tube. The basis of the hydraulic radius approximation equation (2) is that the fluid flow is controlled by a balance between the axial driving force, i.e., GHANBARIAN ET AL. FLOW IN ROUGH-WALLED POROUS MEDIA 2046

3 proportional to the cross-sectional area A and the viscous drag along the side walls, i.e., roughly proportional to the wetted area and therefore the perimeter of the pore [Sisavath et al., 2001]. Comparison of the hydraulic radius approximation with the de Saint-Venant [1879] and Aissen [1951] approximations for tubes of regular cross-sectional shapes, e.g., ellipse, equilateral triangle, square, rectangular, and semicircle, showed that the Saint-Venant and Aissen approximations were typically within 15% of the exact conductance, while the hydraulic radius approximation might be in error by as much as 50% [Sisavath et al., 2001]. However, it was demonstrated that the hydraulic radius approximation predicted the hydraulic conductance of sandstone pores with irregular shapes derived from SEM images much more accurately than either the Saint-Venant or Aissen approximation [see Sisavath et al., 2001]. Øren et al. [1998] and Patzek and Silin [2001] used the Mason-Morrow shape factor G M 5 A/P 2 5 R h /P, introduced by Mason and Morrow [1991] for irregular triangular pore tubes, to obtain simple expressions for the hydraulic conductance in single-phase flow through tubes with triangular, rectangular, and elliptical cross sections. They found that the hydraulic conductance g is nearly a linear function of the shape factor G M and closely approximated by g5c PS A 2 G M ll 5c PS A A 2 P 2 ll (3) where c PS is a constant coefficient whose value depends on the cross-sectional shape, e.g., c PS in triangular and 0.5 in elliptical tubes [Øren et al., 1998; Patzek, 2001; Patzek and Silin, 2001]. The Patzek and Silin [2001] approach may be extended to study two-phase fluid flow along angular tubes by assuming that the nonwetting fluid occupies the central part of the tube, whereas the wetting fluid fills the corners [see e.g., Patzek and Kristensen, 2001]. In a recent investigation of the limits of Poiseuille s law, Mortensen et al. [2005] defined two dimensionless factors: a geometrical correction factor a 5 (A 2 Dp)/(QlL) (where A is the cross-sectional area) and a compactness number C 5 1/G M 5 P 2 /A (in which P is the perimeter of the cross section). By solving the Navier- Stokes equation for steady state flow of a Newtonian fluid for different pore cross-sectional shapes (elliptic, triangular, and rectangular), Mortensen et al. [2005] found a close-to-linear relationship between a and C (i.e., a 5 ac 1 b in which a and b are nonuniversal constants and change with the cross-sectional shape). Using these two parameters (a and C), the hydraulic conductance g may be written as A 2 g5 ðap 2 =A1bÞlL (4) Note that for b 5 0, equation (4) reduces to equations (2) and (3). Mortensen et al. [2005] found that a and b are, respectively, 8/3 and 28p/3 for elliptical, 22/7 and 265/3 for rectangular, and 25/17 and 40 3/17 for triangular cross sections. However, general values for irregularly shaped cross sections have not been estimated. More recently, Sholokhova et al. [2009] computed the hydraulic conductance by analysis of X-ray computed tomography (XCT) images and the lattice-boltzmann method in Fontainebleau sandstone samples with porosities of 7.5, 13, 18, and 22%. They defined the shape factor as G S 5Vd max =A 2 s in which V, d max,anda s are, respectively, the volume, the maximum principle size, and the surface area for the pore. Note that G S is equivalent to G M for cylinders of constant cross section and length d max. Sholokhova et al. [2009]showedthat although the variances in the lattice-boltzmann conductances were large, the dependence of the median value of the hydraulic conductance as a function of shape factor G S exhibited power law behavior as g5c S A 2 ll Gs S (5) where c S is a constant coefficient, and s is an exponent ranged from 2.45 to 4.47 for samples with porosities between 7.5 and 22%; the larger the porosity the greater the s value Pore Radius-Length Relationship in Poiseuille s Law Application of Poiseuille s law equation (1) for either analytical or numerical modeling of porous media intrinsic or relative permeability requires various assumptions of the pore length-radius relationship. For example, Fatt [1956] considered both correlated and uncorrelated scenarios. For the correlated case, pore length was represented as a power law of its radius with an exponent that varied between 22 and 1. For GHANBARIAN ET AL. FLOW IN ROUGH-WALLED POROUS MEDIA 2047

4 Figure 2. Schematic cross section of a rough pore with average radius r. the uncorrelated case, Fatt assumed that the pore length was independent of the pore radius. For permeability modeling, Mualem [1976], Katz and Thompson [1986, 1987], and Hunt [2001] presumed that pore length is linearly a function of its radius (L / R). Such an assumption reduces the exponent 4 in equation (1) to 3 [Hunt, 2001]. Although both direct and inverse relationships have thus been applied, a general theoretical explanation is still required to relate pore length to its radius, if a correlation exists Objectives The study addresses the effect of rough-walled tube on the radius-discharge power of the Poiseuille s law. The resulting relationships will be used to estimate relative permeability in porous media. Additionally, we will systematically evaluate relationships between pore length and the average radius to unify various approximations and provide a consistent formulation for a range of applications. The objectives are therefore: (1) to present an approximation of Poiseuille s law adapted for a tube with irregular cross-sectional area and rough surface, (2) to present theoretical pore length-radius relations in isotropic and anisotropic fractal porous media, and (3) to study the effect of pore roughness on relative permeability modeling using experimental measurements. Since in this study, we address fluid flow in a single pore tube with rough pore-solid interface microscopically, therefore we do not discuss the effect of tortuosity and meandering behavior of flow path, a macroscopic characteristic, in our Poiseuille s approximation derivation. However, tortuosity can be macroscopically described within the percolation theory framework [Hunt and Skinner, 2008, 2010; Ghanbarian et al., 2013; Hunt et al., 2014]. 2. Theoretical Considerations 2.1. Poiseuille s Approximation in a Rough-Walled Pore Tube With a Constant Cross Section Applying either equation (4) or equation (5) to describe the hydraulic conductance in pore tubes with irregular cross-sectional area requires prior knowledge of constant coefficients a, b, and s. Therefore, following Øren et al. [1998], Sisavath et al. [2001], and Patzek and Silin [2001], we assume that g is a linear function of G M 5 A/P 2 and approximated reasonably well by equation (3). We further assume that the pore-solid interface is fractal. Following Mandelbrot [1982], Lovejoy [1982], and Mandelbrot et al. [1984], one may relate the perimeter P of a fractal object to its cross-sectional area A as P 2 / A Ds2 (6) where D s2 is the surface fractal dimension (1 D s2 < 2) quantifying the pore-solid interface roughness of the cross-sectional area (Figure 2). Note that equation (6) has been experimentally evaluated in soils [Pachepsky et al., 1996] and rocks [Schlueter et al., 1997]. By combining equations (3) and (6), one has g / A32Ds2 ll (7) Now the simple assumption A / r 2 may be applied to the cross-sectional area of a pore tube (Figure 2), where r is also the average (equivalent) pore radius. In this study, the area-radius relationship is used to describe the average cross-sectional area for flow. Substituting A / r 2 into equation (7) yields, g / r2ð32ds2þ ll (8) Equation (8) indicates that the hydraulic conductance g of a pore tube with rough cross section is proportional to the average pore radius r to the power 2(3-D s2 ). Since D s2 varies between 1 and 2, the power in equation (8) varies between 4 and 2. Note that equation (8) reduces to the original Poiseuille equation GHANBARIAN ET AL. FLOW IN ROUGH-WALLED POROUS MEDIA 2048

5 Figure 3. Schematic structure of a rough pore in an anisotropic fractal medium. (equation (1)), when the pore-solid interface is smooth (D s2 5 1). The dependence of the exponent on surface roughness (e.g., D s2 ) has been anticipated by Madadi and Sahimi [2003] in fracture networks. We should point out that equation (8) approximates flow discharge in rough cylindrical pore tubes, since we have considered only the effect of surface roughness scaling (surface fractal dimension), neglecting the influence of roughness thickness (root-mean-square surface height). Equation (8), therefore, should provide accurate results only when fluid flow through rough-walled tubes is controlled by the surface fractal dimension factor, or when the ratio of the average pore radius to the rootmean-square surface height is sufficiently large. Equation (8) also indicates that rougher pore-solid interface D s2 results in lower flow rate [Li et al., 2014b], which is due to increased frictional resistance (/ 1/g) [Mala and Li, 1999; Chen et al., 2009]. However, it has not been verified that the frictional losses in the fluid flow are truly in accord with our predicted effects on the total flow rate. Consequently, we seek verification by comparison with experimental measurements. Evaluation of equation (8), however, requires invoking an assumption relating pore radius r to its length L. Thus, in the following section, we propose a relationship by means of fractals describing pore length-radius relationship in porous media The Pore Length-Radius Relation in Fractal Porous Media A fractal object is (statistically) self-similar if its structure is invariant under an isotropic rescaling of lengths. This means that if lengths in different directions are rescaled with the same scaling factor, the same structure is reproduced [Sahimi, 2011]. Consequently, a (statistically) self-similar fractal appears the same at all scales. However, fractal objects in nature may be scaled by different scaling exponents (e.g., fractal dimensions) and factors in different directions. This type of scale-invariance indicates that the fractal object is structurally anisotropic and self-affine and cannot be defined with a single fractal dimension [Sahimi, 2011]. In this section, we consider the case of a pore structure with rough surface area having a geometry quantified with a surface fractal dimension D s3 in three dimensions, 2 D s3 < 3[Avnir et al., 1984], describing the pore-solid interface roughness along the pore structure (Figure 3). When D s3 approaches 3, the surface gets extremely rough, while D s3 close to 2 represents a smooth surface. Sahimi [2003], among many others, indicated that the surfaces of disordered media like fracture networks and pores in sandstones follow more or less a universal roughness (Hurst) exponent, H where H D s3 (0 < H 1) [Mandelbrot, 1982]. However, it was later acknowledged that it was unlikely for H to take on a universal value for all types of porous media [Sahimi, 2011]. Following Mandelbrot [1982] and Mandelbrot et al. [1984], the relationship, for a fractal object, between rough pore structure surface area A s and its volume V (Figure 3) is A 3 s / VDs3 (9) In order to approximate the surface area of such a fractal object (Figure 3), we invoke Mandelbrot s approach [Mandelbrot, 1982], as followed by Wheatcraft and Tyler [1988], in which a fractal length L f is proportional to the apparent (straight-line) length to the power D r (L f / L Dr ) where D r is the roughness fractal dimension. In fact, D r describes the roughness of a line on the pore-solid interface along the pore length (z direction in Figure 3). Thus, the surface area and volume may be given by and, A s / P3L Dr (10) V5A3L (11) GHANBARIAN ET AL. FLOW IN ROUGH-WALLED POROUS MEDIA 2049

6 Note that A is the pore cross-sectional area and A s is the surface area of a rough pore tube along its length. In addition, equation (11) requires the same cross-sectional area along the pore structure. As pointed out earlier, we do not address the converging-diverging geometry of pores here. Given that A / r 2, combining equations (10) and (11) with equations (6) and (9) yields 2D s3 23D s2 L / r 3Dr 2D s3 (12) If one assumes that the roughness along the pore structure is isotropic, D r 5 D s3 1[Mandelbrot, 1982; Sahimi, 2011], and equation (12) reduces to L / r c (13) where c 5 (2D s3 3D s2 )/(2D s3 3). Note that D r is not necessarily equal to D s2, in particular when the medium is anisotropic as we explain in the following. The possible ranges of the values of the surface fractal dimensionalities (D s2 and D s3 ) provide limits to the behavior allowed in equation (13). We show that equation (13) is consistent with some common modeling assumptions in porous media. Since 1 D s2 < 2 and 2 D s3 < 3, 22 < c 1 and c can be negative. As seen therefore, equation (13) is similar to the empirical power law function proposed by Fatt [1956] who assumed c 522, 21, 0, and 1 to relate pore length to its radius in his pore network models. In the case c 5 0, pore length is constant, and therefore independent of its radius r [Fatt, 1956]. Based on equation (13), the pore length can be either an increasing or decreasing function of its radius and values of c 521(D s3 5 3/4(D s2 1 1)) and c 5 1(D s2 5 1) are both possible. The pore length can also be independent of its radius in the case c 5 0, for example, when D s and D s The direct [Fatt, 1956; Mualem, 1976; Katz and Thompson, 1986, 1987] and inverse [Fatt, 1956; Yu, 2008] proportionality between pore length L and radius r are both common assumptions in the literature. However, the general form (equation (13)) has not been reported. Equation (13) is valid for anisotropic fractal media where two fractal dimensions (D s2 and D s3 ) are required to describe the pore-solid interface roughness in different directions. In a (statistically) isotropic self-similar fractal medium, one can set D s2 5 D r 5 D s3 1[Mandelbrot, 1982; Sahimi, 2011]. Thus, in such a medium, c 5 (2 D s2 )/(2D s2 1). Since 1 D s2 < 2, c varies from 1 to 0 indicating a direct relationship between pore length and radius in a self-similar isotropic fractal medium. Combining equations (13) and (8) gives the hydraulic conductance g in terms of the average pore radius r in fractal porous media (L / r c ): g / r 2ð32D s2þ2c 5r k (14) As previously mentioned, in an anisotropic fractal medium 22 < c 1, 1 < k 5 2(3 D s2 ) c < 6. Our results are consistent with the lattice-gas simulation results in single self-affine (anisotropic) fractures studied by Zhang et al. [1996] who found k values larger than 5. Sahimi [2011] also stated that the exponent of Poiseuille s law in fractures with rough surface can be as large as 6. Equation (14) is also consistent with experimental results of Arya et al. [1999] who found (3 D s2 ) c using hydraulic conductivity measurement in soils with different textures (i.e., sand, sandy loam, loam, and clay). We report the average and standard deviation in the exponent from equation (14) for each soil texture class as determined from the data given in Arya et al. [1999, Table 3]: sand, ; sandy loam, ; loam, ; clay, For a special case, if one assumes that pore structure along the z direction is straight with D r 5 1, Figure 3 changes to Figure 4, which is similar to a triadic Koch surface [Feder, 1988]. In this case, the cross-section shape is fixed and does not change along the z direction. However, the pore cross section and structure are still irregular and rough. If one further considers an isotropic self-similar medium in which D s2 5 D s3 1,equation (12) reduces into a simple direct relationship as follows: L / r (15) This linear relationship between pore length and radius has been already assumed in self-similar fractal porous media [Katz and Thompson, 1986, 1987; Hunt, 2001; Hunt et al., 2011; Ghanbarian-Alavijeh and Hunt, 2012; Hunt et al., 2013; Ghanbarian-Alavijeh et al., 2012]. GHANBARIAN ET AL. FLOW IN ROUGH-WALLED POROUS MEDIA 2050

7 Although we address pore roughness in our theoretical framework and adapted Poiseuille s approximation for isotropic and anisotropic structures, we neglect divergence-convergence of the pore path, which may affect the velocity distribution or profile. However, such a geometrical factor, which is out of the scope of this study, may be modeled using a sinusoidal tube [Czachor, 2006; Czachor et al., 2010]. For the hydraulic conductance of a tube with a sinusoidally varying cross section, see e.g., Bernabe and Olson [2000]. Figure 4. Pore structure with D r 5 1. In the next section, the effect of pore roughness and the applicability of the adapted Poiseuille s approximation for prediction of the relative permeability from the drainage branch of a capillary pressure curve are investigated from existing experimental data. 3. Relative Permeability Modeling in Heterogeneous Porous Media: Critical Path Analysis Critical path analysis (CPA) from percolation theory, introduced first by Ambegaokar et al. [1971], is a powerful approach to calculate effective conduction properties of a disordered medium [Sahimi, 2011; Hunt et al., 2014]. In the CPA framework, conduction in a random system is controlled by those conductances with magnitudes greater than the critical conductance g c, the largest possible value of the conductance for which the set of all larger conductances (g > g c ) still forms an infinite connected cluster [Katz and Thompson, 1986, 1987]. In the following, we invoke critical path analysis to model relative permeability k rw in natural porous media with broad pore size distribution and rough pore-solid interface. For the sake of simplicity, we assume that each pore of the porous medium is occupied by wetting or nonwetting fluid. This pore occupancy hypothesis obviously only approximates a more realistic situation in which the nonwetting fluid occupies the central part of the pore tube and the wetting fluid fills the corners. Ten soil samples were selected from the UNSODA database [Nemes et al., 2001]. The capillary pressure and relative permeability curves included more than 9 and 11 measured data points for each sample, respectively. Characteristics and selected properties of each experiment are summarized in Table 1. In the following, the adapted Poiseuille approximation (equation (14)) is compared with the original Poiseuille s law (equation (1)) in the relative permeability prediction. Since experimental details such as 3-D images of the samples are not available, it is assumed that the media is isotropic (D s2 5 D s3 1) and that the surface fractal dimension D determined by the pore-solid fractal (PSF) capillary pressure curve model is D s3 (D 5 D s3 ). We should caution that the pore radius determined from the measured capillary pressure curve is related to the local minimum of the pore (throat) radii by using Young-Laplace s law, rather than to the average (equivalent) pore radius considered in our assumptions earlier. Under the assumption of constant cross section along the rough-walled pore tube, however, the two (i.e., average and local minimum radii) should be more or less identical. Note that modern techniques, such as high-resolution tomography and 3-D image analysis will provide more accurate information, e.g., D s2 and D s3 of the porous medium [see e.g., Katz and Thompson, 1985; H ohr et al., 1988; Radlinski et al., 1999; Sen et al., 2002; Dathe and Baveye, 2003]. However, capturing small pores and their roughness characteristics in rocks, e.g., shale remains challenging. It is further assumed that the effect of surface roughness thickness is negligible compared to that of surface fractal dimension scaling surface roughness. This means that the ratio of the average pore radius to the roughness thickness is large. Following equation (14), we therefore set k 5 2(4 D) (3 D)/(2D 3), which is valid in isotropic porous media. The PSF capillary pressure curve model is [Bird et al., 2000; Ghanbarian- Alavijeh and Hunt, 2012] " S w 512 b / 12 P # D23 c ; P cmin P c P cmax (16) P cmin GHANBARIAN ET AL. FLOW IN ROUGH-WALLED POROUS MEDIA 2051

8 Table 1. Selected Properties of 10 Experiments From the UNSODA Database Used in This Study Soil Code Texture / D b P cmin (kpa) S wc D s2 D s3 k (Equation (11)) 1280 Silt loam Silt loam Sandy loam Sandy loam Silty clay Sandy loam Sandy loam Clay Silt loam Sand where S w is the water saturation, P c is the capillary pressure, D is the pore-solid interface fractal dimension, b is the PSF model parameter ranging theoretically between / and 1 in which / is the porosity [Ghanbarian-Alavijeh et al., 2011], and P cmin and P cmax are the minimum (or displacement) and maximum capillary pressures, respectively. The unknown parameters, such as D, P cmin, and b (/ b 1), are determined by fitting equation (16) to the measured capillary pressure data. Figure 5 shows equation (16) fit, the optimized PSF model parameters, and the correlation coefficient value for the sample We applied critical path analysis from percolation theory to identify the dominant flow paths [Hunt, 2001; Ghanbarian-Alavijeh and Hunt, 2012] and deduce the relative permeability. In this framework, the smallest conductances do not affect the dominant flow through the critical path. A critical water saturation for percolation S wc the minimum water saturation required for the existence of a sample-spanning cluster of interconnected pores is defined. The relative permeability model developed first by Hunt [2001] and revisited by Ghanbarian-Alavijeh and Hunt [2012] and Ghanbarian et al. [2015] is 8 k 32D b=/211s w 2S wc ; Swx S w 1 >< k rw 5 k w k 5 >: b=/2s wc b=/211s wx 2S wc b=/2s wc k 32D S w 2S 2 wc ; S wc S w S wx S wx 2S wc where k w and k are permeability under partially and fully saturated conditions, respectively, and S wx is the crossover point at which fractal scaling from critical path analysis switches to universal percolation scaling from percolation theory. We should point out that the effect of pore connectivity on the relative permeability can be seen in the critical water saturation S wc whose value in three dimensions may be approximated by 3/(2Z), Z is the average coordination number, within bond percolation theory [Stauffer and Aharony, 1994]. (17) Figure 5. The PSF capillary pressure curve model equation (16) fit to the measured data (sample 1380). The S wx value can be determined from [Ghanbarian-Alavijeh and Hunt, 2012; Ghanbarian et al., 2015] S wx 5S wc 1 2ðb2/Þ (18) k=ð32dþ22 Here we roughly approximate the critical water saturation S wc from the last measured data point on the capillary pressure curve. Alternatively, Liu and Regenauer-Lieb [2011] recently proposed a morphological technique to estimate the percolation threshold from microtomography and 3-D image analysis of a porous medium. Here it is assumed that S wc can be estimated from the last measured point of the capillary pressure curve. GHANBARIAN ET AL. FLOW IN ROUGH-WALLED POROUS MEDIA 2052

9 Figure 6. Comparison of the water permeability k w predicted using equation (17) when k 5 2(4 D) (3 D)/(2D 3) (derived from the adapted Poiseuille s law, equation (14), for isotropic porous media where D s2 5 D s3 1) and k 5 3 with the measured one for 10 soil samples from the UNSODA database. Note that we assumed D s3 5 D where D is the pore-solid interface fractal dimension calculated from the capillary pressure curve. GHANBARIAN ET AL. FLOW IN ROUGH-WALLED POROUS MEDIA 2053

10 Following Katz and Thompson [1986, 1987], Hunt [2001], and Ghanbarian-Alavijeh and Hunt [2012], we assume that Poiseuille s law (equation (1); g / r 4 /L) is valid in natural porous media as well if the medium is self-similar (L / r), and thus set k 5 3. We therefore compare k 5 3 with k 5 2(4 D) (3 D)/(2D 3) in prediction of the experimentally measured relative permeability curves. 4. Results and Discussion In general, the adapted Poiseuille approximation greatly improved the relative permeability predictions, although we did not examine its applicability directly to the study of fluid flow in a single rough-walled pore tube. The pore-solid interface fractal dimension D and the Poiseuille equation exponent k ranged from to and to 3.049, respectively (see Table 1). As we demonstrate in Figure 6, setting k 5 2(4 D) (3 D)/(2D 3) derived from our adapted Poiseuille s approximation (equation (14)) led to substantially improved predictions of the relative permeability than k 5 3 assumed in a previous studies [Ghanbarian-Alavijeh and Hunt, 2012; Ghanbarian et al., 2015] in equation (17). Figure 7 shows the predicted log 10 (k w ) versus the measured one when k 5 3 (Figure 7a) and k 5 2(4 D) (3 D)/(2D 3) (Figure 7b) for all 10 experiments. As can be seen, k 5 3 results in relative permeability underestimations up to 6 orders of magnitude, while the adapted Poiseuille approximation (equation (14)) with k 5 2(4 D) (3 D)/(2D 3) renders much more accurate predictions. However, there is still underestimation. Ghanbarian-Alavijeh and Hunt [2012] pointed out that when pore-solid interface fractal dimension D is underestimated, relative permeability is overestimated. Therefore, k w underestimation might be also due to D overestimation derived from capillary pressure data. Another plausible possibility for relative permeability underestimation might be neglecting corner and film flow [see e.g., Or and Tuller, 2000; Tuller and Or, 2001, 2002], which may dominate capillary flow, particularly at low water saturations. Further investigation is, therefore, required to incorporate the effect of corner and film flow in the modeling of the relative permeability. Although in our adapted Poiseuille s approximation it is assumed that the effect of surface roughness thickness is negligible compared to that of surface fractal dimension, as Figure 7 demonstrates, equation (14) predicts relative permeability (unsaturated hydraulic conductivity) more accurately than the original Poiseuille equation. Note that the assumption k 5 2(4 D) (3 D)/(2D 3) is valid for isotropic porous media and media in which the PSF capillary pressure model (equation (16)) represents the pore size distribution accurately. In addition, we estimate D s3 from the capillary pressure curve assuming D s3 5 D. Any failure of these assumptions leads to inaccurate prediction of relative permeability. For example, to our knowledge, Dullien [1975] was the first to point out the discrepancy between capillary pressure curves measured with different methods, e.g., mercury intrusion porosimetry and photomicrography. He noted that the mercury intrusion method gives the entry pore size distribution, while the photomicrography technique provides the distribution of pore volume by all pore sizes. Pore size distribution determined by photomicrography is closer to the true pore size distribution than the entry pore size given by mercury intrusion porosimetry [Dullien, 1975]. In a related study, Hall et al. [1986] measured the pore size distribution of shaly rocks with several methods: small-angle neutron and X-ray scattering (SANS and SAXS), nitrogen adsorption, nitrogen desorption, and mercury intrusion porosimetry. They found that pore size distributions measured by SANS and SAXS techniques were in reasonable agreement with those from the nitrogen adsorption isotherm, but often in disagreement with distributions derived from the nitrogen desorption isotherm and mercury porosimetry. The question Which method of pore size distribution measurement should be applied? thus remains unanswered. Sahimi [2011] concluded that each measurement method has its own strong and weak points. For example, the success of mercury intrusion porosimetry and sorption isotherm methods requires prior knowledge of the pore space connectivity and pore shapes [Sahimi, 2011]. Pore shape also affects the pore size distribution derived from scattering methods, and even if the shape is known, the calculated distribution may still be sensitive to the shape [Sahimi, 2011]. In fact, precise prediction of relative permeability requires accurate characterization of the porous medium, e.g., the pore size distribution and the pore roughness properties. When the medium is anisotropic it is strongly recommended to use modern techniques such as 3-D image analysis to determine the roughness exponents (e.g., D s2 and D s3 ), although the resolution of tomographic images might be a restricting factor. GHANBARIAN ET AL. FLOW IN ROUGH-WALLED POROUS MEDIA 2054

11 Water Resources Research Figure 7. The predicted log10(kw) versus the measured one when (a) k 5 3 and (b) k 5 2(4 D) (3 D)/(2D 3) for all 10 experiments. The dashed line represents 1:1 line. The critical water saturation Swc can be also derived from 3-D images based on the recently proposed method by Liu and Regenauer-Lieb [2011]. Direct measurement of pore space characteristics has advanced substantially with the advent of X-ray computed tomography, producing nondestructive 3-D images of porous media with uniform resolution in different directions [Lindquist et al., 2000]. Although different algorithms [see e.g., Lindquist et al., 1996; Silin et al., 2003; Øren and Bakke, 2003] have been developed to quantify pore space geometrical properties and topological characteristics in porous media, analysis of three-dimensional images and identification of pore radius, its volume, and length are still ambiguous. Nonetheless, over the last decade, pore-scale imaging has become a potential method, particularly in oil and gas engineering, to study fluid flow and contaminant transport in porous media [Blunt et al., 2013]. Despite limitations and challenges such as resolution effect and finding representative samples, there has been remarkable progress in 3-D imaging of porous materials, including image-based modeling of capillary pressure curve [see e.g., Schaap et al., 2007; Yang et al., 2015], intrinsic permeability [see e.g., Manwart et al., 2002; Alyafei et al., 2015], and relative permeability [see e.g., Martys and Chen, 1996; Apourvari and Arns, 2016]. More recent developments and advances in pore-scale tomographic imaging of porous media can be found at Blunt et al. [2013], Wildenschild and Sheppard [2013], and Cnudde and Boone [2013]. Comparison with relative permeability experiments indicates that the approximate general form of Poiseuille s law using scaling theory of fractals has a significant impact on the relative permeability predictions. Further numerical simulations of the Navier-Stokes equations in individual rough-walled pore tubes with different surface roughness characteristics are still needed to evaluate the validity of our adapted Poiseuille s approximation and its applicability in porous media. 5. Conclusion In the present study, we proposed an adapted Poiseuille s approximation for pore structures with rough surfaces. The roughness was defined with surface fractal dimension, which describes the irregular pattern of pore-solid interface. We found that in our approach hydraulic conductance in terms of the pressure gradient is a function of the average pore radius to the power 2(3-Ds2) instead of 4 in Poiseuille s law. We also presented a pore length-radius relation in isotropic and anisotropic fractal porous media. We found that in an isotropic medium, the pore length is directly proportional to its radius. However, in an anisotropic fractal medium, pore length could be either directly or inversely proportional to its radius. Comparing the relative permeability model developed by means of critical path analysis with experiments indicated that the GHANBARIAN ET AL. FLOW IN ROUGH-WALLED POROUS MEDIA 2055

12 adapted Poiseuille s approximation (equation (14)) led to much more accurate predictions than the original one (equation (1)). However, precise predictions require accurate characterization of the medium. Notation A cross-sectional area. A s surface area. a and b constant coefficients in the Mortensen et al. [2005] model. C compactness number (5 P 2 /A 5 1/G M ). c h hydraulic radius constant coefficient. c PS Patzek-Silin constant coefficient. c S Sholokhova et al. constant coefficient. D surface fractal dimension derived from capillary pressure curve. D s2 surface fractal dimension of the pore cross section. D s3 surface fractal dimension of the pore-solid interface along the pore length. D r roughness fractal dimension. g hydraulic conductance. g c critical hydraulic conductance. G M Mason-Morrow dimensionless shape factor (5A/P 2 ). G S Sholokhova et al. dimensionless shape factor. H Hurst exponent. k intrinsic permeability. k w water permeability. k rw water relative permeability. L cylindrical tube length. L f fractal length. p pressure. P pore cross-sectional perimeter. P c capillary pressure. P cmin minimum capillary pressure. P cmax maximum capillary pressure. Q flow discharge. R cylindrical tube radius. R h hydraulic radius. r average (equivalent) pore radius. s pore shape factor. S w water saturation. S wc critical water saturation. S wx crossover water saturation. V pore volume. a geometrical correction factor (5A 2 Dp/lLQ). b PSF model parameter. Dp pressure drop. / porosity. c (2D s3 3D s2 )/(2D s3 3). k 2(3 D s2 ) 2 c. l dynamic fluid viscosity. Acknowledgment The authors acknowledge the Associate Editor, one anonymous reviewer, and Amir Raoof, Utrecht University, for their valuable comments. The UNSODA data sets used in this study are available upon request to the corresponding author. References Adler, P. M., and J.-F. Thovert (1999), Fractures and Fracture Networks, Kluwer Acad., Dordrecht, Netherlands. Ahuja, A. S. (1975), Augmentation of heat transport in laminar flow of polystyrene suspensions. I. Experiments and results, J. Appl. Phys., 46, Aissen, M. I. (1951), Estimation and computation of torsional rigidity, PhD dissertation, Stanford Univ., Stanford, Calif. Alyafei, N., A. Q. Raeini, A. Paluszny, and M. J. Blunt (2015), A sensitivity study of the effect of image resolution on predicted petrophysical properties, Transp. Porous Media, 110, Ambegaokar, V., B. Halperin, and J. S. Langer (1971), Hopping conductivity in disordered systems, Phys. Rev. B, 4(8), GHANBARIAN ET AL. FLOW IN ROUGH-WALLED POROUS MEDIA 2056

13 Apourvari, S. N., and C. H. Arns (2016), Image-based relative permeability upscaling from the pore scale, Adv. Water Resour., doi: / j.advwatres , in press. Arya, L. M., F. J. Leij, P. J. Shouse, and M. Th. van Genuchten (1999), Relationship between the hydraulic conductivity function and the particle-size distribution, Soil Sci. Soc. Am. J., 63, Avnir, D., D. Farin, and P. Pfeifer (1984), Molecular fractal surfaces, Nature, 308, Bernabe, Y., and J. F. Olson (2000), The hydraulic conductance of a capillary with a sinusoidally varying cross-section, Geophys. Res. Lett., 27, Bird, N. R. A., E. Perrier, and M. Rieu (2000), The water retention function for a model of soil structure with pore and solid fractal distributions, Eur. J. Soil Sci., 51, Blunt, M. J., B. Bijeljic, H. Dong, O. Gharbi, S. Iglauer, P. Mostaghimi, A. Paluszny, and C. Pentland, (2013), Pore-scale imaging and modeling, Adv. Water Resour., 51, Brown, S. R. (1987), Fluid flow through rock joints: The effect of surface roughness, J. Geophys. Res., 92, Brown, S. R. (1989), Transport of fluid and electric current through a single fracture, J. Geophys. Res., 94, Brown, S. R., and C. H. Scholz (1985), Broad bandwidth study of the topography of natural rock surfaces, J. Geophys. Res., 90, 12,575 12,582. Brown, S. R., H. W. Stockman, and S. J. Reeves (1995), Applicability of the Reynolds equation for modeling fluid flow between rough surfaces, Geophys. Res. Lett., 22, Bryant, S., and M. Blunt (1992), Prediction of relative permeability in simple porous media, Phys. Rev. A, 46(4), Carman, P. C. (1956), Flow of Gases Through Porous Media, Academic, N. Y. Chen, Y., C. Zhang, M. Shi, and G. P. Peterson (2009), Role of surface roughness characterized by fractal geometry on laminar flow in microchannels, Phys. Rev. E, 80, Cnudde, V., and M. N. Boone (2013), High-resolution X-ray computed tomography in geosciences: A review of the current technology and applications, Earth Sci. Rev., 123, Czachor, H. (2006), Modelling the effect of pore structure and wetting angles on capillary rise in soils having different wettabilities, J. Hydrol., 328, Czachor, H., S. H. Doerr, and L. Lichner (2010), Water retention of repellent and subcritical repellent soils: New insights from model and experimental investigations, J. hydrol., 380, Dathe, A., and P. Baveye (2003), Dependence of the surface fractal dimension of soil pores on image resolution and magnification, Eur. J. Soil Sci., 54, Davidovits, P. (2001), Physics in Biology and Medicine, Academic, N. Y. de Saint-Venant, B. (1879), Sur une formule donnant approximativement le moment de torsion, C. R. Acad. Sci. Paris, 88, Dullien, F. A. (1975), New network permeability model of porous media, AIChE J., 21(2), Dullien, F. A. L. (1992), Porous Media: Fluid Transport and Pore Structure, Academic, San Diego, Calif. Fatt, I. (1956), The network model of porous media. I. Capillary pressure characteristics, Trans. Am. Inst. Min. Metall. Pet. Eng., 207, Feder, J. (1988), Fractals, Plenum, N. Y. Ghanbarian, B., A. G. Hunt, M. Sahimi, R. P. Ewing, and T. E. Skinner (2013), Percolation theory generates a physically based description of tortuosity in saturated and unsaturated porous media, Soil Sci. Soc. Am. J., 77(6), Ghanbarian, B., A. G. Hunt, T. E. Skinner, and R. P. Ewing (2015), Saturation dependence of transport in porous media predicted by percolation and effective medium theories, Fractals, 23, Ghanbarian-Alavijeh, B., and A. G. Hunt (2012), Unsaturated hydraulic conductivity in porous media: Percolation theory, Geoderma, , Ghanbarian-Alavijeh, B., H. Millan, and G. Huang (2011), A review of fractal, prefractal and pore-solid-fractal models for parameterizing the soil water retention curve, Can. J. Soil Sci., 91, Ghanbarian-Alavijeh, B., T. E. Skinner, and A. G. Hunt (2012), Saturation dependence of dispersion in porous media, Phys. Rev. E, 86, Giordano, R., A. Salleo, S. Salleo, and F. Wanderlingh (1978), Flow in xylem vessels Poiseuille s law, Can. J. Bot., 56, Hall, P. L., D. F. Mildner, and R. L. Borst (1986), Small-angle scattering studies of the pore spaces of shaly rocks, J. Geophys. Res., 91, Hillel, D. (2004), Introduction to Environmental Soil Physics, Elsevier, Amsterdam, Netherlands. H ohr, A., H. B. Neumann, P. W. Schmidt, P. Pfeifer, and D. Avnir (1988), Fractal surface and cluster structure of controlled-pore glasses and Vycor porous glass as revealed by small-angle x-ray and neutron scattering, Phys. Rev. B, 38, Hu, Y. D., C. Werner, and D. Q. Li (2003), Influence of three-dimensional roughness on pressure-driven flow through microchannels, J. Fluids Eng., 125, Hunt, A. G. (2001), Applications of percolation theory to porous media with distributed local conductances, Adv. Water Resour., 24, Hunt, A. G., and T. E. Skinner (2008), Longitudinal dispersion of solutes in porous media solely by advection, Philos. Mag., 88, Hunt, A. G., and T. E. Skinner (2010), Incorporation of effects of diffusion into advection-mediated dispersion in porous media, J. Stat. Phys., 140, Hunt, A. G., T. E. Skinner, R. P. Ewing, and B. Ghanbarian-Alavijeh (2011), Dispersion of solutes in porous media, Eur. Phys. J. B, 80, Hunt, A. G., B. Ghanbarian, and K. C. Saville (2013), Unsaturated hydraulic conductivity modeling for porous media with two fractal regimes, Geoderma, 207, Hunt, A. G., R. P. Ewing, and B. Ghanbarian (2014), Percolation Theory for Flow in Porous Media, Lecture Notes Phys., vol. 880, 3rd ed., Springer, Heidelberg, Germany. Katz, A., and A. H. Thompson (1985), Fractal sandstone pores: Implications for conductivity and pore formation, Phys. Rev. Lett., 54, Katz, A. J., and A. H. Thompson (1986), Quantitative prediction of permeability in porous rock, Phys. Rev. B, 34, Katz, A. J., and A. H. Thompson (1987), Prediction of rock electrical conductivity from mercury injection measurements, J. Geophys. Res., 92, Li, S., A. Raoof, and R. Schotting (2014a), Solute dispersion under electric and pressure driven flows; pore scale processes, J. Hydrol., 517, Li, S., J. Wu, S. Yu, H. Li, and P. Qiu (2014b), Influence of roughness fractal dimension of fracture surfaces on hydraulic characteristics, EJGE, 14, Lindquist, W. B., S. M. Lee, D. A. Coker, K. W. Jones, and P. Spanne (1996), Medial axis analysis of void structure in three-dimensional tomographic images of porous media, J. Geophys. Res., 101, GHANBARIAN ET AL. FLOW IN ROUGH-WALLED POROUS MEDIA 2057