Homogenization of Wall-Slip Gas Flow Through

Transport in Porous Media 36: 293–306, 1999. c 1999 Kluwer Academic Publishers. Printed in the Netherlands. ⃝

293

Homogenization of Wall-Slip Gas Flow Through Porous Media ERIK SKJETNE1,⋆ and JEAN-LOUIS AURIAULT

1 Department of Petroleum Engineering, Stanford University, Stanford, CA 94305-2220, U.S.A. 2 Laboratoire Sols Solides Structures (3S), UJF, INPG, CNRS, Domaine Universitaire, BP 53 X,

38041 GRENOBLE Cedex, France (Received: 22 December 1997; in final form: 6 November 1998) Abstract. The permeability of reservoir rocks is most commonly measured with an atmospheric gas. Permeability is greater for a gas than for a liquid. The Klinkenberg equation gives a semi-empirical relation between the liquid and gas permeabilities. In this paper, the wall-slip gas flow problem is homogenized. This problem is described by the steady state, low velocity Navier–Stokes equations for a compressible gas with a small Knudsen number. Darcy’s law with a permeability tensor equal to that of liquid flow is shown to be valid to the lowest order. The lowest order wall-slip correction is a local tensorial form of the Klinkenberg equation. The Klinkenberg permeability is a positive tensor. It is in general not symmetric, but may under some conditions, which we specify, be symmetric. Our result reduces to the Klinkenberg equation for constant viscosity gas flow in isotropic media. Key words: Klinkenberg, wall-slip, Knudsen, gas flow, low pressure, Navier–Stokes, homogenization.

Nomenclature Roman Letters b c, C G H k Kg , Kl , K Kn l L M Ma n p p∗ p∗ pm

Klinkenberg constant. constants. macroscopic pressure gradient. Klinkenberg tensor. microscopic velocity field. gas, liquid, tensorial permeability. Knudsen number. characteristic size of the pores. macroscopic characteristic length. molecular weight. Mach number. unit normal vector to ". pressure. characteristic pressure. characteristic pressure difference over sample. mean pressure.

⋆ Present address: Statoil, N-4035 Stavanger, Norway.

294 Q R Re t T u v, vs , w v∗ W x y Z

ERIK SKJETNE AND JEAN-LOUIS AURIAULT

dimensionless ratio of pressure to viscous forces. gas constant. Reynolds number. unit tangential vector to ". temperature. vector in W. gas velocities. characteristic velocity. Hilbert space. dimensionless microscopic space variable. dimensionless macroscopic space variable. gas compressibility factor.

Greek Letters β ε η " λ µ µ∗ ' 'p ρ ρ∗ τn

Klinkenberg constant. small parameter of scale separation. viscosity. pore surface. mean free path. viscosity. characteristic viscosity. periodic cell. pore volume in the periodic cell. gas density. characteristic gas density. shear rate of deformation on ".

1. Introduction At low or near atmospheric pressure, the gas permeability Kg of a porous sample is greater than its liquid permeability Kl . They are related by the so-called Klinkenberg equation, which was first formulated by Adzumi (1937a, b) and later on by Klinkenberg (1941): ! " b Kg = Kl 1 + , (1) pm where b is a gas dependent constant and pm is the arithmetic mean of the inlet and outlet pressures of the sample. Equation (1) represents a nonlocal law. It is based on the analytical solution for wall-slip flows in tubes and on experiments on gas flows through porous media. A wall-slip velocity is a nonvanishing velocity on the pore walls, which results from a small but nonnegligible Knudsen number, the ratio of molecular mean free path to pore size. Wall-slip is of great importance in petroleum engineering. The vast majority of basic permeability and inertial resistance measurements are done with near atmospheric gas flow.

295

HOMOGENIZATION OF WALL-SLIP GAS FLOW

A review of the work on wall-slip gas flows is given in (Skjetne, 1995). The combination of wall-slip and inertial effects makes the analysis of flow data difficult when only an ad hoc noncoupled Klinkenberg–Forchheimer model is used (Jones, 1972). The coupling between wall-slip and inertial effects has been studied by Skjetne and Gudmundsson (1993) using the average theorem (Whitaker, 1969) and by Skjetne (1995) using an iteration perturbation in periodic media. Models which take into account coupling effects provide better fits to experimental data (Skjetne and Gudmundsson, 1995). In this paper, we analyze the low velocity compressible wall-slip gas flow using the homogenization technique (Sanchez-Palencia, 1980; Hornung, 1997). For simplicity we consider periodic porous media. In Sections 2 and 3, we state the flow problem and we make a perturbation analysis with respect to the ratio of pore to core scale. In Section 4, we make a second perturbation analysis with respect to the Knudsen number. To the lowest order, Darcy’s law is valid with a permeability equal to that of the liquid. The lowest order wall-slip velocity is linear in the macroscopic pressure gradient. In Section 5, we show that our solution is a tensorial form of a local Klinkenberg equation, with a positive Klinkenberg tensor. In Section 6, we explore the symmetry of Klinkenberg tensors. Conclusions of this work are given in Section 7. 2. Wall-Slip Problem Consider the flow of a gas through a porous medium. The porous medium is spatially periodic and consists of repeated unit cells (parallelepipeds), see Figure 1. There are three characteristic length scales in this problem: the molecular mean free path λ, the characteristic microscopic length scale l of the pores and of the unit cell, and the macroscopic pressure loss scale L. We assume that the three length scales are well separated λ ≪ l ≪ L.

(2)

The unit cell has a volume ' and is bounded by ∂', the fluid part of the unit cell has a volume 'p , and the fluid–solid interface inside the unit cell is ". A curvilinear coordinate system is defined on ". The two tangential and the normal unit vectors are t1 , t2 , and n, respectively, where t1 is in the direction of the wall-slip velocity and n is pointing out of the fluid. The local flow problem is governed by the conservation of mass and momentum, boundary conditions, an expression for λ, and the state law for a barotropic gas. Since λ ≪ l, equations for continuum physics are valid in 'p . In order to incorporate a possible Knudsen effect, the boundary condition on " is given by Equation (5): µ∇ 2 v + (η + µ)∇∇ · v − ∇p = ∇ · (ρvv) ∇ · (ρv) = 0 in 'p ,

in 'p ,

(3) (4)

296


Figure 1. Porous medium: (a) macroscopic core sample, (b) periodic cell.

v = −cλ t1 · ∇v · n t1

on ",

p = ZRT ρ,

(5) (6)

where v is the velocity vector, ρ is the density, µ and η are the viscosities, T is the temperature, and Z = Z(p, T ) is the gas compressibility factor. For pressures less than 10 bar, Z is very close to unity. So in the following we assume that Z is a constant. The viscosities µ and η are functions of the pressure: µ = µ(p),

(7)

η = η(p).

(8)

The wall-slip equation (Equation (5)) is specified as follows (see Cercignani, 1988, 1990 for more details): c = 1.1466, √ µ π RT /2 λ= . p

(9) (10)

When p is sufficiently large, λ becomes negligible and Equation (5) becomes the no-slip boundary condition v=0

on ",

(11)

which is also valid for liquids. The shear rate of deformation on " is given by τn = −t1 · ∇v · n = t1i

∂vi nj . ∂Xj

(12)

The equation of state is specified by R=

R0 , M

where R0 is the universal gas constant and M is the molecular weight.

(13)


297

For the momentum and mass equations to be decoupled from the energy equation, the Mach number, defined as the ratio of the characteristic velocity v ∗ to the speed of sound a, must be small Ma =

v∗ ≪ 1. a

(14)

The characteristic velocity v ∗ is the maximum velocity in the pores. The relative variation in temperature must also be small. In this paper, we will only consider isothermal flows. To analyze the above problem, we follow closely the development for steady, compressible flow given by Auriault et al. (1990). Further references can be found in (Ene and Sanchez-Palencia, 1975; Levy, 1987). Compressibility effects are also considered in (Sanchez-Palencia, 1980). Our development differs from that of (Auriault et al., 1990) with respect to the boundary conditions on the pore walls. In this work, the no-slip condition is replaced by a definition for the wall-slip velocity. The conditions at the boundary of a porous medium sample lead to boundary layers in which the flow is not periodic, although the geometry is periodic. Such effects are treated in (Ene and Sanchez-Palencia, 1975; Levy and Sanchez-Palencia, 1975). Although the sample boundary effects may be significant, they are not the subject of this study, and will therefore be ignored. Since the ratio between microscopic and macroscopic length scales is small (Equation (2)), the fundamental perturbation parameter ε is chosen to be ε=

l , L

ε ≪ 1.

(15)

The independent dimensionless numbers which characterize the wall-slip problem may be related to the magnitude of ε. We use the local length scale of a pore l as the characteristic length scale for the variations of the operator ∇. The characteristic pressure p ∗ stands for the average pressure in the sample, and p ∗ is the characteristic pressure increment in the sample. The characteristic density ρ ∗ and viscosity µ∗ are the gas density and viscosity at the average pressure in the sample, respectively. The dimensionless numbers for this problem are the ratio Q of pressure to viscous forces, the ratio Re (the Reynolds number) of inertial to viscous forces, and the ratio Kn (the Knudsen number) of molecular mean free path to pore size: Q=

p∗ l , µ∗ v ∗

(16)

Re =

ρ ∗v∗l , µ∗

(17)

Kn =

cλ . l

(18)

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For completeness we have introduced c = O(1) in the definition of the Knudsen number. The Knudsen number can be seen as an estimate of the ratio of wall-slip velocity to the characteristic velocity v ∗ in 'p : " ! |cλt1 · ∇v · n t1 | cλv ∗ = O(Kn). (19) =O |v| lv ∗ The estimates for the Reynolds number and the Knudsen number are reasonable with regard to the physical processes taking place, while the estimate for the Q number does not reflect the actual relative size between pressure and viscous forces. To see this we apply the phenomenological argument used in (Ene and SanchezPalencia, 1975; Auriault et al., 1990), i.e. the viscous flow is locally driven by a macroscopic pressure gradient ! ∗" µ∗ v ∗ p = O , (20) l2 L so that Q=O

!

p¯ ∗ /L L µ∗ v ∗ / l 2 l

"

= O(ε −1 ).

(21)

For simplicity we use the same names for the dimensionless variables as for the original variables. We apply the so-called macroscopic point of view (Auriault, 1991), and replace ∇ by 1/L times a dimensionless ∇, that is ∇ → (1/L)∇. Note that the dimensionless numbers Re and Kn are still based on the local length scale l. By dividing all terms in the momentum equation with the characteristic viscous force µ∗ v ∗ / l 2 , we obtain the formal dimensionless set ε2 ∇ 2 v + ε2

η+µ ∇∇ · v − ∇p = Re ε ∇ · (ρvv) µ

in 'p ,

(22)

∇ · (ρv) = 0 in 'p ,

(23)

v · n = 0 on ",

(24)

v = −Kn ε t1 · ∇v · n t1 p = ρ.

on ",

(25) (26)

3. Homogenization The next step is to introduce multiple scale coordinates (Sanchez-Palencia, 1980; Bensoussan et al., 1978). The microscopic space variable y is related to the macroscopic space variable x by y = ε −1 x,

(27)


299

and the derivative operator becomes ∇ = ∇x + ε −1 ∇y ,

(28)

where the subscripts x and y mean derivatives with respect to the variables x and y, respectively. Clearly, the flow behavior depends strongly on the evaluation of Re and Kn as functions of the perturbation parameter ε. In this study we focus on the low velocity wall-slip flow and we neglect effects of inertia at the lowest order. Therefore we consider that Re ! O(ε).

(29)

At this stage we assume that ε ≪ Kn ≪ ε −1 .

(30)

Following the multiple scale expansion technique (Sanchez-Palencia, 1980; Bensoussan et al., 1978), the velocity v and the pressure fluctuation p are looked for in the form of asymptotic expansions of powers of ε: v = v(0) (x, y) + εv(1) (x, y) + ε 2 v(2) (x, y) + · · · ,

(31)

p = ρ = p (0) (x, y) + εp (1)(x, y) + ε 2 p (2)(x, y) + · · · .

(32)

Also t1 depends on v. Therefore, the expansion for t1 is in the form t1 = t1(0) (x, y) + εt1(1) (x, y) + ε 2 t1(2) (x, y) + · · ·

(33)

t1(0) is the unit vector in the direction of v(0) . Using the above expansion for p yields appropriate expansions for Kn, µ and η: φ = φ (0) (x, y) + εφ (1) (x, y) + ε 2 φ (2) (x, y) + · · ·

(34)

φ = Kn, µ, η,

(35)

φ (0) = φ(p (0) ).

Substituting these expansions in Equations (22)–(26) gives, by identification of the like powers of ε, successive boundary value problems to be investigated. The lowest order approximation of the pressure verifies ∇y p (0) = 0,

p (0) = p (0) (x),

ρ (0) = ρ (0) (x).

(36)

The first order approximation of the velocity v(0) and the second order approximation of the pressure p (1) are determined by the following set: ∇y2 v(0) − G − ∇y p (1) = 0 in 'p ,

(37)

∇y · v(0) = 0 in 'p ,

(38)

300


v(0) = −Kn(0) t1(0) · ∇y v(0) · n t1(0)

on ",

(39)

where v(0) and p (1) are '-periodic. G is the macroscopic pressure gradient that causes the flow G = ∇x p (0).

(40)

Note that all effects of inertia and compressibility are absent at this level. 4. Small Knudsen Number We now consider the case of small Knudsen numbers. As for low Reynolds numbers in (Firdaouss and Guermond, 1995), we look for v(0) and p (1) in the form of the expansions v(0) = v0 + Kn(0)v1 + (Kn(0) )2 v2 + · · · ,

(41)

p (1) = p 0 + Kn(0) p 1 + (Kn(0) )2 p 2 + · · · .

(42)

The substitution of Equations (41) and (42) into Equations (31) and (32) shows that these expansions are valid up to the term in (Kn(0) )n provided that ε 1/n ≪ Kn ≪ 1.

(43)

Using the above expansion in the formulation of Equations (37)–(39) and identifying like powers of Kn(0) yields, to the zeroth order in Kn(0), ∇y2 v0 − G − ∇y p 0 = 0 in 'p ,

(44)

∇y · v0 = 0 in 'p ,

(45)

v0 = 0 on ",

(46)

where v0 and p 0 are '-periodic. This is a Darcy flow problem. For this case, we define the Hilbert space W of '-periodic, divergence free vectors, where the vectors vanish on ", with the scalar product # (u, v)W = ∇y u : ∇y v dy ∀u, v ∈ W, (47) 'p

where ∇y u : ∇y v = (∂ui /∂yj )(∂vi /∂yj ). Now, let us multiply Equation (44) by u ∈ W and integrate over 'p . By using integration by parts, the divergence theorem, periodicity, and the boundary condition on ", one obtains (Sanchez-Palencia, 1980): ∀u ∈ W,

(u, v0 )W = −(u, G)L2 .

(48)


301

As expected, this is the linear Darcy formulation. Equation (48) shows that v0 is a linear vectorial function of G. The Lax–Milgram theorem ensures a unique 0j solution. Let ki be the particular solution on vj0 when Gi = Iij . We have v0 = −k0 · G,

(49)

which by averaging yields Darcy’s law ⟨v0 ⟩ = −K0 · G, # 1 0 0 K = ⟨k ⟩ = k0 dy. ' 'p

(50) (51)

The second rank permeability tensor K0 is symmetric and positive (SanchezPalencia, 1980; Barenblatt et al., 1989; Dagan, 1989; Matheron, 1967). The first order problem in Kn(0) is the wall-slip correction problem ∇y2 v1 − ∇y p 1 = 0 in 'p ,

(52)

∇y · v1 = 0 in 'p ,

(53)

v1 = −t1(0) · ∇y v0 · n t1(0)

on ",

(54)

where v1 and p 1 are '-periodic. To investigate the above boundary value problem, we first construct a particular '-periodic solenoidal vector vs in 'p which takes 0 the value vs = −t1(0) · ∇y v0 · n t(0) 1 on " (Ladyzhenskaya, 1969). Since v is linear 1 in G, vs is also linear in G. Second, we let v = w + vs and we look for w ∈ W. Multiplying Equation (52) by u ∈ W and integrating on 'p yields the following variational formulation ∀u ∈ W,

(u, w + vs )W = 0.

(55)

The Lax–Milgram theorem ensures the existence of a unique w. Since vs is linear in G, w is also linear in G. The velocity v1 takes the form v1 = −k1 · G,

(56)

⟨v1 ⟩ = −K1 · G,

(57)

where K1 = ⟨k1 ⟩.

(58)

Up to the two first terms, the expansion (41) gives ⟨v(0) ⟩ = ⟨v0 + Kn(0) v1 ⟩ = −(K0 + Kn(0) K1 ) · G.

(59)

302


5. Klinkenberg Law Let us first investigate the properties of the second rank tensor K1 . Multiplying Equation (44) by v1 , using the identity v1 · ∇y2 v0 = ∇y · (v1 · ∇y v0 ) − ∇y v1 : ∇y v0 and integrating on 'p , we obtain 1

0

1

(v , v )W + ' ⟨v ⟩ · G = =

#

'p

#

#"

∇y · (v1 · ∇y v0 ) dy

(v1 · ∇y v0 ) · n dy

τn(0) t1(0) · ∇y v0 · n dy " # = − (τn(0) )2 dy < 0.

=

(60)

(61)

"

The surface integrals on fluid entrances and exits on the unit cell boundary cancel because of the periodicity. After making use of Equation (55) with u = v0 , we are left with −⟨v1 ⟩ · G = G · K1 · G > 0.

(62)

Therefore, K1 is positive. Now, let us consider the flow law Equation (59). After returning to dimensional variables and neglecting higher order correction terms, it can be put in the form ! " K β ⟨v⟩ = − (63) · I + H · ∇x p, µ(p) p where K is the permeability tensor, the second rank tensor H is positive: H = (K0 )−1 · K1 ,

(64)

and β is given by √ cµ(p) π RT /2 β= . (65) l Relation (63) represents the tensorial form of a local Klinkenberg law. For isotropic media, the one-dimensional flow is described by −

dp µ(p) = v, dx K(1 + b(p)/p)

b > 0.

(66)

When the viscosity can be considered as independent of p, the above relation is identical to the Klinkenberg law. On the contrary to Equation (1), Equation (66) represents a local law. It is valid when Re ! ε ≪ Kn ≪ 1.

(67)


303

6. Symmetries Let us check the symmetry of K1 and H, starting with K1 . We introduce two pairs of solutions A and B of the systems of Equations (44)–(46) and of Equations (52)– (54). We insert the first mixed pair v0A and v1B in Equation (61). After making use of Equation (55) where we let u = v0A , we are left with # (0) (0) 1 −' ⟨vB ⟩ · GA = t1B · ∇y v0B · n t1B · ∇y v0A · n dy. (68) "

By using Equation (57) on the LHS of Equation (68) we get

−⟨v1B ⟩ · GA = (K1 · GB ) · GA = GB · (K1 )T · GA = GA · K1 · GB ,

(69)

where T denotes the transpose. Inserting the RHS of Equation (69) in Equation (68) gives # (0) 1 0 0 ' GA · K · GB = t(0) (70) 1B · ∇y vB · n t1B · ∇y vA · n dy. "

For the mixed pair v0B and v1A , we obtain Equation (70), but with A replaced by B and vice versa # (0) 1 0 0 ' GB · K · GA = t(0) (71) 1A · ∇y vA · n t1A · ∇y vB · n dy. "

Subtracting Equation (71) from Equation (70) yields % $ 'GA · K1 − (K1 )T · GB # & (0) ' (0) (0) 0 0 t1B · ∇y v0B · n t1B = · ∇y v0A · n − tt(0) A · ∇y vA · n t1A · ∇y vB · n dy. (72) "

In general, the two terms on the RHS of Equation (72) do not cancel. Hence, K1 is in general not a symmetric tensor. However, if the Darcy flow field v0 is approximately parallel and parallel to " in some neighborhood of ", that is if v0 has small components in the n and t(0) 2 directions, we can write (0) ∇y v0A · n ≃ −|∇y v0A · n| t1A ,

(0) ∇y v0B · n ≃ −|∇y v0B · n| t1B ,

on ".

(73)

For flat surfaces, it can be shown by using Equation (45), as in (Skjetne, 1995), that at least the normal component vanishes: n · ∇y v0 · n = 0.

(74)

Within the approximation (73), Equation (72) becomes ' GA · [K1 − (K1 )T ] · GB # (0) (0) = [|∇y v0B · n| (t1B · t1A ) |∇y v0A · n| − "

(0) (0) −|∇y v0A · n| (t1A · t1B ) |∇y v0B · n|] dy = 0,

(75)

304


and K1 = (K1 )T is symmetric. Note that Equation (73) is a strong criterion. The tensor K1 is close to symmetric when the RHS of Equation (72) is small for all GA and GB . Based on our experience with Darcy flow simulations, it seems that velocity fields typically follow well to the local pore geometry (Skjetne, 1995). This means that Equation (73) may be a good approximation. Let us explore under which conditions H can be symmetric when K1 is symmetric. Since K0 is symmetric, (K0 )−1 is symmetric. ((K0 )−1 )T = ((K0 )T )−1 = (K0 )−1 .

(76)

Using this, we can write HT = ((K0 )−1 · K1 )T = (K1 )T · ((K0 )−1 )T = K1 · (K0 )−1 .

(77)

Then, H is symmetric if and only if (K0 )−1 · K1 = K1 · (K0 )−1 .

(78)

Since, K1 and (K0 )−1 are symmetric, they have real eigenvalues and their eigenvectors can be chosen orthonormal (Strang, 1988), i.e. they are diagonalizable. K1 and (K0 )−1 share the same eigenvector matrix if and only if Equation (78) is fulfilled (Strang, 1988), that is if and only if H is symmetric. The relation between K1 and (K0 )−1 can be investigated. Averaging (see Equation (51)) Equation (44) over 'p and using Gauss’s theorem gives # # 1 1 0 −G = − ∇y v · n dy + p 0 ndy. (79) 'p " 'p " Note that Equation (73) can be reformulated as v1 ≃ −∇y v0 · n on ", so that using Equations (50) and (56) results in # # 1 1 0 −1 0 1 0 −1 0 (K ) · ⟨v ⟩ = k dy · (K ) · ⟨v ⟩ + p 0 ndy. 'p " 'p "

(80)

(81)

If we assume that the first integral on the RHS of Equation (81) is proportional to the volume average K1 of k1 , and that the second term on the RHS of Equation (81) either vanishes (as for flow in a tube) or takes the same form as the first term, we can write C(K0 )−1 · ⟨v0 ⟩ = K1 · (K0 )−1 · ⟨v0 ⟩,

(82)

where C is a constant. Now, we choose ⟨v0 ⟩ to be an eigenvector for (K0 )−1 . Then Equation (82) reduces to C⟨v0 ⟩ = K1 · ⟨v0 ⟩.

(83)


305

Thus, ⟨v0 ⟩ is also an eigenvector of K1 . Since (K0 )−1 and K1 share the same eigenvectors, Equation (78) is fulfilled. Hence, H is symmetric. In brief, H is symmetric if and only if K1 is symmetric and if Equation (81) reduces to Equation (82). Since the symmetry requirements for H are so much more severe than for K1 , we recommend to the experimentalist who applies the tensorial form to first calculate K0 and K1 , then check if K1 is symmetric. If K1 is symmetric, one may check the symmetry of H. It should be noted that when K0 is known, H does not contain more information than K1 . 7. Concluding Remarks The most common way to measure permeability in reservoir cores is to use a gas at atmospheric pressure. We investigated this wall-slip gas flow problem (small Knudsen number) by using a double scale homogenization in spatially periodic media. The Navier–Stokes continuum equations for a compressible gas are valid in the pores except in a thin layer, of size equal to the mean free path of the molecules λ, near the pore walls. The effect of such a Knudsen layer is to change the no-slip Navier–Stokes boundary condition to a wall-slip boundary condition. The fundamental perturbation parameter is the ratio ε = l/L of microscopic (pore) scale l to the macroscopic (core) length scale L. The Reynolds number Re which describes the size of inertial effects was assumed to be small, Re ! O(ε). The Knudsen number Kn = O(λ/ l) was assumed to be greater than the Reynolds number, but still small ε ≪ Kn ≪ 1. The gas was described as nearly ideal (constant compressibility factor). For this problem, both local compressibility and inertial effects are smaller than the wall-slip effects. Under these conditions the main results are – To the lowest order, the flow is described by the Darcy’s law. – The lowest order permeability for a wall-slip gas is equal to that of a liquid. – The lowest order wall-slip effect, Equation (63), is a local tensorial form of the Klinkenberg equation. – The correction Klinkenberg tensor H in Equation (63) is positive. The symmetry of the wall-slip tensors was explored. In general H is not symmetric, but it may, under certain assumptions, be close to symmetric. The tensor K1 = K · H is more likely to be symmetric. If K1 is symmetric, it is an indication that the Darcy velocity field in a small neighborhood out of the pore walls is uni-directional and parallel to the pore walls. Acknowledgements E. Skjetne wants to thank Prof. R. N. Horne for valuable comments and good discussions. Financial support for E. Skjetne to this study was from the Research Council of Norway through the project ‘Non-Darcy flow applications in petroleum science and technology’ (111241/431) of the PROPETRO program.

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