Derivation of the Forchheimer Law via Homogenization - Springer Link

Transport in Porous Media 44: 325–335, 2001. c 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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Derivation of the Forchheimer Law via Homogenization ZHANGXIN CHEN1, STEPHEN L. LYONS2 and GUAN QIN2 1 Department of Mathematics, Box 750156, Southern Methodist University, Dallas, TX 75275-0156, U.S.A. e-mail: [email protected]. 2 Upstream Strategic Research, Mobil Technology Company, Dallas, TX 75244-4390, U.S.A. e-mail: steve l [email protected]; guan [email protected].

(Received: 20 October 1999; in final form: 15 February 2000) Abstract. In this paper we derive the Forchheimer law via the theory of homogenization. In particular, we study the nonlinear correction to Darcy’s law due to inertial effects on the flow of a Newtonian fluid in rigid porous media. A general formula for this correction term is derived directly from the Navier–Stokes equation via homogenization. Unlike other studies based on the same approach that concluded for the nonlinear correction to be cubic in velocity for isotropic media, the present work shows that the nonlinear correction is quadratic. An example is constructed to illustrate our theory. In this example, the analytic solution to the Navier–Stokes equation is obtained and is utilized to show the validity of the quadratic correction. Both incompressible and compressible fluids are considered. Key words: porous medium, Forchheimer law, non-Darcy’s law, homogenization, high flow rate.

1. Introduction The simplest law for describing the flow of a fluid in porous media is the law obtained by Darcy (1856). Derived from empiricism, this law indicates a linear relationship between the fluid velocity relative to the solid and the pressure head gradient. Subsequently, Dupuit (1863) and Forchheimer (1901) gave further empirical evidence that the linearity in Darcy’s law does not hold for high rates of fluid flow and generalized this law in a nonlinear fashion (i.e., Forchheimer’s law). This generalization has been the subject of many experimental and theoretical investigations. These investigations have centered on the issue of providing a physical or theoretical basis for the derivation of Forchheimer’s law. Many approaches have been developed and analyzed for this purpose such as empiricism fortified with dimensional analysis (Ward, 1964), experimental study (MacDonald et al., 1979), averaging methods (Irmay, 1958), and variational principles (Knupp and Lage, 1995). For more reviews of this subject, refer to books (Bear, 1972; Hannoura and Barends, 1981; Scheidegger, 1974) and the references therein. Recently, the mathematical technique of two-scale homogenization (Bensoussan et al., 1978; Dupuit, 1863) has been exploited to derive Forchheimer’s law (Giorgi, 1997; Mei and Auriault, 1991; Wodie and Levy, 1991) from the Navier–

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Stokes equation. This two-scale technique averages the detailed microscopic equations of flow and yields simpler, macroscopic equations. This is achieved by a careful scaling of the microscopic equations by the ratio of two length scales associated with the microscopic and macroscopic phenomena in a periodic porous medium. This technique is simpler and more mathematically rigorous than a similar technique, the volume averaging method (Whitaker, 1996). However, the homogenization analysis in (Mei and Auriault, 1991; Wodie and Levy, 1991) suggested that the nonlinear correction to Darcy’s law due to inertial effects is cubic in velocity in the isotropic case. This is not in an agreement with experiments (MacDonald et al., 1979) and other theories (Irmay, 1958), which showed that the correction term should be quadratic. To recover the quadratic term, a linearized Navier–Stokes equation was used in (Giorgi, 1997), where at the microscopic level the velocity in the convective inertial term was replaced by the macroscopic velocity; that is, the Stokes equation was essentially used. In this paper we derive the Forchheimer law via the theory of homogenization directly from the Navier–Stokes equation. In particular, we study the nonlinear correction to Darcy’s law due to inertial effects on the flow of a Newtonian fluid in rigid porous media. A general formula for this correction term is derived and analyzed. In the case of isotropic media, we show that this correction is quadratic in velocity. This agrees with experimental results and other theories, as mentioned above. An example is constructed to illustrate our theory. In this example, the analytic solution to the Navier–Stokes equation is obtained and is utilized to show the validity of the quadratic correction. The equation in this example is similar to the Bernoulli equation (Bird et al., 1960). Both compressible and incompressible fluids are analyzed in this paper. The paper is organized as follows. In the next section we derive the general Forchheimer law. Then, in the third section we consider the isotropic case. In the fourth section, we construct our example. Finally, in the fifth section we consider a compressible fluid. We end with two remarks. First, one of the differences between the approach in (Mei and Auriault, 1991; Wodie and Levy, 1991) and our approach is the way the Navier–Stokes equation is scaled in . The choice of power of made in (Mei and Auriault, 1991; Wodie and Levy, 1991) led to the absence of microscopic inertial effects in the first order approximated problem of the Navier–Stokes equation; that is, physically the flow is still in the Darcy regime. In our approach, the microscopic inertial effects are reflected in the first order approximation. The outcome of this difference is that a cubic nonlinear correction term was derived in (Mei and Auriault, 1991; Wodie and Levy, 1991), while a quadratic term is derived here, as mentioned above. For more details on this comparison between (Mei and Auriault, 1991; Wodie and Levy, 1991) and the present work, see the discussion after equation (2.7) later. Second, vectors and matrices will be represented by bold face variables, and the rectangular coordinates in 3 are denoted by x = (x1 , x2 , x3 ).

DERIVATION OF THE FORCHHEIMER LAW

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2. The General Forchheimer Law Let be a bounded rigid domain in 3 with a Lipschitz boundary . We assume that is connected but not necessarily simply-connected. The steady-state Navier– Stokes equation and continuity equation for an incompressible Newtonian fluid in are ρ(u · ∇)u = −∇p + µu + ρg in , ∇ · u = 0 in , u = 0 on , where ρ is the constant density, u is the velocity vector, p is the pressure, µ is the viscosity, ρg is an external body force per volume, and ∇, , and ∇· are the gradient, Laplacian, and divergence operators, respectively. The first equation is the momentum equation, the second is the incompressibility condition, and the third is the no-slip condition on the boundary. An incompressible fluid is considered in this and the next two sections, while the compressible case is treated in the last section. The important characteristics of the homogenization procedure is the existence of two vastly different length scales: the microscale l, which characterizes the typical layer thickness, and the macroscale L, which characterizes the global variation of external forces and boundary data. Let = l/L, with 1. We shall consider the above Navier–Stokes problem in a fluid domain f defined as follows. Let the porous medium have a periodic microstructure with period Y , where Y = Yf ∪ Ys , with Yf and Ys being the fluid and solid parts, respectively. Below Yf sometimes denotes the union of the fluid parts of all the periods. Define f = ∩ {x : x ∈ Yf }. In this paper we only consider a formal expansion of the velocity and pressure, and the boundary of does not play a role in this expansion. Consequently, let = 3 , and define f = {x : x ∈ Yf }. We now consider the problem of finding u and p such that ρ(u · ∇)u = −∇p + µ β u + ρg in f , u = 0 on ∂f , ∇ · u = 0 in f ,

(2.1)

where we have scaled the viscosity coefficient through β (β is to be determined below) and ∂f is the fluid–solid interface. Following the custom of homogenization, we assume that any point is described by two coordinates: x ∈ describing the general location of the point and y ∈ Y giving the location of the point within the -cell Y . Obviously, x and y are related

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by the constant : y ∼ −1 x, (up to translation). Consequently, by the chain rule the relation holds ∇ ∼ ∇x + −1 ∇y ,

(2.2)

where ∇x and ∇y represent the gradient operators with respect to x and y, respectively. Assume that the solution to (2.1) behaves as if it was a function of these two coordinates and that it can be expanded in a power series in terms of . Then the velocity and pressure are expanded in the asymptotic form u (x) = α u0 (x, y) + α+1 u1 (x, y) + · · · , p (x) = p 0 (x, y) + 1 p 1 (x, y) + · · · ,

(2.3)

where α is to be determined below and ui and p i are Y -periodic in y, x ∈ , y ∈ Yf . We now substitute (2.3) into (2.1) and apply (2.2) to see that, with x ∈ and y ∈ Yf , ρ{( α u0 + α+1 u1 + · · · ) · (∇x + −1 ∇y )}( α u0 + α+1 u1 + · · · ) = −(∇x + −1 ∇y )(p 0 + p 1 + · · · ) + ρg + + µ β (∇x · + −1 ∇y ·)(∇x + −1 ∇y )( α u0 + α+1 u1 + · · · ),

(2.4)

(∇x · + −1 ∇y ·)( α u0 + α+1 u1 + · · · ) = 0.

(2.5)

and

Also, we have the no-slip condition, with x ∈ and y ∈ Yfs , α u0 + α+1 u1 + · · · = 0,

(2.6)

where Yfs is the interface between Yf and Ys . We now collect terms with like powers of in (2.4). Before this, we need to determine the values of α and β in (2.4). Toward that end, let us rewrite (2.4) as follows: ρ 2α−1 (u0 · ∇y )u0 + + ρ 2α [(u0 · ∇x )u0 + (u0 · ∇y )u1 + (u1 · ∇y )u0 ] + ρ 2α+1 . . . = −( −1 ∇y p 0 + {∇x p 0 + ∇y p 1 } + ∇x p 1 ) + ρg + + µ α+β−2 y u0 + µ α+β−1 (2∇x · ∇y u0 + y u1 ) + µ α+β . . .

(2.7)

The macroscopic pressure gradient ∇x p 0 is of order zero (i.e., 0 ), and the first term in the viscous force, µy u0 , is of order α + β − 2. To balance them, let α + β − 2 = 0. Also, keep in mind our attempt to include the nonlinear inertial effect; that is, we want to have the inertial and viscous forces to play a comparable role. The first term in the inertial force, (u0 · ∇y )u0 , is of order 2α − 1, so we set 2α − 1 = 0. Consequently, it follows that a reasonable choice for α and β is α = 12 , β = 32 .

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As mentioned in the introduction, a different scaling in was used in (Darcy, 1856; Wodie and Levy, 1991). Essentially, the contribution from the inertial force in the zero order (i.e., in order 0 ) was fully ignored. Namely, the macroscopic pressure gradient ∇x p 0 was balanced solely by the viscous force µy u0 . Physically, this implies that we are still in the Darcy regime at this stage. To include the inertial effect, the next order term in (2.7) was utilized in (Mei and Auriault, 1991; Wodie and Levy, 1991), which led to a cubic nonlinear correction to Darcy’s law, instead of the quadratic correction derived here. With the above choice of α and β, the −1 term of equation (2.4) yields −∇y p 0 (x, y) = 0. This implies that p 0 = p 0 (x).

(2.8)

That is, p 0 is independent of y, and thus all terms involving ∇y p 0 vanish. This corresponds to the intuition that the local average of p does not oscillate. Also, the 0 term of (2.4), the −1/2 term of (2.5), and the 1/2 term of (2.6) lead to ρ(u0 · ∇y )u0 = −∇y p 1 + µy u0 + ρg − ∇x p 0 u0 = 0 on Yfs . ∇y · u0 = 0 in Yf ,

in Yf ,

(2.9)

We study the local problem (2.9). The usual Sobolev spaces W m,π () with the norm · W m,π () will be used, where m is a nonnegative integer and 0 π ∞. When π = 2, we simply write H m () = W m,2 (). When m = 0, we have L2 () = H 0 (). Below (·, ·)Q denotes the L2 (Q) inner product (or sometimes the duality pairing). For notational convenience, (H m ())3 will be simply indicated by H m (). Also, introduce the space of Y -periodic functions VY = {v ∈ H 1 (Yf ) : v|Yfs = 0, ∇y · v = 0, and Y -periodic}, with the inner product ∇y v · ∇y w dy, (∇y v, ∇y w)Yf ≡

v, w ∈ VY .

(2.10)

Yf

Note that VY is a Hilbert space and the associated norm is equivalent to the usual H 1 (Yf )-norm. We now write (2.9) in a variational formulation. Towards that end, first observe that, by the definition of VY and the divergence theorem, (∇y p 1 , w)Yf = (p 1 , w · n)Yfs = 0

∀w ∈ VY ,

(2.11)

where n denotes the outward unit normal vector to Yfs . Then (2.9) can be written as follows: −ρ(u0 u0 , ∇y w)Yf + µ(∇y u0 , ∇y w)Yf = (ρg − ∇x p 0 , w)Yf

∀w ∈ VY , (2.12)

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where, with u0 = (u01 , u02 , u03 ) and w = (w1 , w2 , w3 ), (u u , ∇y w)Yf = 0 0

3

u0j u0i

Yf i,j =1

∂wi dy. ∂yj

Conversely, it can be shown that (2.12) yields (2.9). Namely, they are equivalent. Note that (2.12) always admits a solution. Moreover, if ρg − ∇x p 0 VY∗ is not very large, the solution is unique (Girault and Raviart, 1981), where · VY∗ is the dual norm to VY . For large values of ρg − ∇x p 0 VY∗ , the uniqueness fails and bifurcations may arise. In this situation, there is no Darcy’s law. For i = 1, 2, 3, define Ki ∈ VY to be the solution of (∇y Ki , ∇y w)Yf = (1, wi )Yf

∀w = (w1 , w2 , w3 ) ∈ VY .

(2.13)

The tensor K is defined by K = (Kij )i,j =1,2,3 . Also, for fixed v0 ∈ VY we introduce an operator J = J(v0 ) ∈ L(VY , VY ), which is defined as follows: for each v ∈ VY , Jv ∈ VY is the solution of the Stokes problem (∇y (Jv), ∇y w)Yf = (v0 v, ∇y w)Yf

∀w ∈ VY .

(2.14)

The operator J depends on v0 as a parameter. Note that, by the Sobolev imbedding, |(v0 v, ∇y w)Yf | Cv0 L4 (Yf ) vL4 (Yf ) |w|H 1 (Yf ) C|v0 |H 1 (Yf ) |v|H 1 (Yf ) |w|H 1 (Yf ) . Thus (2.14) is well-defined. With the definition of K and J, (2.12) can be written as (−ρJ(u0 ) + µI)u0 = K(ρg − ∇x p 0 ),

(2.15)

where I is the identity matrix. We remark that the homogenization of the Navier–Stokes equation was also studied in (Sanchez-Palencia, 1980), where a fully nonlinear correction term was derived. This is in contrast with the quasi-nonlinear correction derived here, which is made possible by the introduction of the operator J. Also, this quasi-nonlinear correction is suitable for the study of the isotropic case in the next section. For any generic function φ defined on Yf , we introduce its volume average over Yf 1 φ = φ(y)dy, |Yf | Yf where |Yf | indicates the volume of Yf . We now apply the average operator to (2.15): (2.16) −ρ J(u0 )u0 + µ u0 = K (ρg − ∇x p 0 ). When J(u0 ) is zero, (2.16) represents the classical Darcy’s law. In general, (2.16) is the generalized Darcy’s law, i.e., Forchheimer’s law.

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We conclude this section with three remarks. First, as in (Sanchez-Palencia, 1980), it can be easily shown that the matrix K is symmetric and positive definite. Second, by Green’s formula and the definition of VY , we see that (∇y (Jv), ∇y v)Yf = (v0 v, ∇y v)Yf = 0

∀v ∈ VY ,

which means that J is trivially negative semidefinite in the sense (Jv, v)Yf ≡ (∇y (Jv), ∇y v)Yf 0

∀v ∈ VY .

We can solve (2.13) and (2.14) for K and J, respectively; these two equations are numerically solvable. Finally, the convergence of homogenization for the Stokes problem was proven in (Tartar, 1979). A convergence proof for the Navier–Stokes problem considered here is beyond the scope of this paper. 3. The Isotropic Case In this section we treat the case where the porous medium is isotropic, following the treatment presented in (Barak, 1987; Hassanizadeh and Gray, 1988). We first review some classical results on second-order, tensor-valued isotropic functions (Boehler, 1977; Giorgi, 1997). A second-order, symmetric, tensor-valued, isotropic function of a vector v = (v1 , v2 , v3 ), L = (Lij )i,j =1,2,3 , can be expressed as follows: Lij (v) = a0 (|v|)δij + a1 (|v|)vi vj ,

(3.1)

where a0 and a1 are scalar-valued isotropic functions of |v|, δij is the Kronecker symbol, and |v| = (v12 + v22 + v32 )1/2 . On the other hand, a second-order, skewsymmetric, tensor-valued, isotropic function of a vector is identically zero. Apply a series expansion to (3.1) to see that Lij (v) = (a00 + a01 |v| + a02 |v|2 + · · · )δij + + (a10 + a11 |v| + a12 |v|2 + · · · )vi vj .

(3.2)

With u0 T being the transpose of the vector u0 , set H = H(u0 ) ≡ −

ρ J(u0 )u0 u0 T I + µI. |u0 |2

Note that H = 12 (H + HT ) + 12 (H − HT ), where HT indicates the transpose of the operator H (or the adjoint of H with respect to the inner product (2.10)). With the application of (3.2) to (H + HT )/2 and the skew-symmetry property of (H − HT )/2, we see that H = (H00 + H01 |u0 |)I + O(|u0 |2 ),

(3.3)

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for some constants H00 and H01 . Substitute (3.3) into (2.16) and retain the first two terms in (3.3) to see that (H00 + H01 |u0 |)u0 = K(ρg − ∇x p 0 ).

(3.4)

This is the Forchheimer law for an isotropic porous medium, and we see a quadratic correction to the Darcy law. If desired, it is possible to retain higher order terms (e.g., the third order term in u0 ) in (3.4). 4. An Example In this section we consider the Navier–Stokes equation without the gravity term; that is, we consider the following example: ρ(u · ∇)u = −∇p + µu,

∇ · u = 0.

(4.1)

We rewrite (4.1) in terms of cylindrical polar coordinates (r, θ, z), where x1 = r cos θ,

x2 = r sin θ,

x3 = z.

Define the vectors er = cos θe1 + sin θe2 ,

eθ = − sin θe1 + cos θe2 ,

ez = e3 ,

where ei is the unit vector in 3 with the ith component one and other components zero, i = 1, 2, 3. The vectors er , eθ , and ez represent the unit vectors in the direction of increasing r, θ, and z, respectively. With u = ur er + uθ eθ + uz ez , (4.1) can be written in cylindrical polar coordinates as follows: ρ 2 ∂p ur 2 ∂uθ + µ ur − 2 − 2 , ρ(u · ∇)ur − uθ = − r ∂r r r ∂θ ρ 1 ∂p uθ 2 ∂ur + µ uθ − 2 + 2 , ρ(u · ∇)uθ + ur uθ = − r r ∂θ r r ∂θ ∂p + µuz , ρ(u · ∇)uz = − ∂z ∂uz 1 ∂(rur ) 1 ∂uθ + + = 0, (4.2) r ∂r r ∂θ ∂z where uθ ∂ ∂ ∂ + + uz , (u · ∇) = ur ∂r r ∂θ ∂z

1 ∂ = r ∂r

∂ r ∂r

+

1 ∂2 ∂2 + . r 2 ∂θ 2 ∂z2

We now seek the solution to (4.2) in the form u = uθ (r)eθ .

(4.3)

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That is, we consider a circular flow. This form always satisfies the incompressibility condition. Substituting (4.3) into (4.2), the Navier–Stokes equation becomes 2 1 ∂p ∂ uθ uθ ∂p 1 ∂uθ ∂p ρ 2 uθ = , 0=− +µ − . (4.4) + , 0= 2 2 r ∂r r ∂θ ∂r r ∂r r ∂z The third equation of (4.4) implies that p is independent of z. Also, since uθ depends only upon r, the second equation of (4.4) yields that there is a function p1 (r) such that ∂p = p1 (r). ∂θ Consequently, integration leads to p = p1 (r)θ + p2 (r). Note that p1 (r) = 0; otherwise, p would be a multivalued function of θ. The above analysis, together with the first equation of (4.4), implies that dp ρ 2 u (r) = . (4.5) r θ dr Comparing (3.4) and (4.5), we see the validity of the quadratic correction term in the Forchheimer law (3.4). We remark that while the above analysis is analytical and does not involve homogenization, it follows from the equivalence between (2.9) and (2.12) that the homogenization approach would produce the solution of the same form as in (4.5). Also, uθ can be obtained from the second equation of (4.4), i.e., duθ d2 uθ − uθ = 0, + r (4.6) dr 2 dr and an application of the no-slip condition. From uθ we can solve (4.5) for p. Notice that (4.6) is an ordinary differential equation and its analytical solutions can be easily found. Equation (4.5) is similar to the Bernoulli equation (Bird et al., 1960), as mentioned in the introduction. r2

5. A Compressible Case In this section we consider the case of a compressible fluid where the density ρ is assumed to be a given function of the pressure ρ(x) = F (p(x)), with F being an increasing function of p (i.e., density increases as pressure increases). The analysis below follows (Sanchez-Palencia, 1980) for treating the Stokes flow. In the fluid domain f , we consider the steady-state Navier–Stokes equation and continuity equation for such a compressible fluid ρ (u · ∇)u = −∇p + µ β u + η β ∇(∇ · u ) + ρ g u = 0 on ∂f , ∇ · (ρ u ) = 0 in f ,

in f ,

(5.1)

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together with ρ e = F (p e )

in f ,

(5.2)

where η is another viscosity coefficient. As in (2.3), we assume the asymptotic expansion u (x) = α u0 (x, y) + α+1 u1 (x, y) + · · · , p (x) = p 0 (x, y) + 1 p 1 (x, y) + · · · , ρ (x) = ρ 0 (x, y) + 1 ρ 1 (x, y) + · · · .

(5.3)

From (5.2) we see that ρ 0 = F (p 0 ). Also, with the same analysis as for equation (2.7), the reasonable choice for α and β is α = 12 ,

β = 32 .

Next, similar to (2.8) it can be seen that p 0 = p 0 (x). As a result, we have ρ 0 (x) = F (p 0 (x)).

(5.4)

Now, substituting (5.3) into (5.1), applying (2.2), and collecting terms with like powers of , the 0 term of the first equation, the −1/2 term of the second equation, and the 1/2 term of the third equation of (5.1) lead to ρ 0 (u0 · ∇y )u0 = −∇y p 1 + µy u0 + η∇y (∇y · u0 ) +ρ 0 g − ∇x p 0 in Yf , u0 = 0 on Yfs . ρ 0 ∇y · u0 = 0 in Yf ,

(5.5)

Note that since ρ 0 > 0, the above second equation yields that ∇y · u = 0

in Yf .

(5.6)

This means that the flow is incompressible at the local level. Consequently, the η term in the first equation of (5.5) vanishes. Thus, (5.5) is the same as (2.9), and so is the analysis. To analyze the macroscopic behavior, we use the 0 term of the expansion of the second equation of (5.1): ρ 0 (∇x · u0 + ∇y · u1 ) + ρ 1 ∇y · u0 + (∇x ρ 0 + ∇y ρ 1 ) · u0 = 0.

(5.7)

As in (2.11), it can be shown that ∇y · u1 = ∇y ρ 1 · u0 = 0, where · is the volume average, as defined in the second section. Apply this, (5.6), and the volume average operator to (5.7) to obtain ∇x · (ρ 0 u0 ) = 0, where ρ 0 is given by (5.4). This equation and (2.16) (or (3.4)) are the macroscopic equations for the compressible fluid considered in this section.


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Acknowledgement The first author is supported in part by National Science Foundation grants DMS9626179, DMS-9972147, and INT-9901498, and by a gift grant from the Mobil Technology Company. References Barak, A. Z.: 1987, Comments on high velocity flow in porous media by Hassanizadeh and Gray, Transport in Porous Media 2, 533–535. Bear, J.: 1972, Dynamics of Fluids in Porous Media, Dover, New York. Bensoussan, A., Lions, J. L. and Papanicolau, G.: 1978, Asymptotic Analysis of Periodic Structures, North-Holland, Amsterdam. Bird, R. B., Stewart, W. E. and Lightfoot, E. N.: 1960, Transport Phenomena, Wiley, New York. Boehler, J. P.: 1977, On irreducible representations for isotropic scalar functions, Z. Angew Math. Mech. 57, 323–327. Darcy, H.: 1856, Les Fontaines Publiques de la Ville de Dijon, Victor Dalmond. Dupuit, J.: 1863, Estudes Theoriques et Pratiques sur le Mouvement des Eaux, Dunod. Forchheimer, P.: 1901, Wasserbewegung durch Boden, VDIZ 45, 1782–1788. Giorgi, T.: 1997, Derivation of the Forchheimer law via matched asymptotic expansions, Transport in Porous Media 29, 191–206. Girault, V., and Raviart, P.-A.: 1981, Finite Element Approximation of the Navier–Stokes Equations, Springer-Verlag, Berlin, Heidelberg, New York. Hannoura, A. and Barends, F. B. J.: 1981, Non-Darcy flow, a state of art, in: A. Verruijt and F. B. J. Barends (eds), Proc. Euromech 143, 37–51. Hassanizadeh, S. M. and Gray, W. G.: 1987, High velocity flow in porous media, Transport in Porous Media 2, 521–531, 3 (1988), 319–321. Irmay, S.: 1958, On the theoretical derivation of Darcy and Forchheimer formulas, J. Geophys. Res. 39, 702–707. Knupp, P. M. and Lage, J. L.: 1995, Generalization of the Forchheimer-extended Darcy flow model to the tensor permeability case via a variational principle, J. Fluid Mech. 299, 97–104. MacDonald, I. F., El-Sayed, M. S., Mow, K. and Dullien, F. A. L.: 1979, Flow through porous media: the Ergun equation revisited, Indust. Chem. Fundam. 18, 199–208. Mei, C. C. and Auriault, J.-L.: 1991, The effect of the weak inertia on flow through a porous medium, J. Fluid Mech. 222, 647–663. Sanchez-Palencia, E.: 1980, Non-homogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer-Verlag, Heidelberg. Scheidegger, A. E.: 1974, The Physics of Flow Through Porous Media, 3rd ed., University of Toronto Press, Toronto. Smith, G. F.: 1971, On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors, Int. J. Engng. Sci. 9, 899–916. Tartar, L.: 1979, Compensated compactness and applications to PDEs, in: J. Knops (ed.), Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, vol. IV, Research Notes in Math. Pitmann, London, pp. 136–212. Ward, J. C.: 1964, Turbulent flow in porous media, J. Hydr. Div. ASCE 90, 1–12. Whitaker, S.: 1996, The Forchheimer equation: A theoretical development, Transport in Porous Media 25, 27–61. Wodie, J.-C. and Levy, T.: 1991, Correction non lineaire de la Loi de Darcy, C. R. Acad. Sci. Paris, Serie II 312, 157–161.