Darcy Law-Theoretical Derivation

Acta ~echanica 25, 153--170 (1977)

| by Springer-Verlag 1977

Theoretical Derivation of Darcy's Law

By

S. P. Neuman~ Tucson, Arizona

With 4 Figures

(Received June 23, 1975)

Summary- Zusammenfassung

Theoretical Derivation of Darey's Law. Darcy's law for anisotropic porous media is derived from the Navier-Stokes equation by using a formal averaging procedure. Particular emphasis is placed upon the proof that the permeability tensor is symmetric. In addition, it is shown that there is a one-to-one relationship between the local and macroscopic velocity fields. This leads to the interesting phenomenological observation that the local velocity vector at any given point must always lie either on a fixed line or in a fixed plane. All of this holds true for an incompressible homogeneous Newtonian fluid moving slowly through a rigid porous medium with uniform porosity under isothermal and steady state conditions. The question whether Darcy's law is applicable under nonsteady or compressible flow conditions, or when the medium has nonuniform porosity, is also discussed. Finally, it is shown that the Hagen-Poiseuille equation, as well as the expression describing Couette flow between parallel plates, can be derived from the equations presented in this work and may thus be viewed as special cases of Darcy's law.

Eine theoretisehe Ableitung des Darcysehen Gesetzes. Das Darcysche Gesetz fiir aniso- trope porSse Werkstoffe wird yon den Navier-Stokes-GIeichungen durch formale ~V[ittelwert- bildung abgeleitet. Insbesondere betont wird der Beweis der Symmetric des Permeabilit~ts- tensors. Welter wird gezeigt, dab eine eineindeutige Beziehung zwischen lokalen and makro- skopischen Geschwindigkeitsfeldern existiert. Dies fiihrt zur interessanten ph~nomenologischen Beobachtung, da$ in jedem Punkt der lokale Geschwindigkeitsvektor entweder auf einer festen Geraden oder in einer festen Ebene liegt. All dies gilt fiir inkompressible homogene Newtonsche Flfissigkeiten, die sich langsam, stationer unter isothermen Bedingungen durch einen starren porSsen KTrper gleichfTrmiger Por6sit~t bewegen. Die Frage, ob das Darcysche Gesetz fiir instationEre StrSmungen oder kompressible Fi~lle oder ffir ungleichfOrmige PorOsit~t gilt, wird ebenfalls diskutiert. AbschlieBend wird gezeigt, dab die Hagen-Poiseuille-Gleichung und der Ausdi'uck fiir die Couette-StrTmung zwischen parallelen Platten yon der in dieser Arbeit angegebenen Gleichung abgeleitet und daher als Spczialfglle des Darcyschen Gesetzes betrachtet werden kTnnen.

Nomenclature

The text is wr i t t en in an indicial no ta t ion whereby quant i t ies with a single subscript (or index) represent components of vectors, and quant i t ies with two subscripts are components of second rank tensors. W h e n an index appears twice in any given term, this te rm mus t be summed over all admissible values of tha t par t icular index.

scalar ratio between v i and projection of (vi)* at centroid of R in the direction of v i half distance between parallel plates, L

Acta Mech. 25/3--4 ~[1

154 S.P. N~U~A~:

d diameter of straight tube, L ] any local property of the fluid defined in the pores and vanishing in the solids </} bulk-volume average of / (/}* pore-volume average of / g acceleration due to gravity, L T -~

gi (0, O, --g), L T -2 G Green's function of porous space h(]) probability density of / on Psi ki i permeability tensor, L ~ I i unit vector lying in the plane of v i (or m 0 and (v~}* and perpendicular to m i m i unit vector parallel to v i n i unit vector normal to -Psi and pointing into the solids p local fluid pressure, M L - 1 T -2 (p}* macroscopic (pore-volume average) pressure, M L - 1 T -2 r radius of straight tube, L R representative elementary volume of porous medium, L ~ t time, T v i local fluid velocity vector, L T -1 (vi} bulk-volume average of v i (darcy velocity or specific flux vector), L T -1 xi vector of Cartesian coordinates, xa being the vertical, L a # - - a m i m j or - - a ( 6 i j - - Iil~) /'m Dirichlet boundary segment of ~9, L ~ l 'sl solid-fluid interfaces in R, L 2 6~ Kronecker delta # dynamic fluid viscosity, M L - 1 T -1 o fluid density, M L -~ ~b porosity of R q5 local force potential, defined as - - x i 9 i d- P---, L 2 T -2

<05}* macroscopic (pore-volume average) force potential, defined as - - x i 9 i d- \ P / * , L e T -." / \

q)m prescribed values of ~D at external boundaries of porous medium, LZT 2 ~o V 2 Laplacian operator, L -2

Introduction

D a r c y ' s law is genera l ly accep ted as the macroscopic equa t ion of mo t ion for Newton ian fluids in porous med ia a t smal l Reyno lds numbers (Stokes flow). The law was es tab l i shed empi r ica l ly b y DAI~C:~ [6] in 1856 and has since then been verif ied expe r imen ta l ly on numerous occasions. A genera l iza t ion of these pre- d o m i n a n t l y one-dimensionM and s t e a d y s ta te expe r imen ta l resul ts leads to the following homogeneous l inear re la t ionship ,

(v~} = ~ e ~@>*. i, i = 1, 2, 3 (1) # Oxj '

which is recognized as Da rcy ' s law for anisot ropic porous med ia (all symbols as well as the index n o t a t i o n are expla ined in the Nomencla ture) .

Eq. (1) is founded essent ia l ly on empir ica l evidence (a l though some people view i t as a n a t u r a l ex tens ion of the Hagen-Poiseui l le equat ion) and i t cannot be de r ived ana ly t i ca l ly b y per forming a m o m e n t u m balance in t e rms of macroscopic quant i t i es on a smMl e lement of the porous medium. As such, i t m u s t be v iewed mere ly as a cons t i tu t ive re la t ion which does not y ie ld much in format ion a b o u t the proper t ies of the pe rmeab i l i t y tensor , k~]i F o r example , is k~i symmet r i c or nonsymmet r ic? Is Eq. (1) va l id under n o n s t e a d y s ta te condit ions? Is i t va l id

Theoretical Derivation of Darcy's Law 155

under conditions of nonisothermal or compressible flow? Some of these and other related questions cannot be answered on the basis of experimental work alone, and a recognition of this fact has led researchers to look for alternative methods of deriving Darcy's law from theoretical considerations.

Numerous theoretical analyses of fluid flow through porous media have been reported in the literature during the past few decades. Some of these analyses treat the flow as a stochastic process [2], [21], [22] while other rely on various simplified models to represent the geometry of the porous medium [2], [7], [12]. PI~AGEI~ [19] considered an isotropie porous medium and, by using a variational approach, arrived at an upper bound for the permeability in terms of a function describing the geometry of the pore space. MOKADA~ [14] derived Darcy's law on the basis of 0nsager's theory and concluded that it cannot be obtaT"med from momentum balance. I~AATS and KLUTE [20] arrived at the macroscopic equations of continuity and motion by utilizing the theory of mixtures. A more complete review of some of these and other related works can be found in SCI~IEII)a~GGEI~ [21] and BEAI~ [4].

HUBBEI~T [11] and HALL [9] were the first workers who tried to derive Darcy's law in a mechanistic fashion by integrating the Navier-Stokes equation over a representative volume element of the porous medium, without specifying the geometry of the pore space. Since the Navier-Stokes equation is an expression of momentmn balance for a Newtonian fluid, the mechanistic theory regards Darcy's law as an average (macroscopic) momentum equation. Therefore, if the averaging process is performed correctly, it should provide significant insight into the nature of the permeability tensor as well as the physical nature of Darcian flow. This had been recognized by PO~EH and ELATA [18] and WHITAXEa [29] who, following the earlier work of HUBBE~T and HALL, tried to derive Darcy's law from the Navier-Stokcs equation in a somewhat more rigorous fashion. How- ever, their results are presented in terms of areal averages of velocity and pressure which are not the quantities usually measured in practice. Moreover, Whitaker 's result differs from the traditional form of Darcy's equation and it does not answer the question whether lc~i is symmetric or not. The analysis of PO~EH and ELATX does indicate that/~ij is symmetric but their conclusions are restricted to a uniform macroscopic flow field in which the average pressure does not depend on the orientation of the averaging area. AItMED and SUNADA [1] used a volume averaging technique to derive a macroscopic equation of motion for Reynolds numbers greater than those encountered in the linear Darcian flow regime. Their development is limited to isotropic porous media and therefore, it does not yield information about the nature of the tensor/~j.

A breakthrough in this trend of analysis occurred in 1967 with the discovery of a mathematical relationship which, in this text, will be referred to as the Slattery-Whitaker averaging theorem. This theorem, discovered simultaneously and independently by SLATTERY [25] and W~IT~KE~ [30], enables one to express the volume averages of space derivatives in terms of the space derivatives of volume averages, thereby making it possible to proceed with the integration of differential equations from one scale of measurement to another in a mathe- matically rigorous fashion. Armed with this new tool, SLATTE~Y [26] and WHITAKE~ [31] both made renewed attempts at deriving Darcy's law from the Navier-Stokes

156 S.P. 5 T ~ u ~ :

equation. Unfortunately, neither of the two developments presented by these authors seem to be quite satisfactory. The result obtained by SLATT]~Y [26, pp. 197--204] differs from the usual form of Darcy's law and it contributes little to one's understanding of the relationship between permeability and pore space geometry. WmTAK]~ [31] in his analysis went into considerably greater efforts in discussing the concept of permeability and yet, he also was not able to relate the permeability tensor directly to pore space geometry. Due to this latter diffi- culty, Wt~ITAK~ was forced to embark on an indirect and laborious argument in trying to demonstrate that permeability must be a symmetric tensor. How- ever, his mathematics involves a fundamental error (WmTAK]~, 1975, personal communication) and his conclusion is therefore not valid. Furthermore, Whitaker's entire derivation of Darcy's law rests on several basic assumptions some of which he could justify only for a linfited class of porous media. The most important of these assumptions states that if the local velocity of the fluid in the pores satisfies the biharmonic equation together with no-slip conditions along the solid-fluid interfaces then, given the average (macroscopic) velocity at each point, the local velocity field is uniquely determined. In his paper WHITAK]~ mentioned that this assumption can be proven for the case of flow past a sphere provided the velocity remains uniform at infinity, and for the case of a spatially periodic flow field. However, the author expressed doubts that a general proof of this assumption will ever be given, " . . . for in each case where we know it holds, there is always one additional requirement placed on the flow -- i.e., the flow is one dimensional, or uniform at infinity, or spatially periodic" [31, p. 21]. This point will be discussed again later in the text.

An at tempt to derive Darcy's law for nonisothermal conditions was reported recently by SC~WEITZE~ [24]. His analysis follows the heuristic approach of Ht~BBW~T [11] and is restricted to isotropic porous media.

The purpose of the present work is to show how Darcy's law can be derived from the ~avier-Stokes equation by using a formal averaging procedure. Partic- ular emphasis is placed upon the proof that kii is symmetric. In addition, it is shown that there is a one-to-one relationship between the local and average velocity fields, such that if either one of them is given, the other is uniquely determined everywhere in the porous medium. This proof also leads to the interesting phenomenological observation that, at any given point inside the pore space, the local velocity vector v~ must always lie either on a fixed line or in a fixed plane passing through that point. The question whether Darey's law is applicable when the flow is nonsteady, the fluid is compressible, and the porous medium has nonuniform porosity is also discussed in the text. Finally, it is shown in the appendix that the Hagen-t)oiseuille equation, as well as the expression describing Couette flow between parallel plates, can be derived from the equations presented in this work and may thus be viewed as special cases of Darcy's law.

Averaging Procedure

Let R be a representative elementary volume of the porous medium defined in the sense of BEAI~ [4, p. 19] and let ~ be the effective porosity (excluding isolated pores) of R. WI~ITAKEa [31] demonstrated that the averaging procedure


used below will lead to meaningful results if the characteristic length of R is much greater than the characteristic length of the pores, and is much smaller than the characteristic length of the entire porous medium. An obvious requirement is that R be large enough to provide a fair representation of all the statistical properties of the pore space. The present analysis is restricted to media which are homogeneous with respect to porosity, qb (i.e., qb remains fixed as R moves from one position in the medium to another; however, k~j is allowed to vary from one R to another), Furthermore, for the Slattery-Whitaker averaging theorem to apply, the size and shape of R must be constant and its orientation relative to any inertial frame must remain unchanged.

Let [(xi) be some property of the fluid (e.g., density, viscosity, etc.) which, by definition, is zero in the solids. Then the "pore-volume average" of J is defined a s

f/dR. (2) <i>* =

Similarly, the "bulk-volume average" of ] is defined as

1 f / dR = qb{/}* (3) <1> = ~ ~n

These averages can be viewed as point macroscopic quantities associated with the eentroid of R, which may lie either in a pore or in a solid portion of the medium. In other words, {]} and @* are defined at each point in a fictitious continuum representing the fluid-saturated porous medium and their vMues may thus change from point to poirLt within a given R. For example, Fig. 1 shows four different points within a single R (solid lines) each of which is the eentroid of another R (solid and broken lines) and thus corresponds to a different value of {[} and {/}*. I t therefore makes sense to talk about the average value of {/} or {/}* over a given R, and the question is how to evaluate it. If the characteristic length of R is chosen according to the requirements mentioned earlier then ([} and (/}* are continuous

~r_-_- 2 I I 1 I I I [ L

~ _ _ _ [ . . . .

I I

_ _ 1 _ _ _ _ . I I I

I �9 [

i I

. . . . I

J

- - 7 I I I I

I I

.1

]Pig. 1. Overlapping representative elementary volumes and their corresponding eentroids

158 S.P. NtlUNANN:

and "smooth" functions. WmTAKER [31] showed that under these circumstances

<</>> = </> (4)

i.e., the average of the average is equal to the average. This result will be used routinely throughout the text.

The Slattery-Whitaker theorem [31], [27, pp. 192--196] relates the average of the gradient to the gradient of the average according to

F s.e

(5)

where F# is the solid-fluid interface in R, and ni represents the components of a unit normal vector pointing in the direction of the solids. This relationship is closely related to Leibnitz's rule for differentiation under an integral sign and can also be viewed as the spatial analog of the general transport theorem. The theorem applies to multiply connected domains of the type represented by the effective void space of a porous medium.

Derivation of Darey's Law

Hydrodynamic Consideration8

The following analysis is restricted to the case of an incompressible homogeneous Newtonian fluid moving slowly through the voids of a rigid porous medium under isothermal and steady state conditions. The fluid is viewed as a single-phase continuum (thus excluding molecular or Knudsen flow) completely filling the effective void space of the medium. The Knudsen number in each pore is assumed to be sufficiently low to prevent slippage at the solid-fluid interface from taking place. The solid and fluid phases are chemically inert and there are no molecular forces of attraction acting between them. Under these conditions, the local equation of continuity at a point in the pore space is simply

~ = 0 (6) ~ x i

and the local equation of motion (Navier-Stokes equation) can be written as

D@ ~V~v~ = o ox~" (7)

Here @ is a force potential (energy per unit. mass) defined as [10]

q5 = --x~g~ + p (8)

and g~ = (0, 0, --g). I t is important to recognize t'hat Eq. (7) does not include inertial terms and therefore it is restricted to relatively small Reynolds numbers (Stokes flow).

Theoretical Derivation el Darcy's Law 159

Taking the divergence of Eq. (7) and utilizing Eq. (5) one finds that the left hand side of the Navier-Stokes equation vanishes, and thus the force potential satisfies Laplaee's equation

F20 = O. (9)

Consider an arbitrary porous medium, ~9, as shown in Fig. 2. The potential at any point within the pore space of ~9 is then uniquely determined by (8)

O - - l ~ f o m ~ 4~ ,,=1 ~ n~ dF (10)

Fm

where G is the Green's function for the pore space of/2, O m is the value of O prescribed at the/~m segment of the boundary (all other segments of the boundary being impermeable), M is the total number of such segments (M = 6 in Fig. 2),

/-- Prescribed ~b boundory [, / (Dirichlel type)

~ ~,/--No flow boundory / ~ ~' ~ i (Neumonn type)

I"2

&

�9 r3

&

Fig. 2. Porous medium, D, with external boundary conditions and representative elementary volume, R

n~ is the unit outward normal vector at the boundary, and the summation is taken over all M Dirichlet type boundary segments surrounding the medium. Eq. (10) indicates that if all the values of Om are now multiplied by a constant, c, the potential at each point in the pores will simplychange from O to cO. Owing to the linear nature of Eq. (7), each eomponent of the local velocity vector will change by a constant factor, c, and therefore the geometrical configuration of the

160 S.s NEV~A~:

streamlines will remain unchanged. However, the total flow rate across any given cross-section through the medium will increase by the factor c.

Consider a particular representative elementary volume R in s as shown in Fig. 2. Let v~ ~ be the local velocity at a given point in R corresponding to the original boundary conditions, and let v~ c be the local velocity at the same point after all M values of r have been multiplied by c. Since the configuration of the streamlines remains fixed, one has

v~ c = cvi ~ (11)

as mentioned earlier. But c is a constant and therefore the corresponding average velocities over R are related by

<v,~} * = (cv~~ * = c(v,~ * . (12)

Let mi represent a unit vector p~ralle] to vi (see Fig. 2) and note that the components of this vector are constant for a given configuration of streamlines and a given set of coordinate axes. The projection of {v~)* parallel to m~ (see :Fig. 3) is then given by m~mi(vj)* where m j ( v l ) * is the scalar (dot) product of the vectors m~ and {v~) *~. As long as the orientations of m~ and (v~}* remain fixed in space (which is the case when the configuration of the streamlines is fixed), the vector m i m j ( v j ) * remains directly proportional to (vi}* and titus Eq. (12) implies

m~mj(v jC} * = c m i m ~ ( v j ~ * . (13)

Eqs. (11) and (13) show that each component of the local velocity vector, v , is directly proportional to the corresponding component of the projection vector m i m j ( v j > * . :Furthermore, since these two vectors are colinear, the constant of proportionality is the same for all three components (i = 1, 2, 3) and remains invariant under a rotation of coordinates. One can therefore write

v~ = a m ~ m j ( v j } * , (14)

where a, the constant of proportionality, depends only on the relative locations of v~ and (v~}* in R.

m~mj<vj~ / " - ,

Fig. 3. Decomposition of vector {vi)* parallel and normal to unit vector mi (or vi)


Let l~ be defined as a unit vector perpendicular to ms and lying in the same plane as m~ (or vi) and (vi}*. The vector (@* can then be decomposed into two orthogonal component vectors, mimi(vj}* and /~/i(vj}*, as shown in Fig. 3. This means that

(v~)* = ~ ( v ~ ) * + h~Xvj)* or (15)

where (~ij is Kronecker delta. Thus, an alternative way of writing Eq. (14) is

v~ = a ( ~ j - l~lj) (vj}*. (16)

Note that the components of l~ (just like those of mi) are constant for a given configuration of streamlines and a given set of coordinates.

Defining

OQj ~ - - a m i I t ~ j

o r

one can write

(17a)

(17b)

where ~ j is obviously a symmetric tensor of second rank (this fact will be used later to prove that the permeability tensor is also symmetric), l~eeognizing that the average vector @~}* varies much more slowly inside R than the local tensor ~ii does (recall that the characteristic length of R is much greater than that of a single pore), Eq. (18) can be integrated over the pore space of R to give

so that the average of ~ i must satisfy

- ( ~ ) * = ~ i , (20)

where d~j is Kronecker delta. This finding will prove to be useful later in the text. Eq. (18) indicates that, for any pattern of streamlines in the pores, there

exists a one-to-one linear transformation mapping the average velocity field into the local velocity field. Having shown that such a transformation exists, one must now ask the question whether or not ccij remains unique when the geometry of the streamlines changes due to a change in boundary conditions (i.e., the question whether or not c%- is an intrinsic local property of the porous medium). To answer this question, it is sufficient to recognize that any potential field in the pore space of P- (see Fig. 2) can be constructed by the linear superposition of M distinct fields, the m-th field being due to a prescribed potential Om on /'m and a zero potential at all other Dirichlet type segments of the boundary.

The local velocity due to the m-th field is related to the average velocity due to that same field by the unique (and invariant) relationship (see Eq. (18))

v ~ = -~,~<v?~}*. (21)

(19)

v~ = - ~ & j ) * , (18)

162 S.P. NEUh'IANN :

The Navier-Stokes and Laplace equations (7) and (9) must be satisfied for the m-th field as well as for any linear combination of these fields. Since both of these equations are linear and Eq. (9) is homogeneous, it follows immediately that

M

v~ = Z vi m (22) m = l

which, together with Eq. (21), gives

M

- X (23) m - - 1

Integrating over the pore space of R and remembering that --@ij}* = dis one finds that

M M

= = X (24) m = l m = l

Since Eqs. (18) and (23) must be equal, one has

M

2 = (25) m = l

and when Eq. (24) is substituted into the right hand side of Eq. (25) the result becomes

M

(c~]} - - ~ i ) (vim) * = 0. (26) m = l

Here, each (vim) * is due to an independent set of boundary conditions and therefore, the vectors (vi~)* are linearly independent. Thus, the only way that Eq. (26) can be satisfied for every possible choice of the vectors (v~-m) * is if

~ = ~ i j for all m = l , 2 , . . . , M (27)

indicating that ~ i is independent of boundary conditions (or streamline configuration) and is thus an intrinsic local property of the porous medium. This together with Eq. (18) proves that there is a one-to-one relationship between the local velocity field, vi, and the macroscopic velocity field, (@*, such that if either one of these fields is given, the other is uniquely determined. I t will be recalled from the introduction that W~TAK]~ [31, p. 21] expressed doubts that a general proof such as this will ever be given.

The uniqueness,of c%- also leads to an interesting phenomenological observation regarding the orientation of the local velocity vector, vi. I~eference to Eqs. (17a) and (17b) will reveal that for cr to be unique, at least one of the two unit vectors, mi or l~, must have a fixed orientation in space (except for sign); the other unit vector is free to rotate in a plane perpendicular to the fixed vector. If m~ is fixed then ~ j in Eqs. (iTs) and (17b) will be constant for any vector l~ lying in a plane perpendicular to the vector m~ (this happens in a pipe-shaped porous channel where m~ is fixed because the orientation of the local velocity is restricted by the surrounding walls). If mi is not fixed, then ~ii in Eq. (17a) will no longer be constant; however, ocij in Eq. (17b) will still be constant provided that the vector


l~ is fixed (except for sign) and m~ is restricted to rotate in a plane which is normal to li. In this latter case, a~. must be represented by Eq. (17b) instead of Eq. (17a). All of this leads to the inevitable conclusion that the local velocity vector, vi, must either have a fixed orientation in space (except for sign) or it must be restricted to rotate in a fixed plane. Intuitively, it appears that this plane will tend to be subparallel to the solid surface nearest the point of interest.

Having proven that the linear transformation implied by Eq. (17) exists and is unique, it is now possible to integrate the Navier-Stokes equation over the porous space of R. Considering the left hand side of Eq. (7)first, and recalling that (v~)* varies slowly in R relative to ~ j (i.e., the characteristic length of (v~)* is much greater than that of v~) one can use Eqs. (4) and (18) to write

(~%)* = -~ (P~ j (v~)* )* = - ~ ( ~ % j ) * (vj)*. (28)

Taking the pore-volume average of the right-hand side of Eq. (7) and using the Slattery-Whitaker averaging theorem (Eq. (5)) together with Eq. (8), one obtains

-~g~ ~- ~ -{- ~ pn~dF. (29)

Fsf

Equating Eqs. (28) and (29) and replacing (v~)* by (v~)/r leads to

a(r --re (72~ i')-----~* (v,)~-@ ax----~- + ~ f pn, dF, (30)

Fsf

where (v~) is Darcy velocity (specific flux) and the pore-volume average of the force potential is given by

(~b)* = --x~g~ -~ (p}* (31) o

The term (V~i)*/~ can be appropriately called the intrinsic macroscopic resistance of the porous medium. To see whether or not it has a unique inverse, one simply needs to substitute Eq. (18) into Eq. (17) and find that

~O -~(p2or (vj)* = e a~" (32)

This shows that V ~ j must be nonsingular wherever the local driving force differs from zero which, under dynamic conditions, must be the case almost everywhere in the porous space of J~. Thus, (V2a~j)*/r cannot be singular and it therefore must have a unique inverse,/q~., such that

(V2er kli = ~ . (33) r

The tensor /q~ is termed the "intrinsic permeability" of the porous medium. Recalling that a~i is symmetric and uniquely defined at each point in the pore space, it is obvious that k~. is also symmetric and must be an intrinsic (unique) macroscopic property of the porous medium (the fact that k~] is a unique property

164 S.P. NEv~A~:

of the medium also follows in a more direct manner from the linear nature of the governing equations).

I t is interesting to note that Eqs. (18), (20), and (33) enable one to determine k~j for porous media with a relatively simple geometry in which the Navier-Stokes equation can be solved either analytically or numerically. This is illustrated in the appendix for two simple eases, one involving Poiseuille flow in a pipe and the other describing Couette flow between two parallel plates.

Substitution of Eq. (33) into Eq. (30) leads to

_ ~ < ~ ) * k~j f pnjdF (34) f F~f

which is a nonhomogeneous linear relationship between the macroscopic darcy velocity, (v~}, and the macroscopic potential, (~)*. The purpose of the following section is to investigate the conditions under which Eq. (34) will reduce to Darcy's law as represented by Eq. (1).

Statistical Considerations

Eq. (34) is identical with Darcy's law except for the surface integral on the right hand side which does not appear in Eq. (1). Since Eq. (34) was derived with the assumption that q~ = constant, it is clear that homogeneity of the porous medium with respect to porosity does not necessarily guarrantee that Darcy's law applies. In fact, it is easy to demonstrate that for certain types of porous media with constant qb, the surface integral in Eq. (34) will not vanish [30, p. 422].

As will be shown below, a sufficient condition for the surface integral in Eq. (34) to vanish is that each component of the surface orientation vector, n~, be symmetrically distributed about a zero average value. This means that the probability of encountering any particular value of n~ on the solid-fluid interface in R, /"8I, is exactly equal to the probability of encountering --n~. All porous media consisting of flow channels or solid particles which possess a point, an axis, or a plane of symmetry automatically satisfy this requirement. This latter class includes practically all the geometrical models that have been used to represent porous media in the past, for example sphere packs, various networks of capil- laries with straight axes, as well as solids cut by systems of planar fractures. The requirement of a symmetrically distributed orientation vector is also satisfied by all porous media having a random arrangement of solid particles or solid-fluid interfaces. This is true regardless of whether or not the material is isotropic with respect to permeability, so that most of the porous media encountered in nature will probably fall within this category.

Let h(n 0 be the probability density function of n~ in R and let h(p) be the probability density of p on Fsf. Obviously, the probability of encountering a given pressure at any point on Fs/is independent of the orientation of F8I at that point. For example, Fig. 4 shows two sets of n~ vectors in R, each set representing points at which the solid-fluid interface has a given orientation. Clearly, within the class of porous media discussed here, there is no physical reason why these two sets of points (indicated by dark and open circles in :Fig. 4) should correspond to two distinct values of pressure, so that h(n~) and h(p) are statistically indepen-


dent (i.e., n~ and l0 are uncorrelated) and therefore h(p, ni) = h(p) h(n~). Thus

c~ 1 c~ 1

f p,~, dF : f f p~h(p, ~)a.~ dp = f #(p)dp f n i h ( n , ) d n i = 0 Fsf 0 --1 0 --1

(35)

owing to the symmetric nature of h(n~), and Eq. (34) reduces to Darcy's law.

Fig. 4. Two sets of n i vectors in R, each set having a different orientation

Extension to 1Nonsteady State

Under nonsteady state conditions the Navier-Stokes equation for low Reynolds numbers (neglecting the higher order inertial term and thus remMning within the Stokes regime) becomes

~v~ (36) #172v~ ~x~ -- ~o -~ .

Taking the divergence of Eq. (36) and utilizing Eq. (6) one finds that the force potential satisfies Laplace's equation just as it did under steady state conditions. This means that one can follow exactly the same argument as that outlined in Eqs. (9)--(27) to demonstrate that c~j in Eq. (18) is symmetric and unique irre- spective of whether or not the velocity varies with time.

This can be understood intuitively by recalling that in steady flow, c~- is independent of the streamline configuration as caused by different external boundary conditions. Since the discussion is limited to incompressible fluids and rigid porous media, temporal changes in the flow pattern can be caused only by changes in the external boundary conditions, which in turn do not affect the values of conj.

Since the medium is rigid and the fluid incompressible, one can write

~v~\* ~(v~}* (37) ~ ~t l = ~

and with this modify Eq. (i) for nonsteady flow according to

# ~x i dp# ~t

166 S.P. NEu~A~:

which is I)arcy's law for nonsteady state flow. Since (V~c~) * is an intrinsic property of the medium, so is /c~i , indicating that the steady state definition of permeability applies also when the flow regime becomes nonsteady. On the other hand, the form of Darcy's law is no longer the same as it was for steady state, and this latter fact has also been noted by other workers [13], [15], [16], [17], [28], [31].

Since in practice Darcy's law is always expressed in the form given by Eq. (1), it is important to know under what conditions can one justify neglecting tile last term in Eq. (38). According to WnITAK]~ [31, p. 25] " I f a tube filled with fluid is subjected to a sudden change in the pressure drop, essentially steadyflow occurs for time greater than t o where t o ~ d~/dv. Here d is the tube diameter and v is the kinematic viscosity. For the purpose of estimating microscopic transient times in porous media, a practical lower bound on v is 10 -2 cm2/sec, and an upper bound on d might be on the order of 10 -1 cm. This gives a microscopic transient time on the order of l sec, and if the transient time for the macroscopic process is much larger (say on the order of minutes), we should be able to treat the flow as quasisteady . . ." . A similar conclusion was drawn earlier by P~rmr~ [17] who showed analytically that the transient time in a saturated porous medium (assum- ing that both medium and fluid are incompressible) is proportional to the permeability and inversely proportional to the kinematic viscosity. PmJ,~ found that the error in using Darcy's law without the transient term will usually be unimpor- tant for the case of a suddenly applied potential gradient. However, he predicted that significant deviations from Darcy's law may occur when the applied potentiM at the boundaries of the porous medimn is periodic in time, even if the frequency is quite low.

(Jompressible Fluids and Nonuniform Media

The foregoing analysis relies heavily on the linearity of the Navier-Stokes equation with respect to the local velocity, vs. When the fluid is compressible,

is no longer constant, and the Navier-Stokes equation becomes nonlinear. However, if one is willing to assume that ~ varies slowly so that within t h e representative elementary volume it is essentially a constant, then the previous results would apply locally within each volume, R. In other words, one may be justified in applying I)arcy's law to compressible Newtonian fluids on a local basis, provided pressures are large enough (or Knudsen numbers are small enough) to ensttre that the no-slip condition at the solid-fluid interface will still hold.

A similar argument can be employed to justify the application of I)arey's law to porous media with nonuniform porosity on a local basis. If the porosity of the porous medium varies slowly relative to the characteristic length of R, then it can be treated as a constant during the averaging process and the results will apply locMly within each representative elementary volume.

Acknowledgements

The author is grateful to T. N. Narasimhan, University of California, Berke- ley; S. Whitaker, University of Californivo, I)avis; J . l~. Philip, CSII~O, Canberra, Australia; and V. J. Gupta, The University of Arizona, Tucson, for commenting


on various aspects of this work. The derivation leading from Eq. (9) to Eq. (12) is similar in principle to that offered by Professor P. IL Day in his graduate course, Soil Science 220, at the University of California, Berkeley.

Appendix

A. Reduction o I Darcy's Law to Hagen-Poiseuille Equation

Consider a porous medium consisting entirely of a single straight tube having a constant circular cross-section. Let r be the radius of the tube and suppose that the averaging volume corresponds to a unit length of the tube. Since flow in the robe is one-dimensional, one can easily solve the Navier-Stokes equation and find that velocity varies parabolically with radial distance from the axis, ~. I~emembering also that the velocity must vanish at the walls of the tube, one can immediately write Eqs. (17) and (18) for the tube as

and

__~x~s(~ ) = c(r ~ _ ~2) m~mS

v@) = - ~ @ ) (vs>.

(39)

(40)

Here (v~)* has been replaced by (v~) because q~ = 1 and therefore these two quantities are identical. Also, e is an unknown constant characteristic of the tube, and mi is a unit vector parallel to the axis of the tube.

From Eq. (20) one has --(~s) = ~s where the star symbol has again been dropped because r = 1, and this together with Eq. (39) leads to

r

~r 21 f c(re __ ~2) m~ms2~r~ d~o = ~S"

0

After evaluating the integral this becomes

i cr~mim j = ~ij. 2

(41)

According to Eq. (33) the permeability of the tube is obtained from (V2~k} kkS = ~iS- But from Eq. (39) it is seen that

so that

(172~s) - ~r ~ c -~ -~ 0 (r 2 _ ~2) m~ms2~ d e =4creams

0

4cm~mkkkS = ~s" (42)

Comparing Eqs. (41) and (42) one finds that

i r2mk~nj. ~ks = T (43)

168 S.P. N~u~A~:

In the particular situation where the tube is oriented parallel to one of the coordinate axes, say x~, then all the components of the permeability tensor will vanish except k~l which will take the form

1 rZ" (44) kll = -~

This is exactly the permeability term appearing in the Hagen-Poiseuille equation, indicating tha t this latter equation may be viewed as a special case of Darcy's law. This result has been anticipated earlier by HVBBE~T [11] on the basis of intuitive reasoning.

B . Reduc t i on to Couette F l o w between Para l l e l P la tes

As another example consider a porous medium consisting of a void space completely enclosed between two parallel plates. Let b be half the distance between the plates and suppose that the averaging volume corresponds to a unit surface area of the plates times the distance between the plates. I f the flow is one-dimensional, one can easily solve the Navier-Stokes equation and again find that the velocity varies parabolically in a direction normal to the plates. Con- sidering the no-slip conditions at the solid-fluid interfaces, one can immediately write Eqs (17) and (18) for the medium as

- ~ j ( x ) = c(b 2 - x ~') m~mj (45)

and

v~(~) = - ~ ( x ) (vj}. (46)

Here x is a variable measured normal to the plates and vanishing at mid-distance between the plates, c is an unknown constant, mi is a unit vector parallel to the direction of flow, and (vi} = (vi}* because qb = i.

From Eq. (20) one has --(c~]} --~ 5~i where the star symbol has been dropped because ~b ~- L and this together with Eq. (45) leads to

b

1 f c(b: x 2) mira s dx ~ 6ii . ~g --b

After evaluating the integral this becomes

2 eb2mim j =_ (~i" (47) 3

According to Eq. (33) the permeability is obtained from (V2c~k} kk~- = dij. But from Eq. (45) it is seen that

b

1 f ~ (V%c~j) - - 2b C ~x--- ~ (b 2 - - x 2) m i m ] dx = 2 c m i m i

--b

so that

2cm~mkkkj = ~i . (48)


Compar ing Eq. (47) and (48) one f inds t h a t

1 b~mkmj" (49)

If li is a un i t vec tor no rma l to the p la tes then Eq. (49) can be wr i t t en as

i b2 (~ ] _ l~lj) (50) #~i = y

ind ica t ing t h a t pe rmeab i l i t y depends on ly on the o r ien ta t ion of the p la tes and is i ndependen t of the d i rec t ion of flow. I n the pa r t i cu l a r s i tua t ion where the p la tes are o r ien ted pe rpend icu la r ' to one of the coord ina te axes, s ay x3, then all the components of the pe rmeab i l i t y tensor will van ish except kl~ and k22 which will t ake the form

1 b2" (51)

This is exac t l y the pe rmeab i l i t y t e rm appear ing i n the well known equa t ion for Couet te flow be tween para l le l plates , ind ica t ing t h a t this l a t t e r equa t ion m a y also be v iewed as a special case of D a r c y ' s law.

References

[1] AI-nVIED, N., and D. K. Sg~AD.~: Nonlinear flow in porous media. Jour. Hydr. Div., ASCE 95 (1969).

[2] B~CE~{.r Y.: Basic transport coefficients as aquifer characteristics. International Association of Scientific Hydrology Symposium on Fractured Rocks, Dubrovnik, 1, 63, (1965).

[3] B~AWTAe~AR.~YA, R.N. , V. I. Gvrwx and G. SPOSITO: On the stochastic foundations of the theory of unsaturated flow. Water Resources Research (1976), in press.

[4] BE~CR, J.: Dynamics of Fluids in Porous Media. New York: Elsevier. 1972. [5] CAg~A~, P. C.: Flow of Gases through Porous Media. London: Butterworths. 1956. [6] DAisY, H. : Les FontMnes Publiques de la Vi]le de Dijon. Paris: Victor Dalmont. 1856. [7] FERRX~DO_~, J. : Les lois de l'6coulement de filtration. Le G6nie Civil 125, 24 (1948). [8] GA/CABEDIAN, P. R.: Partial Differential Equations. New York: J. Wiley. 1964. [9] HALL, W. A.: An analytical derivation of the Darcy equation. EOS Trans., AGU, 37,

185 (1956). [10] HU]3BERT, M. K.: The theory of ground water motion. Journal Geology 48, 785 (1940). [11] HUBBERT, M. K.: Darcy's law and the field equations of the flow of underground fluids.

Amer. Inst. Mining Engin., Petrol. Trans. 207, 222 (1956). [12] I~AY, S.: On the theoretical derivation of Darey and Forchheimer formulas. EOS

Trans., AGU, 39, 702 (1958). [13] LAPWOOD, E. R.: Convection of a fluid in a porous medium. Proc. Cambr. Philos. Soc.

44, 508 (1948). [14] 1VIOKADA~L R. G. : Thermodynamic analysis of the Darcy law. Trans. Amer. Soc. Mech.

Engrs. 83, 208 (1961). [15] OgovYA~, T., and E. S~'vYA~: O povyshenii davleniya v skvazhine posle ee zakrytiya.

Rev. M6ean. Appl. 5, 215 (1960). [16] PAa~A, G. : Flow of fluids through porous deformable solids. Appl. Mech. Rev. 16, 421

(1963). [17] Pm~rP, J. t~.: Transient fluid motions in saturated porous media. Australian Jour.

Phys. 10, 43 (1957). [18] PonEs, M., and C. EL~T~: An analytical derivation of Darcy's law. Israel Jour. Tech-

nology 4, 214 (1966).

Acta Mech. 25/3--4 12

170 S.P. NEU~_~: Theoretical Derivation of Darcy's Law

[19] PR~OE~, S. : Viscous flow through porous media. Phys. of Fluids 4, 1477 (1961). [20] RAATS, P. A. C., and A. KLv~E: Transport in soils: The balance of momentum. Soils

Sci. Soc. Am. Proc. 32, 452 (1968). [21] SCHEIDEGG~, A.E.: The Physics of Flow through Porous Media, 2nd ed. Toronto:

University of Toronto Press. 1960. [22] SC~EIDEOGER, A. E. : Statistical hydrodynamics in porous media, in: Advances in

Hydroscience, Vol. 1. New York: Academic Press. 1964. [23] SC~EIDEGGER, A. E. : Statistical theory of flow through porous media. Trans. Soc. l~heoh

9, 313 (1965). [24] ScHwErrz~a, S. : On a possible extension of Darey's law. Jour. Hydrology 22, 29 (]_974). [25] SL~TTE~Y, J. C. : Flow of viscoelastic fluids through porous media. Amer. Inst. Chem.

Engin. Jour. 13, 1066 (1967). [26] SLATTERY, J. C.: Single-phase flow through porous media. Amer. Inst. Chem. Engin.

Jour. 15, 866 (1969). [27] SL&TTEICY, J.C.: Momentum, Energy, and Mass Transfer in Continua. New York:

McGraw-Hilh 1972. [28] TRUESDELL, C., and W. NOLL: The non-linear field theories of mechanics, in: Handbuch

der Physik, Vol. 3, pt. 3. Berlin--Heidelberg--l~ew York: Springer. 1965. [29] WHrr~XER, S.: The equations of motion in porous media. Chem. Engin. Sci. 2], 291

(1966). [30] W~LKER, S.: Diffusion and dispersion in porous media. Amer. Inst. Chem. Engin.

Jour. 13, 420 (1967). [31] WHITAKE~, S.: Advances in theory of fluid motion in porous media. Industr. Engin.

Chem. 61, 14 (1969).

Pro]. Shlomo P. Neuman Department o/Hydrology and Water Resources

The University of Arizona Tucson, AZ 85721, U.S.A.

Darcy Law-Theoretical Derivation

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