Stochastic Hydrology and Hydraulics 8, 109-116 © Springer-Vcrlag t994

Nonequilibrium statistical mechanical derivation of a noniocal Darcy's Law for unsaturated/saturated flow X. H u and J. H . C u s h m a n 1150 Lilly Hall, Purdue University, West Lafayette, IN 47907-1150, USA

A b s t r a c t : As illustrated variously by wetting and drying scanning curves, flow in unsaturated porous media is inherently nonlocal. This nonlocality is also manifest in hysteresis in the classical Darcy conductivity. It is the authors' belief that most current theories of unsaturated/saturated flow are often inadequate, as they do not a~eonnt for spatial nonlocality and memory. Here we provide a fundamental theory in which nonlocality of the flow constitutive theory is a natural consequence of force balances. The results are derived from general principles in statistical physics and under appropriate limiting conditions, the classical Darcy's Law is recovered for saturated flow. A notable departure in this theory from other nonlocal flow theories is that a classical Darcy type equation on a small scale need not exist.

Key words: Unsaturated, nonlocal, memory, statistical physics. 1 Introduction In 1856, Henry Darcy investigated water flow through vertical homogeneous sand filters in connection with the fountains of Dijon, France. From these experiments Darcy stated the law which bears his name: Q = KA(hl - h2)/L

(1)

where Q is the flow rate (volume per unit time), A is the cross sectional area, and hl-h~ is the difference in piezometric head across the filter length L. This basic law has since been heuristically modified to account for anisotropy and unsaturated flow. A number of tools have since been developed that provide a rational theory for anisotropie saturated flow assuming at least one discrete scale of motion exists (eg., Bear, 1972; Sanchez-Palencia, 1980). Model evidence abounds that flow in unsaturated media is inherently nonlocal. Graphic evidence in this regard can be found in percolation models (el., Grimmett, 1989; Kestin, 1982). and "micromodels". Two-fluid systems at the individual pore level and for small domains have been studied by Chen and Koptik (1985), and Lenormand and Zareone (1984) using micromodels. Related studies can be found in Li et.al., (1986), Li and Wardlaw (1986), Williams and Dawe (1988), Chatzis and Morrow (1984) and Lenormand and Zarcone (1988). Most recently three-phase studies illustrating nonlocal effects have begun to appear (Soil et al., 1993). However, no consistent theory has ever been stated that accounts for the inherent memory and spatial nonlocality associated with unsaturated flow. Invariably, when attempting to derive a flux vs force relationship, even for saturated flow, some sort of homogeneity (asymptotic limit) is assumed on at least one scale, commonly called the REV or Darcy scale (a noteable exception is Neuman and Orr, 1993). When there is no Darcy scale this assumption of local homogeneity brings into question the validity of Darcy's Law, even for saturated

110 flow. Fractal porous media are good examples where a REV does not exist (Cushman, 1993). We will show later that in such cases, even saturated flow may require a nontocal Darcy type law. This is consistent with Neuman and Orr (1993) who provide a nonlocal theory for saturated Steady-flow under the assumption that at some "small" scale a Darcy type linear flux versus force relation holds. The theory presented here is slightly more general in that no small-scale Darcy type equation is assumed appiori. The classical conductivity, being a material coefficient, is only computable indirectly. Consequently, it is in general dependent on the scale of the particular set of observations (Cushman, 1984; 1990). In a heterogeneous environment, observations on one scale will produce one number, while observations on a different scale may give a totally different value. What then is the "correct" value of the conductivity, or more generally, is the classical hydraulic conductivity well defined? The focus of this article is on answering, or at least attempting to address this question. About a decade ago Baveye and Sposito (1984, 1985) attempted to derive balance and constitutive laws from what they called the "relativist" point of view. They argued that an REV may not exist (or if it does exist, it is never measured) and consequently it is an overly restrictive and unnecessary construct. They showed, as had others earlier (e.g., Anderson and Jackson, 1967; Marie, 1967), that the standard averaging analysis could be carried out using a weight function formalism ( the weight function replacing the REV volume). The novelty with their approach was the interpretation of the weighting function as the instrumental window and then assigning meaning to each term in the equations only with respect to the measurement process .... hence the term "relativist". As pointed out by Cushman (1984), the relativist concept was important and novel, but it suffered from a lack of appreciation of scale separation. If a natural scale separation exists, then REV's can be defined on each of the separate scales. Instruments should then be designed with support contained in the maximal REV on each discrete scale. A hierarchy of instrumental windows then leads to a hierarchy of decoupled transport and constitutive laws (as in dual porosity models); the smallest scale constitutive laws will be local in space and time, while the higher scale laws will be local in space. If an REV can not be defined on any scale, the relativistic concept is applicable. But Bayeye and Sposito failed to realized that in such cases constitutive theories are often inherently uonlocal, contrary to what they imply in there 1984 and 1985 articles. It is for this reason that in the formulation presented here our point of departure is nonlocal. In a sequence of articles, Cushman and colleagues (Cushman, 1991; Cushman and Ginn, 1993; and Cushman et al., 1994) have provided a framework based on statistical physics for developing very general theories of dispersive transport without relying on an assumption of Fickian transport on any scale. The resulting equations are nonlocal and applicable to porous media which possess evolving heterogeneity. Cushman and colleagues' results have the same form as those derived using other techniques with other assumptions by Neuman (1993). Based on the success of the statistical physics methods, we apply a related approach to the flow problem in the following section.

1.1 Theory of a dynamic disturbance Suppose at time t = 0 a force field is applied to an unsaturated porous medium which was originally in equilibrium. This force field causes fluid motion (velocity v(r,t)) in the porous medium and an associated gradient in chemical potential. Assume the force depends only on position. In the presence of such a field of force, the force exerted on particle j is represented by Fj(rj ,t), where rj is the position of particle j. Instead of the usual Hamilton's equations, one has /'j = VpjHo(rl . . . . . rN; Pl . . . . . PN)

(2)

and f)j = - V r j H 0 ( r l . . . . . rN; Pl . . . . , PN) + F j ( r i , t )

(3)

where pj is the momentum coordinate of particle j at rj and H0(rl, ..., rN, Pl, ..., PN) is the Hamiltonian of the unperturbed fluid. (It includes fluid-fluid as well as fluid-solid interactions.) Equation (2) is the usual Hamilton equation which states the relationship between the momentum and particle velocity; equation (3) is simply the Newtonian equation of motion (Jackson and Mazur, 1964), Throughout the discussion we consider the system to be classical so that the equation of continuity in phase space for the density function f(rl, ..., rN, pl, ..., pN;t) is Of

N

b~ = - ~ j=l

N

v,~. (~jf) - Z j=l

v , ~ . (/,if)

(4)

111 As from equations (2) and (3) one still has Vrj'rj

(~)

+ V m'lSj = 0

Inserting (5) in (4) gives N

Of

0-T = - ~

(rJ . v , j

+ pj . v p ~ ) f ( r , . . . . . ,'N, p~ . . . . ,

pN;t)

(6a)

j----1

or after using (2) and (3) Of

N

~--~ ---- --iL0f - Z

(6b)

F j ( r j , t ) . Vpj f(rl, ..., pN;t)

j=l

Here we have introduced the Liouville operator for the unperturbed system, L0, defined by N

iLo = Z

( V p j H o ' V r j - Vr~Ho . V p j )

(7)

j=l

Equation (6b) was initially derived by Jackson and Masur (1964) for a pure liquid bulk system. They define Fj(rj,t) to be an external force field (not necessarily derived from a potential). It drives the fluid away from equilibrium. We use a simple perturbation method to study (6b). If one assumes the external field is small, one may write the probability density as the sum of an equilibrium component, f0, and a nonequilibrium perturbation, Af, f = f0 + A f

(8)

so that 0f0 c~-t = -iL0f0

(9)

By substituting (8) into (6b) and using (9) we obtain OAf 0t

N N ---- - i L s A f - Z F j ( r j , t ) • Vpjf0 - Z Fj(rj, t ) . VpjAf j----1

(10)

j=l

Equation (10) is the essential starting point for our theory. Using a different approach, Kubo N

(1959) obtained a similar result. Classical first order perturbation theories assume ~ Fj(rj ,t).Vpj Af j=l

is second order relative to the other terms in (10). For reasons explained later, we will not make this assumption here. Instead, in future analysis we keep all terms in (10). This equation can be solved formally as a linear operator equation whose general solution is

[;-ex.{j

".,]'.} ..,'o.,'..]

According to the initial condition A f = 0 at t = 0 (i,e.,the system is in equilibrium at t = O) we find

112 Fj(rj, t'). Vpjf0dt' J

,{

,

0

}

m t

(12) J

Next assume all the water molecules are identical so that Fj (rj,t) is only dependent of particle positions, not on the particles themselves, i.e., Fj(rj,t) _= F(rj,t). Finally, setting r = t-t'~ we arrive at our principle result Af = - i e x p

-iLor-

t-

F(rm,w)dw-Vp=

(13)

~Vpjfo-F(rj,t-r)drj

To the authors' knowledge (13) is novel.

1.2 GeneralizedDarcy'slaw In this section we use (13) to derive a nonlocal generalization of Darey's law. The average velocity of the fluid at point r and time t is (e.g., Kirkwood, 1967; McQuarrie, 1976) f ~i

V(r,t) = ( ~ i i ' ~ ( r i - r ) ) n(r, t)

-

i'6(ri - r)f(R, P; t)ditdP

n

(14) n(r, t)

where t t = (r~ .... , rN), P : (Pl ..... PN) and n(r,t) is the average number density at r and t. Substituting (13) into (14) and using the fact that the equilibrium expected velocity is zero, one obtains V(r,t)--

{

J

}

n(r-~/dr/i~ri~(ri-r)ex p -iLor-~m F(rm,w)d~v.Vp.

Vpjfo- F(rj,

t-r

r)dRdP

t -

J

V(r,t) : - 1 . , f d, fdr' f ~ i ~ ( ~ - r ) e x p ntr ~J J

0

V p j f 0 ~ ( r j --

J

Ra

J

•

-iLo--

F(r~,~)d~.'~,~ t

r ' ) d R d P . F(r', t - r)

]

= /

dr /

dr/K(F,r,r',t,r)-F(r',t-r)

(15)

where

K(F,r,rt,t,r)=-n~,tQ/~i ri6(ri-r)exp{-iLor-~mtL jF(r",cv)6(rm-r")dr"dw'V£p~ Vp~f06(rj - r ' ) d R d P J

(16)

113 All the physical and mathematical complexities are buried in the tensorial functional, K. From (15) and (I6) we see that K is not only space and time-dependent, but also dependent on F. For fixed time F(r~,t-r) only depends on the position in real space, not on positions of individual particles. In the sense of Gibbs we assume F can be written locally as the gradient of chemical (or gravichemical) potential. Under certain limiting conditions this reduces to a gradient of total head and under even more restrictive conditions, to a gradient in moisture content. We write F(r, t) = - n V / ~ ( r , t)

(17)

where #(r,t) is the chemical potential at time t and location r and where ~ is a constant. Substituting (17) into (16) we get V(r~t) ---- --

d r ' K ( F , r, r', t, r ) . Vp(r', t - T)

(18)

/V ( r , t ) = - i d r / dr'K(/J, r, r', t, r ) . V#(r', t - r)

(19)

dr i 0

Ra

or

0

Ra

where we have used

t

R~

dr V,,,/~(r , w ) 5 ( r m - r " )

....

-- -

fS

ds /~(r ,w)/f(r - r m ) +

......

J

R ,~

.... ,w)Vrm_,,,5(rm-r" )

dr # ( r

(20) to replace F by # in (18) with S the boundary of the flow domain. W e call K the <
1.3 Simplifications Equation (19) is the general form of Darcy's law for an unsaturated porous medium. A number of questions present themselves: i) W h a t is the general form of Darcy's law in a heterogeneous saturated porous medium? ii) If a saturated porous medium is homogenous, what form will the law take? iii) Under what conditions will we get the classical form of Darcy's law? We a t t e m p t to answer these questions in the following. The general outline of what follows can be found in Jackson and Mazur (1964). Let us reconsider equation (10). Unlike the unsaturated system where the perturbation in probability Af may be large relative to f0, in a s a t u r a t e d system this perturbation is small (assuming laminar flow). Consequently, in a saturated porous medium the last term in P~HS is a higher order perturbation when compared to the other terms in (10). For a saturated system we omit this term and approximate OAf as -~y(OAf u 0 t = - i L o A f - E F j ( r j , t ) . Vp)f0 j=l

(21)

This equation can be easily solved to obtain formally t Af = - / exp{-iL0r} E J 0 J

Vpjf0 . F ( r j , t - r ) d v

(22)

If we substitute (22) into (14), we find

v(r,t) = ] aT fdr'K'@,r',t,r).r@',t-T) 0

R3

(23)

114 where

K'(r, r', t, r)

=

n(r,

lci~(r i - r)exp{-iL0r} Z J

VPif°6(rJ - r ' ) d R d P

(24)

The main difference between (23) and (15) is that K' in (23) is no longer a function of F. Identifying Vkt with fiVh (where h is head) in (17) we can also write (23) in a more familiar form t

V(r, t) = /

dr / dr'K(r, r', t, r ) - V h ( r ' , t - r)

0

(25)

P~

where K(r,r',t,r)

= ~K'(r,r',t,r)

(26)

Equation (25) is the general form for Darcy's law under saturated conditions. Inserting (24) into (26), and tinearizing we find (Jackson and Mazur, 1964, equation 19) K(r,r',t,r)

-

n0kBT ~ j i~

i'i~(ri - r)exp{-iL0r} ZJ /'j~(rj - r')fodRdP

(27)

Here no is the uniform equilibrium number density, kB is Boltzmann constant, and T is temperature. The operator exp{-iL0r} is the translation operator for the unperturbed system, displacing the coordinates and momenta upon which it acts backward by the time r. Noting that f0 is invariant with respect to this operator, (27) can be written as K ( r , r ' , t, r) -

n0kBT

(sO~, O)s(,", - " ) ) o

(28)

where we have defined the current vector N

s(r,t) = E

i'i(t)~(ri(t) -- r)

(29)

i=l

If we assume that surface effects may be neglected (a reasonable assumption for saturated systems), and that the solid is statistically homogeneous, then the ensemble average in equation (28) is a function of only the distance between the two points at which the two current vectors are taken (and the difference of the times). Thus if one introduces a new variable y = r'-r, from (25) and (29) one obtains V(r, t) = --

dr 0

dy ~

(s(O, 0)s(y, r))0. Vh(r - y, t - r)

(301

Ra

In equation (30) we have also replaced -r by r and -y by y in the correlation function. This is justified as a consequence of the property of microscopic reversibility and statistical homogeneity. We define the conductivity as K(y, v) -

fl nokBT (s(0, 0)s(y, r))0

(31)

where ( )0 is the equilibrium expected value. Inserting (31) into (30) and replacing r by x we obtain

v(x,t) = - / 0

[ p..a

(321

115 or t

V(x,t) = - /

dr /

0

dyK(x-y,t-r).Vh(y,r)

(aa)

Ra

Equations (32) and (33) are the general forms of Darcy's law in a statistically homogeneous medium with time-dependent hydraulic gradient. If we further assume there is no retardation between the exerting force -Vh and fluid current (or the system is in steady state) and the medium is classically homogeneous ( in the sense of Bear, 1972), then K(x-y,t-r)

~ K~(x-y)~f(t-r)

(34)

Inserting (34) into (33) we obtain the classical Darcy law

v(x, t) = - K . Vh(~, t)

(35)

1.,~ Resume of constitutive equations In this section we summarize the different forms of the generalized Darcy's law. In an unsaturated statistically nonhomogeneous porous medium t

V(r, t) = - /

dr /

0

dr'K(p,r,r', t, r). Vp(r',t - r)

(36)

Ra

where K is a functional tensor of #. In a saturated statistically nonhomogeneous porous medium V(r,t) = - /

dr /

0

d r ' K ( r , r ' , t , r ) . V h ( r ' , t - r)

(37)

Ra

In a saturated statistically homogenous porous medium t

0

Ra

In a saturated classically homogeneous porous medium with no delay response to the external force V(x,t) = - K - Vh(x,t)

(39)

2 Conclusions 1. A nonlocal Darcy's law for unsaturated flow is derived from general principles in statistical physics. In this law conductivity depends on not only time and space, but also chemical potential. 2. For a general saturated heterogeneous porous medium, conductivity is time- and space dependent, but does not depend on head or external force. Under statistical-homogeneity, Darcy's law is of convolution form. 3. The classical Darcy's law is recovered in a saturated, homogeneous, no-retarded medium. Acknowledgements This work was supported under DOE/OIIER contract DE-FG-02-86ER-60310.

116 Reference

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