Rend. Sem. Mat. Univ. Poi. Torino Fascicolo Speciale 1989

Nonlinear POE's

W. O. Williams

ON FLOW OF A VISCOUS FLUID THROUGH A POROUS MEDIUM

Introduction

I discuss here some results for a set of equations which I devised seve-ral years ago as appropriate to describe the flow of an incompressible viscous fluid through a porous medium. There are many models for flow through porous medium, based on various constitutive models and even on various versions of the balance equations; my excuse for adding to the group was that I wanted to fìnd the simplest model which was consistent with the balance equations for mixtures, obeyed the laws of thermodynamics, and yet had at least some approximation of terms which model the two fundamental effects of the medium on the fluid: the drag which is exerted on the fluid due to the impedence of the small and tortuous flow channels and the capillarity force which tends to move the fluid from more-filled into less-fìlled channels. My primary goal was not to obtain new equations but to place in context whate-ver the resulting simplified equations were, so that their status was clear and it might be more clear how to remedy by generalization whatever defects of prediction the resulting equations would have. In fact, these equations dupli­cate those derived by several others. At the same ti me they are sufrìciently general, as verifìed below, to include the classical models of flow in porous media, and simple enough that one can do some analysis with them, so that it is feasible to fìnd out how well they model the situation which they are designed to represent. I present some results, largely due to Pericak-Spector, which demonstrate at least in simplified geometry or special situations, that the equations have many of the right properties, and which suggest strongly

160

that the full equations merit further study.

Discussion of the physics; relation to classical forms

The equations which we will discuss were developed from a formula-tion of the thermodynamics of diffusing mixtures developed by R. Sampaio and me [SW]. It wàs assumed that the energies, the stresses and body forces, and the other fluxes for the fluid were functions of the fluid velocity u , its gradient VM, the volume fraction y?, its gradient Vy? and of various thermody-namical arguments, with the force terms assumed to be linear in u, VM, and Vip. The resulting balance equations as derived in [W] are very similar to equations considered by Brinkman [B1,B2], Aifantes [A], Nunziato & Walsh [NW], Bedford and Drumheller [Dru, BD] and Kenyon [K]. Thus, they seem to constitute a well-accepted model for this situation; below we will observe that they generalize the classical equations.

We consider a rigid porous medium. Its porosity is specified by ^>m(tf), the fraction of the total volume dedicated to the pores and thus available to the fluid. The volume fraction of the fluid in any flow thus must lie in the interval [0, >m(a:)] at each point x. The mass density for the flow representing the averaged motion of the fluid is p<p, where p is the actual fluid density, so that balance of mass is expressed by

(p<p)t +dìv(p<pu) = 0.

Silice p is presumed Constant, it can be divided out of the equation. Neglecting external body forces and introducing an averaged stress tensor T, we find that balance of forces takes the standard forni

div T + b = p<p(ut + u • Vu) .

The constitutive equations reduce to

T = —pi + 2p(ip)sym Vu ,

6 = — a(<p, |u|)u — (3(<p)V<p .

Thus the stress consists of the usuai arbitrary pressure plus a viscous term, modulo effects due to <p which we will discuss later. In the body force appear the direct effects of the interaction with the medium: au is the drag term and (3V<p is the capillary force. It is worth noting that the latter appears in the

161

thermodynamical theory as as the gradient of a free energy closely related to the capillary potential which often is introduced in the classical theory.

Introducing these equations into the balance equations, we find it natu­rai to distinguish between two cases. In the region in which <p = <pm and the flow is saturatedy we find [PWl] that the indeterminant pressure is modified due to the constraint of incompressibility, and we have

div u = 0 ,

P<Pm(ut + u • Vw) = -<pmVp + div (pmsymVu) - a(<pm,\u\)u ,

which is a modified version of the Navier-Stokes system for an incompressible fluid, (e/. [B2]).

In the unsaturated region where <p < <pm, the pressure p is zero (atmo-spheric) and we have

tpt + div(y?u) = 0 ,

pip(ut + u • Vw) = div (fisyrriVu) — au — (3V(p ,

which is a modified version of the compressible Navier-Stokes system. It is interesting to relate these equations to the classical versions. The

reduction is efFected by the assumption that the inerti al and viscous terms can be neglected. First, for the saturated case (Darcy flow) one obtains

div u = 0

-<pmVp = au ,

so that Vp = u ,

and we have the classical form of Darcy's Law. We deduce a form for a from the usuai form for the coefficient:

with A' the specific permeability and i] the (bulk) viscosity of the fluid. A form for this equation which is regarded as more accurate experimentally (e/. [BF]) is the Forcheimer law, which yields a more elaborate form for a:

162

Here 6 is a Constant. For the unsaturated case, neglect of the inertial and viscous terms leads to

<pt + div (fu) ,

which combine to yield, if a is a function only of >, the classical diffusion equation

(pt = div f - Vy?) .

If we let D((p) be the usuai difFusion coefficient,

yieids a value for /?; the standard form assumed for D is e + <pp, with e and /? Constant. One should note the other classical form of this balance equation: if we introduce the capillary potential

HV) = JD(-).

and set y = V*"1? then we have the porous medium equation

(T(0))i = A 4, ,

The only coefficient function not yet identified with a classical coefficient is the viscosity //. In [W] it is argued 011 the basis of experimental forms determined for other types of mixtures that an appropriate approximate form would be

/2 = \<p3i] ,

where A is a Constant. Comparison of predictions with experimental data support this form.

Finally, we sometimes may reduce the equations to simpler form even in the case that inertial or viscous terms are left in: in particular, for the unsaturated case if we make the equations quasilinear [W], we arri ve at

- <Ptt + <Pt - Dùnp - - A<pt = 0 , a a

an equation devised by Barenblatt, Kochina and Zheltov [BKZ] to model flow through fissured porous media. This equation, arising in other contexts, was also studied by Ting [Ti] and by Chen and Gurtin [CG]. Of particular

163

interest is the demonstration that solutions to the equation admit stationary discontinuities which decay in time.

Some results for the full system

In [P1,P2] Pericak-Spector adapted methods of fluid dynamics to esta-blish some results for these systems. We outline her results here. Let us consider first the unsaturated equations,

(pi + div (<pu) = 0 ,

(U) <p(ut + u - Vu) = div (fj.syw.Vu) — au — /?V^> ,

on a smooth bounded region B and time interval [0,T], and suppose the coeffi-cients a, /?, TJ to be Cl functions of <p and |t/|. We set initial condìtions w(a;,0), ^>(a;,0) on B and set u(x,t) on the entire boundary of B and (p(x}t) on the portion of the boundary where un< 0, ie., where there is an influx of fluid. Then one can establish a uniqueness theorem by considering two solutions, subtracting the equations for the tvvo and finding estimates for the energy of the difference of the two solutions (using methods of Graffì [G]). The results are:

THEOREM: Ift](<p) > e > 0, or if /3 is strictly isotone, then for each T there is at most one classical solution of (U) with the indicated initial and boundary values.

For the saturated flow equations,

div u = 0 ,

(S) p<pm(ut + w • Vii) = -<fmVp + div (pmsymVu) - a(<pm, \u\)u ,

on a smooth bounded region B and time interval [0,T], we set initial data u(x, 0) on B and boundary values u(x,t) on the boundary of B. Slight modifìcations of the standard proof of uniqueness for the incompressible Navier-Stokes system (e/. [Se]) then yield

THEOREM: For each T there is at most one classical solution of(S) saiisfying the indicated boundary and initial values.

164

In order to consider the stability of solutions to (5), we make the equa-tions non-dimensionai. The dimensionless constants which remain in the equa-tions are

R = ^ , and Q = pU

Pm am L

The first is the usuai Reynold's number, while the second measures the drag effect against other values: a small value corresponds to a high drag force.

Let us introduce the quantity

f(QÌ + Vu)v • v r0 = max max —— r ,„ , ,

[0,T)vtV /|Vv|a

where V denotes the set of divergence-free vector fields with zero values on dB. We let Amm denote the maximum in time of the minimum eigenvalue of symVu, and let A be the Poincaré Constant in

A f\u\> < / |Vw| a .

Then one can establish the following.

THEOREM: Suppose that on [0,T] one of the following holds:

(1) Q < \mm, or (2)R < min(A/Amr0) .

Then the solution to (S) is stable in the nomi of Li. Naturally, this extends to a global (in time) result if the bounds extend

for ali times. The result verifies the naturai prejudice that the drag must have a stabilizing effect, reinforcing the stabilization due to the viscosi ty. In fact,in the usuai applications one may expect the drag efTect to dominate the viscous efTect. (Using experimental values corresponding to the flow of water in foametal from [BJ] we fmd that the ratio R/Q corresponding to a normalizing length of L = 10 inches is IO6.)

Finally, we can remarle that the equations (S) are so similar to the Navier-Stokes system that existence results for the latter (e/. [Te]) easily can be modified to apply to them.

Results for one dimensionai flow with zero viscosity

Consider unsaturated flow in one dimension with zero viscosity:

165

ip(ut + uux) = - O M - (3(<p)<px •

Let us convert these to referential form, so that

x(X,t)= f v(Xr) + Xt ti (*(*,*)) = »(*.*) , Jo

in the usuai way. The equations become

H>\ - vx = 0 ,

Vt = V + (T(<p)<fX •

If we assume a(<p) = a >, with a Constant, the equations become the ones studied by MacCamy [M], Slemrod [SI], and Drew [Dre], the only difFerence being that we must ensure that our function <p does not exceed (pm. It is'not difficult to verify that the results established in those works remain valid even with this restriction [PW2]. We consider ini ti al-vaine problems on (-00 , oo) with smooth data.

THEOREM: Givernp* e [0,y?m], ?/sup |i;(-,0)|+sup |v?(-,0)-y?*| and sup |vx(-,0)|+ sup|y>A"(»0)| Jore sujjiciently small then there exists a solution for ali Urne. The more interesting question concerns instabilities.

THEOREM: / / sup \v(-,0)| + sup |y?(-,0) - <p*\is sujjiciently small and if eitlier inf t>A'(vO) is negative and sujjiciently large or sup |v?x(->0)| sujjiciently large then the solution has unbounded X-derivatives within a finite Urne. Given the problem on a finite interval [0,L], if we have the naturai boundary conditions v(0,<) = v(L,t) = 0 and (for balance of mass) ^>x(0,0 = <px(l,i) = 0, we may extend the initial data as periodic functions to (-00 , oo) and note that the corresponding solution on (-oo , oo) will restrict to become a solution on [0,L].

These results are established by constructing solutions by use of Itie-mann invariants. The method is closely tied to the form of the equation: a modification of the term a to any form but linear seems to rule out use of these techniques. Nontheless, one may expect that this behavior should occur in the more general case. Partially verifying this statement, one can show easily that for the equations with general a, acceleration waves have a criticai amplitude: if the amplitude is larger than this value, then the wave grows to infinite amplitude in a finite time, while if the amplitude is smaller then it decays to zero. This sort of instabilìty of derivatives should correspond to

166

the situation in which the fluid, because of capillary eifects, forms intó smal-ler globs which propagate or remain at rest as a unii, which is an observed phenomenon. A careful study of this situation must await, however, a more careful examination of the interface conditions which obtain in the limiting (globular) case.

Concluding Remarks

It must be emphasized that these equations are not complete in the phy-sical sense; the phenomena of flow in a naturai porous medium can be expected to be modeled in detail only by mudi more involved systems. Unfortunately, such a system, when it is devised, predictably will not be amenable to mathe-matical study. Thus the equations presetited here represent a compromise: while they may be expected to be more precise in prediction than the clas-sical porous medium equations, they remaiii sufFiciently simple that analysis can be made of them.

REFEI tENCES

[A] E. C. Aifantes, On the problem of diffusion in solids, Ada. Medi. 37:265-296 (1980).

[BZK] G.I. Barenblatt, I.P. Zheltov, & I.N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, J. Appi. Math. Medi., PMM 24:1286-1303 (1960).

[BJ] G.S. Beavers & D.D. Joseph, Boundary conditions at a naturaiiy permeable wall, J. Fluid Medi. 30:197-207 (1967).

[BS] G. Beavers & E. Sparrow, Non-Darcy flow through fibrous porous media, J. Appi Medi. 36:711-714 (1969).

[BD] A. Bedford h D. S. Drumheller, A variational treatment of immiscible mix-tures, Ardi. Rational Medi. Anal. 68:37-51 (1978).

[Bl] H. C. Brinkman, On the permeability of media consisting of closely packed porous particle, Appi. Sci. Res. Al:81-86 (1947).

[B2] H. C. Brinkman, Problems of fluid flow through swarms of particles and through macromolecules in solution, Research 2:190-194 (1949).

167

[CG] P. J. Chen & M. E. Gurtin, On a theory of heat conduction with two temperatures, Z. Angew. Math. 19:614-627 (1968).

[Dre] D. A. Drew, Stability of a Stokes layer of a dusty gas, Phys. Fluids 22:2081-2086 (1979).

[Dru] D.S. Drumheller, The theoretical treatment of a porous solid using a mixture theory, Int. J. Solids Struct., 14:441-456 (1978).

[G] D. Graffi, Il teorema di unicità nella dinamica dei fluidi compressibili, J. Rational Medi. Anal. 2:99-106 (1953).

[K] D. E. Kenyon, The theory of an incompressible solid-fluid mixture, Arch. Rational Medi. Anal. 62:131-147(1976).

[M] R. C. MacCamy, An integro-difrerential equation with applications in heat flow, Q. Appi Math. 35:1-19 (1977).

[NW] J. W. Nunziato &; E. K. Walsh, On ideal multiphase mixtures with chemical reactions and difTusion, Ardi. Rational Medi. Anal. 73:285-311 (1980).

[PI] K. A. Pericak-Spector, On stability and uniqueness of fluid flow through a rigid porous medium, Q. Appi Math. 42:165-178 (1984).

[P2] K. A. Pericak-Spector, On fluid flow through a rigid porous medium, in Physical Mathematics and Nonlinear Partial Differential Equations, ed. J. H. Lightboume & S. M. Rankin, Dekker (1985).

[PWl] K. A. Pericak-Spector & W. O. Williams, On work and constraints in mix-ture theory, ZAMP 35:401-412 (1984).

[PW2] • K. A. Pericak-Spector k, W..O. Williams, Existence and non-existence of global solutions for flow in a porous medium, unpublished.

[Se] J. Serrili, Mathematica! principles of classical fluid mechanics, in Handbuch der Physik v.8/1, Springer-Verlag (1959).

[SI] M. Slemrod, Instability of steady shearing flows in a non-linear viscoelastic fluid, Ardi. Rational Medi. Anal. 68:211-255 (1978).

[SW] R. Sampaio &; W. O. Williams, Thermodynamics of di finsi ng mixtures, J. de Mecanique 18:19-45 (1979).

[Te] R. Teman, Navier-Stokes Equations, North-Holland (1977).

[Ti] T. W. Ting, Certain non-steady flows of second order fluids, Arch. Rational Mech. Anal. 14:1-26 (1963).

168

[W] VV. O. Williams, Constitutive equations for flow of an incompressible viscous fluid through a porous medium, Q. Appi Math. 36:255-267 (1978).

W.O. WILLIAMS Department of Mathematics Carnegie Mellon University Pittsburgh, PA - 15213 - U.S.A.

Finito di stampare nel mese di novembre 1991 presso la M. S./Litografia s.r.l. - Torino

per conto del Seminario Matematico dell'Università e del Politecnico di Torino Via Carlo Alberto, 10 - 10123 TORINO