Some global properties of flow in a heterogeneous isotropic porous medium

Mechanics Research Communicalions, Voi. 26, No. 2, pp. 161-166, 1999 Copyright © 1999 Elsevier Science Ltd Printed in the USA. All fights reserved

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Some Global Properties Of Flow In A Heterogeneous Isotropic Porous Medium M. S. Ghidaoui Department of Civil and Structural Engineering, H o n g K o n g U n i v e r s i t y of Science a n d Technology, C l e a r W a t e r Bay, H o n g K o n g A. A. K o l y s h k i n D e p a r t m e n t of E n g i n e e r i n g M a t h e m a t i c s , R i g a Technical University, Riga, L a t v i a LV 1010 (Received 30 July 1988; acceptedfor print 5 January 1999)

Introduction The hydraulic conductivity of a heterogeneous porous media varies with position. In an inspiring paper, Sposito [4] applied a dynamical system approach to the study of the global properties of flow paths that result from applying the Darcy's law to steady flow in a heterogeneous porous media. He showed that vorticity is non-zero if the iso-conductivity and equipotential surfaces are not parallel. In addition, Sposito [4] showed that the flow paths produced by applying Darcy's law to steady flow in an arbitrary heterogeneous porous media are not closed paths and have zero helicity. That is, trajectories never cross themselves and the flow paths are not looped. Sposito [4] concluded that this flow do not exhibit chaotic behavior. The understanding of whether or not fluid trajectories in an arbitrary heterogeneous porous media cross themselves is important for proper water quantity and quality modeling. For example, looped trajectories induce transverse transport of mass, momentum and energy. The length scale of this transverse transport is defined by the width of the loop. Therefore, looped motion tends to accentuate the transverse spread of a contaminant in a porous media--explaining why looped motion can be important to the understanding of water quality modeling. Moreover, looped motion is important to the understanding of the flow dynamics because this motion can enhance the mixing of mass, momentum and energy across the main flow direction. The objective of this paper is to establish global dynamic features of fluid trajectories in steady flow in heterogeneous and isotropic porous media, where C 1 continuity is the only condition imposed on the spatial variation of the hydraulic conductivity. In particular, it is shown that the three dimensional flow paths, which result from using Bachmat's equation to model steady state flow in heterogeneous and isotropic porous media, are not closed paths and have zero helicity. That is, a fluid particle 161

162

M.S. GHIDAOUI and A. A. KOLYSHKIN

cannot re-visit any of its previous locations and a trajectory never crosses itself implying that flow paths are not looped. Hence, the motion is not chaotic. This proof is accomplished by the Lyapunov and the vorticity functions. Moreover, it is shown that flow paths which result from applying either Bachmat's equation or Darcy's equation to steady state flow in heterogeneous and isotropic porous media are unconditionally stable to time dependent perturbations. This proof is accomplished by the classical result of Sturm-Liouville.

Governing Equations When the flow in a porous media is laminar the three dimensional Darcy's equation is a good approximation for the momentum equation [2]. That is,

= -KVh,

(1)

where ~"= ¢k d~t ~ d~ ~ is the velocity vector of the fluid; (x, y, z) are the three spatial dt' dr/ coordinates; t is the time coordinate; K ( x , y, z) is the function defining the spatial variability of the hydraulic conductivity of a heterogeneous isotropic porous media; Vh is the gradient of the hydraulic head. It must be noted that Darcy's law may give satisfactory results for Reynolds number, Re, as high as 10 [2]. Hence, the Darcian model for three-dimensional saturated flow in an isotropic heterogeneous porous medium with no sources or sinks is [2]

ot-ox\ g / + N \ oy/+g\ Oz]'

(2)

where S is the effective storage coefficient. As Re increases beyond 10 inertial forces become important and Darcy's law fails to produce acceptable results. Forchheimer [3] is believed to have been the first to derive a nonlinear head-velocity relationship for turbulent flow in porous media. Bachmat [1] derived an equation for turbulent flow in heterogeneous porous media in three dimensions. Bachmat's momentum equation is as follows

g

"~v

g=-KVh,

(3)

where v is the kinematic viscosity of the fluid; n is the porosity of the porous media; fl is a geometrical coefficient; and V = I]71 is the norm of the velocity vector. Hence, Bachmat's model for three-dimensional saturated flow in an isotropic heterogeneous porou s medium with no sources or sinks is [2]

ot= whereKb=gK/u

1+ ~

\

oz)' .

(4)

GLOBAL PROPERTIES OF POROUS MEDIUM FLOW

163

Extending Sposito's Work to Bachmat's Model For Steady Flow In this section, the approach used in [4] (i.e., the Lyapunov function and the vorticity function) is adopted to study the global dynamics of fluid motion that results from applying the nonlinear Bachmat's law to steady flow in an arbitrary heterogeneous porous media and determine whether or not the motion is looped and the flow is chaotic. The main advantage of the approach used in [4] is that a great deal can be learned about the dynamics of the flow in porous media (i) without having to solve the governing equation and (ii) without any assumption about the spatial variability of the hydraulic conductivity of the porous media except that the hydraulic conductivity is C 1 continuous function. Study

of Global

Dynamics

via the Lyapunov

Function

Before applying the Lyapunov function L to the study of the global dynamics of fluid motion that result from applying the nonlinear Bachmat's law to steady flow in an arbitrary heterogeneous porous media, the properties of this function L are introduced below.

1. L(x, y, z) is continuous in some neighborhood U of a fixed point (x0, Y0, z0), differentiable in U - (x0, Y0, z0) and L(xo, yo, z0) = 0, where (x0, y0, z0) is the d~ 0 point where ~ = ( ~d~, d~ d r , ~ ) = (0, O, ). 2. L(x, y, z) > 0 for all (x, y, z) in V - (x0, Y0, z0). 3.

dL OL dx OL dy A- OL d z " ~ = ~ ' d T "~- " ~ dt - - ~ z ' ~ < 0

in U - (x0, Y0, z0).

T h e o r e m : If L is a Lyapunov function in some neighborhood U of a fixed point (x0,Y0, z0) satisfying the above stated conditions 1, 2 and 3, then (xo, Yo, Zo) is an asymptotically stable fixed point. Physically, the Lyapunov function is "bowl" shaped with its minimum at zero and its tangent the velocity vector ~7= ~.[ ~dt ~ ~d r ' ~dt J ' Therefore, this surface represents the streamlines of the flow near an equilibrium point. The fact that the material

dL

derivative, -~-, of the Lyapunov function along the surface defined by L is negative means that if a fluid particle is disturbed (moved away from the equilibrium point) this fluid particle will tend to the equilibrium point as t ---, oo. This is due to the fact that the streamlines are all directed towards the equilibrium point in the neighborhood of this point. In fact, the Lyapunov function is simply a mathematical representation of the "bowl" example that engineers use to conceptually view stability. Taking the inner product of Bachmat's equation with the velocity vector gives the following expression: ~(I+V~.~=-KVh-~,

g

nv/

(5)

164

M.S. GHIDAOUI and A. A. KOLYSHKIN where • is the dot product operation. Using the total derivative concept and rearranging reduces equation (5) to the following form

dh d"Y =

u (I+VP) gg ~

v2.

(6)

The fixed (equilibrium) point for the flow is obtained by setting the velocity vector equal to the zero vector. This implies that Vh is also equal to the zero vector. Let this fixed point be (xo, Yo, Zo). The head at this fixed point, h(xo,Yo, Zo), is minimum. Let L(x, y, z) = h(x, y, z) - h(xo, Yo, zo) then it is simple to check that L is a Lyapunov function. Therefore, any fluid particle in the neighborhood of (x0, Y0, z0) will eventually find itself at the fixed point. To explain, consider a fluid particle that has a head hi at time tl and a head h2 at time t2. Integrating equation (6) from tl to t2 gives the following: -

h: = -

: +

:.

:dr.

(7)

The right hand side of equation (7) is always negative and, thus, the head h2 is always smaller than the head h:. As a result, a fluid particle moves from high head towards low head and eventually becomes infinitely close to the equilibrium point when t2 tends to infinity. Another legitimate question is: can a fluid particle return to any of its previous locations and thus form a closed trajectory? To answer this question consider the following closed path integration:

Vh.d:=

Vh•:dt=

1 dt

=h2-h:.

(8)

If a fluid particle can return to a previous position then the left hand side of equation (8) is zero. However, from equation (7) the right hand side of equation (8) is always negative. This contradiction implies that a fluid particle cannot return to any of its previous positions. Instead, flow paths are directed towards minimum head points at all time. Thus, looped flow paths cannot materialize (i.e., a point moving along a flow path cannot return to any of its previous positions) in this case and the flow field cannot exhibit chaotic behavior [4]. Recall that a necessary condition for chaotic flow paths (trajectories in the phase space) is that a trajectory must cross itself but do not repeat itself [4]. Study

of Global

Dynamics

via the Vorticity

Function

A path of a fluid particle is not helical if the vorticity vector and the velocity vector are orthogonal [4]. To determine whether or not fluid motion that result from applying the nonlinear Bachmat's law to steady flow in an arbitrary heterogeneous porous media is helical, consider the curl of Bachmat's equation V x

1+ ~

= -V

x

(gVh).

(9)

GLOBAL PROPERTIES OF POROUS MEDIUM FLOW

165

Recognizing that ~7 x Vh is the zero vector and that the curl operation is distributive fl~-VVx~+ ng

1+~

Vx

=-VKxVh.

(10)

Both the first term on the left hand side of equation (10) and the term on right hand side of equation (10) are orthogonal to the plane containing the velocity vector. Thus, vorticity V x ,7 is orthogonal to the velocity vector. As a result, fluid particles do not wobble as they move from point to point in the flow field. That is, the flow paths are not helical. The absence of interaction between the voricity vector and the velocity vector is the main reason looped paths do not occur and the flow cannot become chaotic.

Perturbation Analysis of the Dynamics of Darcian and Bachmatian Flows The ~im of this section is to study the stability of non-looped base flow which results from the Darcian or Bachmatian model in an steady flow in a heterogeneous and isotropic. This is accomplished as follows. A steady base flow whose flow paths are not looped is established and unsteady perturbations are imposed on this base flow. To show that the flow paths never become looped, it suffices to show that the amplitude of the perturbations decay with time (i.e., the base flow is stable).

Darcian Flow Let ll(x,9, z) be a steady-state solution of equation (2), that is,

0-'~ K-~x + N \

-~y]+~z K-~z =0,

(Ii)

whereH is the base flowhead. Considerthe followingperturbed field h(x, y, z, t) = H(x, y, z) + ~(x, y, z) e -~t,

(12)

where A is an eigenvalue and ~o(x, y, z) is the amplitude of the normal perturbations. Substituting (12) into (2) and using (11) we obtain

+~y K

K

+ AS~ = 0.

(13)

Equation (13) togetherwith zeroperturbation boundaryconditions(i.e., the boundary conditionon the head are assumedto be known) ~(0) = 0,

~(1) = 0

(14)

form a Sturm-Liouville problem. According to the Sturm-Liouville theory [5] the fact that K(y) > 0 and S > 0 implies that all the eigenvaiues A are discrete positive real values. Thus, the base flow is stable and the flow paths cannot become looped.

166

M.S. GHIDAOUI and A. A. KOLYSHKIN

Bachmatian Flow The analysis for the Bachmatian flow is identical to that of the Darcian flow once Kb is substituted for K. Since Kb > 0, then all conditions of the Sturm-Liouville problem are also satisfied in this case. As a result, the flow paths do not become looped.

Conclusions This paper shows that Bachmat's model for steady flow in porous media produces trajectories that are not looped and flow field that is not chaotic. This conclusion is arrived at through the Lyapunov function and the vorticity function. Moreover, the non-looped base flow trajectories obtained by applying either Darcy's equation or Bachmat's equation to steady flow in porous medium is unconditionally stable. This conclusion is arrived at by showing that the perturbation equation is a SturmLiouville problem. All the conclusions of this paper are valid for arbitrary spatial variation of the hydraulic conductivity with C 1 smoothness being the only required condition. In addition, the present analysis makes no assumption about the form of the boundary condition.

Acknowledgments Financial support for this work is provided by the Hong Kong University of Science and Technology, Hong Kong under the UPGC Research Infrastructure Grant Program, Project number RIG 94/95.EG14. This financial support is gratefully acknowledged.

References [1] Bachmat, Y. (1965). "Basic transport coefficients as aquifer characteristics," I.A.S.H Symp. Hydrology of Fractured Rocks, 63-67. [2] Bear, J. (1988). "Dynamics of fluids in porous media," Dover Publications, Inc., New York, N.K. [3] Forchheimer, P. (1901). "Wasserbewegung durch boden," Z. Ver. Deutsch. Ing., Vol 45, 1782-1788. [4] Sposito, G. (1994). "Steady groundwater flow as a dynamical system," Water Resources Research, Vol. 30, 2395-2401. [5] Tikhonov, A.N. and Samarskii, A.A. (1990). "Equations of mathematical physics", Dover Publications, New York.