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Limitations of a finite mean free path for simulating flows in porous media
G. Kohring
To cite this version:
G. Kohring. Limitations of a finite mean free path for simulating flows in porous media. Journal de Physique II, EDP Sciences, 1991, 1 (6), pp.593-597. �10.1051/jp2:1991192�. �jpa-00247543�
Classification
PhysicsAbstracts
47.tKl
Limitations of a finite mean free path for simulating
flows in porous media
G.A~ Kohring
Institut fur Theoretische Physik,Universitat zu KbIn, Zblpicherstr 77, D-5000 KoIn 41, Germany
Abstract. &Vhen the mean free path, I, of fluid particles for Hydrodynamic Cellular Automata is not much smaller than the characteristic length of the system, then hydrodynamic correlations do
not have a chance to develop and true hydrodynanuc flow will not be obtained liy studying a simple
two dimensional system, it is concluded that the finite size corrections to the permeability, «, for pore size, R, are of the form: + 7 ii/R). The hnutations this result places on the use of such methods for studying flows in porous media is discussed,
The use of hydrodynamic cellular automata [1 4] to study the properties of fluid flow in porous media is very attractive because of the relative ease with which complicated geometries can be
introduced [1,2]. Recently, however, researchers have raised questions about the appropriateness
of this technique when the pore size of the porous medium is small [1, 3]. When the mean free
path of the cellular automata particle is of the same order of magnitude as the pore size, then
hydrodymmic correlations do not have a chance to fully develop, hence, the resulting flows will not be truly hydrodynamic. Studying this effect quantitatively has proven difficult, because of the large sample to sample fluctuations which occur when dealing with random realizations of porous media [3,~. In a previous work [3], it was only possible to make qualitative statements about a possible
crossover effect from a region where the permeability of the porous media, ~, is proportional to the pore size, R, to the expected asymptotic region where
~ c~ R~. In that work~ the crossover value of R was estimated to be on the order of R
~- 100, requiring lattices with tens of millions of sites.
The present paper examines this problem more systematically for a large, but simplified geometry
so as to avoid effects due solely to the complicated geometries needed for realistic studies of flows in porous media and to avoid the sample-to-sample fluctuations associated with chasing a purely
random geometry.
The systems m the present work are two-dimensional channels of length, L, and height, H, and
they contain a solid block located in the center half of the channel. In the center of this block~
and running its entire length, there is a hole of heigh~ R. WE setup is similar to that used for previous studies of flow in porous media, hence, the edge effects near the ends of the hole should be comparable. Essentially, then, this allows one to study the properties of a fluid flowing in a
single pore. The values of H, and L are fixed by the ratios: H/R
= 10 and L/R
= 32. Fixing
these ratios is similar to fixing the porosity in simulations of porous media with different pore sizes.
594 JOURNAL DE PHYSIQUE II N°6
The cellular automata model osed here has been discussed in detail elsewhere[3], and b a simple
extension of the original 6-bit FFIP model [4] to include local conservation of angular momentum
[5]. The fluid is forced through the system as before with a Poiseuille condition at the left end of the channel and periodic boundary conditions at the fight end.
A fluid floving vith small velocity through the hole will be characterized by the following gen-
eral equation if the edge effects are not substantial and the mean free path, I, is vanishingly small 181:
ivlP
~
i j~j
bP/j16 R) vi
where (v) ~ the average velocity of the fluid, p b the density, u is the kinematic viscosity, SF ~ the pressure d#ierence at the ends of the hole and f is a constant. For an infinitely long, frictionless channel, f
= 12, but for a porous medium f would be a function of the porosity, in which case
equation I is generally known as Darcy's law.
lvhen the mean free path, I, can not be neglected, then corrections to equation (I) are to be
expected. As a first step, the mean free path of a fluid particle was measured as a function of the
particle density. In this case, I is estimated by dividing the total number of particles by a sum all different types of collbions, vith each type of collision being multiplied by the number of particles
involved in that collision. The measurements were done for a fluid being driven with the same
driving force as used for studies of flow properties in porous media. This will yield a slightly larger
value of I than that obtained by the Boltzmann approximation for a fluid at rest, since the driving
force correlates the motion of the particles. The results are shown in figure I. The filled circles
correspond to measurements of I in which all of the incoming particles change direction during
the collision [2]. The squares correspond to measurements that also count collision in which the outgoing particles may simply continue in the same direction as the incoming particles, e.g., six
body collisions. From this plot it can be seen that for the regime most commonly studied, p m 0, the mean free path is on the order of 1
~-
9. In order to have true hydrodynamic flow throughout
a porous system, we would then expect that the size of the smallest pores must be much greater than 9 lattice units [9].
A more precise estimate of the minimum pore size can be found by attempting to verify equa-
tion (I) for the present geometry. Figure 2 then shows a plot of ~/R~ versus AlR, where:
~ =
ivlP
~bP/j16 R) ~~~
The data was obtained using the geometry described above, with 40 000 initial sweeps through
the lattice being used for thermalization and another 40 @XU sweeps used for the actual mea-
surements. 400 steps are skipped in between measurements, to minimize time correlations. The
small system sizes were simulated on the NEC-SX3 at the University of KoIn and the larger system sizes, vith up to 200 million sites, were simulated on the Cray-YMP at the KFA Jolich and the Connection Machine CM2 at the GMD in St. Augustin. (Details of the programs can be found elsewhere [2,7].)
One major problem in these simulations is to determine the pressure gradient. Although some
researchers impose a pressure gradient by imposing a density gradient [2], such methods could, due to the variation ofviscosity with density, lead to problems unless the density gradient is small.
Previously [3], we have measured the pressure by measuring the momentum tratlsfer with the channel walls; however, this leads to quite large fluctuations which need very long time averages in order to be damped out. In this paper, the pressure gradient in the system was measured by
'~
"
o
D D
a a
a a
°aa 1
596 JOURNAL DE PHYSIQUE II N°6
at unity, as predicted by eq(I), using u m 2 0 from [3]. For other densities, we rescale u with the viscosities calculated from the Boltzmann approdmation (see eg., [9]). From this graph it is clear that as R - cc, the following scaling relationship holds numerically:
(~(P)/R~)
~ ~ ~ j~/R) (3)
j~jp)/R2)n~
4
w
f3
a
CL-
~ .
~2
' _
~ .
m
ltS, CL
~
~~0
AIR
size is at least R ~- 7 1. This would implya lower limit of about I million sites for rough simulations of such flows. lb obtain results accurate to within 109b of the asymptotic value of the permeability
would seem to require simulations with approxbnately lx million sites.
Acknowledgemenb.
It is a pleasure to thank D. Staufler and B. Chopard for many useful discussions. A grant from the BMFT (# 0326657D) for partial support of this project is gratefully acknowledged as well as a grant of about 100 hours of computer time divided between the University of Cologne's NEC- SX3, the HLRZ'S Cray-YMP/832 at the WA Julich and the HLRZ'S Connection Machine CM-2 at the GMD in St. Augustin.
References
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SuccI S., CANCELLIERE A., CHANG C., FOn E., GRAMIGNANI M. and ROniMAN D
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Methods tn Suhsulfiace Hydiology, eds. GAMBOLAn G., RINALDO A~, BREBBIA C.A~, GRAY WG. and PINDER G.E Spnnger-Verlag, Berlin) 1990.
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[7j KOHRING G.A~ (in preparation)
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[9] LIM HA., Phys. Rev 40 (1989) 968.
