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Dynamic permeability of porous media by cellular automata
J.A.M.S. Duarte, Muhammad Sahimi, João Marques de Carvalho
To cite this version:
J.A.M.S. Duarte, Muhammad Sahimi, João Marques de Carvalho. Dynamic permeability of porous media by cellular automata. Journal de Physique II, EDP Sciences, 1992, 2 (1), pp.1-5.
�10.1051/jp2:1992101�. �jpa-00247608�
Classification Physics Abstracts
47.55M 5.50
Short Communication
Dynamic permeabflity of porous media by cellular automata
J.A.M.S. 0uarte(~>*), Muhammad Sahimi(~'**) and Joio Marques de Carvalho(~)
(~) HLRZ, c/o KFA, D-5170 Jiilich I, Germanj,
(~) Departamento de Fisica, Faculdade de CiAncias, 4000 Porto, Portugal
(Received 28 October 1991, accepted 5 Novembei 1991)
Abstract The step response of
a porous medium with quenched disorder is investigated by
cellular automata. The main conclusions of this dynamic study on samples of up to 2000 x 660 sites were that the response times are a linear function of porosity in the f~ee channel limit and exhibit also a linear dependence on the lattice size, for identical disorder distributions. The porous section response is found to vary according to its length as a modified exponential with
a power dependence on the time that increases monotonically with the porous length.
1 Introduction.
Fluid flow in porous media is of great theoretical and practical interest [1,2]. Problems such as
enhanced recovery of oil, drainage and inbibition in soil, multiphase flow through trickle-bed reactors, mercury porosimetry for determining the pore size distribution of a porous catalyst,
and groundwater flow are but a few processes of interest to hydrologists, soil scientist, and chemical and petroleum engineers.
An important problem is the determination of the effective transport properties of a porous medium. A wide variety of methods have been proposed in the past [ii, a list of which is too
long to be given here. Most. of t.hose methods rely on a part.icu [at model of pore space. These models are often very simple, hecause otherwise cot iiputation~ ivould be prohibitive. As a result, depending on the complexity of a given problem, i,arious models have been invented which, although they may yield reasonable results for the specific problem under study, they often fail to provide insight into another iiirelated problem. Recent advances [3] in computer simulation
techniques have now made it possible to devise highly efficient algorithms for simulating flow in
(*) Present address: C.I.U.P.-Fac. C16ncias, 4000 Porto, Portugal.
(**) Present address: Dep. Chemical Engineering, University of Southern California, Los Angeles,
CA 90089-1211, U-S-A-
2 JOURNAL DE PHYSIQUE II N°1
porous media. Among these are cellular automata (CA) methods [4-6] which, in the context of flow problems, are discrete solutions of the Navier-Stokes equations. In the present paper, we
employ the CA method to calculate a dynamic property of a porous medium, namely, dynamic permeability of the system. Calculating the dynamic permeability of a porous medium has
recently attracted considerable attention [7-9]. It has been argued that this quantity can provide insight into the stucture of a porous medium. However, most previous studies of the
dynamic permeability were restricted to simple geometrical models of the pore space. With the advent of CA methods, we are now in a position to calculate this quantity for any configuration
of the pore space, since this is the main advantage that a CA method oiers one
over previous methods.
All of the previous applications of CA methods to flow iii porous media [5,6] were restricted to
the steady state properties. The only way time was actually involved waJ~ during the relaxation process before the steady state waJ~ reached. In the present paper, how>ever, we are interested in the time variations of the permeability. The dynaiiiic periueability is usually defined by the
dynamic version of Darcy's law
VIW) = -%VP(W) (I)
where V is the average fluid velocity, k(w) the dynamic permeability at frequency w, q the fluid viscosity and P the pressure. Dynamics is introduced by setting l7P(w) = l7Pe~~~~ as the AC pressure gradient between two opposite faces of the medium. Instead of introducing
an oscillatory pressure gradient, we look at the siiupler problem in which the response of the system to a unit step input is studied. In a future paper, we shall treat the problem completely by studying the response of the system to an oscillatory pressure gradient. In the next section
we first describe the model pore space we used arid the details of the siiuulations and, then, present and discuss the results.
2. The model and simulations.
In this paper we use a two-dimensional model porous iuedium. Circular obstacles of a given size
are distributed at random between two parallel plates. The porosity of the system dictates the number of circles, and it would pose no difficulty to choose the circks sizes front a distribution
function. Thus, our model is the same as that used previously by our group [10-14]. Reflecting boundary condition on the surface of the obstacles and the parallel plates are used to insure that a no-slip boundary condition is satisfied. A st.andard solution with a parabolic Poiseuille profile will develop and, at sufficiently low Reynolds numbers, the flow will be laminar with no
obstacles in the channel or even behind a single cylinder.
With the introduction of a great number of obstacles, it is the pressure loss on the complex boundary of the medium that represents virtually all of the momentum transfer. In agreement with the range of validity of Darcy's law, average velocities were chosen at 0.25, 0.20 and 0.10 for the maximum value of the unperturbed profiles distributed at
= 0 and kept at the left
opening of the channel. The distributions were allowed to relax under this initial condition, after which a step was imposed to another velocity level, previously generated and stocked
(following a procedure described iii [10]). For most of the siiuulations the step transitions were
from zero to u and from i> to zero, although in some cases we have also simulated transitions
from +u to u. In this case, relaxation times were typically 20 %> of the overall time spent on
the simulation. This ranged from 3000 to 6000 time st.eps, which allows for full propagation of the velocity variations in the charnel on even a il1011 x 660 lat.( ic~.
Channel dimensions were 300 x 100, 900 x 300,1200 x 400 and 2000 x 660, filled with the
same realization of disorder. The response front propagates through the porous region keeping
a sharply rising profile which then evolves somewhat slowly to an evenly distributed velocity (within the normal statistical fluctuations). Propagation times, measured from the different
transitions available, were within 7 ~ of the average, which seems in acceptable agreement with the level of precision found in CA [10]. Figure shows a linear relation between the time of propagation and thi horizontal length of the porous section of the channel, for L (total length) between 300 and 2000.
437 29+099471 L R-o 9990 t
0 400 800 1200 1600 2000
L
Fig. i. Effective linear fit in the range L
= 300 to 2000 of the times of propagation of a step response with fixed disorder. (Porosity 0.97).
The response times taken at a given size can be extrapolated reasonably to the values
obtained for a complete free channel for high values of porosity (see Fig. 2 for the 900 x 300
lattice). This range of values is still far away from the precolation threshold but concentrations much higher than 100 circles took too long to settle to a final state.
t 96382 85021 pomity R-09953
t
o
0.92 0.93 0.94 0.95 0.96
Porosity
Fig. 2. Dependence of the propagation time on the porosity for a 900 x 300 lattice close to the free channel limit.
In order to probe the porous section response to a step change we have also used it on the 1200 x 400 lattice in which u is measured at transversal cuts. Figure 3 shows the evolution
JOURNAL DE PHYSIQUE II N°1
of the velocity at four sections on the sample, two of them in the middle of the porous region,
another at the edge (L = 1020), and the very last one in the obstacle free region at the end. Velocity evolutions in the first half of the length sulser from overshoot and ripples after
a virtually instantaneous response (the velocity step was introduced at time t = 300). We have tried to fit the time-variations of the velocity and its approach to a steady regime, to an
exponential of the type (Fig.3)
Vii) = M (1 exp (A(t to)~j (2)
with t > to- Here to is the time at which the exponential variations of V start to set in. We found a value of ib ci 1.6, which can be attributed to the effect of the porous section of the channel. Having determined the variation of the velocity V with t, the permeability is easily
determined from equation (I).
V=0.0493°(1.exp(-1.6e-3°(t-I,12e+31~'~ll R=0.9986 L=660 V=0.0543°(1-exp(.0.5e-3°(t.1.46e+3)~'~~ll R=0.9993 -- L=g00 V=0.0520°(1.exp(.0.3e-3°(t-1.66e+3)~'~~l) R=0.9992~ L=iQ2Q V=0.0507°(1-exp(-0.5e.4°(t-1.80e+31~'il R=0.9992 L=i 40
-0
0 500 1000 1500 2000 2500 3000 3500
Fig. 3. Dynamic response, V(t), as measured by the step response at distances L
= 660, 900, 1020 and i140 (lattice units) from the beginning of the channel. The curves correspond to equation (2). R
(the correlation coefficient) is an indicator of the quality of the fitting.
3. Conclusions.
Although we did not determine the response of the system to an oscillatory pressure gradient, figure 3 already indicates the possible richness of the behavior of the system. Given that we can now use very large system sizes [13, 14] and vary the porosity in practically any way we wish, it would seem interesting to expand the data on one hand, to cover the region near the
percolation threshold and, on the other to clarify the behavior for low values of L. On the other
hand, and for a comparison with the values obtained in [7], a number of samples for various
porosities shquld be used and from them, by Fourier transformation, a frequency response of the velocity and, hence, permeability, derived. This work is now in progress. This would then allow us to check the proposal of Sheng and Zhou [7] and Charlaix et al. [8] regarding the
relation
~~~~~f ~ (~)
~ ~
o
~
where ko " k(w = 0), wo a characteristic frequency and f a scaling function which is apparently universal.
Acknowledgements.
Two of us (J.A.M.S.D. and M-S-) would like to thank Dietrich Staulser for useful discussions, and him and the Supercomputer Center HLRZ at KFA Jfilich for w,arm hospitality during
summer 1990, when this work began.
References
iii SAHIMI M., Rev. Mod. Phys., to be published.
[2] SAHIMI M. and YORTSOS Y-C-, SPE Reservoir Engineering 6 (in press).
[3] BINDER K., Applications of Monte Carlo Methods (Springer Berlin, 1987).
(4j FRISCH U., HASSLACHER B. and POMEAU Y., Phys. Rev. Lent. 56 (1986) 1505.
[5] ROTHMAN D-H-, Geophysics 53 (1988) 509.
[6] G. Doolen Ed., NATO ARW
on Lattice Gas Methods for PDE'S, Physica D 47 (January 1991).
[7] SHENG P. and ZHOU M-Y-, Phys. Rev. Lett. 61 (1988) 1591.
(8j CHARLAIX E., KUSHNIK A-P- and STOKES J-P-, Phys. Rev. Lent. 61 (1988) 1595.
(9j JOHNSON D-L-, KOPLIK J. and DASHEN R., J. Fluid Mech. 176 (1987) 379.
[10] BROSA U. and STAUFFER D., J. Stat. Phys. 57 (1989) 399; 63 (1991) 405.
[11] DUARTE J-A-M-S- and BROSA U., J. Stat. Phys. 60 (1990) 501.
[12j SAHIMI M. and STAUFFER D., Chem. Ellg. Sci. 46 (1991) 2225.
[13] KOHRING G., J. Stat. Phys. 63 (1991) 411.
[14] IIOHRING G., J. Phys. II France1 (1991) 593.
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