Transport in Porous Media32: 187–198, 1998.c© 1998Kluwer Academic Publishers. Printed in the Netherlands. 187

Automaton Simulations of Dispersion in Porous Media

D. YANG1, N. UDEY2 and T. J. T. SPANOS31AMOCO, 708, 700-9 Street S. W, Calgary, Alberta, Canada2Advanced Computing, Alberta Research Council, 6815-8 Street NE, Calgary, Alberta, Canada,T2E 7H73Department of Physics, University of Alberta, Edmonton, Alberta, Canada, T6G 2J1

(Received: 18 June 1997; in final form: 8 April 1998)

Abstract. A thermodynamic lattice gas (automaton) model is used to simulate dispersion in porousmedia. Simulations are constructed at two distinctly different scales, the pore scale at which capillarymodels are constructed and large scale or Darcy scale at which probabilistic collision rules areintroduced. Both models allow for macroscopic (pore scale) phase separation. The pore scale modelsclearly show the effect of pore structure on dispersion. The large scale (mega scale) simulationsindicate that when the pressure difference between the displacing phase and displaced phase isproperly chosen (representing the average pressure gradient between the phases). The simulationresults are consistent with both theoretical predictions and experimental observations.

Key words: diffusion, dispersion, miscible, automaton.

Nomenclature

B blue particle.B0 segregated blue particles.B1 mixed blue particles.HO homogeneous model.HE heterogeneous model.m particle mass.n particle number at present time.n′ particle number at the previous time.R red particle.R0 segregated red particles.R1 mixed red particles.T temperature.v particle velocity.

Greekβ ′ Coefficient in dynamic pressure difference equation (A4).

1. Introduction

In order to describe dispersion in porous media one must address at least threedifferent scales; the molecular scale (microscale, at which diffusion is occurring), the

188 D. YANG ET AL.

pore scale (macroscale, at which the continuum equtions are firmly established andthe scale at which dispersion is occurring) and the Darcy scale or scale of hundredsof pores (megascale, at which Darcy’s equation holds for single phase flow). Inaddition it is possible to introduce additional structure at the megascale or to considerintermediate scales (a mesoscale) at which continuum equations cannot be consideredvalid. In the present paper we will exclude such large scale structure or intermediatescales in order to focus on megascopic simulations of dispersion in porous media.Pore scale modeling requires massive computation, but does allow us to see the detailsof the pore structure and its effect on dispersion. In contrast, large scale modelingovercomes some of the computational difficulties but does not provide the detailsof the pore structure. Nevertheless, the two scales should provide consistent results.That is, large scale simulation should integrate pore structure effects on dispersionand reflect the ‘message’ imbedded at the pore scale.

Gao and Sharma (1994) carried out a large scale simulation on dispersion witha no-slip boundary condition for the solid. The displacing and displaced fluid wereassumed to be totally mixed at the ‘particle level’ and thus macroscopic phase sepa-ration was not described. In the present study, a thermodynamic lattice gas model isused and macroscopic phase separation is incorporated into the model. The principalcharacteristic of this model is that the momentum of a particle residing on the latticeis a continuous variable. However, in a single time step a particle can either moveto an adjacent lattice site or remain at it’s current site. This discrete particle motionis related to the particles continuous momentum through a probabilistic propagationrule. Udeyet al.(1998) have shown that if Lorentz invariant elastic collisions are usedthen the collision and propagation rules of this model can be used to rigorously derivethe energy momentum tensor and the relativistic Boltzman equation. Furthermore,since the lattice is only used for book-keeping purposes its shape is unimportant tothe final result (Udeyet al., 1998). We therefore choose to use a square lattice inthe present model. Furthermore, we use nonrelativistic elastic scattering rules, themodel therefore becomes Galilean invariant and has previously been used to modelPoiseuille flow, single phase flow in porous media and diffusion in porous media(Yang et al., 1998). In the present model the effect of viscosity on dispersion isincorporated by indexing fluid particles as to whether they are in the mixing zone ornot, and the particle collision probability can be adjusted accordingly. Since a tracerused in the dispersion simulations has the same viscosity as the displaced fluid, theeffect of viscosity on dispersion does not arise for tracers. A theoretical study (seeAppendix) indicates that there is a pressure difference between the displacing and thedisplaced fluids within a single megascopic volume element (which is representedby a lattice site).

In this study, dispersion in porous media is first simulated at the pore scale using acapillary tube model and the effect of the pore structure on dispersion is demonstrated.The rules in the large scale model incorporate pore scale information, including phaseseparation, pore structure effects and the pressure difference between the displacingand displaced fluids.

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2. Theoretical Background

Recently, the laboratory experimental results of Sternberget al. (1996) providedevidence indicating that the conventional convection diffusion equation fails to ad-equately predict dispersion in porous media. The limitations of the convection dif-fusion equation result from the interpretation of concentration. In the convectiondiffusion equation the concentration appears as the mass fraction of the phasesmixed at the molecular scale. In a displacement process in a porous medium theconcentration, as described at the megascale, also incorporates the relative motionsof the phases at the pore scale. Thus, when the convection diffusion equation isused to describe flow in porous media dispersion is incorporated into a ‘dispersiontensor’. This theory then predicts a time dependent base state, thus the dispersiontensor becomes a variable. From a physical point of view this simply means that theequation is not a valid physical representation of the process.

Theoretically, two approaches (equations) have been used to replace the convec-tion diffusion equation when describing dispersion. One approach involves the use ofnonlocal equations (Edelen, 1976) which allow information from a region of space tobe included to determine the effect at any particular point in the system. This approachincorporates memory of the past history of the flow. The other approach utilizes theequations derived by Udey and Spanos (1993) under the condition of negligiblediffusion in which an additional degree of freedom (i.e. an additional equation andvariable; a dynamic megascopic pressure difference between the phases) is obtained.This pressure difference results from the fact that an average pressure differencemust exist between the displacing and displaced phases during flow, thus a differencebetween the averaged pressures is also obtained.

3. The Model and Simulation Experiments

A pore scale model is constructed by considering a number of two-dimensional,parallel capillaries of varying thickness. The effect that this well controlled porestructure has on miscible flow in a porous media is compared with a megascalemodel which incorporates the pore scale information through the collision rules.

The pore scale model can be constructed by introducing actual solid structures intothe lattice. When a fluid particle encounters the solid matrix, a thermal solid boundarycondition is applied. That is, when fluid particles collide with the solid, they simplybounce back under the condition that particle speeds fit the Boltzmann distribution.Here, the fluid particles are not allowed to stay in solid cells. Following conventionalpractice, a ‘tube model’, i.e. a bundle of tubes is used to represent a porous medium.The tube model is chosen so that one can control permeability by adjusting tuberadii. These models, although simple, do reflect some effects of pore structure ondispersion. Two ‘tube models’ with the same intrinsic permeability (41.67 latticeunits squared) but different pore structures (channel thicknesses) are designed asillustrated in Figure 1. (Here it is important to note that, in the nonrelativistic model,

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Figure 1. A schematic diagram of tube model A, homogeneous (HO) and tube model B,heterogeneous (HE). The permeability of each layer can be calculated using lubricationtheory and the total permeability of each porous medium is the same. The numbers on theleft hand side indicate the position of the pore walls. The length of the capillaries is 500lattice units.

a lattice unit is a dimensionless unit of length that is very small in comparison to thescale at which the continuum theory is modeled, which is also dimensionless.)

Model A consists of five tubes with the same diameter (10 lattice units). It is ho-mogeneous at both the macro-scale and the megascale. Model B consists of six tubeswith different diameters (three tubes which are 5 lattice units wide, two tubes whichare 10 lattice units wide and one tube which is 15 lattice units wide). At the megascaleit has the same permeability as Model A. The two models are used to represent twocores with the same permeability but different pore structures. For simplicity, modelA is called a homogeneous model (HO) and model B a heterogeneous model (HE).

For a thermodynamic automaton model constructed to simulate a porous mediumat the megascale, viscosity related effects can be introduced through the collisionrules. Namely, an increase in the number of fluid–fluid particle collisions resultsin an increase in momentum transfer. Pore structure effects can be introduced byadjusting the particle velocity directions after collisions. To incorporate the porescale information in the large (mega) scale modeling, the rules are as follows:

(1) B (blue) and R (red) are used to represent displacing and displaced particles,respectively. Further, B0 and B1 are used to represent the displacing fluid particle inthe segregated and mixing zones, respectively, and the same for R0 and R1.

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(2) Since B0 and R0 are in a segregated region, B0 is not allowed to collide withR0. Thus, B0 can only collide with B0, B1, and R1. Moreover, B0 becomes B1 whenB0 collides with R1. The same rules are applied to R0 accordingly.

(3) When B0 collides with R1 and in the central mass frame and the rotation angleis greater than 90 degrees but less than 270 degrees, a random number is generated.When the random number is less than a flipping probability, the directions of the twoparticles are reversed. Pore structure effects are incorporated by changing the flippingprobability. In this model, the flipping probability is set to be either 0 or 1. Here thebasic physical statements used to construct the collision rules (e.g. conservationof momentum) are not altered by this choice but we argue that dispersion may beinfluenced and dispersion is effected by the pore structure.

(4) For fluid–solid collisions, the distribution of displacing (B) and displaced (R)fluids, (i.e. the displaced fluid surrounds the solid matrix and thus collides with thesolid), is reflected by setting the priority of the fluid–solid collision as R0 > R1 >

B1 > B0. This is implemented at the time at which the particles are selected forcollisions.

(5) To maintain the same permeability, in each cell, the total number of fluidparticles colliding with the solid should be the total number of fluid particlesN (R0+R1 + B1 + B0), multiplied by the solid collision probabilityP in that cell.

(6) To implement the pressure difference between the displacing and displacedfluids, instead of adding the same amount of momentum to both of the displacing anddisplaced particles, different amounts of momentum are partitioned to the differenttypes of particles according to the following equations:

m2v2 − m1v1 = β ′(n2 − n′2), (1)

n2m2v2 + n1m1v1 = (n2 + n1)mv. (2)

Herem is the mass,v is the velocity,n is the particle number at the present time ina cell andn′ is the particle number at the previous time. The parameterβ ′ describedin Appendix A can be determined by comparison with experimental results. Thesubscripts 1 and 2 refer to fluids 1 and 2 (or B and R), respectively. Equation (1)comes from the dynamic pressure difference between the displacing and displacedfluids (refer to Equation (A4) in Appendix A). That pressure difference is associatedwith the concentration (particle number) changing with time. Equation (2) indicatesthat the total momentum added in a cell is the same as if only one phase existed inthe cell.

Two main types of simulation experiments were conducted:

1. dispersion in pore scale models (tube models),2. dispersion in megascale models with the enhanced rules.

Simulations of Experiment 1 are carried out to analyze the effects of pore structureand temperature on dispersion. Four simulations are run at the same pressure drop(dp/dx = 0.005), and the same permeability (41.67) but at a different initial particle

192 D. YANG ET AL.

velocity(v0), temperature(T ) and with models having different pore structures. Theparameters of the simulations are as follows:

(1.1) v0 = 0.1, T = 1/200, HO;(1.2) v0 = 0.1, T = 1/200, HE;(1.3) v0 = 0.01, T = 1/20000, HO;(1.4) v0 = 0.01, T = 1/20000, HE.

Here units of length are a lattice unit, time is expressed in time steps and the particleshave unit mass.

The initial set up for the above runs was the same i.e. a 500× 50 lattice wasused with 20 particles in each cell. Blue particles were assigned to the left part(Columns 0–199) and the right part (Columns 301–500) and red particles to themiddle part (Columns 200–300). Thermal boundary conditions were applied for thesolid boundary (tube walls).

For Experiment 2, six simulations are presented at the same pressure drop(∂P/∂x = 0.005), temperature(T = 1/200), initial particle velocity(v0 = 0.1) andmean probability of solid collisions(SP= 0.5), but with different values of BF (hereBF is the value ofβ ′ specified in Appendix A by Equation (A4). It relates the averagepressure gradient between the displacing and displaced phases to the changing con-centration), flipping probabilities (FP) and with models having homogeneous (HO)and heterogeneous (HE) SP distributions. In this model the heterogeneous mediaare constructed by randomly selecting a solid collision probability at each latticesite from a probability distribution composed of a double Boltzmann distribution.In this case the SP distribution fits a function obtained by adding two Boltzmannfunctions with peaks at 0.385 and 0.588, this distribution is intended to model aporous medium which is extremely heterogeneous at the pore scale. The parametersof the simulations are as follows:

(2.1) FP= 0, BF = 0, HO;(2.2) FP= 0, BF = 0, HE (DB function);(2.3) FP= 0, BF = 0.001, HO;(2.4) FP= 0, BF = 0.001, HE (DB function);(2.5) FP= 0, BF = 0.01, HO;(2.6) FP= 0, BF = 0.01, HE (DB function);(2.7) FP= 1, BF = 0.01, HO;(2.8) FP= 1, BF = 0.01, HE (DB function).

The initial configuration for Experiment 2 was the same, i.e. a 200× 10 lattice wasused with 100 particles in each cell. The B0 particles were assigned to Columns0–85 and 115–200, and the R0 particles to Columns 94–106. In order to mimic themixing zone, the B1 particles were assigned to Columns 86–89 and 111–114 andthe R1 particles to Columns 90–93 and 107–110. Thermal boundary conditions wereapplied when fluid particles hit a solid.

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The structure of the automaton model allows it to be solved efficiently on parallelcomputers. This should enable large three-dimensional simulations to be accom-plished in the future.

4. Results and Discussion

In this section, the simulation results for dispersion at the pore scale is first presentedand discussed. Then, the simulation results for dispersion at megascale are presentedand discussed.

4.1. pore scale dispersion simulation

Figure 2 depicts the simulation results at 300 iterations for Experiment 1. The fourcurves in Figure 2 show the distribution of particle numbers of tracer for Experiments(1.1)–(1.4), respectively. It is observed that under the same temperature condition,dispersion in a heterogeneous model is larger than that in a homogeneous model.The reason for this is that in the heterogeneous model, the tracer moves faster inthe larger tubes and slower in the smaller tubes. This causes more dispersion in theheterogeneous model than in the homogeneous model because of the different localfluid flow velocities in the heterogeneous model. The double peaks observed in theconcentration profiles for the heterogeneous model results from the separation of theflow in the high permeability tubes from those in the low permeability tubes. If agradual change in tube diameters is used, then this effect should vanish.

Figure 2. Pore scale dispersion in both the homogeneous and heterogeneous porous mediashown in Figure 1. The concentration profiles are plotted after 300 iterations and at twodifferent temperatures.

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When comparing the simulation results at different temperatures (Figure 2), itis observed that the tracer moves faster at the low temperature (highβ ′ (B)) due tothe low fluid viscosities (here it is important to recall that one is dealing with thethermodynamics of a gas model, for which viscosity increases with temperature).However, the tracer at the high temperature (lowβ ′) spreads at a faster rate becauseof the high diffusion coefficient.

In summary, pore scale dispersion simulations clearly demonstrate the pore struc-ture effects on dispersion. That is, a core with continuous paths connecting large pores‘promotes’ dispersion. It also indicates that dispersion is affected by the diffusioncoefficient and flow rates.

4.2. megascale dispersion simulation

As mentioned in Section 3, the initial configuration for the dispersion experiments isthat the red particles (tracer) are initialized in the center of the lattice (Columns 90–110) and the blue particles in the rest of the cells of the lattice. When flow starts, thereare two miscible displacement fronts, i.e. a left front with blue particles displacingred particle, and a right front with red particles displacing blue particles.

Figures 3–6 show the simulation results (distribution of red particle number at1500 iterations) for Experiments (2.1) and (2.2), (2.3) and (2.4), (2.5) and (2.6), and(2.7) and (2.8), respectively. It can be seen that dispersion in the heterogeneous modelis larger than that in the homogeneous model. The additional degree of freedom which

Figure 3. Megascale dispersion in both a homogeneous and a heterogeneous porous mediumwith both the flipping probability andβ ′ equal to zero. The concentration profiles are plottedafter 1500 iterations.

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Figure 4. Megascale dispersion in both a homogeneous and a heterogeneous porous mediumwith the flipping probability being zero andβ ′ = 0.001. The concentration profiles are plottedafter 1500 iterations.

Figure 5. Megascale dispersion in both a homogeneous and heterogeneous porous mediumwith the flipping probability being zero andβ ′ = 0.01. The concentration profiles are plottedafter 1500 iterations.

allows for the flipping probability in the propagation step is a small effect for thetwo-dimensional simulations considered.

To study the effects of the value ofβ ′ on the dispersion, Figure 3 is comparedwith Figures 4 and 5. It is observed that an increase in the value ofβ ′, increasesdispersion. Moreover, whenβ ′ is increased to 0.01, the concentration profiles are not

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Figure 6. Megascale dispersion in both a homogeneous and a heterogeneous porous mediumwith the flipping probability being 1 andβ ′ = 0.01. The concentration profiles are plottedafter 1500 iterations.

symmetric around the displacement velocity which allow an asymmetry in the break-through curves and an early breakthrough. This simulation result is consistent withthe theoretical predictions (Udey and Spanos, 1993), experimental results (Brigham,1974) and the predictions of the pore scale tube model.

A comparison of Figures 5 and 6 indicates that the value of the flipping probabilitydid not play an important role in reflecting the effect of pore structure. The only notice-able effect is that the curves are slightly smoother when a flipping probability is used.However, the effect of the flipping probability may be limited by two-dimensionalmodeling and thus it may play a larger role in three-dimensional simulations.

5. Conclusions

1. Dispersion in porous media was simulated at both the pore scale and themegascale.

2. Pore structure effects on dispersion are clearly demonstrated by pore scale mod-eling. That is, dispersion increases in a porous medium which consists of continuouspaths connected by large pores.

3. Additional rules were added in megascale modeling to allow macroscopic phaseseparation, these rules reflect the fact that the solid has a preference to collide with thedisplaced fluid, they incorporate pore structure in porous media and they allow fora pressure difference between the displacing and displaced fluids. The simulationresults indicate dispersion in the heterogeneous model is larger than that in thehomogeneous model. Further, when the value ofβ ′ is sufficiently large(β ′ = 0.01),

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the concentration profile is not symmetric which indicates an early breakthrough. Thissimulation result is consistent with both theoretical predictions (Udey and Spanos,1993) and experimental results (Brigham, 1974).

Appendix A. The Pressure Difference Between Two Miscible Phases

Starting from the macroscopic dynamic capillary pressure equation (cf. Equation(33) in de la Cruzet al., 1995)

∂(p1 − p2)

∂t= −β

∂S1

∂t, (A1)

where the parameterβ vanishes when surface tension is zero, and is in generaldifferent for different processes. Herep1 is the average pressure in the displacingphase,p2 is the average pressure in the displaced phase,S1 is the volume fraction ofspace occupied by phase 1 andβ is a parameter defined in de la Cruzet al. (1995).Equation (A1) completes the system of equations for slow, incompressible, two phaseflow (de la Cruzet al., 1995).

If the assumption of quasi-static flow is relaxed then an acceleration term of theform β ′(∂2S1/∂t2) must also appear in Equation (A1) yielding

∂(p1 − p2)

∂t= −β

∂S1

∂t+ β ′ ∂2S1

∂t2. (A2)

Thus, in the case of miscible flow,β = 0 and one obtains

∂(p1 − p2)

∂t= β ′ ∂

2S1

∂t2, (A3)

or integrating with respect to time

p1 − p2 = β ′ ∂S1

∂t. (A4)

Note that in the present case, the saturation of phase 1(S1) can be replaced by theconcentration of phase 1(C1). Here the physical origins of Equation (A4) are clear.If the concentration of one phase is increasing in a volume element during segre-gated, incompressible flow, then an average pressure gradient must exist between thedisplacing and displaced phase. As a result the average pressure of the displacingphase must be greater than that of the displaced phase.

References

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Academic Press, New York.

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