Two-Dimensional Network Simulation of Diffusion-Driven

JOURNAL OF COLLOID AND INTERFACE SCIENCE 179, 357–373 (1996)ARTICLE NO. 0227

Two-Dimensional Network Simulation of Diffusion-Driven Coarseningof Foam Inside a Porous Medium

DAVID COHEN,* T. W. PATZEK,† AND C. J. RADKE*,1

Lawrence Berkeley National Laboratory, *Department of Chemical Engineering, and †Department of Materials Scienceand Mineral Engineering, University of California, Berkeley, Berkeley, California 94720

Received May 19, 1995; accepted October 25, 1995

makes foam a promising blocking fluid for underground gasIn order to use foams in subsurface applications, it is necessary storage (3) and for prevention of gas coning in production

to understand their stability in porous media. Diffusion-driven wells (4) . To facilitate such applications, it is crucial tocoarsening of a stationary or nonflowing foam in a porous medium characterize the long term stability of foam in porous media.results in changing gas pressures and a coarsening of the foam

Foam in a porous medium is stable below a limiting capil-texture. A two-dimensional network simulation has been createdlary pressure, which depends on several factors, includingthat predicts the behavior of foam in a porous medium by physi-the surfactant properties, the gas velocity during flow, andcally specifying the locations of all the lamellae in the systemthe permeability of the porous medium (5). When the lim-and by solving the complete set of Young–Laplace and diffusioniting capillary pressure is exceeded, the liquid in the foamequations. An hourglass approximates the shape of the pores, and

the pore walls are considered to be highly water wet. A singularity films cannot withstand the capillary suction which thins andarises in the system of differential algebraic equations due to the eventually destabilizes the films. Our study looks at the co-curvature of the pore walls. This singularity is a signal that the alescence of stationary foams under conditions where allsystem must undergo oscillations or sudden lamellar rearrange- lamellae are exposed to capillary suction pressures signifi-ments before the diffusion process can continue. Newton–Raph- cantly below the limiting capillary pressure and, are there-son iteration is used along with Keller’s method of arc-length fore, unconditionally stable. Diffusion is the only mechanismcontinuation and a new jump resolution technique to locate and

which leads to foam coalescence, as gas diffuses across la-resolve bifurcations in the system of coupled lamellae. Gas bubblesmellae between adjacent gas bubbles.in pore throats are regions of encapsulated pressure. As gas is

Even when there is a net flow of foam in a porous medium,released from these bubbles during diffusion, the pressure of theexperiments show that foam flows in a small fraction of thebubbles in the pore bodies increases. When the pressure increasepores at any given time. The rest of the pores contain trappedis scaled by the characteristic Young–Laplace pressure, the equi-

librium time for the diffusion process is scaled by the ratio of the foam, which effectively blocks the flow. Blockage leads tosquare of the characteristic length to the gas diffusivity and two a permeability that is orders of magnitude lower than fordimensionless groups. One describes the ease with which gas can flow of a single phase (6) . In fact, studies have shown thatdiffuse through a lamella, and the second represents the amount the trapped gas saturation of foams in porous media can beof gas encapsulated within the pore throats initially. Given this as high as 80% (7, 8) . Understanding the diffusion processesscaling, the resulting plots of pressure versus time and normalized within trapped foams in porous media can aid in understand-lamellae positions versus time are universal for all system sizes

ing the overall behavior of flowing foams in porous media.and characteristics. This is true as long as the initial lamella distri-Several methods have been developed to describe the dif-bution is the same in each case. q 1996 Academic Press, Inc.

fusive decay of static foam in the bulk. Experimental tech-Key Words: foam; porous media; diffusion; network simulation.niques exist for measuring the rate of decay of bulk foams(9, 10) and a comprehensive scaling model describing thecoarsening of bulk foam was recently proposed by Patzek

INTRODUCTION(11). However, coarsening of bulk foam is quite differentfrom the coarsening of foam confined in a porous medium.

Foam can significantly reduce the mobility of a gas flow- In this work, we present the first comprehensive simulationing through a porous medium. Therefore, foam is a useful that predicts the diffusive coarsening of static foam in afluid for improving the recovery of oil in steam or CO2 porous medium.flooding processes (1, 2) . In addition, the low mobility

A model is needed to help understand how diffusion af-fects flowing foam systems, and to predict how stationary

1 To whom correspondence should be addressed. foam evolves with time. Below the limiting capillary pres-

357 0021-9797/96 $18.00Copyright q 1996 by Academic Press, Inc.

All rights of reproduction in any form reserved.

AID JCIS 4104 / 6g0e$$$741 04-02-96 18:33:58 coida AP: Colloid

358 COHEN, PATZEK, AND RADKE

FIG. 1. Photograph of a bulk foam. Notice that the lamellae are curved toward the larger bubbles. The arrows represent the direction of gas diffusion.Large bubbles grow at the expense of small bubbles.

sure and in the absence of external effects, such as tempera- polyhedral foam, small bubbles are usually at higher pressurethan the larger bubbles surrounding them. The pressure dif-ture fluctuations, evaporation, and seismic activity, foam tex-

ture coarsens as a result of diffusion of gas from bubble to ference between adjacent gas bubbles is a function of theinterfacial tension s of the liquid–gas interface and of thebubble. The current study predicts the effect of gas diffusion

and the time scales over which this effect is felt. A two- curvature of the film (2H) , as expressed by the Young–Laplace equation,dimensional network is adopted to simulate the motion of

the foam films and the variation of pressures in the foam asthe diffusion process occurs. A two-dimensional network DP Å 2s(2H) Å 4s /r , [1]with enough connectivity does a good job of capturing all thebehavior of a real system, while not being computationally

where r is the equivalent radius of curvature of the lamella.limiting.

In the absence of external forces, such as gravity, a lamellaThe mathematics in the simulation offer a formidable chal-

is a surface of constant curvature, and the equation 2H Ålenge. Prediction of the coarsening of foam in porous media

2/r Å (1/r1 / 1/r2) holds everywhere on the surface, whererequires the solution of a series of nonlinear, coupled differ-

r1 and r2 are the principal radii of curvature. The additionalential–algebraic equations. These equations exhibit multiple

coefficient 2 appears in Eq. [1] because a lamella is com-singularities, at which the system passes through a series of

posed of two liquid–gas interfaces. The pressure differencelimit points, and demonstrate multivalued solutions. In spe-

acts as a driving force for gas to diffuse from one bubble tocial cases, we discover oscillatory behavior of foam films

the next. Therefore, large bubbles increase their volumes atas they collectively rearrange due to diffusional coarsening.

the expense of smaller bubbles. Eventually, the systemTo resolve the regions of mathematical difficulty, Keller’s

reaches equilibrium when the pressure is uniform every-method of arc-length continuation is used, followed by a

where. This occurs either when all bubbles are of the samenew technique that allows resolution of a solution on a sepa-

size and topology, or more commonly when one large bubblerate branch.

remains.Foam injected into a porous medium behaves quite differ-PHYSICAL PICTURE

ently than in bulk, due to geometric constraints at pore walls.In a water-wet porous medium, the projections of the lamel-Foam is a dispersion of gas bubbles surrounded by thin

liquid films, or lamellae, that are stabilized by surface active lae intersect the walls at a 907 angle (12, 13). Therefore,the curvature of a given lamella is completely determinedagents. The curvature of a lamella indicates the direction of

the pressure difference between adjacent bubbles. The gas by its location within the pore. It is no longer necessarilytrue that small bubbles are at higher pressure than largeon the concave side of a lamella is at higher pressure than

the gas on the convex side. As shown in Fig. 1, in a bulk bubbles. This characteristic of foam in a porous medium is


359SIMULATION OF DIFFUSION OF FOAM IN POROUS MEDIA

law relates the number of moles, n , to the pressures in thebubbles. The specific pore geometry chosen determines thefunctional dependence of radii of curvature and bubble vol-umes on the positions of lamellae in the porous medium.

TWO-DIMENSIONAL NETWORK MODEL

In the simulations, the porous medium is approximatedby a network consisting of a two-dimensional array of pores.Figure 3 shows a 41 4 network, which is made up of 16 porebodies, numbered 1 through 16. Four pores come together ateach pore body; each set of four pores and one body is

FIG. 2. Schematic of foam in a porous medium. The cross-hatched referred to as a cluster. The clusters join together to createregions represent the grains of the porous medium. The dotted regions are

the network. For convenience, we divide each pore into twothe liquid phase, which form thin films along the solid and lamellae spanninghalf-pores, so there are 64 half-pores in a 4 1 4 network.the pores. Shaded regions indicate the gas phase.

Each half-pore in the network is referenced by a two indexsystem, as highlighted in pore body 1 in Fig. 3. The first

illustrated by the schematic in Fig. 2. The solid is water- index ( i Å 1, NC) refers to the cluster number; the secondwet, so the liquid phase forms liquid films along the grains index ( j Å 1, 4) refers to the half-pore number; and NC isin addition to the thin foam films. the total number of clusters in the system. Cluster 1 is in

Just as in a bulk foam, gas diffuses through the lamellae the lower right corner of Fig. 3, and cluster NC is in the topfrom high pressure bubbles to low pressure ones. As a result, left corner. Figure 3 also illustrates how the pores sur-the lamellae rearrange as the system strives to achieve a rounding each pore body are numbered. We define a loopstate of minimum free energy, at which point the pressures in the system as the path one follows to connect four clustersbecome uniform. The lamellae, which are assumed to be and return to the starting point. For example, in the 4 1 4infinitely stable, move by sliding along the wetting films system of Fig. 3, a loop starts in pore body 2, and passeswhich coat the surface of the porous medium. This process through bodies 6, 7, and 3, traversing four pores on the way.proceeds slowly enough to allow the liquid phase to re- The multiple connectivity of the pores in the network leadsarrange so the lamellae remain at nearly constant thicknessduring the process. At the equilibrium position, the lamellaeall rest in the pore throats, where they have no curvatureand thus sustain no pressure drop. This configuration is ther-modynamically stable, and gas diffusion stops. In contrastto bulk foam, which always collapses ultimately, the equilib-rium configuration of foam in a porous medium remainsindefinitely in the absence of external disturbances.

The rate of transfer of gas across a lamella is describedby the diffusion equation, derived from Fick’s law,

dn

dtÅ 0kADP ; k Å SD

h, [2]

where dn /dt is the number of moles of gas per unit timepassing through the lamella, A is the surface area of thelamella across which the transfer occurs, and k is the masstransfer coefficient, given by the product of the solubility ofthe gas species in the liquid phase, S , and the diffusivity ofthe gas in the liquid phase, D , divided by the thickness ofthe film, h . Interfacial resistance to mass transfer can usuallybe neglected, as the transfer through the film is much slowerthan the initial dissolution of gas in the lamella (14).

Gas diffusion in a foam in a porous medium must satisfy FIG. 3. A 4 1 4 network illustrating the indexing system. The solidboth the Young–Laplace equation [1] and the diffusion matrix of the porous medium is represented by dashed regions, and the

lamellae are represented by curved lines.equation [2] for each lamella in the system. The ideal gas



the foam coarsens. Coarsening occurs only when there aretwo lamellae located in a pore, as in Fig. 4.

Knowing the pore geometry, and given the position, x , ofa lamella in a pore and the 907 contact angle of the lamellawith the pore wall, the foam geometry in the system can bedetermined (see Appendix B). Each pore is divided concep-tually into two half-pores, separated at the pore throat. Eachhalf-pore may contain a lamella, which has a radius of curva-ture of

r

Rg

Å B 0 (1 0 x 2) 1/2

x, [3]

where B is a shape parameter, related to the aspect ratio.For an hourglass, B is defined as

FIG. 4. Translationally invariant hourglass pore with a lamella at nor-B Å S1 / rt

RgD , [4]malized position x .

where rt is the radius of the pore throat, and Rg is the radiusto interesting dynamic behavior. The system is closed, withof the grain used to generate the hourglass shape. The vol-a constant mass of gas in the system at all times. Pores alongume of gas inside the bubble bounded by a lamella and athe edges of the network are treated as if there were a rigidpore throat iswall at the pore throat.

To complete the network model described here, we mustpick a pore geometry. The shape used in the simulator is a V

R 3g

Å 2Bx 0 arcsin(x)translationally invariant ‘‘hourglass.’’ Two parallel cylindersof radius Rg determine the pore cross section, as illustrated 0 x(1 0 x 2) 1/2 / (B 0 (1 0 x 2) 1/2 ) 2

in Fig. 4. The pore width is equal to Rg , which is called thegrain radius. The narrowest point at the center of the pore

1 (arccos(1 0 x 2) 1/2 0 x(1 0 x 2) 1/2 )x 2 . [5]is the pore throat, where the cylinders are separated by a

distance 2rt . The space connecting pores is called the porebody. The geometry of the pore body is not important. Each Likewise, the surface area of the lamella can be calculated,pore body’s volume is defined as part of the initial condi-tions.

All pores are considered to be water-wet. Thus, when aA

R 2g

Å 2r

Rg

arccos(1 0 x 2) 1/2 . [6]lamella is located at the pore throat it has an infinite radiusof curvature (zero curvature) and sustains no pressure drop.

A fundamental difference between bulk foam and foamThe curvature of a lamella increases as it moves away fromconfined in a porous medium is that the equilibrium configu-the pore throat, and passes through a maximum between theration in the porous medium is a foam which still has apore throat and the pore body. This maximum in curvaturesubstantial number of bubbles that do not necessarily haveoccurs at the ‘‘critical position.’’ The variable x representsequal volumes, but do have equal pressures and zero curva-the position of the intersection of the lamella with the poreture lamellae. As mentioned earlier, this is a stable configu-wall, normalized by the characteristic length of the poreration that remains indefinitely, assuming there are no exter-(Rg) ; x is zero at the throat and unity at the intersection ofnal effects.the pore with the body.

To start the simulation, we specify a distribution of poreSince a lamella contacts the wall of the pore at 907, it issizes. For example, the grain size Rg and shape parameter Balways convex away from the pore throat. Thus, the gas inmay be varied using a Gaussian distribution. The distributionthe pore throat is always at a higher pressure than that in thein grain size is defined as a function of the standard deviationpore body and diffuses out of the pore. Lamellae rearrange asin Rg to bediffusion proceeds and ultimately recede toward the pore

throats during the coarsening process. When two lamellaeapproach one another at the pore throat, they merge. In this c Å RU g / SD

RU g

, [7]way, bubbles, as defined above, completely disappear, and



where RV g is the average grain radius and SD is the standard MATHEMATICAL DESCRIPTIONdeviation in Rg . When c x 1, there is a distribution of porevolumes. In addition, we set the pore body volumes in the As described above, the position of each lamella in thenetwork such that each pore body is of the same size. Finally, system must satisfy Eqs. [1] and [2]. Equation [1] is thewe randomly distribute the lamella positions between preset Young–Laplace equation for a lamella which is spherical. Inlimits, e.g., 0.1£ x£ 0.99. However, these positions cannot our translationally invariant geometry, lamellae are actuallybe random across the entire system. Each half-pore may approximated by sectors of cylinders. Therefore, the curva-contain a lamella, resulting in up to eight lamellae that make ture 2H is 1/r and DP is 2s /r, as the second principal radiusup a loop in the system, as in Fig. 3. Each lamella of a of curvature is infinite. Application of the ideal gas lawfoam in a porous medium must satisfy the Young–Laplace relates the pressure P to the number of moles n and theequation. This means that in a loop which contains eight volume of gas V in the Young–Laplace equation and leadslamellae, only the positions of seven of these lamellae are toindependent. The position of the eighth lamella is completelydetermined once the positions of the other seven are known. ni ,jRT

Vi ,j( xi ,j)0 niRT

Vi (xi ,kÅ1,4 )Å 2s

ri ,j

, [9]To establish an initial configuration in the simulation, sevenlamellae in each loop are randomly positioned, and thenthe necessary radius of curvature of the eighth lamella is

where R is the universal gas constant and T is temperature.calculated from the Young–Laplace equation. From this ra-A number of moles or a volume with two subscripts refersdius of curvature, we calculate the corresponding lamellato a bubble contained in half-pore ( i , j) . When there is oneposition,subscript, the variable refers to the value in pore body ( i) .Substituting the expressions for lamellar surface area, Eq.[6] , and pressure drop (DP Å 2s /r) into the diffusionxi ,j Å

Bi ,jri ,jRg,i ,jRU g {√

R 4g,i ,j 0 B 2

i ,j R4g,i ,j / r 2

i ,j R2g,i ,j RU

2g

(r 2i ,j RU

2g / R 2

g,i ,j),

equation [2] results in

[8]dn

dtÅ 04skRgarccos(1 0 x 2

i ,j)1/2 , [10]

where ri ,j Å ri ,j/RV g is the dimensionless radius of curvatureof lamella ( i , j) . Equation [8] gives two possible solutions

where dn /dt represents the rate of gas transfer across afor the lamella position. As long as the calculated positionlamella. The equations are nondimensionalized using theis between 0 and 1, we can choose either solution.average grain radius (RV g) as the characteristic length, andAt times, Eq. [8] has no admissible solutions. Since thethe surface tension over the characteristic length (s /RV g) topressure difference is inversely proportional to the radiusfind the dimensionless pressure P (ÅPRV g /s) . Equations [9]of curvature and there is a minimum achievable radius ofand [10] then become the two primary residual equations,curvature for any pore, sometimes it is not possible to

achieve a pressure drop across the eighth lamella that isnecessary to complete the loop. When this situation arises, hi ,j

gi , j( xi , j)0 hi

ai (xi ,kÅ1,4 )0 2

ri , j

Å 0 [11]only six of the lamellae can be randomly placed, while thetwo remaining lamellae positions are calculated from theYoung–Laplace equation. In fact, sometimes the only ad- dh

dt/ 4 arccos(1 0 x 2

i , j)1/2 Å 0, [12]

missible solution is that with an empty half-pore, leavingseven lamellae to complete the loop.

Once an acceptable initial configuration of lamellae has where h (ÅnRT /sRU 2g) is the dimensionless number of moles,

been created, the complete distribution of gas moles and g is the dimensionless volume of a bubble in a half-pore, apressures can be calculated, given the total number of moles is the dimensionless volume of a bubble in a pore body,of gas in the system. At this point, the set of equations is r is the dimensionless radius of curvature, and t is thesolved explicitly in time and the motion of all the lamellae dimensionless time,is tracked during the coarsening. Throughout the process,the system is closed. No gas enters or leaves the network,and the total volume remains constant. When lamellae ap- t Å tD

RU 2g b

. [13]proach a pore throat from opposite sides, the bubble in thepore disappears as the two lamellae merge into one. In thisway, there is a coarsening of foam texture from the initial The parameter b is the inverse of the lamella conductivity

to gas transport,configuration to the final, equilibrium state.



different form to account for bubbles that span more thanb Å h

RTSRU g

. [14] one pore.

NONLINEAR DYNAMICSBy way of illustration, we use the 4 1 4 system in Fig.3, where cluster 2 connects to cluster 6. More specifically,

The nine unknowns per cluster are calculated by solvinghalf-pore [2, 2] connects to half-pore [6, 3] . In this example,the residual equations [15] – [19], which in vector notationone lamella resides in each half-pore. When solving theareproblem, Eqs. [12] for adjacent half-pores are added together

to give an expression which defines the total transfer ofR[u(t)] Å 0. [20]moles out of a bubble into the two adjacent pore bodies:

The unknowns are the lamella positions (xi ,j) , the numberdh2,2

dt/ dh6,3

dt/ 4 arccos(1 0 x 2

2,2)1/2

of moles in each half-pore (hi ,j) , and the number of molesin each pore body (hi ) . These variables make up the solutionvector u Å [xi ,j , hi ,j , hi ] . Traditionally, (t) is the indepen-/ 4 arccos(1 0 x 2

6,3)1/2 Å 0. [15]

dent variable in the problem, and the residual equations [20]are solved explicitly by marching forward in time with anSince we divided each pore into two half-pores for conve-increment Dt. At each time step, the nonlinearities in thenience, we must constrain the pressures in the two adjacentresidual equations are resolved using a Newton–Raphsonhalf-pores to be equal:iteration (15). Because of the extensive coupling among thenumerous lamellae, the Jacobian matrix contains a signifi-h2,2

g2,2 (x2,2 )0 h6,3

g6,3 (x6,3 )Å 0. [16] cant amount of off-diagonal elements.

The high degree of nonlinearity of the problem can leadto several solution branches. When there is a bifurcation orIn addition, lamellae [2, 2] and [6, 3] must each satisfy thelimit point in the solution space [u 1 t] , the determinantYoung–Laplace equation:of the Jacobian (G Å ÌR /Ìu) passes through zero. TheNewton–Raphson technique then fails because the Jacobiancannot be inverted at the singularity. Singularities arise phys-

h2,2

g2,2 (x2,2 )0 h2

a2(x2,kÅ1,4 )0 2

r2,2

Å 0 [17]ically due to the critical position in the relationship betweenradius of curvature and x , as noted previously. Figure 5h6,3

g6,3 (x6,3 )0 h6

a6(x6,kÅ1,4 )0 2

r6,3

Å 0. [18] illustrates this effect of the geometry for an hourglass withshape parameter B Å 1.2. Since the pressure drop across alamella is inversely proportional to the radius of curvature,For every pair of half-pores, Eqs. [15] – [18] must bea maximum occurs at the critical position. In some situations,satisfied. A final constraint is the equation of mass conserva-the system of lamellae in the porous medium network dic-tion, one for each pore body, which requires that the totaltates a higher pressure drop across a given lamella than cantransfer of moles into the pore body is equal to the sum ofbe physically achieved, given the pore geometry. When thisthe moles transferred out of each of the surrounding pores:occurs, a limit point may result, and the lamellae suddenlyadjust or jump to an admissible position.

We use Keller’s method (16) of arc-length continuationdh2

dt0 ∑

4

jÅ1

4 arccos(1 0 x 22, j) Å 0. [19]

to resolve the solution around limit points. By making t anadditional dependent variable, the new solution vector is v

For each cluster in the network, there are eight governing Å [u , t] , and the new independent variable is the differentialequations [15] – [18] and one conservation equation [19] arc-length ds of the solution branch:for a total of nine equations per cluster. Thus, in the 4 1 4network, there are 144 total equations which completely (ds)2 Å \du \ 2 / (dt)2 . [21]specify the system. This system of equations forms a set ofdifferential–algebraic equations, with first-order differential The system of lamellae is solved by marching explicitly inequations coupled with nonlinear algebraic equations. The arc-length with an increment Ds , and with an additionalnonlinearity of the algebraic equations results from substitut- residual equation to account for the new dependent variable,ing the expressions for volume, Eq. [5] , and radius of curva-ture, Eq. [6] , into the denominators of Eqs. [16] – [18]. A M Å u\u(s) 0 u(s0) \ 2 / (1 0 u)Ét(s) 0 t(s0)É2

system with one or more lamellae missing has fewer equa-tions, and some of the remaining equations have a slightly 0 (s 0 s0) 2 Å 0, [22]



FIG. 5. (a) Critical position in radius of curvature and (b) pressure difference. The inflection point of the radius of curvature corresponds to amaximum allowable pressure drop across a lamella in the pore.

where u(s0) is the solution at the previous step, and u √ from the pore throat, the adjacent lamella in half-pore (2,4) must follow it and move toward the throat. In addition,{0, 1} is an adjustable parameter. The new residual vector

is Q Å [R , M] , and the resulting Jacobian needed for a because a jump occurs at an instant in time, there is nodiffusion of gas, and the number of moles of gas in everyNewton–Raphson solution ispore body must remain constant. Therefore, some of thelamellae surrounding each pore body move toward the body,

J(s) Å S G Rt

Mu MtD , [23] and some move away.

We define a diagonal matrix F which has elements thatare all close to zero, but which are seeds for the adjustments

where the new Jacobian matrix J is nonsingular even when that are made to each lamella position value in the solutionG is singular. vector u in order to find the new solution branch. Since the

Given the arc-length continuation, the tangent to the solu- number of moles is constant during the adjustment, we musttion vector at a given step can be calculated. The tangent solve for the vector of lamella positions, x . The adjustmentvg

(s) Å dv(s) /ds satisfies is the dot product of the matrix F and a coefficient vector1 , whose elements set the directions in which to make theadjustments of each lamella position and the magnitude ofJ(s)vh (s) Å 0S 0

MsD . [24]

each adjustment. To resolve the first point on the new branch(uh) , we use

The solution vector is found by iterating JDv Å 0Q untilxh Å x(sc ) / 1F [25]Q Å 0.

At any given time, there are multiple branches in solutionto express the lamella positions in the Young–Laplace equa-space that satisfy the physical laws and contain the sametions, where x(sc ) is the solution vector at the singular point.number of moles within the same total volume. All theThe Jacobian is made up of x-derivatives of the Young–branches eventually yield equilibrium lamella configura-Laplace equations multiplied by diagonal elements of thetions. However, not all branches can be reached from thematrix F . The unknown vector is now 1 , which is foundinitial condition because the system would have to follow aby the Newton–Raphson iteration and substituted into Eq.nonphysical path. To jump to the new branch, it is necessary[25] to establish the new lamella positions. Once the pointto pick a direction to move from the point of singularity.on the bifurcated branch is found, the rest of the branch isThe distance of the jump must be large enough so that theresolved using arc-length continuation.solution vector does not remain on the old branch, but yet

small enough so that the new branch can be resolved.Since the direction of the jump is not known a priori, LAMELLA MOTION

we must devise a technique to choose the proper direction.Adjacent lamellae must move in opposite directions during The net motion of all lamellae in a pore network must be

toward the pore throats. However, at any given moment, anthe jump, because the mass inside every bubble is conserved.For example, if the lamella in half-pore (1, 1) moves away individual lamella may move away from the throat, de-



FIG. 6. Motion of eight lamellae in a simple 2 1 2 network with B Å 1.2 and c Å 1. All lamellae migrate toward the pore throat, but the lamellaein pores (2, 4) and (3, 1) , marked by the arrow, move in the opposite direction initially.

pending on the influence of the lamellae around it. Figure 6 is illustrated in Fig. 7. Here, all eight lamellae in the loopstart at x near 0.7. Figure 7a shows that in general, theshows the motion of the eight lamellae in a simple 2 1 2

network, as a function of dimensionless time (t Å lamellae migrate toward the throats. A closer look, however,shown in Fig. 7b, reveals an oscillation just after t Å 48.7.Dbt /RU 2

g) . There is a random distribution of initial lamellaFigure 8 graphically illustrates on an exaggerated scalepositions ranging from x Å 0.33 to x Å 0.99. Eventually,

what occurs as the system passes through the oscillation.these lamellae rest at the pore throats, where x Å 0. Six ofThe actual magnitude of the oscillation is very small, as seenthese lamellae move in toward the pore throats at all times.in Fig. 7. In Fig. 8, we see four snapshots in time duringHowever, two of them, (2, 4) and (3, 1) , marked with anthe oscillation of the loop. The motions resemble the rotationarrow in the figure, move away from the throats initially andof a wheel. The dotted regions represent the solid grains.then turn and head toward the throats.The lamellae enter the oscillation sequence in frame a, moveWhen the lamella positions span the critical position, in-in the direction of the arrows to the configuration in frameteresting behavior, such as the local reverse motion in Fig.b, turn and move back in the other direction to the configura-6, results. For an hourglass with a shape parameter B Å 1.2,tion in frame c, and finally go through one more jump tothe critical position occurs when the lamella resides at x Åthe configuration in frame d. This sequence is for 48.8 £ t0.55. When the arithmetic average of the lamella positions£ 49.5.exceeds 0.55, the lamellae clearly span the critical position.

When the initial lamella distribution is broader, the lamel-In addition to the changing direction of lamella motionlae do not oscillate as the ones shown in Figs. 7 and 8.described above, there can be oscillatory behavior near aInstead, the system approaches a singular point and directsingular point. This type of behavior is evident in a lamellae-Newton–Raphson iteration fails. If the original solutionensemble where the initial lamella distribution is nearly uni-branch is followed using arc-length continuation, it is possi-form. In other words, each lamella is nearly the same dis-ble to continue past the singular point. It turns out that thetance from its corresponding pore throat as all the others.singular point is a turning point in the solution, and a de-When the singular point is reached, near xi Å 0.55, thecreasing time solution can be found. Clearly, this is a non-lamellae jump to new positions in order to satisfy the diffu-physical solution to the problem. As the lamellae-ensemblesion process. In a given loop, this motion is analogous tomoves in the direction of decreasing time, gas diffuses intothe rotation of a ‘‘wheel.’’ All eight lamellae in the loopthe pores from the pore bodies, in the direction against therotate in the same direction, either clockwise or counter-pressure drop. As the system follows the nonphysical branch,clockwise. The loop behaves like a rigid body, because themany of the lamellae move away from the pore throats to-rotation happens so quickly that negligible diffusion occurs.ward a second equilibrium position, in the pore bodies,Several rotations may occur sequentially until the systemwhere the lamellae have zero curvature, and diffusion acrosssettles at a new point, at which time diffusion-driven motion

continues smoothly. A 2 1 2 system with such an oscillation the lamellae stops. This is an unstable equilibrium state be-



FIG. 7. (a) Lamella positions in a 2 1 2 network with a narrow lamella distribution. Looking closely at the circled region of the graph, (b) showsthe oscillation that occurs.

cause any perturbation of a lamella causes that lamella to bodies as a function of dimensionless time, showing thedecreasing time solution when the arc-length continuationmove away from the body toward the pore throat. Figure 9

is a plot of the dimensionless pressure in one of the pore is used to pass through the singular point.

FIG. 8. Schematic illustration of oscillation in Fig. 7 on an exaggerated distance scale.



these lamellae during the jump. The mass of gas in thebubble in the pore body marked with an asterisk is conservedbecause the motion toward the body from the bottom isbalanced by the sum of the motions out of the body to thetop and right. The lamella on the left in Fig. 12b does notmove during the jump because it is on an outer boundaryand thus does not interact with other lamellae.

As a lamellae-ensemble decays to its final equilibriumstate, several transitions or turning points may be encoun-tered. In fact, the 4 1 4 network system illustrated in Figs.9–11 reaches a second transition point where the systemundergoes a second rearrangement shown in Fig. 13. Afterthis point, the system decays until reaching the low freeenergy equilibrium configuration.

It is useful to verify that the free energy of the systemFIG. 9. Dimensionless pressure in a pore body versus dimensionless decreases with time during the diffusion-driven rearrange-

time for a 4 1 4 network. The solution continues around the turning pointment of the lamellae. To describe a system of constant vol-and heads back in time. The lower branch is nonphysical.ume and temperature, we use the Helmholtz free energy.The Helmholtz free energy is

Since the resolved continuation of the original branch can-not occur physically, the lamellae must jump to a new con- F 0 F 7figuration that allows continuation of the physical diffusionprocess. In general, there is no complete oscillation as in Å 0∑

i

piVi / s ∑i

Ai / RT ∑i

nilnS Pi

NRT /VtD , [26]

Fig. 7, rather an instantaneous jump to a new solution branch.The jump appears to be instantaneous because our modelneglects viscous losses in the system. In reality, the jump is where N is the total number of moles of gas in the system,a very rapid rearrangement of lamellae, with nearly negligi- and Vt is the total system volume. The summations in theble gas diffusion. The lack of a complete oscillation is due equation indicate that we must sum the contributions of ev-to the fact that the compressibility of the gas inside the ery bubble in the system, both in the pore throats and thebubbles has a damping effect on the system. pore bodies. The reference energy F7 is the equilibrium

Figure 10 shows the jump from the original branch to the Helmholtz energy. Also, we must remember that each la-new branch for the system in Fig. 9. Instead of continuing mella contributes twice to the surface area term, one foraround the turning point, the system passes through the jump each liquid–gas interface. After nondimensionalizing Eq.described in the previous section. The bubbles move as [26] using the scaling presented above, we verify in Fig. 14nearly rigid bodies because the gas does not redistribute. that the Helmholtz free energy indeed decreases for a singleTheir pressures do change slightly, so the bubbles compress lamella in a pore as it moves toward equilibrium.or relax during the rearrangement. In fact, in a pore with Rg Figure 15 shows the overall behavior of the Helmholtz freeÅ 200 mm and a system with a surface tension of 35 dyn/cm, the dimensionless pressure jump shown in Fig. 10 corre-sponds to an actual change in pressure of 21.2 Pa (2.1 11004 atm), which is very small. In general, the loops moveas rotating wheels, as illustrated in Fig. 11, where the arrowsshow the direction of the jump from frame a to b.

Systems larger than 2 1 2 exhibit more complicated be-havior at a singular point, as not all of the loops behave likea wheel. The criterion of no mass transfer out of each bubbleduring the sudden rearrangement must still be met. Figure12a shows a 3 1 3 network with arrows sketched in theshaded grains showing the direction of wheel rotation foreach of the four loops during the jump. Notice that the loopat the top left actually divides into two halves which rotatein opposite directions. Figure 12b shows a close up of the FIG. 10. Dimensionless pressure in a pore body versus time for a 4 1cluster marked with an asterisk in Fig. 12a, with the lamellae 4 network, the same system as in Fig. 9. The solution jumps to a new

branch at the turning point.sketched and arrows showing the direction of motion of



FIG. 11. Rotation of wheel in a 2 1 2 network at a singular point. (a) The lamellae hit the turning point and jump in the direction of the arrows tothe configuration in (b) .

energy in a 3 1 3 network during the rearrangement of the pore bodies may experience decreases in pressure initially,depending on how they interact with the other pore bodieslamellae-ensemble. The energy decreases throughout the entirein the network, but they always end up at a higher pressureprocess. This system passes through one transition point, andthan they were at initially. In fact, all pore bodies end up atF decreases during the jump to the new configuration as well.the same pressure. However, the average pore body pressuremust increase at all times. These results are verified in Fig.RESULTS AND DISCUSSION16 for a 5 1 5 system, where the diffusion process reaches

The equilibrium pressure in a pore body must be higher equilibrium. Recall that the pressure is nondimensionalizedthan its initial pressure. This is true because the lamellae by (s /RV g) , which is the characteristic pressure for a Young–that surround the pore bodies encapsulate gas at higher pres- Laplace problem and represents the degree of encapsulationsure in the pore throats than in the pore bodies. As the gas of pressure in the system.escapes from the pore throats across the lamellae during For diffusion collapse of a single spherical foam bubble,diffusion, the pressures in the pore bodies increase. This is Chambers (19) derived from the diffusion equation [2] theanalogous to the increase in pressure of a headspace above characteristic diffusion time. The length of time it takesa bulk foam, which has been verified experimentally (17, a spherical bubble, in contact with an infinite medium, to

disappear is approximated as18) and is predicted by available models (9, 11). Individual

FIG. 12. (a) Lamella motion during jump at a singular point in a 3 1 3 network. (b) Closer look at the behavior of the lamellae around the porebody marked with an asterisk. The mass in each region is conserved.



FIG. 13. Dimensionless pressure in a pore body versus time for the 4FIG. 15. Helmholtz free energy versus time for a 3 1 3 network system.1 4 network of Figs. 9 and 10. The solution passes through two transition

F decreases at all times.points, as marked by arrows in the figure.

single bubble with an initial radius of curvature of 150 mmtdif Å hP0R 2

0 /8DSRTs, [27] in a pore with a throat radius of 43 mm (Rg Å 216 mm) atatmospheric pressure, the simulation gives an elapsed time

where R0 is the initial bubble radius and P0 is the initial to reach equilibrium of about 11 min. Accordingly, a bubblebubble pressure. The initial bubble pressure is approximately in a porous medium takes an order of magnitude longer toequal to the pressure outside the bubble, since the Young– decay than a bubble in bulk with the same initial radius andLaplace correction (4s /R0) is much less than one atmo- pressure.sphere. For a single bubble of radius 150 mm at atmospheric The characteristic time for diffusion of a bubble in a po-pressure, the diffusion time is about 1 min. The constants rous medium is given byused in this calculation are the ones used throughout thispaper: D Å 1 1 1009 m2/s, S Å 5 1 1006 mol/(m3 Pa),

tc ÅRU 2

g

DbPU c , [28]and s Å 3.5 1 1002 N/m.

We seek to establish the characteristic time for the diffu-sion process in a porous medium. This is the time for a

where PU c describes the amount of gas initially encapsulatedsingle lamella to move from its initial position to its finalin the pore throats. The measure of encapsulation is theresting place in the pore throat, with zero curvature. For a

FIG. 16. Pressure change in the pore bodies during diffusion in a 5 15 network. One individual pore body pressure is plotted, as is the averageFIG. 14. Helmholtz free energy and lamella position versus time for a

single lamella sitting in a pore (B Å 1.2) . value of all the pore body pressures.



the final system pressure. We define the decay time, teq , asthe time at which the diffusion process is 90% of the wayto equilibrium, based on the average lamella position beingat 10% of its initial value. Figure 17 is important for theapplication of foam at reservoir conditions, which are typi-cally at very high pressures. Thus, the diffusion time forfoam is long, as much as several hours for systems at 10atm and with an average pore size of 100 mm.

The decay time for diffusion of foam in a porous mediumdepends strongly on the size of the pores. Smaller size poresresult in more tightly curved lamellae; thus, the driving forcefor diffusion is greater and the foam completes the diffusionprocess faster. In fact, Eq. [28] shows that the characteristictime for diffusion is proportional to the characteristic poresize for constant pressure.

FIG. 17. Linear dependence of completion time on system pressure. The most important consequence of scaling time by PU c isthat the time for the system to reach equilibrium is indepen-dent of network size. For example, a 2 1 2 system, which

average initial number of moles of gas nV

in the bubbles has 4 clusters, takes the same reduced time to complete thespanning the pore throats, converted to a pressure and nondi- diffusion process as a 7 1 7 system, with 49 clusters, asmensionalized by the surface tension over characteristic long as the lamella distribution is the same. In addition, thelength, dimensionless pressure increase is the same for different size

networks with identical lamella distributions. Therefore, wecan understand the behavior of a megascopic porous mediumPU c Å

nVRT

RU 3g

RU g

s. [29]

by scrutinizing the smaller systems. This results in signifi-cant computational savings.

Equation [14] defines the parameter b, which describes End effects have a slight impact on the quality of thethe resistance to transfer of gas across a lamella from the results for smaller systems. End effects arise because thepore throat to the pore body. The lower the value of b, the pores in between two pore bodies may have two lamellaefaster is the gas transport across the lamellae. For a thicker in them and definitely have interactions with multiple porelamella or lower solubility, the rate of diffusion will de- bodies. The pores on the sides of the network only interactcrease. with one pore body. Smaller networks clearly have a higher

The characteristic time for a bubble with a radius of curva- fraction of the system made up of pores which only connectture of 150 mm in a pore with a throat radius of 43 mm is to one pore body. An even bigger disadvantage of a smallcalculated from Eq. [28] to be almost 10 min. This illustrates network is the inability to set a truly random initial lamellathe order of magnitude difference in time scale between distribution with the smaller number of pores. For example,diffusion of bulk foam and foam in a pore and is very close when there are only 36 or 64 lamellae, like in a 3 1 3 or 4to the 11 min decay time found in the simulation. 1 4 network, the distribution is less random than it is in a

The characteristic time, and thus the time for the system system with 1000 lamellae.to reach equilibrium, scales as PU c , which is in turn linearly The scaling arguments for pressure and time can be summa-proportional to the system pressure. The system pressure is rized in ‘‘universal’’ curves of pore body pressure increase aschanged in the simulation by changing the total number of a function of time. Pore body pressure increase represents howmoles of gas in the network. The dimensionless variable much the pore body pressure deviates from the initial valuewhich represents the total number of moles of gas is (P0). For identical lamella position distributions, a universal

curve for all absolute pore sizes, all system pressures, and allvalues of the parameter b can be achieved by normalizingn Å NRT

RU 2gs

. [30]pressure by s/RV g and normalizing time by tc . The universalcurve for one particular lamella distribution, a uniform poresize distribution, and shape parameter B Å 1.2 is given for aFigure 17 shows how the decay time changes with varying3 1 3 network in Fig. 18. The plot shows the universality ofsystem pressures for 3 1 3 networks. Each of these systemsthe curve, even for systems with varying pressures (n), porehas the same total network volume and the same initial la-sizes (Rg), and solubility parameters (b).mella configuration, the only difference being the total num-

ber of moles of gas distributed in the network (n) , and thus Increasing the pore body volumes decreases the final value



FIG. 18. Universal curve of pressure, normalized by s /RV g, plotted against FIG. 20. Universal curve of pressure, normalized by s /RV g , plottedtime, normalized by tc . All four systems have a uniform pore size distribution against time, normalized by tc , for various distributions of geometric param-and the same initial lamella distribution. The absolute grain size varies in these eters.systems by 5 orders of magnitude, as does the parameter b.

for various pore size standard deviations and different valuesof normalized pressure in Fig. 18 because the ratio of encap- of B . The normalized lamella plot is given in Fig. 21. Thesulated pressure to pore body pressure decreases. However, agreement is not as good here as it was before. This isthe effect is minimal, and variations in the distribution of because changing these distributions results in slightly dif-pore body volumes are not important in the overall behavior ferent lamella distributions, and therefore the initial diffusionof the system. driving forces and amounts of encapsulated pressure are dif-

If we plot the average lamella position as a function of ferent. Also, the final pressures differ slightly in Fig. 20the reduced time, the resulting curve is also universal. Figure because the total system volumes differ as the B values19 shows this result for the same data presented in Fig. 18. differ. Nevertheless, we have discovered a way of univer-

The universal curve is the same for different pore size sally determining the diffusion behavior of foam in porousdistributions. The pore size distribution is changed by ad- media by using scaling constants, as long as the lamellajusting the standard deviation of the pore characteristic distribution for the system is fixed. This behavior is similarlengths (Rg) and by changing the range within which B falls. to that in the numerical solutions of Herdtle and Aref (20)Figure 20 shows the normalized plot of pressure versus time where different initial bubble size distributions in bulk foam

FIG. 21. Universal curve of average lamella position, normalized byFIG. 19. Universal curve of average lamella position, normalized byits corresponding pore size, plotted against time, normalized by tc . Same its corresponding pore size, plotted against time, normalized by tc , for

various distributions of geometric parameters. Same systems as in Fig. 20.systems as in Fig. 18.



resulted in different rates of convergence to the universal nitude of the pressure increase is inversely proportional tothe characteristic pore size. When scaled properly, the dataself-similar solution.

The final value of pressure is different in Fig. 20 than in for pressure as a function of time and average lamella posi-tion as a function of time fall on a universal curve. ThereFig. 18. This is because the initial lamella configuration is

different. The final pressure value depends on the ratio of is one universal curve for each initial lamella configuration.The curvature of the pore walls leads to interesting behav-volume inside the bubbles spanning the pore throat to the

volume in the pore bodies. The higher this ratio, the higher ior caused by the presence of a critical position, where alamella has a zero derivative in its radius of curvature withthe final pressure.

The requirement that the lamella distribution be fixed is, respect to position. This position corresponds to a minimumvalue of the radius of curvature and therefore a maximumof course, hard to meet when trying to simulate a real system

since we never know the actual lamella distribution. In order value of the pressure drop across the lamella. The criticalposition leads to singular behavior of the system. When thisto predict accurately the behavior of real systems, we must

use large networks so that differences in the random genera- occurs, the lamellae must either jump from their currentconfiguration to another solution branch or they oscillate. Attion of lamella positions will be smaller. Then we must

average the results obtained from several different runs, each a singular point, all the lamellae in a loop must move in acooperative way, since the jump happens so quickly thatwith different, random configurations of lamellae.

The network simulation can be solved with a coordination negligible gas diffusion occurs.The techniques presented in this paper can be used tonumber of 4 or 6, representing the number of pores that

connect at a single pore body. Both give the same overall predict how foam bubbles in a porous medium of arbitrarysize change as a result of diffusion. While this diffusiveand local behavior. Furthermore, it is easy to substitute the

equations which define a three-dimensional hourglass in behavior does not have a major effect on the overall lifetimeof the foam, it does result in a rearrangement of foam in aplace of those for a translationally invariant hourglass.

Again, this substitution does not change the results of the porous medium. The texture of foam changes in that thenumber of lamellae remaining trapped in the porous mediumsimulation. Both of these tests make us comfortable with

the use of the two-dimensional model with a coordination is reduced. The equilibrium state is one where all bubblesare at equal pressure. Understanding the diffusion evolutionnumber of 4 to simulate foam in a full porous medium.

Clearly, the two-dimensional case is less computationally is particularly important if one is trying to characterize theregion of trapped foam in a system where foam is flowingintensive, and its use is, therefore, desirable.through a porous medium.

SUMMARY

APPENDIX A: NOMENCLATUREA two-dimensional network simulation model was created

to predict the behavior of foam in a porous medium. The Symbolsmodel generates a nearly random initial configuration oflamellae in the pores, which are defined as water-wet transla- a height of a segment of a circle in Appendix B (m)

A lamellar surface area (m2)tionally invariant hourglass shapes. Four pores are connectedtogether in a cluster at a single pore body. Once the initial B pore characteristic [1 / rt /Rg]

d length of base of triangle used to derive the formulasconfiguration is generated, the simulation tracks the motionof the individual lamellae as they rearrange in the porous in Appendix B (m)

D diffusivity of gas through liquid phase (m2/s)medium. During this process, all the bubble pressures arecalculated and the change in the average system pressure is F Helmholtz free energy (J)

h lamella thickness (m)investigated. The pressure trends predicted can be comparedwith experimental measurements of gas pressures in porous k mass transfer coefficient (mol/N s)

K surface area of a segment of a circle in Appendix Bmedia.The time scale for diffusion is an order of magnitude (m2)

M arc length normalization residual equationgreater for a bubble at atmospheric pressure in a porousmedium than for the same size bubble in bulk. The time to n moles of gas (mol)

N total number of moles of gas in the system (mol)reach equilibrium is a quadratic function of the characteristicpore size and a linear function of pressure. At high pressure, NC number of clusters in the network

P gas pressure (Pa)the foam takes a much longer time to decay. The time scaleis independent of network size. r radius of curvature (m)

rt pore throat radius (m)The lamellae in a porous medium recede toward the porethroats during the diffusion process. The pressures of the Rg radius of cylinder used to generate hourglass [grain

radius] (m)bubbles in the pore bodies increase to equilibrium. The mag-



RV g characteristic length [average grain radius] (m) rt , where rt is the throat radius. The dimensionless equationfor the circle isR gas constant (8.314 N m/mol K)

s arc lengthS solubility of gas in liquid phase (mol/N m) y(x) Å B 0 (1 0 x 2) 1/2 , [B1]t time (s)tc characteristic time (s) where B is the shape parameter defined in Eq. [4] . Thetdif characteristic diffusion time for a bulk bubble (s) expression for radius of curvature of a lamella at position xteq time at which diffusion process is 90% toward equilib- is derived by drawing a right triangle whose base is the

rium (s) center line of the pore and whose hypotenuse is the tangentT temperature (K) line to the pore wall at the point where the lamella intersectsV bubble volume (m3) the pore wall. The third side of the triangle is the line seg-Vt total system volume (m3) ment of length y(x) connecting the point where the lamellax dimensionless distance of a lamella from the pore intersects the pore wall with the center line. The length of

throat [0 £ x £ 1] the hypotenuse of the triangle is the radius of curvature ofy function of x defining the shape of the pore wall in the lamella. The expression for radius of curvature is given

Appendix B in Eq. [3] .The surface area of a lamella per unit width of transla-Greek letters

tionally invariant hourglass, Eq. [6] , is calculated using thea dimensionless pore body volume formula for the length of a segment of a circle,b solubility parameter [h /RTSRV g]g dimensionless volume [V /RU 3

g] A Å 2r arccos(d /r) , [B2]n dimensionless total number of moles of gas

[NRT /sRU 2g] where d is the length of the base of the triangle described

h dimensionless number of moles of gas [nRT /sRU 2g] above.

P dimensionless pressure [PRV g /s] There are two parts that make up the expression for vol-PU c ume per unit length of a bubble in a pore. The first is thedimensionless measure of encapsulation [n

VRT /sRU 2

g]surface area of the pore itself from 0 to x . The second isr dimensionless radius of curvature [r /RV g]the surface area of the circular cap which is defined by thes liquid/gas interfacial tension (N/m)lamella. To get the first part, it is necessary to integrate Eq.t dimensionless time [ tD /RU 2

gb][B1] from 0 to x . The second part involves the mensurationu arc length normalization parameterformula for the surface area of a segment of a circle,c grain size distribution factor—sum of RV g and its stan-

dard deviation over RV g

K Å r 2 arccos[(r 0 a) /r] 0 (r 0 a)(2ra 0 a 2) 1/2 ,Subscripts

[B3]i cluster index { i e(1, NC)}j pore index { j e(1, 4)}

where a is the height of the segment. For our geometry, a0 initial valueis the distance along the center line from the position x to

Vectors and Matrices the lamella. These two parts are summed together to givethe expression for volume in Eq. [5] .G Jacobian matrix—derivative of R with respect to

elements of uACKNOWLEDGMENTSJ augmented Jacobian matrix (arc-length continuation)

Q augmented residual vector (arc-length continuation)This work was partially funded by the Assistant Secretary for Fossil[R , M]

Energy, Office of Oil, Gas, and Shale Technologies of the U.S. DepartmentR residual vector of Energy under Contract DE-AC03-76SF00098 to Lawrence Berkeley Na-u solution vector [xi ,j , hi ,j , h0,i ] tional Laboratory. Additional funding was provided by the U.S. Bureau ofv augmented solution vector [u , t] Mines through a fellowship received by David Cohen from the California

Mining and Mineral Resources Research Institute, Grant G1114106.e coefficient adjustment vectorF diagonal matrix, seed for jumping to new branch

REFERENCESAPPENDIX B: DERIVATION OF THE

MENSURATION FORMULAS 1. Fried, A. N., ‘‘The Foam-Drive Process for Increasing the Recoveryof Oil,’’ U.S. Bureau of Mines, 1961.

We start from the equation for a circle of radius Rg . The 2. Patzek, T. W., SPE Res. Eng. II (1996).3. Persoff, P. J., Radke, C. J., Pruess, K., Benson, S. M., and With-center of the circle is above the origin by a distance Rg /



erspoon, P. A., Paper presented at SPE California Regional Meeting, 12. Bikerman, J. J. (Ed.) , ‘‘Foams.’’ Springer-Verlag, Berlin, 1973.13. Chambers, K. T., and Radke, C. J., in ‘‘Interfacial Phenomena inBakersfield, CA, 1989.

4. Hanssen, J. E., J. Pet. Sci. Eng. 10, 117 (1993). Petroleum Recovery’’ (N. R. Morrow, Ed.) , pp. 191–253. Dekker,New York, 1990.5. Khatib, Z. I., Hirasaki, G. J., and Falls, A. H., SPE Res. Eng. 3, 919

(1988). 14. de Vries, A. J., Recueil 77, 209 (1958).15. Press, W. H., Vetterling, W. T., Teukolsky, S. A., and Flannery,6. Prieditis, J., and Flumerfelt, R. W., in ‘‘Surfactant-Based Mobility

Control’’ (D. H. Smith, Ed.) , Vol. 373, pp. 295–326. Am. Chem. B. P., ‘‘Numerical Recipes in Fortran,’’ 2nd ed. Cambridge Univ.Press, New York, 1992.Soc., Washington, DC, 1988.

7. Gillis, J. V., Ph.D. Thesis, University of California at Berkeley, 1990. 16. Keller, H. B., in ‘‘Applications of Bifurcation Theory’’ (P. H. Rabi-nowitz, Ed.) , pp. 359–384. Academic Press, New York, 1977.8. Friedmann, F., Chen, W. H., and Gauglitz, P. A., SPE Res. Eng. 6,

37 (1991). 17. Nishioka, G., and Ross, S., J. Colloid Interface Sci. 81, 1 (1981).18. Schechter, R. S., and Monsalve, A., J. Colloid Interface Sci. 97, 3279. Nishioka, G., Ross, S., and Whitworth, M., J. Colloid Interface Sci.

95, 435 (1983). (1984).19. Chambers, K. T., M.S. Thesis, University of California, Berkeley,10. Monsalve, A., and Schechter, R., J. Colloid Interface Sci. 97, 327

(1983). 1990.20. Herdtle, T., and Aref, H., J. Fluid Mech. 241, 233 (1992).11. Patzek, T. W., AIChE J. 39, 1697 (1993).


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Two-Dimensional Network Simulation of Diffusion-Driven