Effect of Diffusion on Dispersion

March 2011 SPE Journal 65

Effect of Diffusion on DispersionRaman K. Jha, Steven L. Bryant, and Larry W. Lake, The University of Texas at Austin

Copyright © 2011 Society of Petroleum Engineers

This paper (SPE 115961) was accepted for presentation at the SPE Annual Technical Conference and Exhibition, Denver, 21–24 September 2008, and revised for publication. Original manuscript received for review 7 July 2008. Revised manuscript received for review 11 February 2010. Paper peer approved 3 May 2010.

SummaryIt is known that dispersion in porous media results from an inter-action between convective spreading and diffusion. However, the nature and implications of these interactions are not well under-stood. Dispersion coefficients obtained from averaged cup-mixing concentration histories have contributions of convective spreading and diffusion lumped together. We decouple the contributions of convective spreading and diffusion in core-scale dispersion and systematically investigate interaction between the two in detail. We explain phenomena giving rise to important experimental observa-tions such as Fickian behavior of core-scale dispersion and power-law dependence of dispersion coefficient on Péclet number.

We track movement of a swarm of solute particles through a physically representative network model. A physically representa-tive network model preserves the geometry and topology of the pore space and spatial correlation in flow properties. We developed deterministic rules to trace paths of solute particles through the network. These rules yield flow streamlines through the network comparable to those obtained from a full solution of Stokes’ equa-tion. Paths of all solute particles are deterministically known in the absence of diffusion. Thus, we can explicitly investigate purely convective spreading by tracking the movement of solute particles on these streamlines.

Then, we superimpose diffusion and study dispersion in terms of interaction between convective spreading and diffusion for a wide range of Péclet numbers. This approach invokes no arbitrary parameters, enabling a rigorous validation of the physical origin of core-scale dispersion. In this way, we obtain an unequivocal, quanti-tative assessment of the roles of convective spreading and diffusion in hydrodynamic dispersion in flow through porous media.

Convective spreading has two components: stream splitting and velocity gradient in pore throats in the direction transverse to flow. We show that, if plug flow occurs in the pore throats (accounting only for stream splitting), all solute particles can encounter a wide range of independent velocities because of velocity differences between pore throats and randomness of pore structure. Consequently, plug flow leads to a purely convective spreading that is asymptotically Fickian. Diffusion superimposed on plug flow acts independently of convective spreading (in this case, only stream splitting), and, consequently, dispersion is simply the sum of convective spread-ing and diffusion. In plug flow, hydrodynamic dispersion varies linearly with the pore-scale Péclet number when diffusion is small in magnitude compared to convective spreading.

For a more realistic parabolic velocity profile in pore throats, par-ticles near the solid surface of the medium do not have independent velocities. Now, purely convective spreading (caused by a combina-tion of stream splitting and variation in flow velocity in the transverse direction) is non-Fickian. When diffusion is nonzero, solute particles in the low-velocity region near the solid surface can move into the main flow stream. They subsequently undergo a wide range of inde-pendent velocities because of stream splitting, and, consequently, dispersion becomes asymptotically Fickian. In this case, dispersion is a result of an interaction between convection and diffusion. This interaction results in a weak nonlinear dependence of dispersion on Péclet number. The dispersion coefficients predicted by particle tracking through the network are in excellent agreement with the literature experimental data for a broad range of Péclet numbers.

Thus, the essential phenomena giving rise to hydrodynamic dispersion observed in porous media are (1) stream splitting of the solute front at every pore, causing independence of particle veloci-ties purely by convection; (2) velocity gradient in pore throats in the direction transverse to flow; and (3) diffusion. Taylor’s disper-sion in a capillary tube accounts only for the second and third of these phenomena, yielding a quadratic dependence of dispersion on Péclet number. Plug flow in the bonds of a physically representative network accounts only for the first and third phenomena, resulting in a linear dependence of dispersion on Péclet number. When all the three phenomena are accounted for, we can explain effectively the weak nonlinear dependence of dispersion on Péclet number.

IntroductionTraditionally, dispersion coefficient is quantified from the flow-averaged (cup-mixed) effluent-concentration history obtained for a core sample (Lake 1989). The dispersion coefficient has contribu-tions (in inseparable form) from (1) convective spreading, caused by variations in path lengths and velocities of solute particles traveling along different streamlines, and (2) molecular diffusion.

The nature and consequences of interaction between convective spreading and diffusion are often not well understood. For exam-ple, longitudinal dispersion in porous media is often modeled as

D D vL o L= + � , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

where �L is a fundamental property of the medium called dispersiv-ity. This equation expresses dispersion as a sum of diffusion and convective spreading.

Eq. 1 implies independence of diffusion and convective spread-ing. Under what conditions is this approximation valid? Because convective spreading is orders of magnitude larger than diffusion, dispersion often is modeled as a purely convective process. Is dispersion in porous media predominantly an artifact of convective spreading (Coats et al. 2004), or are there serious implications of its interaction with diffusion?

The objective of this paper is to explain core-scale dispersion from pore-scale physics. We decouple the contributions of convec-tive spreading and diffusion on core-scale dispersion and system-atically investigate the interaction between the two in detail.

We investigate dispersion by tracking motion of a swarm of solute particles through a granular porous medium. We extracted a physically representative network model from a dense random packing of spheres to represent the pore space accurately. We developed deterministic rules to trace a solute particle’s path from the inlet of the medium to its outlet. Because the rules are deter-ministic, the paths of solute particles are completely known in absence of diffusion. It also ensures that the flow paths are revers-ible—that is, upon reversal of the flow direction, each particle will exactly retrace its path back to the inlet, a characteristic of purely convective spreading. Upon flow reversal, convective spreading gets canceled and echo dispersion is zero (Hiby 1962; Taylor 1972; John et al. 2008; Jha et al. 2009). Thus, our algorithm allows us to investigate convective spreading explicitly and rigorously in a realistic pore space. To the best of our knowledge, this has not been reported previously.

Next, diffusion is superimposed and movement of solute par-ticles because of the combined effects of convection and diffusion is monitored. Dispersion is quantified from spatial and temporal statistics of solute particles. With this framework, we can quanti-tatively investigate the influence of increasing diffusion on disper-sion. We explain the origin of core-scale dispersion as a result of interaction between convection and diffusion.

66 March 2011 SPE Journal

Many previous studies [see, for example, Safmann (1959) and Sahimi et al. (1986)] likewise have shown strong arguments for the fundamental importance of this interaction. In that sense, our conclusions are not surprising, but they are reassuring. If one accounts for solute-transport mechanisms rigorously in the pore space of a disordered granular material, the correct dispersion coefficient emerges when the pore-scale results are averaged. Our predictions of dispersion coefficients are consistent with the experimental results reported in the literature for a broad range of Péclet numbers, including near independence of dispersion coef-ficient on Péclet number for a diffusion-dominated regime. Models more sophisticated than ours [e.g., a direct solution of the Stokes (or Navier-Stokes) equation in the pore space of a granular mate-rial coupled with a solution of the convection/diffusion equation] would also predict core-scale dispersion correctly. We propose that ours is the simplest model that captures the essential physics.

Model DevelopmentPhysically Representative Network Model. Pore-network mod-eling is an important tool that provides a link between continuum (core) -scale properties of a porous medium and the pore-scale physics. It has been used widely to estimate dispersion coeffi cient [see, for example, Bruderer and Bernabe (2001), Bijeljic et al. (2004), and Acharya et al. (2007)]. In a network model, the pore space is discretized into a set of pores (nodes) connected by pore throats (bonds). Because it is very diffi cult to capture the details of pore geometry explicitly, most of the network models reported in the literature make some simplifying assumptions. Common assumptions include same length for all the throats and regular network lattice. Throat radii often are picked randomly from an assumed distribution. All of these assumptions are found to be

invalid in realistic pore space, even in relatively simple porous media. Mellor (1989) showed that the topology of the pore-space network in Finney’s dense, random packing of equal spheres is completely disordered. Picking bond radii randomly from a dis-tribution disregards spatial correlation in the bond conductances. Neglecting spatial correlation may fail to account for physically signifi cant features of porous media and affects macroscopic prop-erties of the networks (Bryant et al. 1993).

We adopt the approach of physically representative network models, which replicate the pore space more closely. In this work, we use a computer-generated, dense, random, periodic packing of 10,000 spheres as a model porous medium (Thane 2006). Such packings capture essential geometric and topologic features of sediments. The periodicity eliminates artifacts from boundary walls. The radius and the center coordinates of each sphere are known, which completely determines the microstructure of the medium (Fig. 1). The radius of each sphere is 2.1918×10−4 m. The medium is approximately 34 sphere diameters long in the z direction and 17 sphere diameters long in the x and y directions. The porosity of the medium is 36%.

From these data, we extract a physically representative network model that has pore bodies located at the same spatial positions as the pores in the actual medium. Moreover, the bonds con-necting neighboring pores have the same conductances as in the actual medium. Thus, a physically representative network model preserves the geometry, topology, and spatial correlation in flow properties. It is 3D and unstructured.

Evaluating Bond Conductances and Obtaining Flow Rates. Delaunay tessellation is an unambiguous way of dividing the sphere packing into cells by grouping together sets of four nearest spheres. Thus, a Delaunay cell is a tetrahedron (Fig. 2a) whose vertices lie at the centers of the four spheres forming that cell.

The interior of the cell encloses a region of void space identi-fied as the pore body. The geometric center of the Delaunay cell can be considered as the pore center. Each face of a Delaunay cell is a plane of maximum constriction and represents a narrow entrance to the pore body (Fig. 2b). Because each cell is a tetra-hedron, every pore has four faces or throats connecting it to four neighboring pores.

Two pores are connected by a path of converging/diverging cross section (shown schematically in Fig. 3a). The flow conduc-tance of the path is governed by the narrowest constriction in the path. Fig. 3b shows a cell face shared by two neighboring cells (pores). The cell face is the narrowest constriction in the flow path connecting the two pores. Fig. 3b also shows two circles that can approximate the narrowest constriction. rc is the radius of the larg-est circle that can be fit in the narrowest constriction, and re is the radius of the circle having the same area as the narrowest constric-tion. If the constriction radius is approximated by rc, the path (or bond) conductance is underestimated because some area available for flow in the constriction is not taken into consideration. On the other hand, if the constriction radius is approximated by re, the path (or bond) conductance is overestimated because, for a given

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Fig. 1—A dense, random periodic packing of 10,000 spheres. The radius of each sphere is 2.1918×10−4 m. The coordinates on the x, y, and z axes are in units of 10−4 m.

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Fig. 2—(a) Delaunay tessellation grouping together four nearest neighboring spheres. The vertices of the tetrahedra correspond to the centers of the spheres. (b) Delaunay cell in a pack of equal spheres defines pore bodies (cell interior) and throats (void area in cell faces). S, T, U, and V are the centers of four spheres. The pore body is centered at X, while the throat between grains U, V, and T is centered at W.

March 2011 SPE Journal 67

area, a circle provides the least resistive path for viscous flow. The arithmetic average of the two radii is a good estimate of the effec-tive radius of the bond connecting the two neighboring pores, reff = (re+rc)/2 (Bryant et al. 1993). For the purposes of computing flow, we replace the converging/diverging geometry of each throat with a cylindrical bond of radius reff. This idealization proves to preserve the essential features of the local flow field while enormously simplifying the computation of flow dynamics.

The conductance of the bond connecting the two neighboring pores is given by g r l= � �eff

4 8 , where � is the fluid viscosity and l is the distance between the pore centers.

A conceptual schematic of the network is shown in Fig. 4a as a network of electrical resistances. The location of every pore body (bond junction) and conductance of bonds connecting it to its neighbors have been calculated. We apply a potential gradient across the network. The side boundaries of the network are sealed. Then, we write the mass-balance equation at each pore. Imposing steady state (no mass accumulation at any pore) results in a set of linear equations, and we can solve for potential at each pore (Fig. 4b). After knowing the potentials, the flow rate through any bond can be calculated easily as

q g P= � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

Rules for Particle TrackingParticle Movement in the Network in the Absence of Diffusion. Bonds in a physically representative network model connect centers of pairs of adjacent pores. Because the bonds have nonzero radii,

the four bonds originating at the center of a pore necessarily overlap (Fig. 5). Overlapping bonds may seem unphysical. However, bonds are not to be considered in a completely literal sense. It is a way to model fl ow from one pore to another. We track particle movement through the network of overlapping bonds. We neglect momentum loss and mixing in the overlapping region and discount bond lengths in the overlapping region in calculation of the local Péclet numbers. This simple model yields an accurate a priori prediction of the permeability of the sphere packing (Bryant et al. 1993).

In the absence of diffusion, a solute particle moves in a bond parallel to the bond’s axis. After reaching the outlet face of the bond at a pore center, the particle will enter one of the out-flowing bonds originating at that pore center. One of the most common sim-plifying assumptions made in particle tracking through a network model is the probabilistic choice of an out-flowing bond. A solute particle arriving at a pore body (junction of bonds) is assigned to an out-flowing bond randomly with a flow-rate-weighted probabil-ity. However, probabilistic choice of out-flowing bond introduces irreversibility of dispersion caused by purely convective spreading. That is, solute particles will not return to the starting location if the flow direction is reversed. Thus, probabilistic choice of out-flowing bonds causes mixing, even in the absence of diffusion, that is unphysical (Jha 2008). Moreover, it ignores spatial correlation in bond conductances, which is one of the key features of the physically representative network models (Sahimi et al. 1986; Jha 2008). Therefore, dispersion from convective spreading cannot be modeled correctly unless the choice of out-flowing bond is (1) deterministic and (2) symmetric with respect to flow reversal.

)b( )a(

Pore 1Pore 2

Pore throat

Pore 1Pore 1Pore 2Pore 2

Pore throatPore throat

cr

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Fig. 3—(a) A converging/diverging path connecting two neighboring pores. (b) The faces are areas of narrowest constriction (throats) that connect the cell to neighboring cells. rc is the radius of the largest circle that can fit in the constriction. re is the radius of the circle having the same area as that of the void space. The arithmetic average of rc and re is a good estimate of the equivalent radius of the bond that describes its hydraulic conductivity.

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Fig. 4—(a) A schematic for the physically representative network as a network of resistances. (b) The flow potential at each pore in the network for steady-state, single-phase flow in the z direction, with no-flow boundaries on the side faces. The coordinates on the x, y, and z axes are in units of 10−4 m.

68 March 2011 SPE Journal

In order to model convective spreading realistically, we devel-oped deterministic rules to decide an out-flowing bond and to map the entrance point of the solute particle on this bond. The rules are based on the patterns followed by streamlines and are described briefly in Appendix A. A similar concept has been implemented previously in 2D regular networks (Bruderer and Bernabe 2001; Sahimi et al. 1986). Our algorithm can be seen as a generalization of their approach for 3D, unstructured networks. The principal logical features of the rules are (1) at every pore center, the rules establish a one-to-one mapping from points on the outlet face(s) of in-flowing bond(s) to points on the inlet faces of out-flowing bond(s), and (2) they follow the patterns expected from actual streamlines. Thus, when the direction of flow is reversed, a par-ticle retraces its flow path exactly in the absence of diffusion. A direct calculation of the flow field would eliminate the need for the rules. However, it was not practical to resolve the flow field so accurately for the entire domain, which contains approximately 35,000 pores.

After reaching the outlet face of the bond at a pore body, the particle immediately jumps to the inlet face of the next out-flowing bond (decided from the deterministic rules). This causes a small discontinuity in the path of the particle because of overlapping of bonds in the region near the pore body. Discontinuities are small compared with the path length. Moreover, the effect of discontinui-ties on particle statistics accumulated over several pores tends to get cancelled. Thus, the particle statistics are not affected by these discontinuities (Jha 2008).

A particle continues its motion through successive bonds to the outlet of the network.

Particle Tracking in a Bond With Convection and Diffusion. The motion of a solute particle in a bond in presence of diffusion is divided into small timesteps of equal duration dt. We split each step into a convection-only displacement followed by an instan-taneous diffusive jump, r D dtodiff = 6 . The magnitude of convec-tion-only displacement is equal to the product of fl uid velocity at that location and the duration of the timestep. The direction of the convection-only step is parallel to the bond axis. In the diffusive step, a particle moves in a random direction on the surface of a sphere with radius r (Bruderer and Bernabe 2001), where Do is the diffusion coeffi cient. The particle’s diffusive displacement in a Cartesian coordinate system is given by (Bijeljic et al. 2004): �x = rdiff cos� sin�, �y = rdiff sin� sin�, and �z = rdiff cos�, where � is uniformly distributed between 0 and 2� and cos� is uniformly distributed between −1 and 1. For every timestep, we pick a value for � and �. � is a random number between 0 and 2�. For generat-ing �, we pick a random number between −1 and 1 and take the inverse cosine of that.

If, during any timestep, the particle hits the bond wall, it is reflected back into the bond.

The algorithm for tracking particle movement through a capil-lary tube is validated by comparing simulations in a cylindrical tube with results of Taylor’s theory and experiments (Jha 2008). A close agreement of simulated results with Taylor’s theory veri-fies the algorithm.

Algorithm for Particle Tracking Through the Network

1. Starting positions of a swarm of 15,000 particles at the inlet face of the network are decided in advance. Particles are distributed to the inlet bonds in proportion to their flow rates. In an inlet bond, particles are distributed uniformly over its inlet face.

2. We track one particle at a time. A particle moves with convec-tive and diffusive steps through a bond until it reaches a pore body (bond junction). Its position is mapped to an out-flowing bond on the basis of deterministic rules discussed earlier and described in Appendix A.

We used the same rules for particle tracking for all the Péclet numbers. The position of a particle at the entrance of an outlet bond depends only the flow configuration and relative flow rates at a pore center. This rule is strictly valid when solute transport is dominated by convection. A better rule to decide the out-flowing bond in a diffusion-dominated regime (NPe < 4) would be based on cross-sectional areas of bonds rather than the flow rates. However, the point of the rules is to capture the convective transport from one bond to the next in a deterministic way and thereby account for the contribution of convective spreading. Thus, in this work, we use the same outflow rules for all values of N Pe.

3. The particle continues its movement through successive bonds until it reaches the outlet of the network. Particle positions are scanned at regular time intervals. We also record the residence time of the solute particle. The dispersion coefficient is calculated from both spatial and temporal statistics.

Results We use the model described to focus on the three physical features of pore-level flow and solute transport. The first is stream split-ting of the solute front at every pore. This causes a sequence of independent and random velocities of solute particles in successive bonds. The second is velocity gradient in pore throats in the direc-tion transverse to flow, the consequence of viscous momentum transfer with no-slip boundary conditions. The third is diffusion, which results in the random movement of solute particles between streamlines. We examine these phenomena individually and in combinations. Our deterministic algorithm of particle tracking (which is conceptually equivalent to solving Stoke’s equation for the entire domain) in a realistic porous medium enables us to inves-tigate systematically the effect of coupling of these phenomena and gives an unambiguous explanation of the origin of core-scale dispersion. The combination of the first two gives rise to convec-tive spreading. The combination of the second and the third gives rise to Taylor’s dispersion in a capillary tube. We show that only when all three phenomena are accounted for do we obtain model predictions consistent with the experimental observations.

Convective Spreading as a Diffusive Process. We fi rst apply the algorithm to investigate convective spreading caused purely by stream splitting. We impose a plug-fl ow velocity profi le in each bond (i.e, all particles in a bond move at the same velocity). The network fl ow calculation yields volumetric fl ow rates through each bond. The direction of movement is parallel to the bond axis, and the speed is v q r= / � eff

2 . Diffusion is neglected. Thus, only one of the physical features of interest, stream splitting, affects this set of simulations.

Fig. 6a shows paths of five different pairs of particles. The particles in each pair initially were close to one another. The rules determining the exit bond taken by a particle entering a pore constitute a mapping. The mapping is unique for each pore in the network. At the level of pore throats, the mapping relates geometric

Entrance on out-flowing bond

Exit on in-flowing bond

Fig. 5—Intersecting bonds at a pore body. An exit point on an in-flowing bond is mapped on an entrance point on an out-flow-ing bond on the basis of deterministic rules.

March 2011 SPE Journal 69

regions; well-defined segments of the exit faces of in-flowing bonds are connected to well-defined segments on inlet faces of out-flowing bonds. At the level of particles, the mapping connects a single point within a geometric region on an exit face to a point within a corresponding region on an inlet face. Thus, if a pair of particles reaches an exit face and is in the same geometric region, it will enter the same out-flowing bond (details in Appendix A). As the pair of particles proceeds through the network, eventually it will arrive at an exit face on opposite sides of a line separating two geometric regions on that face.

When the particles fall on opposite sides of the split stream on an exit face, they take different paths and their movements become independent of each other. They are unlikely to come together again, and their positions become uncorrelated. This is true even in the absence of diffusion. It is simply a consequence of the asymmetric splitting and joining of flow paths around ran-domly arranged obstacles—that is, the grains forming the porous medium. This is the nature of convective spreading. It is useful to contrast the situation in ordered arrangements of grains. There, the splitting/joining of the flow stream is symmetric and positions of a pair of particles remain correlated regardless of their initial separation (Jha 2008). If plug flow occurs in bonds and there is no diffusion, then no independence of particle paths occurs by means of convective spreading in an ordered packing.

Fig. 6b shows paths of several solute particles (moving without diffusion with plug flow in bonds), starting at the same inlet pore. A particle’s displacement becomes independent of other particles very quickly because of frequent splitting of flow passages. The randomness of convective spreading in a porous medium is inher-ent in the morphology of the pore space (Sahimi et al. 1986).

In the conventional Fickian representation of dispersion, con-vective spreading is considered to be diffusion-like—that is, a sta-tistically random process. The sufficient conditions for convective spreading because of splitting at pore junctions can be treated like diffusion in a continuum transport equation and can be stated in terms of the central-limit theorem (Chandrashekhar 1943; Sahimi et al. 1986). The central-limit theorem states that the sum of a large number of independent and identically distributed random variables will be distributed approximately normally (Bear 1972). After a particle has traveled a distance greater than the correlation

length in velocity, its total displacement can be considered as the sum of independent and random convective steps. Consequently, after a few steps, the spatial distribution of solute particles’ dis-placements is expected to be normal (Gaussian), as per the central-limit theorem, and convective spreading caused purely by stream splitting can be considered like a diffusive process. Cenedese and Viotti (1996) and Moroni and Cushman (2001) show by 3D par-ticle-tracking velocimetry experiments in beadpacks that velocity components quickly become independent. Correlation lengths are of the same order as the grain dimensions. Longitudinal dispersion coefficient becomes Fickian after the solute front has traversed five to six pore diameters (Manz et al. 1999). Though the experiments are not diffusion-free, the agreement with our simple model is encouraging.

Particle Tracking Without Diffusion. With Plug Flow in Net-work Bonds. Having seen the role of convective spreading caused by stream splitting with small groups of particles, we now track movement of a swarm of 15,000 particles through the same physi-cally representative network model. Again, we impose a plug-fl ow velocity profi le in each bond of the network and neglect diffusion. The particles move from bottom to top of the domain (positive z direction). The average interstitial velocity is 5.12×10−5 m/s, which roughly corresponds to 0.12 grain diameters per second.

The particle positions are scanned at various times. A prob-ability-distribution plot of particles’ spatial positions is shown at several different times in Figs. 7a through 7c. For comparison, a normally distributed probability-distribution plot having the same mean and standard deviation as the particle-position statistics is also plotted in each case.

The dispersion coefficient at each time is calculated from spatial statistics using D tL z= � 2 2/ , where � z

2 is the variance of solute-particle positions in the z (longitudinal) direction at time t. For verification, we also compute the dispersion coefficient using the particles’ residence-time statistics and using a solution to the 1D convection/diffusion equation (Lake 1989). The dispersion coefficient calculated from spatial statistics increases with time (and travel distance) and approaches an asymptotic value (Fig. 7d). The asymptotic dispersion coefficient is very close to that obtained from temporal statistics in all the cases studied. For consistency,

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Fig. 6—(a) Paths of five different pairs of neighboring particles, each beginning in the same pore but not at the same math-ematical point. Flow is from bottom to top, with sealed sides. Bonds have plug-flow velocity profiles (no gradient in velocity in the transverse direction), and diffusion is zero. Particles split their paths after traveling together for some number of pores. (b) Paths of several particles starting at different positions in the same pore. Particles’ paths are independent of each other. If each convective step is independent and has same global statistics, solute-particle displacements will be normally distributed. The coordinates on the x, y, and z axes are in units of 10−4 m.

70 March 2011 SPE Journal

the dispersion coefficient obtained from temporal statistics is taken as the dispersion coefficient.

It can be seen from Figs. 7a through 7c that particles are nor-mally distributed at all the times. The particle velocities become independent quickly because of splitting of flow stream at each pore and randomness of pore structure. Solute particles become normally distributed as expected from the central-limit theorem. The convergence of the dispersion coefficient to an asymptotic value is governed by correlation in the local pore structure. After the particles have traveled far enough, dispersion coefficient reaches an asymptotic value and convective spreading because of stream splitting becomes Fickian. The asymptotic dispersion coefficient for purely mechanical dispersion, which is caused by stream splitting, is evaluated to be 9.25×10−9 m2/s (for our case, with grain radius of 2.1918×10−4 m and average interstitial velocity of 5.12×10−5 m/s).

With Parabolic Velocity Profile in Bonds. Using the same steady-state solution for flow in the network as in the preceding subsection, we impose a parabolic velocity profile in each bond of the network. As in the preceding subsection, we neglect diffu-sion. Thus, two of the physical features of interest that give rise to convective spreading (stream splitting and velocity variation) affect this set of simulations. The velocity profile is taken from the classical Hagen-Poiseuille analysis of flow in a cylindrical bond of radius reff:

v r vr

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where r measures the radial distance of the particle from the bond axis.

Fig. 8 shows the particle-position statistics for a swarm of 15,000 particles moving through the network. At early times, there

are two peaks in the probability-distribution function. The peak at smaller displacement corresponds to solute particles initially located in the slower zone near the wall of the bonds (i.e., at r ≈ reff). At longer distances traveled, a second peak emerges in the distribution. It corresponds to particles initially located in the faster-moving core of a bond (i.e., at 0 < r < areff, where a ≈ 0.9). These particles are free to move at early times and have traveled a sufficiently large distance to experience a wide range of flow velocities (Lebon et al. 1997). At longer times, the second peak is closer to normal distribution. However, the first peak persists at the longest time observed.

It may be argued that the dispersion coefficient is converging to an asymptotic value (Fig. 8d). However, the concentration profiles make it clear that the transport process is not Fickian. This behavior is general. If a stochastic velocity field contains regions of zero velocity, there is no purely hydrodynamic mechanism by which solute particles in these regions can reach the rest of the pore space. Particles in these regions cannot have any random velocity from the velocity distribution. Therefore, the central-limit theorem is not applicable and the dispersion coefficient is not well defined in this case. The effect of diffusion must be considered for the transport process to become Fickian, even in the limit of high Péclet number (Koch and Brady 1985; Duplay and Sen 2004).

Steady-state single-phase flow in the network is a linear pro-cess. If the overall pressure drop across the network is changed, the pressure difference between the extremities of each bond of the network changes in proportion. It follows that the velocities, flow rates, and transit times in each bond also change in proportion. Hence, the ratio between flow rates in different bonds that connect to the same node remain unchanged, and, therefore, the mapping rule at the pore junction remains unchanged if the average veloc-ity changes. Consequently, the dispersion coefficients caused by purely convective spreading are proportional to the mean velocity. Therefore, in absence of diffusion (or with negligible diffusion), dispersion depends linearly on velocity (Sahimi et al. 1986).

0 5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

Distance, 0.1 mm

Pro

babi

lity

Pro

babi

lity

Pro

babi

lity

Concentration Profile at 25 seconds

0 10 20 30 40 50 60 70 800

0.01

0.02

0.03

0.04

Distance, 0.1 mm

Concentration Profile at 100 seconds

0 20 40 60 80 100 1200

0.005

0.01

0.015

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0.025

0.03

0.035

Distance, 0.1 mm

Concentration Profile at 150 seconds

0 50 100 150 200

2

4

6

8

10x 10–9

Dis

pers

ion

Coe

ffici

ent,

m2 /

s

Dispersion Coefficient

From Spatial StatisticsFrom Temporal Statistics

)b()a(

(c) (d)

0 5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

Simulated dataNormal distribution

0 10 20 30 40 50 60 70 800

0.01

0.02

0.03

0.04

0 20 40 60 80 100 1200

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 50 100 150 200

2

4

6

8

10x 10

Time, seconds

From spatial statisticsFrom temporal statistics

Simulated dataNormal distribution

Simulated dataNormal distribution

Fig. 7—Scanned spatial distribution of solute particles traveling through the physically representative network for plug flow in the network bonds and Do = 0 m2/s [(a) through (c)]. Normally distributed curves having the same mean and standard deviation as the actual data are also shown for comparison. (d) Dispersion coefficient as evaluated from spatial statistics (dots) for several times. The dotted line represents the dispersion coefficient obtained from temporal statistics. The asymptotic dispersion coef-ficient obtained from spatial statistics is very close to that obtained from temporal statistics.

March 2011 SPE Journal 71

Particle Tracking With Diffusion. With Plug Flow in Network Bonds. Next, we study the infl uence of diffusion on particle statistics and dispersion coeffi cient. A plug-fl ow velocity profi le is imposed in the bonds of the network. Thus, two of the three physical features of interest (stream splitting and diffusion) affect the solute transport.

Particles move with convective and diffusive steps as described previously. The timestep should be small enough to prevent dif-fusive jumps from being larger than the bond diameter (Bruderer Bernabe 2001). In the simulations reported here, the timestep was taken to be one-tenth of this value (evaluated for the average bond diameter) to reduce numerical error.

Fig. 9 shows the effect of diffusion on solute-particle statistics. It is evident that diffusion has negligible effect on particle distribu-tion for diffusion coefficients ranging from 0 to 10−9 m2/s (for our case, with grain radius of 2.1918×10−4 m and average interstitial velocity of 5.12×10−5 m/s, resulting in a dispersion coefficient of 9.25×10−9 m2/s if stream splitting is accounted for) because diffusion is very small in magnitude compared with convective spreading. The dispersion coefficient remains constant in this range of diffusion coefficients. For higher diffusion coefficients, diffu-sion becomes significant in magnitude compared with convective spreading, and, therefore, the variance of particle displacements increases and this increases the dispersion coefficient.

When diffusion is very small, transport is dominated by con-vection. However, at very low Péclet numbers when diffusion is large, our flow-rate-weighted rule for deciding an out-flowing bond captures only part of the particle dynamics at a pore center. Therefore, in this case, the particle statistics deviate from normal distribution at longer times.

Fig. 10 shows the a priori prediction of dimensionless disper-sion coefficient vs. pore-scale Péclet number for this case. The agreement with the experimental data from the literature is good for small and moderate Péclet number, but it is clear that the scal-ing is incorrect for large Péclet numbers. The experimental data suggest a power-law relationship between dispersion coefficient

and Péclet number with the power-law coefficient in the range of 1.1 to 1.3. The simulations yield = 1.

The cause of the incorrect scaling is the plug-flow velocity profile in the bonds. Under this assumption, there is no stretching of a solute front as it moves along a bond. Therefore, flow velocity within a bond has no interaction with diffusion; a particle making a diffusive jump to another streamline within a bond still travels at the same velocity after the jump. Thus, diffusion acts indepen-dently of convective spreading, and the dispersion coefficient is just the sum of the two. Consequently, for small diffusion (large Péclet number), we find a linear dependence of dimensionless dis-persion coefficient on pore-scale Péclet number. The longitudinal dispersion coefficient for the full range of Péclet numbers can be expressed as

D

D F

vD

DL

o

p

o

= +10 877

�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

The first term in Eq. 4 represents the contribution of diffusion, and the second term is the result of convective spreading (stream splitting).

Thus, Eq. 1, commonly used to describe dispersion in porous media, considers mechanical dispersion caused by stream split-ting only. It neglects the effect of variations in fluid velocity in the transverse direction and, thus, interaction between convective spreading and diffusion.

With Parabolic Velocity Profile in Bonds. Finally, we examine the influence on dispersion of all three physical features (stream splitting, velocity variation, and diffusion) simultaneously. We repeated the simulations of the preceding subsection but with a parabolic velocity profile in each network bond, as described.

If a bond has a velocity gradient within it, then the interaction between convection and diffusion is no longer trivial. Now, a solute front stretches as it travels in a bond, and diffusion becomes a much more effective mixing mechanism. (This interaction is the basis of

0 5 10 15 20 25 30 350

0.01

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0.005

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0.03

0 20 40 60 80 100 1200

0.005

0.01

0.015

0.02

0 50 100 150 2000.5

1

1.5

2

2.5x 10–8

From spatial statistics

Particles in low velocity layer

Particles in the main stream

Non-Fickian behavior

Distance, 0.1 mm

Pro

babi

lity

Pro

babi

lity

Pro

babi

lity

Concentration Profile at 25 seconds

Distance, 0.1 mm

Concentration Profile at 100 seconds

Distance, 0.1 mm

Dis

pers

ion

Coe

ffici

ent,

m2 /

s

Time, seconds

Simulated dataNormal distribution

Simulated dataNormal distribution

Simulated dataNormal distribution

)b()a(

(c) (d)

Concentration Profile at 150 seconds Dispersion Coefficient

Fig. 8—Scanned spatial distribution of solute particles for parabolic flow in network bonds and Do = 0 m2/s [(a) through (c)]. The first peak in the distribution at small distance corresponds to particles in the slow-velocity regions near the pore walls. Other particles that are free to move form a second peak. Dispersion is not Fickian in this case. (d) Dispersion coefficient as evaluated from spatial statistics.

72 March 2011 SPE Journal

the familiar Taylor’s dispersion.) Because velocity now influences dispersion within a bond, the dispersion coefficient will not be simply the summation of convective spreading and diffusion.

Fig. 11 shows a comparison of the dimensionless dispersion coefficient obtained from simulations with the experimental data. The match is excellent for the whole range of Péclet numbers. From

curve fitting, we obtain a power-law coefficient of 1.23. This a priori quantitative prediction of dispersion coefficient and its scaling behavior indicates that the essential physics of dispersion through the pore space of a disordered granular material has been captured.

We investigate the interaction of convection and dispersion more closely. Fig. 12 shows the effect of diffusion on particle

0 10 20 30 40 50 600

0.02

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0.1

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0.005

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0.035

01×10–13

1×10–11

1×10–10

1×10–9

1×10–8

1×10–7

Distance, 0.1 mm

Pro

babi

lity

Pro

babi

lity

Concentration Profile at 25 seconds

Distance, 0.1 mm

Concentration Profile at 50 seconds

Distance, 0.1 mm

Pro

babi

lity

Pro

babi

lity

Concentration Profile at 100 seconds

Distance, 0.1 mm

Concentration Profile at 150 seconds

)b()a(

(c) (d)

Fig. 9—Effect of diffusion on spatial statistics of solute particles for plug-flow profile in bonds. Diffusion has negligible effect unless its magnitude becomes significant compared with that of mechanical dispersion. All diffusion coefficients shown in the legend are in m2/s.

0.1

1

10

100

1,000

10,000

100,000

1,000,000

0.001 0.01 0.1 1 10 100 1,000 10,000 100,000 1,000,000

Pfannkuch (1963)

Seymour and Callaghan (1997)

Kandhai et al. (2002)

Khrapitchev and Callaghan (2003)

Stohr (2003)

Perkins and Johnston (1963)

Jha (2005)

Simulation, Plug Flow

ν Dp /Do

DL /D

o

Fig. 10—Comparison of dimensionless dispersion coefficients simulated with plug flow in bonds in a physically representative network (beadpack with a bead radius of 2.1918×10−4 m) and experimental data in the literature.

March 2011 SPE Journal 73

statistics for a physically representative network with a parabolic velocity profile in the bonds. Even a vanishingly small amount of diffusion of 10−13 m2/s (for our case, with grain radius of 2.1918×10−4 m and average interstitial velocity of 5.12×10−5 m/s) starts moving solute particles from low-velocity regions near the wall to high-velocity regions. After moving out of the low-veloc-ity zone, solute particles can sample all the regions of pore space because of stream splitting. The first peak at small displacement

corresponding to particles in the slow-moving region near the walls starts decreasing and almost disappears by 150 seconds. We obtain a normal distribution of solute-particle positions.

This is consistent with experimental observations by Lebon et al. (1997). They studied dispersion at short times using a pulsed-field-gradient/nuclear-magnetic-resonance technique. At short times, the displacement of the molecules is small enough that the local displacement is proportional to the local velocity component along

ν Dp /Do

DL /D

o

0.1

1

10

100

1,000

10,000

100,000

1,000,000

0.001 0.01 0.1 1 10 100 1,000 10,000 100,000 1,000,000

Pfannkuch (1963)

Seymour and Callaghan (1997)

Kandhai et al. (2002)

Khrapitchev and Callaghan (2003)

Stohr (2003)

Perkins and Johnston (1963)

Jha (2005)

Simulation, Parabolic Flow

Fig. 11—Comparison of dimensionless dispersion coefficient simulated with parabolic flow in bonds and experimental data in the literature. The simulated data quantitatively match with the experimental data for the whole range of Péclet numbers.

From Spatial StatisticsFrom Temporal Statistics

Falling due to

diffusion

From Spatial StatisticsFrom Temporal Statistics

Falling due to

diffusion

0 5 10 15 20 25 30 350

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1.2

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2.2x 10 -8

From spatial statisticsFrom temporal statistics

Falling due to diffusion

Distance, 0.1 mm

Pro

babi

lity

Pro

babi

lity

Pro

babi

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Concentration Profile at 25 seconds

Distance, 0.1 mm

Concentration Profile at 100 seconds

Distance, 0.1 mm

Dis

pers

ion

Coe

ffici

ent,

m2 /

s

Time, seconds

Simulated dataNormal distribution

)b()a(

(c) (d)

Concentration Profile at 150 seconds Dispersion Coefficient

Simulated dataNormal distribution

Simulated dataNormal distribution

Fig. 12—Scanned spatial distribution of solute particles for parabolic flow in the network bonds and Do = 10−13 m2/s [(a) through (c)]. The first peak corresponds to particles in the slow-velocity regions near the pore walls and starts falling because of diffu-sion. Dispersion is asymptotically Fickian at large times. (d) Dispersion coefficient as evaluated from spatial statistics.

74 March 2011 SPE Journal

the magnetic-field gradient. At mean displacements larger than five bead diameters, the displacement distribution was found to be Gaussian. At intermediate displacements, the measured distribu-tion displayed two peaks. With increasing diffusion coefficient, the peak at small displacements disappears quickly (Jha 2008).

Fig. 13 shows the effect of increasing diffusion on dispersion. For zero diffusion, there is a wide range of particle positions because of velocity gradient in the bonds. As diffusion is increased, solute particles move in the transverse direction in the bonds and the effect of velocity gradient in bonds is reduced. This reduces the spread in solute-particle positions, and, therefore, the dispersion coefficient is reduced. This trend continues with increasing diffusion and reverses when the magnitude of the diffusion coefficient becomes large compared with convective spreading. The dispersion-coefficient-vs.-diffusion-coefficient plot goes through a minimum.

Discussion Dispersion in porous media results from an interaction between convective spreading and diffusion. Convective spreading occurs because of variations in path lengths and velocities of solute particles traveling along different streamlines. There are two com-ponents of convective spreading: (1) stream splitting caused by splitting and joining of flow streams at every pore and (2) velocity gradient in pore throats in the direction transverse to flow. In this study, we decouple all three key phenomena giving rise to disper-sion in porous media.

For dispersion to be Fickian, all solute particles must undergo a range of independent velocities. If we consider the effect of stream splitting only (by having plug flow in bonds where there is no variation in particle velocities in the transverse direction), different bonds carry different flow rates and there is no long-range correla-tion of these flow rates. Thus, particle positions eventually become uncorrelated as flow continues, even in the absence of diffusion. Consequently, as per the central-limit theorem, convective spread-ing caused by stream splitting is Fickian and grows like NPe.

However, with a parabolic velocity profile in the bonds, when the effect of variation in velocity in the transverse direction is considered along with stream splitting, solute particles near the pore walls are not free to move. Thus, those particles do not have independent velocities. Consequently, the central-limit theorem is not applicable and disper-sion because of convective spreading becomes non-Fickian.

Dispersion caused by convective spreading is also called mechanical dispersion. Steady-state single-phase flow in the net-work is a linear process. If the overall pressure drop across the network is changed, the flow rate in each bond changes propor-tionately. Consequently, mechanical dispersion is proportional to the mean velocity.

When diffusion is superimposed on convection, solute particles in the low-velocity layer near solid surfaces can move out and enter

the main flow stream. Subsequently, solute particles can encounter a wide range of velocities because of convective spreading, and this gives rise to the Fickian behavior of dispersion (Koch and Brady 1985; Duplay and Sen 2004). Therefore, diffusion, even though small in magnitude, is essential for Fickian behavior of dispersion. Now, dispersion is a result of an interaction between convective spreading and diffusion, and this interaction leads to nonmechani-cal dispersion, as termed by Koch and Brady (1985).

The plug-flow velocity profile does not have regions of zero velocity. Therefore, nonmechanical dispersion is absent there and dispersion scales linearly with NPe. However, in the case of a parabolic velocity profile in the network bonds, there are regions of zero velocity near pore walls. Therefore, we find a nonmechani-cal dispersion that becomes important at high Péclet numbers and results in a mild nonlinear dependence of dispersion coefficient on Péclet number.

Taylor (1953) showed that, in a single capillary tube, the inter-action of diffusion and the parabolic velocity profile yields a much stronger dependence of dispersion coefficient on Péclet number ( = 2). Our model invokes tubes, though of short length, in which a parabolic velocity profile exists. Why then does transport in the physically representative network—and in experiments—exhibit a much smaller value of than Taylor’s analysis? The reason is the asymmetric splitting and joining of flow paths around grains in the porous medium. The splitting/joining process in a disordered medium causes convective spreading to dominate the dispersive behavior at large values of Péclet number.

Interaction of convective spreading and diffusion is also the reason of irreversibility of dispersion. Purely convective spreading is reversible (i.e., all solute particles retrace their path back to the inlet when the flow direction is reversed). Thus, convective spread-ing is canceled and echo dispersion is zero. However, an interaction of convective spreading with diffusion causes the fluids to mix (as opposed to pure spreading). This makes dispersion irreversible and makes echo dispersion as large as forward (or transmission) disper-sion. Mixing and irreversibility of dispersion in terms of interaction between convective spreading and diffusion have been explained in detail by Garmeh et al. (2007) and Jha et al. (2009).

Summary and Conclusions We study single-phase solute transport through porous media and investigate the three key features giving rise to dispersion: (1) splitting and joining of flow streams at every pore, (2) velocity gradient in pore throats in the direction transverse to flow, and (3) diffusion. The first two of these features collectively give rise to convective spreading.

We track movement of solute particles through a physically representative network model. We infer dispersion from particle-displacement statistics. We introduce deterministic (nonstochastic)

11×10–9

11×10–8

11×10–7

1×10–13 1×10–12 1×10–11 1×10–10 1×10–9 1×10–8 1×10–7 1×10–6

DL ,

m2 /

s

DO, m2/s

Fig. 13—Change in dispersion coefficient with diffusion coefficient for parabolic velocity profile in bonds. Dispersion coefficient decreases with increased diffusion unless diffusion becomes significant in magnitude compared with mechanical dispersion.

March 2011 SPE Journal 75

rules to map particle position from an in-flowing bond to a point on an out-flowing bond. The rules are essentially geometric and depend on the network flow field (rates in individual bonds and the local configuration of inlet and outlet bonds at each pore body). These rules provide a simple way to obtain particle paths through the network equivalent to those obtained from Stoke’s law. Thus, the particle path is completely known in the absence of diffu-sion. These rules also yield the exact cancellation of convective spreading if the flow direction is reversed. Thus, they enable us to attribute correctly the contribution of convective spreading to core-scale dispersion without requiring a detailed (subpore) solu-tion of the flow field through the porous medium. Subsequently, we superimpose diffusion and study the effect of the interaction between diffusion and convective spreading.

Because our methodology invokes no arbitrary assumptions either about the geometry of the porous medium or about the solute-trans-port mechanism, the simulations provide a priori (no adjustable parameters) predictions of dispersion coefficient as a function of pore-scale Péclet number. The predicted trends quantitatively match the experimental data found in the literature for a wide range of Péclet numbers, including the well-known empirical observation that the scaling exponent has a value of approximately 1.2. We also obtain near independence of dispersion coefficient for small Péclet numbers. The agreement indicates that the key features of the model correspond to the key physical phenomena causing dispersion in porous media.

The model permits rigorous attribution of the contribution of the phenomena individually and of the interaction between combinations of phenomena. Fickian behavior of solute transport is asymptotically observed when solute particles’ displacements are independent, identically distributed, and random. If we ignore variation in fluid velocity in pore throats in the transverse direction (i.e., plug flow occurs in network bonds), Fickian behavior can occur without diffusion. Convective spreading and diffusion act independently of each other, and dispersion coefficient is the sum of the two. In the more-realistic case of parabolic velocity profile in the bonds, purely convective (i.e., no diffusion) spreading is not asymptotically Fickian. Diffusion is required to move solute particles from low-velocity regions near pore walls. Subsequently, stream splitting causes independent, random movement of solute particles and gives rise to Fickian behavior of dispersion.

In the absence of diffusion, convective spreading in porous media results in a linear dependence of DL on NPe. Interaction between convective spreading and diffusion results in a weak nonlinear dependence of DL on NPe, in agreement with the experi-mental observations.

Nomenclature DL = longitudinal-dispersion coeffi cient, m2/s Do = molecular-diffusion coeffi cient, m2/s Dp = particle diameter, m F = formation resistivity factor g = hydraulic conductivity of a bond, m3/s/Pa l = length of a bond connecting two neighbors, m NPe = pore scale Péclet number, NPe = vDp /Do

q = fl ow rate through a bond, m3/s rc = radius of the largest circle to fi t in a pore throat, m rdiff = magnitude of diffusive jump, m re = radius of a circle having same area as a pore throat, m reff = effective radius of a bond, m = power-law coeffi cient characterizing dependence of dis-

persion coeffi cient on Péclet number � z

2 = variance of solute particles in z direction, m2

� = fl uid viscosity, Pa·s v = interstitial fl uid velocity, m/s = average fl uid velocity through a bond, m/s � = porosity

AcknowledgmentsWe would like to thank Cynthia Thane for providing coordinates of random packing of spheres. We would also like to thank Branco

Bijeljic for helping us with a particle-tracking algorithm in a tube. This work was conducted with the support of the National Petroleum Technology Office of the US Department of Energy through contract DE-PS26-04NT15450-3F. Larry W. Lake holds the W.A. (Monty) Moncrief Centennial Chair at The University of Texas at Austin.

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Appendix A—Deterministic Rules for Mapping an Entrance Point of a Particle on an Out-Flowing BondA particle enters the pore through an in-flowing bond, and it leaves through an out-flowing bond. The task is to develop rules for calculat-ing which out-flowing bond will be the exit and to map the entrance point of the solute particle on the out-flowing bond. Here, we describe a simple and computationally tractable approach for this purpose.

We identify six flow configurations feasible at a pore body. In each case, we determine splitting of streams on the basis of the flow configuration and flow rates. This tells us which segment of which out-flowing bond a solute particle exiting an in-flowing bond at a particular point will enter. In each case, we calculate some reference points that serve as guiding points to map the exact entrance position of solute particle on the out-flowing bond.

For ease of illustration and for calculating reference points, we displace all the bonds along their axes by equal distance. The dis-placement should be enough to remove overlap between the bonds. After identifying flow configuration and calculating reference points, all the bonds are moved back to their original location.

The bonds are numbered according to their flow rates. Inflow is assigned positive sign and outflow negative. Then, flow rates are sorted in descending order along with their sign. Thus, the first bond is the one carrying maximum inflow and the fourth one is carrying maximum outflow. The first bond is taken as the reference bond. The distances of the face centers of all the bonds from the face center of the first bond are calculated. The closest bond will have the biggest influence on flow.

Confi guration 1: One In-Flowing Bond and Three Out-Flowing Bonds. In this case, one in-fl owing bond feeds all the out-fl owing bonds. Therefore, the incoming stream splits into three segments on the basis of the fl ow rates of the three out-fl owing bonds (Fig. A-1a).

For example, in Fig. A-1a, the fourth bond is closest to the in-flowing bond and the third bond is the farthest. First, we find (numerically) the pair of points on the first and fourth bond faces that are closest to each other. These are reference points (Reference Point 1 on Bond 1 and Reference Point 6 on Bond 4). A solute particle exiting Bond 1 at Reference Point 1 on Bond 1 will enter Bond 4 at Reference Point 6. These reference points also guide mapping of other points.

Starting with Reference Point 1, we mark a segment of in-flowing bond that carries the same flow rate as the out-flowing Bond 4. A solute particle exiting Bond 1 from this segment will enter Bond 4.

Similarly, we mark a middle segment of the in-flowing bond that carries the same flow rate as the farthest out-flowing bond. A solute particle exiting the in-flowing bond on the middle segment will enter the farthest bond. Solute particles exiting Bond 1 on the remaining third segment of the in-flowing bond will enter the second bond.

Reference Point 2 on the in-flowing face is taken as the point where the line joining Reference Point 1 to the center of the face intersects the boundary of the first segment. Reference Point 3 is diametrically opposite of Reference Point 1. Reference Point 4 is the point on Tube 2 that is closest to Reference Point 3. Similarly, Reference Point 5 is the point on Tube 3 that is closest to Refer-ence Point 2.

Confi guration 2: Three In-Flowing Bonds and One Out-Flow-ing Bond. This confi guration is exactly the opposite of Confi gura-tion 1 (Fig. A-1b). Three in-fl owing bonds feed to one out-fl owing bond. Therefore, the out-fl owing stream consists of three segments, each segment receiving fl ow from one of the in-fl owing bonds. The procedure described for connecting segments and calculating refer-ence points in Confi guration 1 is applicable in this case, also.

There is more than one possible flow configuration when we have two in-flowing and two out-flowing bonds at a pore junction.

Confi guration 3: Third Bond Is the Farthest, and Flow in First Bond Is Smaller Than That in the Fourth Bond. Because fl ow rate in the fi rst bond is smaller than that in the fourth bond, the fourth bond must receive fl ow from both the in-fl owing bonds. Because some streamlines move from Bond 2 to Bond 4 and streamlines cannot cross, no streamline can move from Bond 1 to Bond 3. Therefore, Bond 3 will get all its fl ow from Bond 2 and all fl ow from Bond 1 will go to Bond 4.

Marking of segments and calculation of reference points are similar to that described for Configuration 1.

Confi guration 4: Third Bond Is the Farthest, and Flow in First Bond is Greater Than That in the Fourth Bond. This confi gu-ration is similar to Confi guration 3, except that, in this case, the fl ow in the fi rst bond is larger than that in the fourth bond. In this case, the fourth bond receives all its fl ow from Bond 1 and all of the second bond’s fl ow enters the third bond. The fi rst bond feeds to both the out-fl owing bonds.

Confi guration 5: Fourth Bond is the Farthest. Because fl ow rate in the fourth bond is greater than that in the second bond, the fourth bond receives fl ow from both the infl owing bonds. Because stream-lines cannot cross, fl ow from the second bond cannot enter the third bond. Hence, all the fl ow from the second bond enters the fourth bond. Also, the third bond receives all its fl ow from the fi rst bond. As evident from Fig. A-1e, Bond 1 feeds to both the out-fl owing bonds and Bond 4 receives fl ow from both the in-fl owing bonds.

Confi guration 6: The Second Bond is the Farthest. In this confi gu-ration, the farthest bond is in-fl owing. Or, in other words, the second bond is the farthest. In this case, both the infl owing bonds will feed to both the out-fl owing ones, as shown in Fig. A-1f. Flow segments and reference points are calculated as previously described.

The reader is referred to Jha (2008) for calculation details.

Mapping an Incoming Point to an Out-Flowing Bond. After decid-ing the incoming and outgoing segments of the bonds, we must map

March 2011 SPE Journal 77

the incoming point to a corresponding point on the outgoing section. Incoming and outgoing sections are arbitrary sections of a circle. We impose geometric rules that are physically reasonable: (1) Reference point on the in-fl owing segment will connect to the corresponding reference point on the out-fl owing segment, and (2) the center of the incoming segment connects to the center of the outgoing segment. We take the center of the circular segment as its geometric center (or center of gravity). We call the vector joining the reference point to the center of the segment the reference vector. We mark reference vectors on incoming as well as outgoing segments (Fig. A-2).

In a polar coordinate system, two parameters are sufficient to describe the outgoing point: (1) the angle the position vector of the outgoing point (here, defined with reference to the center of the segment) makes with the reference vector and (2) its relative radial distance from the center (defined as distance of the point from the center normalized by distance to boundary in that direc-tion). We impose another geometric rule: These parameters for the outgoing point have the same value as those for the incoming

point. Therefore, we evaluate the angle �, the angle that the posi-tion vector of the incoming point makes with the reference vector on the incoming segment; and relative radial distance of the point from the boundary, r/R. Here, r is the distance of incoming point from the center and R is distance from the center to the boundary of the segment. We place the outgoing point at the same � and r/R on the outgoing section (Fig. A-2). This rule provides a deterministic, one-to-one mapping between exit and entrance points.

Raman K. Jha is a research engineer with Chevron Corporation in Houston. He holds MS and PhD degrees from The University of Texas at Austin and a BTech degree from the Indian School of Mines, Dhanbad, India, all in petroleum engineering. His research interests include heavy oil, enhanced oil recovery, reservoir simula-tion, and hydrocarbon-phase behavior. Steven L. Bryant is an asso-ciate professor in the Department of Petroleum and Geosystems Engineering at The University of Texas at Austin, where he holds the J.H. Herring Professorship and the George H. Fancher Centennial Teaching Fellowship in petroleum engineering. He worked in research laboratories of BP and Eni before joining The University of Texas faculty. Bryant holds a BE degree from Vanderbilt University and a PhD degree from The University of Texas at Austin, both in chemical engineering. He is the director for the Geological CO2 Storage Joint Industry Project. His current research efforts include grain-scale models for unconventional gas reservoirs and the role of methane hydrates in the Earth’s carbon cycle. Bryant has pub-lished more than 70 papers and one textbook with applications in production engineering, reservoir engineering, and formation evaluation. He served as SPE Distinguished Lecturer in 2001–2002. Larry W. Lake holds the W.A. (Monty) Moncrief Centennial Chair in the Department of Petroleum and Geosystems Engineering at The University of Texas at Austin. Lake has published more than 100 peer-reviewed journal articles and has taught industrial and professional short courses in enhanced oil recovery and reservoir characterization around the world. He is the author or coauthor of four textbooks and the editor of three bound volumes. Lake has been teaching at The University of Texas for 30 years, before which he worked for Shell Development Company in Houston. He holds BS and PhD degrees in chemical engineering from Arizona State University and Rice University, respectively. Lake is a member of the US National Academy of Engineering.

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Fig. A-2—Marking the outgoing point on the basis of reference angle and relative radial distance of the incoming point.

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