Investigation of Mixing in Field-Scale Miscible Displacements Using Particle-Tracking Simulations of Tracer Floods With Flow Reversal

Transcription

1 Investigation of Miing in Field-Scale Miscible isplacements Using Particle-Tracking Simulations of Tracer Floods With Flow Reversal Abraham K. John,* SPE, Larry W. Lake, SPE, Steven L. Bryant, SPE, and James W. Jennings,* SPE, The University of Teas at Austin Summary ispersivity data compiled over many lengths show that values at typical interwell distances are approimately two to four factors of larger than those measured on cores. Such large dispersivities may represent significant miing in the reservoir, or they may be a result of convective spreading driven by permeability heterogeneity. The work in this paper uses the idea of flow reversal to resolve the ambiguity between convective spreading and miing. We simulate flow-reversal tests for tracer transport in several permeability realizations using particle-tracking simulations (free from numerical dispersion) on 3, high-resolution models at the field scale. We show that convective spreading, even without local miing, can result in dispersion-like miing-zone growth with large dispersivities because of permeability heterogeneity. But, for such cases, the dispersivity estimated on flow reversal is zero. With local miing (diffusion or core-scale dispersion), the dispersivity value on flow reversal is nonzero and much larger than typical core values. Layering in permeability, while increasing the convective contribution to transport, also enhances miing by providing larger area in the transverse direction for diffusion to act. This suggests that in-situ miing is an important phenomenon affecting the transport of solutes in permeable media even at large scales. ispersivity values increase with scale mainly because of the increase in the correlation in the permeability field, but they could also apparently appear to do so because the Fickian model fails to capture the miing-zone growth correctly at early times. The results and approach shown here could be used to differentiate between displacement and sweep efficiency in field-scale displacements, to ensure accurate representation of dispersive miing in reservoir simulation, and to guide upscaling workflows. The flow-reversal concept motivates a new line of inquiry for laboratory- and field-scale eperiments. Introduction ispersion is the in-situ miing of chemical components as they are transported through porous media. It results from the combined effects of molecular-diffusion and fluid-velocity gradients (Taylor 953). The recovery efficiency of processes such as miscible-gas or chemical flooding depends partly on the miing that an injected slug undergoes. For eample, Solano et al. (2), performing a range of and 2 simulations of enriched gasfloods, show that the recovery difference between the cases studied could be up to 8% of the original oil in place, depending on the degree of dispersion. Similar observations have been made by others (Stalkup 99; Haajizadeh et al. 999; Jessen et al. 24; Moulds et al. 25). ispersion is also an important effect in water injection * Now with Shell International Eploration and Production. Copyright 2 Society of Petroleum Engineers This paper (SPE 3429) was accepted for presentation at the SPE/OE Symposium on Improved Oil Recovery, Tulsa, 2 23 April 28, and revised for publication. Original manuscript received for review 2 February 28. Revised manuscript received for review 25 September 29. Paper peer approved 8 October 29. where mineral scales are formed by the miing of injected and reservoir brines (Sorbie and MacKay 2; elshad and Pope 23), in the underground storage of gases where miing of the injected and in-situ gas changes the quality of the stored gas (Verlaan et al. 998), and in proposed methods of enhanced natural-gas recovery by injecting anthropogenic CO 2 (Oldenburg et al. 2). Modeling any of these processes requires an accurate estimation of the degree of miing and its effect, relative to other transport mechanisms, at the length scale under consideration (core, gridblock size, or interwell distance). This is straightforward for a homogeneous system (e.g., a single bounded flow unit), where the results from the core scale are easily translated to the field scale. In heterogeneous media, despite considerable work performed over the years, significant ambiguity eists in addressing this problem. ispersivities measured using interwell tests show values at the field scale that are two to four factors of larger than those at the core scale. But these values were obtained by treating the flow unit as a homogeneous medium. analytical models were used to match the effluent concentrations because the permeability description was unknown or because computational considerations necessitated an upscaled description. So, such large dispersivity values could be a result of convective spreading because of permeability heterogeneity (which causes widely varying arrival times of injected chemical at the well, followed by miing within the well). Alternatively, they could be the consequence of significant miing or dilution in the reservoir (driven by diffusion). In this paper, we present concepts and results that address fundamental aspects of this issue. Using high-resolution simulations of tracer transport in 3 heterogeneous permeable media, we emonstrate the application of flow reversal to distinguish between spreading and miing. The concept is described in the Convection, ispersion, and Flow Reversal subsection. In the Miing and Spreading subsection, we show that, in certain cases, permeability variations alone can cause dispersion-like behavior (even in the absence of diffusion). But, with a flow-reversal test, the dispersivities estimated in such cases are zero, indicating no miing. Show the influence of permeability heterogeneity (variance and autocorrelation lengths), permeability anisotropy, core-scale dispersivity, and residence time on in-situ miing (Effect of Correlation Structure, Effect of Local Miing, Effect of Permeability Variance, Effect of Permeability Anisotropy, and Effect of Travel Time subsections). We compare the dispersivities estimated from our simulations to those measured from field tests (Comparison With Measured ispersivities subsection). We argue that a large value of dispersivity is not always the result of permeability heterogeneity or conformance effects [e.g., Coats et al. (29)]. A significant amount of miing could occur in the reservoir. Show that the dispersivity estimated using the traditional (forward-flow) approach is not much affected by changes in degree of diffusion or core-scale dispersion. But this does not imply that the effects of diffusion are negligible. In such cases, the dispersivity values obtained on flow reversal are always nonzero and also much larger than typical core-scale values (Effect of Local Miing subsection). Show that dispersivity values increase with an increase in the correlation lengths of the permeability field (Effect of Travel Time 598 September 2 SPE Journal

2 and Comparison with Measured ispersivities subsections). This also increases the amount of miing as indicated by the dispersivities estimated from flow-reversal tests. Show that dispersivity values can also appear to increase with scale because of the inadequacy of the Fickian dispersion model in capturing the early-time behavior of the injected solute (Effect of Travel Time and Comparison With Measured ispersivities subsections) We consider tracer transport in a porous medium with uniform porosity and spatially varying permeability under single-phase, incompressible flow at constant density. The tracer is assumed to be volumeless and without adsorption. The transport calculations are free from numerical dispersion. The Concepts and efinitions section epands on the problem description and provides the conceptual basis of our approach. The computational scheme used is described in the Computational Approach section. The Results section shows the evolution of longitudinal dispersivities for various cases. In the iscussion section, the implications of our results are discussed and compared with results in the literature. Concepts and efinitions iffusion and ispersion. iffusion is the spontaneous net movement of particles driven by a concentration gradient. The mechanism of diffusion is Brownian motion and is described by Fick s law: c j= mol () Here, j is the diffusive flu, c is the local (point-scale) concentration of the injected chemical, and mol is its diffusion coefficient. iffusion increases entropy, decreases Gibbs free energy, and is the fundamental mechanism responsible for miing. By miing, we mean dilution or reduction in local concentrations. Convection (or spreading) is the movement of particles by the motion of the carrier fluid. This is sometimes called advection. Spreading implies redistribution of local concentrations. Both miing and spreading involve the distribution of solute in a solvent, but, in miing, the volume occupied by the solute increases, whereas, in spreading, the volume occupied by the solute remains the same. Taylor (953) analyzed the miing of a solute introduced into a solvent slowly flowing in a circular capillary tube (Fig. ). He showed that, when certain conditions are met, a quasiequilibrium is established where convection and diffusion interact in a manner with the net effect appearing as if the solute were in plug flow despite the radial variations in velocity. The radial concentration variations are almost zero, making the local and averaged concentrations the same. This phenomenon could be characterized as an unsteady diffusion process governed by the same equation (Eq. ) but with an effective diffusion coefficient whose value depends on the flow profile and is much larger than mol. This phenomenon is called dispersion. It must be noted that, in the absence of diffusion, such a condition would never be reached. In this paper, we use the term dispersion to imply enhanced diffusion or miing. The same idea applies to transport in porous media (Aronofsky and Heller 957), with the dispersion coefficient for an isotropic medium now being a tensor: where is the dispersion coefficient of the porous medium, I is an identity matri, and V is the average-fluid-velocity vector. The value of mol is adjusted for the porous medium. l and t are the longitudinal and transverse dispersivities, respectively, that are assumed to be fundamental constants for the medium (Bear 96). In eperiments performed on homogeneous sand columns, it was found that l is of the order of magnitude of the average grain size and t is approimately to 3 times smaller than l (Perkins and Johnston 963). The longitudinal dispersivities of consolidated media are approimately a centimeter (Arya et al. 988). ispersion entails averaging. In using Eq. for dispersion, we have tacitly invoked the continuum hypothesis. Local scale now implies the scale of the representative elementary volume (Bear and Verruijt 987). At the continuum scale, the dispersive flu captures the combined effect of diffusion and velocity variations. The dispersion coefficient applies to averaged concentrations. Its magnitude is governed largely by variances in the velocity field. Without them, the coefficient of dispersion would be the same as that of molecular diffusion (adjusted for the geometry of the porous medium). But, without diffusion, there would be no dissipation occurring at the local scale and the dispersive flu would simply be representing the convective spreading caused by local velocity variations. So, following Lake (989), the transport equation for a chemical component in a homogeneous porous medium, epressed in dimensionless terms, is C t = N Pe 2 C 2 C, (3) where C is the concentration (limited between and ) at any point normalized to the injection concentration. The chemical is injected uniformly over the entire inlet face. The dimensionless time (pore volumes injected) is qt t = Vt AL = (4) L imensionless distance (limited between and ) is = L (5) The dimensionless measure of the degree of convective to dispersive transport is the Péclet number: VL N = Pe, (6) l where l is the longitudinal dispersion coefficient. efining the dimensionless miing-zone length as ( ) = C = 9. C =. / L, (7) the analytical solution to Eq. 3 (to its first-order approimation) can be manipulated to give (Lake 989) l t = ( t V + mol ) I + VV, (2) V = t N Pe, (8) V Convection/iffusion Convection/ispersion Fig. Sketch showing Taylor s (953) analysis of convection/diffusion in a capillary tube. At early times, the solute transport is dominated by convection and it takes up the parabolic velocity profile of the fluid. At late times, a quasiequilibrium is established where convection and diffusion interact such that the net effect appears as if the solute were in plug flow despite the radial variations in velocity. September 2 SPE Journal 599

3 which, using Eqs. 6 and 2, can also be epressed as = l L t (9) A high Péclet number (>5) yields a sharp displacement front (smaller miing zone), and a low number (<) yields a more-spread-out one (larger miing zone). Hence, an eperiment where the miing zone grows with the square root of time can be matched using the Fickian model for dispersion. In a discrete sense, considering the solute to be an ensemble of particles, the miing zone represents the standard deviation of the particle locations. So, another way of stating that is that an eperiment where the particle variance grows linearly with time can be matched with the Fickian model for dispersion. ispersion and Reservoir Heterogeneity. Consider a miscible displacement in a homogeneous-flow unit of -m length. If the dispersivity were, say, cm (core-measured value), the Péclet number would be,. This would result in a dimensionless miing zone of.3625 at pore volume injected (Eq. 8), an insignificant fraction of the length. For a heterogeneous medium of the same length, composed of many such rock types, the key question is what value of dispersivity should be used to estimate the miing-zone size if Eq. 3 were to be used. ispersivities are inferred for such cases in the field using interwell tracer tests [see, for eample, Chrysikopoulos et al. (99)]. ata compiled from results of such tests published over the past few decades (Schulze-Makuch 25) are shown in Fig. 2. This data set represents many formation types and length scales. espite the scatter, dispersivities appear to increase with system length. At m, the range of dispersivities is 5 5 m. This means that the smallest possible dimensionless miing-zone size at pore volume would be.8, which is quite large. But does this mean that there is significant miing taking place? It probably does not, because the displacements during the tests are unlikely to be entirely in the dispersive miing regime. The use of Eq. 3 to interpret them is probably incorrect because this counts the contribution of convective spreading (because of permeability variations) as a dispersive miing effect. But, it must be noted that some of the measurements used point samples from sources injecting at small flow rates (Arya et al. 988). Also, velocities in typical field displacements are approimately ft/ (approimately 3 4 cm/s). At this rate, the effects of molecular diffusion, acting over the long travel times between wells, could compound and lead to significant reduction in local concentrations. Layering increases the convective contribution to transport but, by increasing the area available in the transverse direction, also increases the dilution of the solute (Lake and Hirasaki 98). So, the question of how much miing takes place in a heterogeneous medium at the field scale has no clear answer. If all the heterogeneity could be represented eplicitly and local miing could be modeled only by diffusion or core-scale dispersion, then the question would be irrelevant. But, even if the computational resources were available for such an undertaking, one must consider the cost of collecting and assimilating data at such a fine scale. As long as upscaled descriptions are in use, one must address this issue. Our work is set in this contet. Convection, ispersion, and Flow Reversal. We use the idea of flow reversal to distinguish between miing and spreading. For an incompressible fluid, in the absence of temperature effects (which cause changes in local fluid viscosity) and turbulence effects (which cause temporal variations in the local flow field), the local velocities reverse themselves on reversing the boundary conditions (Flekkøy 997). With no diffusion, the concentration history observed on flow reversal at the inlet would be same as the original input. For this case, particles return back on the same streamline. Convective reversibility has been well illustrated in unmiing demonstrations for circular Couette flow (Heller 96; Taylor 972); for slow, laminar flows between disordered arrays of cylinders (Hiby 962; Flekkøy et al. 996); and for neutrally buoyant particles in dilute suspensions (Co and Mason 97). ispersivity, m Reliability I (High) Reliability II Reliability III (Low) istance, m Fig. 2 ispersivity data compiled from results of interwell tracer tests performed over several formation types and a range of length scales [Schulze-Makuch (25) and references therein]. The reliability classification is from Gelhar et al. (992). ispersivities increase with distance and reach values approimately four factors of larger than laboratory-measured ones. The effect of heterogeneity (layering) in such large values is unknown. The trend at large distances suggests a leveling off. We assume convective reversibility at low-reynolds-number flows in porous media. Numerical simulations have validated this assumption (Jha et al. 29), but it has not been demonstrated eperimentally. If miing occurs, then particles jump across streamlines and the concentration history on flow reversal is dispersed. Any initial solute distribution, once mied, cannot be reconstructed by reversing the direction of the flow. Referring to Fig. 3, the traditional approach of estimating the dispersivity of a medium is using a slug or a step change in concentration (as shown) with a constant flow direction. This is called a transmission eperiment and can be interpreted using Eq. 3. All the values in Fig. 2 are transmission dispersivities. One could perform the same eperiment but reverse the flow direction before the injected solute reached the forward outlet. This is called an echo test. The effluent is recovered at the inlet face. Assuming no eperimental artifacts, if miing were not occurring, then the effluent concentration history would also be a step change (same injection). If miing were occurring, then, on flow reversal, the fluids would continue miing (not unmi) and a variation in concentrations would be seen in the effluent history. In this paper, we use Eq. 3 to estimate echo dispersivities also. A zero echo dispersivity (or dispersion coefficient) would imply no miing and only convection. An echo dispersivity value equal to the transmission value would imply well-mied transport (effective diffusion). Hulin and Plona (989) and Rigord et al. (99) studied the reversibility of tracer dispersion in careful eperiments on bead packs and sandstones. They observe that homogeneous porous materials do not ehibit reversibility: The echo dispersion (after flow reversal) is the same as transmission dispersion over a wide range of Péclet numbers. In heterogeneous materials, they observe echo dispersion to be less than transmission, implying that the equilibrium condition to ensure the use of Eq. 3 was not met. They argue that a small amount of diffusion is sufficient to cause irreversibility. Heller (972) also points out the influence of diffusion on miing and reversibility. Mahadevan et al. (23), estimating echo dispersivities from single-well tracer-test data, observed values in the range of. m at a length scales in the range of 5 m. These are much larger than the range of corescale values (Fig. 4). This suggests that miing could be significant 6 September 2 SPE Journal

4 Injection TRANSMISSION TEST Production Injection REVERSIBLE TRANSPORT Production ECHO TEST : SPREAING Injection Production ECHO TEST : MIXING IRREVERSIBLE TRANSPORT Fig. 3 Sketch showing the use of flow reversal to distinguish between spreading (convective transport) and miing (dispersive transport). If the flow is laminar and slow (low Reynolds number), then ideal reversibility in the flow field can be assumed; in absence of diffusion, the tracer concentration history on flow reversal will be the same as that originally injected. even at the field scale. There is some uncertainty in the use of the tracer data because of fluid drift and transient flow effects during the test. Also, the corresponding transmission values for their cases are not available for comparison. In this work, we use numerical simulations to investigate the significance of dispersive miing at the field scale and test their observations over a range of reservoir descriptions. Computational Approach Overview. We simulate transmission and echo tests for tracer injection in field-scale heterogeneous media. The degree of local miing is varied from zero (purely convective) to core-scale dispersion (well mied). The dispersivities inferred from the simulations ispersivity, m 5 4 Transmission ata Echo ata (Lab) Echo ata (SWTT) istance, m Fig. 4 Comparison of field-measured dispersivities with echo dispersivity data estimated from laboratory tests (Hulin and Plona 989; Rigord et al. 99) and those history matched from single-well tracer tests (Mahadevan et al. 23). Echo dispersivities fall on the overall trend, and the ones at the field scale (> m) are much larger than laboratory-scale values, suggesting the possibility of significant miing in field-scale displacements. are compared with core-scale dispersivities (input values) and the field data (Fig. 2) to make our conclusions. A 3 Cartesian grid is used. The grid spacing is uniform, and grid cells are equidimensional. We controlled the system size and aspect ratio by changing the number of cells in each direction. Primary input variables are the medium dimensions, the permeability field, the flow boundary conditions, and the degree of local (now meaning at the scale of a grid cell) miing (diffusion or core-scale dispersion). The permeability field is characterized by a correlation structure (having a short-range and a long-range component), variance, and anisotropy. We assume that subcell permeability variations are negligible and that local dispersion, when introduced, implies miing. For each case, we generate the permeability field, compute the flow field, and then use the flow velocities in the transport calculations. A Lagrangian particle-tracking scheme is used to model the transport of tracer through the computed flow field. The details of each individual step are described in the following sections and in Appendi A. Permeability-Field Generation. The correlation structure is modeled with a sum of short- and long-range structures, ( )= short ( short )+ long ( long ) h s h, r s h, r, () where is the semivariance of the permeability at distance h (a vector), s is the variance or sill parameter, and r is the range parameter (a vector). Each structure is described by an eponential semivariogram, h ( hr, )= s ep r () We used this model because measurements on outcrops typically show multiple scales of variability (Tidwell and Wilson 2; Willis and White 2; Jennings et al. 2). A stochastic simulation technique based on spectral transforms is used to generate permeability-field realizations. The input correlation structure is converted into a covariance function, which is sampled on the grid used for the flow calculations. The fast-fourier-transform (FFT) algorithm is used to produce the amplitude spectrum of the permeability covariance function. This is multiplied by the FFT of an uncorrelated Gaussian noise and inverted to produce a realization that has a standard normal distribution. These data are transformed to a log-normal distribution of the desired mean and variance. The z-direction permeabilities are multiplied by the desired anisotropy September 2 SPE Journal 6

5 ratio. All the points in the permeability realization thus produced are conditioned globally. The method is fast, even for large number of gridpoints simulated. This approach is detailed in Appendi C of Jennings et al. (2). Flow-Field Generation. The flow equation solved is u =, (2) where the arcy velocity is k u = p (3) Constant pressures are specified on the boundary cell faces in the direction, and the transverse boundaries are closed to flow. Thus, the principal direction of flow is aligned with the ais. The fluid viscosity is cp, and density is g/cm 3 (both are constant). The permeabilities are assigned directly to the cell faces instead of cell centers to avoid a change in variability because of harmonic averaging with neighboring cells (Romeu and Noetinger 995). The overall pressure gradient of psi/m is imposed, and the mean permeability is set at 7 md to produce an average velocity close to m/d for all cases. The code we used to compute the flow field is Parssim (Arbogast 998), developed by the Center for Subsurface Modeling of the Institute for Computational Engineering and Sciences at The University of Teas at Austin. It is designed to simulate incompressible, single-phase flow and reactive transport of chemical components through porous media accurately (Arbogast et al. 996). Parssim uses a logically rectangular cell-centered finite-difference procedure to discretize the flow equation. For large problems, it has the option of using multiple processors to compute the solution on the basis of a domain-decomposition approach. This was the main advantage of using Parssim apart from the fleibility of easily specifying pressure-boundary conditions on entire boundaries. The velocities at each cell face obtained from Parssim are written to separate files, which are used as an input in the transport calculations. At the desired time of flow reversal, the velocity direction is reversed instantaneously. The magnitudes of the velocities are constant throughout the transport simulation. Transport Calculations. We use a random-walk particle-tracking method to simulate tracer transport. This method has been used successfully by the hydrology community to simulate mass transport (Tompson and Gelhar 99). Convection is modeled by integrating particle displacement in the local velocity field over a given timestep. ispersion is modeled by an appropriately scaled particle Brownian motion. Further details of this approach and our implementation are found in Appendi A. The code s solution was in good agreement with known analytical solutions: the convection/ dispersion equation and Taylor s dispersion in a rectangular duct (John 28). Compared to finite-difference and finite-element methods, this method has certain features that made it an appropriate choice for this problem: It is free from numerical dispersion, is stable in problems dominated by advection, and is much more computationally efficient because each particle calculation is performed independently. The mean distance traveled by the particle ensemble in our setup is limited by the arrival of the particles at the outlet boundary. This meant that a flow domain much larger than the desired distance of investigation was required, especially in highly heterogeneous cases. Also, for the flow-reversal cases, it proved convenient to choose the center of the domain as the origin of the particles. This eliminated the possibility of particles eiting through the inlet boundary. Results Overview. We set up several cases by varying the following parameters: permeability variance, correlation lengths, anisotropy, and tracer travel time (mean penetration distance). The base-case dimensions are 248 m with cell size of m. This is populated with an uncorrelated permeability field (range of m in all directions) with variance (natural logarithm of permeability) of. and no anisotropy (k z = k ). The longitudinal dispersivity in each cell is set to be. m (following the trend in Fig. 2), and the transverse dispersivity (in y and z directions) is set to be one-tenth of that (Perkins and Johnston 963). These values are assumed to be constant and the same in all cells, for simplification. The coefficient of molecular diffusion is. 4 m 2 /d (approimately the same as that of brine water at room temperature). Ten thousand particles were used for the transport simulations. This number was chosen following a sensitivity study that showed differences of less than % compared to a simulation with, particles (John 28). Miing and Spreading. For the case with uncorrelated permeabilities, Fig. 5 shows the particle positions at selected times during forward and reverse flow. The particle cloud spreads out in the direction as it moves forward, and, on flow reversal, the cloud returns to the origin (in terms of its mean position). Statistics of the particle positions are processed first to obtain the rate of change of variance with time (Fig. 6), from which we obtain t t t2 t3 t t t2 t Transmission - with local miing Transmission - no local miing t6 t5 t4 t3 t6 t5 t4 t3 z z z z -3 3 Echo - with local miing -3 3 Echo - no local miing Fig. 5 Eample of particle-tracking calculations. The figures in the top row show the progression of the ensemble of particles for the uncorrelated permeability case during forward flow (transmission) at successive times (t is the initial condition). The figures in the bottom row show the particle positions during flow reversal (echo) with increasing time. In the left column, local miing is included ( mol =. 4 m 2 /d, l =. m, and t =. m within each grid cell), and, on the right, there is no local miing ( mol =, l =, and t = ). With no local miing, particles return to their initial positions (compare t6 and t). The transmission profiles (top row) are quite similar, whereas the echo profiles (bottom row) are different. 62 September 2 SPE Journal

6 variance, m2 Fig. 6 Evolution of the variance of the particle positions (taken over an ensemble of, particles) with travel time for the uncorrelated permeability case. uring forward-flow period (up to 3 days), the variance increases almost linearly with time. After flow reversal, for the case with no local miing ( mol =, l = t = ), the particles retrace their paths and the variance returns to zero. With local miing, there is a short period of convective reversal, after which the particle cloud grows again. the evolution of the longitudinal dispersivity with mean position of the particle cloud (Fig. 7). In forward flow, the dispersivity for the case with local miing ( mol =. 4 m 2 /d, l =. m, and t =. m within each cell) grows and reaches an asymptotic limit of m (transmission value) at a distance of m. For this case, on flow reversal, the particle variance decreases first but then grows again with time. The particles do not return to their origin, and there is a finite variance of the particle cloud at the end of the flow-reversal eperiment (the mean position of the cloud is the same as the initial position). The dispersivity estimated on flow reversal (echo value) is approimately.6 m (much larger than the input value of. m). The same simulation sequence (forward and reverse flow) was repeated with the same velocity field but without local miing ( mol =, l =, and t = in each cell). The behavior during forward flow is the same as the case with local miing, as can be seen by the evolution of particle variance (Fig. 6). But, on flow reversal, the particles now return to their origin and the echo dispersivity estimated is zero. The dispersivity values estimated after Longitundal ispersivity, m With local miing No local miing With local miing Reverse flow No local miing Time, days Flow reversed Fig. 7 Evolution of the longitudinal dispersivity with distance traveled for the uncorrelated permeability case. This is computed from the statistics of the particle ensemble. The mean velocity is approimately.75 m/d. The dispersivity estimated in the forward-flow (transmission) case is almost the same with and without local miing. On flow reversal, the echo dispersivity for the case with local miing is approimately.6 m, which is much larger than the input value (. m). flow reversal do not retrace the transmission values because, for the same particle variance (spread), the time spent in the medium is greater (measured from start of injection and not from start of flow reversal). If only the transmission results were available, then one would interpret both scenarios the same way (i.e., one would conclude that both cases were in the transport regime where miing was occurring). Both cases would be characterized by a dispersivity value of. m. But, in the case with no local miing, it is the spreading of the tracer because of the velocity field that has resulted in dispersion-like behavior. There is no dissipation or dilution occurring within each cell. It is the behavior on flow reversal that shows the fundamental difference in the transport mechanism. Crucially, the echo dispersivity for the case with local miing is also much larger than the input dispersivity. This suggests that the variations in the velocity field significantly enhance miing. If this were not so, then the echo dispersivity should have been close to the core-scale (input) value. The transmission and echo dispersivities would lie in the overall trend of measured dispersivities (Fig. 2). Effect of Correlation Structure. We continue with the same reservoir size and introduce correlation in the permeability fields. Two new cases are simulated by introducing a long-range component in increasing proportions (refer to Eq. ) to the uncorrelated case described earlier. In the first, the proportion of permeability variance in the long-range structure is 2%; in the second, it is 8%. The long-range component has a range of 25 m in the direction and 5 m in the transverse directions. Sample cross sections are shown in Fig. 8. The increase in layering as the long-range structure becomes dominant is apparent. To conserve space, the corresponding histograms are not shown but are available elsewhere (John 28). The results of the particle-tracking simulations are shown in Fig. 9, plotted on a logarithmic scale. The short-scale result is the same as described in the Miing and Spreading subsection. With the introduction of more of the long-range component, the transmission dispersivity increases from m to approimately 5 m at a length of 2 m. Neither long-range case seems to have reached an asymptotic limit at this length scale because the dispersivities are still increasing at the end of the forward-flow simulation. The transmission Long 52 Long+Short 52 Short md 52 Fig. 8 Cross sections of a single realization of the permeability field showing the effect of changing correlation structure. As the proportion of the long-range component is increased, the amount of layering is more pronounced. The short-scale structure has a range of m in all directions. The long-range structure has a range of 25 m in the direction and 5 m in y and z directions. The contribution of the short-scale structure to the overall variance is, 8, and 2% from bottom to top. The figures are plotted with no vertical eaggeration. The actual reservoir dimensions used in the flow-field simulation are 248 m. The size of each grid cell is m. z September 2 SPE Journal 63

7 Longitudinal ispersivity, m. Long scale Short scale Long+short scale Input dispersivity. Fig. 9 Evolution of the longitudinal dispersivity for cases with changing correlation structure (Fig. 8). Layering significantly increases the transmission dispersivity and enhances the miing, as seen by the increase in echo dispersivities. The dispersivities on flow reversal are much larger than the input value ( mol =. 4 m 2 /d, l =. m, and t =. m). dispersivity grows almost linearly with time. The striking feature of the flow reversal is the similar increase in magnitudes of echo dispersivities of the long-range cases. Compared to a value of.6 m for the uncorrelated case, we now observe values that are as large are 5 m for the same total travel time. This reinforces the point made in the Miing and Spreading subsection: Flow-field variations strongly enhance the macroscopic effect of local miing. However, the ratio of echo to transmission dispersivities is much less than that of the short-range case. For the uncorrelated case, it was.6 (.6 m/. m); for the long-range cases, it is approimately. (5. m/5. m). This means that the convective contribution to particle variance is much larger than that for the uncorrelated case. Effect of Local Miing. To estimate the sensitivity of miing at large scales to local miing, we simulated the cases again with three levels of input (within cell) miing levels: (a) only diffusion, mol =. 4 m 2 /d; (b) diffusion and core-scale dispersion, mol =. 4 m 2 /d, l =. m, and t =. m in each grid cell; and (c) diffusion and larger core-scale dispersion, mol =. 4 m 2 /d, l =.5 m, and t =.5 m in each grid cell. The dispersivities estimated are shown in Fig.. The transmission behavior is almost unaffected by the degree of local miing (within each cell) because, in this direction, the contribution of spreading dominates. The echo dispersivities increase with increasing local miing, but Longitudinal ispersivity, m.. Long scale Short scale Fig. Comparison of the effect of local miing on longitudinal dispersion for two cases: uncorrelated permeability (lower set of curves in shades of blue) and long range correlation (upper set of curves in shades of pink). The three results around each case correspond to increasing levels of local miing: only diffusion ( mol =. 4 m 2 /d), diffusion+dispersion ( mol =. 4 m 2 /d, l =. m, t =. m), diffusion+dispersion ( mol =. 4 m 2 /d, l =.5 m, and t =.5 m). There is almost no change in transmission dispersivities and little change in echo values. If diffusion were absent, all echo dispersivities would return to zero, as in Fig. 7. the change is not substantial. This does not mean that local miing can be neglected; without local miing, the echo dispersivities would be zero, as in Fig. 7. No dilution would be taking place, and local concentrations would be large. Effect of Permeability Variance. The results from two cases, both having the same correlation structure (2% long-range and 8% short-range variance) but one having a smaller variance (.3 measured on the natural logarithm of the permeability) compared to the other (.), are compared in Fig.. The low-variance case had a transmission dispersivity of 5.4 m, which is smaller than that of the high-variance case (3.5 m), but its echo dispersivity is much lower at.27 m (5% of transmission) compared to.88 m (4% of transmission). This is because the larger hetereogeneity results in larger variance in local velocities, which leads to more enhancement of local miing (diffusion). Effect of Permeability Anisotropy. For the same correlation structure (2% long-range and 8% short-range variance), we compare the results of two cases in Fig. 2. One case has no permeability Longitudinal ispersivity, m High variance. Low variance Long+short scale. Longitudinal ispersivity, m k z / k =. k z / k =.. Long+short scale. Fig. Comparison of dispersivities with change in permeability heterogeneity. The low-variance case had a transmission dispersivity lower than that of the high-variance case, but its echo dispersivity is much lower (5% of transmission) compared to that of the high-variance case (4% of transmission). Fig. 2 Comparison of dispersivities with change in permeability anisotropy (lowered from to.). Both have identical transmission dispersivities, but the echo dispersivity of the case with anisotropy is slightly smaller than that without anisotropy. September 2 SPE Journal

8 Longitudinal ispersivity, m.. Long+short scale Longitudinal ispersivity, m.. Short scale Fig. 3 Comparison of dispersivities for change in travel time (penetration distance). Even for the shortest penetration distance (approimately m), an echo dispersivity value (approimately. m) much larger than the input value is obtained. The same data for the uncorrelated field are presented on the right for comparison. In both cases, echo dispersivities increase with penetration distance. anisotropy (k z = k ), and the other has a low value of vertical permeability (k z =. k ). Both have similar transmission dispersivities, but the echo dispersivity of the case with anisotropy is less than that of the case without anisotropy. Note that we have not changed the values of the input dispersivities with change in anisotropy. The weak effect of anisotropy suggests that the longitudinal dispersivity is governed largely by the -velocity variations. Effect of Travel Time. All of the simulations presented so far have been for a fied simulation time (number of days injected). The cases were repeated with shorter injection periods, giving results that sample shorter lengths of the model. In Fig. 3, we combine the results of a set of simulations performed for the case of long-range correlation in the permeability field. The transmission dispersivities from all the cases overlap and are merged. Echo dispersivities increase with penetration distance. Even for the shortest penetration distance (approimately m), the echo dispersivity (approimately. m) is much larger than the input value. The same data for the uncorrelated permeability case are presented alongside for comparison. ispersivity, m 5 Transmission 4 Echo Input istance, m Fig. 4 Comparison of dispersivities estimated from simulations (Figs. 9 through 3) with the field-measured values (Schulze-Makuch 25). Both echo and transmission values fall within the measured trend. The ambiguity in intepreting a transmission dispersivity value on the plot is now reduced by having the corresponding echo dispersivity. The simulation results also help eplain the large values of echo data estimated from single-well tracer tests (Fig. 4). The echo-dispersivity value at short penetration distance is approimately the same. The trend of increasing echo dispersivities is seen, but, here, the value at the largest penetration distance is much closer to its corresponding transmission value. Comparison With Measured ispersivities. The simulated dispersivities are compared with the overall trend of field dispersivities in Fig. 4. Both echo and transmission values fall well within the trend. Given only the transmission value and the unknown reservoir description, it would be difficult to infer whether the transport regime was miing or spreading. Three possibilities arise: First, the transport is in the preasymptotic regime and the value of dispersivity has not plateaued. Here, the interplay between convection and diffusion has not reached equilibrium. Typical field cases would be in this regime. If the tracer test were run for a longer distance on the same medium, then a (different) plateau value might be reached. Second, transport is asymptotic but is locally convective. Here, the dispersivity value is unique but it does not represent miing or dilution. Third, transport is in the asymptotic regime and is well mied. Here, the dispersivity value is unique and represents large miing zones. The ambiguity in interpreting a transmission-dispersivity value on the plot is reduced if the corresponding echo dispersivity is known. All echo dispersivities are greater than core-scale values, implying that miing is significant at the field scale. The case with the longest correlation length has the highest dispersivity value (both echo and transmission). This indicates that convection is enhancing the degree of miing. iscussion The most important observation in our work is that significant miing may be occurring in field-scale miscible displacements. Field-scale miscible displacements need not be entirely in the convective regime, as usually assumed. All the echo dispersivities in our simulations are much larger than input values of core- or gridblock-scale miing and are comparable to field-measured values. This corroborates the observations of Mahadevan et al. (23). The key learning is that heterogeneity, while increasing the convective contribution, also increases the amount of miing in the transverse direction. This effect, while smaller than in the longitudinal direction, is not negligible because it is compounded over the long travel times, resulting in significant dilution of the injected concentration. This suggests that numerical methods based on streamline formulations should be used with caution for miscible transport problems. Moreover, common practice in reservoir simulation uses numerical dispersion as a surrogate for physical dispersion. This can overestimate or underestimate the degree of miing, depending on how the numerical dispersion compares the physical dispersion occurring in that volume. Another observation is that dispersion-like behavior can be caused by heterogeneity alone, but this does not imply miing. Other works [e.g., Coats et al. (29)] have made similar observations. September 2 SPE Journal 65

9 This is the effect of the travel-time distribution caused by the random geometry of a disordered porous medium in three dimensions. Furthermore, transmission dispersivities are affected weakly by the degree of local miing. But this does not imply that local miing can be neglected. If there were no local miing, then echo dispersivities would be zero. We never observe this because diffusion is always present and, for homogeneous isotropic media, it becomes significantly enhanced by the local velocity variations, resulting in a larger dispersion coefficient. Even with only diffusion as the local miing mechanism, we observe echo dispersivities comparable in magnitude to transmission values. Most of the work performed on modeling solute transport using stochastic theories is based on the assumption of no local miing or infinite local Péclet number (Gelhar 986; agan 99; Rubin 23). While they might match averaged concentration histories, such theories would not be a good predictor of local concentrations (Kitanidis 994; Kapoor and Gelhar 994). Flow reversal provides an unambiguous test to distinguish between spreading and miing. Another approach to resolve this ambiguity is to measure local concentration histories at multiple points and compare them with the averaged wellhead concentration history (Cirpka and Kitanidis 2a). This approach has been used to interpret an intermediate-scale laboratory eperiment based on artificially created heterogeneous media (Jose et al. 24). Cirpka and Kitanidis (2b) show an approach to use the results of streamtube-based tracer transport for the study of miing-controlled reactive transport. We still have not addressed directly the practical issue of how to model miing at the field scale when using an upscaled description. Our results show the need to capture not only the effect of heterogeneity on miing but also the preasymptotic (non-fickian) transport behavior. Miscible displacements in typical reservoirs are not epected to reach an asymptotic limit that can be described by a unique dispersivity value. Approaches based on continuous-time random walks and fractional derivative formulations of the dispersive flu term have been proposed to address this issue (Benson et al. 2; Berkowitz et al. 2). Echo dispersivities do indicate the degree of miing, but we do not propose to use them as an input value. So, how to best represent the subgrid effects in coarse-scale simulation? Of the many approaches that have been proposed to this end [Barker and Fayers (994), Efendiev et al. (2), and Leonormand and Wang (995) to name a few], the most promising may be by Berentsten et al. (27). They generalize Taylor s description and derive a upscaled model for tracer transport in single-phase stratified flow that captures the evolution of the dispersive flu in both early and late time scales. Conclusions ispersive miing is significant in field-scale miscible displacements even in heterogeneous formations. Permeability layering increases the area available for transverse miing, and this effect becomes compounded over large travel times. Flow-reversal (echo) tests can be used to distinguish between convective spreading and dispersive miing. Echo dispersivities estimated from our simulations are comparable in magnitude with the corresponding transmission values. They follow the overall measured trend. This suggests that the large value of dispersivities observed in Fig. need not be the result of averaging unknown permeability heterogeneity, as commonly understood. Transport in typical reservoirs is usually in the preasymptotic regime, where the Fickian model for dispersion fails to capture the miing-zone growth accurately. Purely convective transport, in certain cases, can also appear to have a dispersion-like behavior. It would be incorrect to model this using a dispersive flu term. Nomenclature A = cross-sectional area of the medium, L 2 c = local concentration, M/L 3 C = average concentration, M/L 3 = dispersion coefficient, L 2 /T mol = molecular-diffusion coefficient, L 2 /T g = velocity gradient, /T h = lag distance, L k = permeability, L 2 L = length of the medium, L N Pe = Péclet number p = pressure, M/LT 2 q = average flow rate, L 3 /T r = semivariogram range, L s = semivariogram sill t = time, T u = flu vector, L/T = local fluid velocity, L/T V = average fluid velocity, L/T = distance in direction, L X = particle-position vector, L Z = vector of standard normal random numbers = dispersivity, L = semivariance, L 2 = increment = fluid viscosity, M/LT = mean penetration distance, L = standard deviation = porosity Superscripts and Subscripts = dimensionless l = longitudinal (to mean flow direction) o = cell face p = particle Pe = Péclet number t = transverse (to mean flow direction) = direction z = z direction Acknowledgments We would like to thank Todd Arbogast of the epartment of Mathematics at the The University of Teas at Austin for access to the flow-simulator code used in this work. We appreciate the assistance of the technical support staff members at the Teas Advanced Computing Center for assistance with running our codes on their platforms. This work was conducted with the support of the National Petroleum Technology Office of the epartment of Energy through contract E-PS26-4NT545-3F. Larry W. Lake holds the W.A. (Monte) Moncrief Chair, and Steven L. Bryant holds the J.H. Herring Centennial Professorship, both in the epartment of Petroleum and Geosystems Engineering at The University of Teas at Austin. References Arbogast, T User s Guide to Parssim: The Parallel Subsurface Simulator, Single Phase. Version: Parssim v. 2., May 998. Center for Subsurface Modeling, Teas Institute for Computational and Applied Mathematics, The University of Teas at Austin, Austin, Teas, 84. Arbogast, T., Bryant, S., awson, C., Saaf, F., Wang, C., and Wheeler, M Computational methods for multiphase flow and reactive transport problems arising in subsurface contaminant remediation. Journal of Computational and Applied Mathematics 74 ( 2): doi:.6/ (96)5-5. Aronofsky, J.W. and Heller, J.P iffusion Model to Eplain Miing of Flowing Miscible Fluids in Porous Media. Trans., AIME, 2: SPE-86-G. Arya, A., Hewett, T.A., Larson, R.G., and Lake, L.W ispersion and Reservoir Heterogeneity. SPE Res Eng 3 (): SPE-43-PA. doi:.28/43-pa. Barker, J.W. and Fayers, F.J Transport Coefficients for Compositional Simulation With Coarse Grids in Heterogeneous Media. SPE Advanced Technology Series 2 (2): 3 2. SPE-2259-PA. doi:.28/2259-pa. 66 September 2 SPE Journal

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11 Stalkup, F.I. 99. Effect of Gas Enrichment and Numerical ispersion on Enriched-Gas-rive Predictions. SPE Res Eng 5 (4): 7 655; Trans., AIME, 289. SPE-86-PA. doi:.28/86-pa. Taylor, G.I ispersion of soluble matter in solvent flowing slowly through a tube. Proceedings of the Royal Society of London: Series A, Mathematical and Physical Sciences 29 (37): doi:.98/rspa Taylor, G.I Low Reynolds number flows. VHS produced by Educational Services Incorporated under the direction of the National Committee for Fluid Mechanics Films. Chicago, Illinois: Encyclopædia Britannica Educational Corporation. Tchelepi, H.A Viscous fingering, gravity segregation and permeability heterogeneity in two-dimensional and three dimensional flows. Ph dissertation, Petroleum Engineering epartment, Stanford University, Stanford, California. Tidwell, V.C. and Wilson, J.L. 2. 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Appendi A Particle-Tracking Calculations The idea of applying the random-walk particle-tracking method to solute-transport problems is based on an analogy between the convection/dispersion equation for solute transport and the Fokker-Planck equation for diffusion (Uffink 99; Risken 996; Hassan and Mohamed 23). The mass of the injected solute is represented by a finite number of particles. The particles are displaced in space by the action of a velocity field (convection) and Brownian motion (dispersion). The tracking in three dimensions is based on the equation ( )= ()+ ( ) + ( ) X p t+ t Xp t A Xp, t t B Xp, t Z t, (a-) where X p is the particle-position vector [L], A is a drift vector [L/T], B is a tensor [L/T /2 ] defining the strength of diffusion, and Z is a vector of independent normally distributed random numbers with zero mean and unit variance. To simulate tracer transport, choose A and B such that the mass density of the particles satisfies the convection/dispersion equation ( c) + ( vc) ( c)=, (a-2) t where is the local dispersion tensor given by l t = ( t v + mol ) I + vv (a-3) v and l and t are the longitudinal and transverse dispersivities, respectively. mol is the effective molecular diffusivity of solute in the given porous medium. It has been shown that Eq. A- satisfies Eq. A-2 by setting A and B as (LaBolle et al. 996) A= v (A-4) and 2= B B T (A-5) When computing A, we neglect the gradients in and the gradients in porosity are zero because of the constant-porosity assumption. The drift term (Eq. A-4) hence consists of only the velocity term. The velocities are obtained from the flow-simulation step (Flow-Field Generation subsection). To compute the convective displacement, we integrate over the local velocity field for a given timestep. Only the face-centered velocities and cellcentered gradients are used in this calculation. To be consistent with the fluid-flow equation, we take each component of the velocity to vary linearly within a cell in its respective direction (Pollock 988). v = g+ vo, (a-6) where is the distance of the particle from the cell face, v o is the velocity at the cell face, and g is the gradient in the velocity. At any particle location, d = v (a-7) dt Starting with the particle at Location (distance from the cell face in the direction being ), the new location 2 can be found by integrating Eq. A-7 over a timestep size of t or 2 d = g+ v o t g 2 + v ln g g + v This gives dt (a-8) o o = t (a-9) g vo e t g t 2 = ( ) e g (A-) Similar epressions apply for the y and z components of the particle position inside a cell. If the particle crosses a cell boundary in a given timestep, then these epressions are modified to account for the change in cell velocities and gradients (John 28). We found this approach to be more accurate and efficient in highly heterogeneous realizations compared with approaches that treat the velocity as constant over a given timestep. To compute the dispersive displacement, we first estimate using the approach suggested by Tchelepi (994). An orthogonal coordinate aes system, aligned in the direction of v, is set up temporarily at the current particle location. In this system, reduces to its diagonal form, making it much simpler to solve Eq. A-5 for B. The dispersive displacements are computed in this system and transformed to the global coordinate system. Boundary conditions were enforced using a reflection condition on the y and z faces and by ensuring that the particles do not eit the system in the direction. This ensures global mass conservation. To generate uniform deviates, we use a random-number-generator routine adapted from Press et al. (992) that has a long period (> 9 ). The timestep size is chosen to limit the particle displacement to /,th of the cell size. In all our simulations, we use a uniform grid with equidimensional cells. Particles are introduced uniformly over the entire y z face at the center of the flow domain. As they are transported, statistics of particle positions are computed at desired time intervals. The dispersion coefficient is estimated as l = 2 2 t, (a-) where 2 is the variance of the particle ensemble. To calculate the dispersivities, we assume that Eq. 2 holds true: V l l =, (A-2) 68 September 2 SPE Journal

12 where the mean velocity of the ensemble is computed from its mean position, V = (A-3) t Our computing environment was a Linu cluster at the Teas Advanced Computing Center at The University of Teas. Each computer node is a ell Power Edge 955 blade containing two Xeon Intel uo-core -bit processors running at 2.66 GHz with 8 GB of memory. The peak performance is rated at approimately GFLOPS/processor core. Most of our particle-tracking simulation runs were completed in approimately 3 hours for the cases described. Abraham K John is a reservoir engineer working for Shell International Eploration and Production in Houston. He holds a BS degree from the Indian School of Mines, hanbad, and MS and Ph degrees from The University of Teas at Austin, all in petroleum engineering. He worked previously as a field engineer for Halliburton Logging Services India Limited. Larry Lake is the W.A. (Monty) Moncrief Centennial Chair holder in the epartment of Petroleum and Geosystems Engineering at The University of Teas at Austin and director of the Center for Petroleum Asset Risk Management. He has published more than peer-reviewed journal articles and has taught industrial and professional society short courses in enhanced oil recovery and reservoir characterization around the globe. He is the author or coauthor of four tetbooks and the editor of three bound volumes. He has been teaching at The University of Teas at Austin for 28 years. Before joining the university, he worked for the Shell evelopment Company in Houston. Lake is a member of the US National Academy of Engineers. He holds BSE and Ph degrees in chemical engineering from Arizona State University and Rice University, respectively. Steven Bryant is associate professor in the epartment of Petroleum and Geosystems Engineering at The University of Teas at Austin, where he holds the J.H. Herring Professorship and the George H. Fancher Centennial Teaching Fellowship in Petroleum Engineering. He served as SPE istinguished Lecturer in 2 2, and he directs the Geological CO 2 Storage Joint Industry Project at The University of Teas at Austin. Bryant holds a BE degree from Vanderbilt and a Ph degree from The University of Teas at Austin, both in chemical engineering. His current research interests include the grain-scale models of unconventional gas reservoirs and the role of methane hydrates in the Earth s carbon cycle. He is a member of SPE and has published more than 7 papers and one tetbook with applications in production engineering, reservoir engineering, and formation evaluation. James Jennings Jr. is principal reservoir engineer with Shell Eploration and Production in Houston, where he has been providing internal consulting on carbonate reservoir modeling. He has 23 years of previous eperience at The University of Teas at Austin, Arco, and BP, primarily in characterization, geostatistics, modeling, and scale-up for carbonate reservoirs. Jennings has been a technical editor for SPE Res Eval & Eng journal and chairman for the SPE reprint volume Advances in Reservoir Characterization. He holds a BS degree from the University of Wyoming and MS and Ph degrees in petroleum engineering from Teas A&M University. September 2 SPE Journal 69