Multiscale modeling of permeability and non-darcy factor in propped fractures

Transcription

1 Peer Reviewed Multiscale modeling of permeability and non-darcy factor in propped fractures Ali Takbiri-Borujeni, West Virginia University, Mayank Tyagi, Louisiana State University Abstract Based on the principles of fluid dynamics and using lattice Boltzmann method (LBM), permeability and non-darcy factor of an imaged proppantfilled fracture system (Berea sandstone/fracture/proppant system) under 4, 12, and 20 kpsi stress levels are computed. The images were taken using high-resolution x-ray computed tomography (XCT). Images of the packing show sliding of the proppant grains, embedding of the grains at the fracture walls, and crushing of individual proppant grains. Velocity profiles of the fluid in the system at each stress level show that the flow pathways become narrower as loading stress increases, which causes reduction of permeability and increase in non-darcy factor. Compared with the experimental values (reported by the vendor) that show a smooth change of permeability and non-darcy factor by increasing the loading stress, LBM simulation results show that these properties slightly change at loading stresses prior to grain crushing (at 12 kpsi, z-direction permeability is reduced almost 15 percent and non-darcy factor increased by 28 percent compared to their computed values with no loading) and then a sharp change at the stress level at which proppant grains crush (at 20 kpsi, z-direction permeability is reduced 63 percent and non-darcy factor increased by 181 percent compared to their computed values with no loading). Conductivity of the packing, which is calculated by measuring the width of the images of system at each stress level, shows similar trend as permeability. One of the possible reasons for this behavior is that the packing structure is close to ideal (i.e., packing with least porosity and maximum grain-to-grain contact points), where the grains are carefully put together compared to the standard tests for measuring hydraulic properties of proppants with more room for proppant movement. Predicted stress-dependent permeability and non- Darcy factors corresponding to the effective stress fields around the hydraulic fractured completions are included in a two-dimensional gas reservoir simulator to calculate the productivity index ratios. Productivity index ratios with permeability and non-darcy factors kept constant at initial effective stress (6,000 psi) are less than 0.2 percent higher than those with stress-dependent permeability and non-darcy factors for a gas rate of 30 MMscf/D. Introduction Fracture permeability and non-darcy factor are two key hydraulic properties of proppants that control the effectiveness of fracturing stimulation treatments. These transport properties depend on proppant material type, size, shape, and concentration (or pack porosity). The proppant particles should be large enough and smoothly shaped to provide high fracture conductivity and low non-darcy factors, and be strong enough to withstand the closure stresses (Donaldson et al., 2014). Stress-related changes in permeability and non-darcy factor can be attributed to the alteration of pore space topology resulted from grain movement, changes in proppant grain shape, grain embedding, and grain crushing. Although there have been numerous investigations of the impact of loading stress on the properties of the proppant-filled fracture, there is still a need for an in-depth analysis to investigate how these systems respond to changes in loading stress and their permeability and non-darcy factor, and fracture conductivity. This work tries to fill this need by a look at what happens to the proppants under stress. The permeability and non-darcy factor of proppant packs are commonly estimated using standardized American Petroleum Institute (API) tests of proppants in a linear flow cell at a specific proppant concentration, closure stress, and temperature. In 2003, a committee was formed to write procedures on measuring long-term conductivity of proppants used for hydraulic fracturing (Kaufman et al., 2007). The new procedures have helped engineers evaluate and compare proppants. In 1973, Cooke measured the fracture conductivity and non-darcy factor of propped fractures under stress. He performed the experiments on mesh sand under loading stresses from 4,000 psi to 10,000 psi for various temperatures and fluids (brine, oil and gas). The author reported two orders of magnitude reduction in permeability under stress. He also reported two orders of magnitude increase in non-darcy factor for the same sand under the range of loading stresses. Much and Penny (1987) performed long-term tests on mesh Jordan Sand and Intermediate Strength Ceramic Proppant (ISP). They reported more than 50% decrease in conductivity of Jordan sand while the decrease for ISP was around 30%. Moreover, ISP presented conductivities over 10 times greater than sand proppant at equal concentrations and conditions. Fredd et al. (2001) conducted conductivity measurements on mesh sintered bauxite and Jordan sand at 0, 0.1, and 1.0 lbm/ft2 concentrations at effective pressures ranging from 1,000 to 7,000 psi. For Jordan sand the fracture conductivity rapidly decreased from approximately 800 md-ft at 1,000 psi to less than 1 md-ft at 7,000 psi. They contributed this rapid decrease to crushing of the proppant. However, the conductivity of the fracture with sintered bauxite remained near 1,000 md-ft up to approximately 5,000 psi and then decreased as the proppant began to crush. Barree et al. (2003) analyzed many laboratory conductivity measurements on similar proppant samples at similar stress, temperature, and flow conditions, finding that small variations in proppant properties (less than 15 to 20 percent difference in conductivity) may not be statistically significant when applied to performance measurements in the field. They argued that conductivity estimates of a granular pack should be based on a sufficient number of observations under similar packing conditions and along similar loading paths. The statistical analyses of pack width and permeability for different loading stresses by Barree et al. (2003) indicated that conductivity variations (product of variations of pack width and permeability) of +/- 20 percent are within laboratory accuracy. 64 Volume 3 - Number 2

2 For computing the hydraulic properties of porous media such as porosity, permeability, and non-darcy factor, numerical simulation using digital images has become a credible alternative, enabled by improvements of 3D imaging techniques, numerical methods, and computing power. Application of this approach enables researchers to probe pore-scale physics at a level not possible with traditional experiments and the ability to perform an endless set of numerical tests without degrading or altering the sample. Considerations that can limit this digital approach include whether the imaging technique can resolve all relevant characteristic scales in the pore space and whether numerical algorithms are able to accurately model the physical processes (Takbiri-Borujeni et al., 2013). LBM has been extensively used to model the fluid flow in many types of porous media. Succi et al. (1989) were one of the first groups that employed LBM to compute permeability of porous media. Ferréol and Tothman (1995) simulated single phase flow through Fontainebleau Sandstone and found approximately the same permeability values as equivalent finite-difference calculations and laboratory measurements. Jin et al. (2004) also presented an integrated procedure for the estimation of the absolute permeability of unconsolidated and consolidated reservoir rock. Challenges for applying LBM to real problems include finite-size effects and relaxation time dependence of no-flow boundaries (Takbiri-Borujeni et al., 2013). In image-based simulations, the accuracy of the computed flow properties is dependent on the spatial resolution of the rock image (Manwart et al., 2002; Ginzbourg and Adler, 1994). However, there is always a trade-off between the image resolution (and the related lattice spacing) versus computational power. Furthermore, in all digital samples, there is a resolution threshold, below which certain flow characteristics, such as recirculation, are not resolved (Maier and Bernard, 2010). In this work, segmented images of 20/40 CarboHSP high-strength sintered bauxite proppant sample placed between two Berea sandstone cores under different stress levels of 0, 4, 12, 20 kpsi are used to compute permeability and non-darcy factor using LBM simulations. Conductivity of the packing is also computed by visual inspection of the width at each stress level. Computed permeability and non-darcy factor values at different stress levels are included in a two-dimensional hydraulically fractured gas reservoir simulator to evaluate the effects of stress-dependent permeability and non-darcy factor on well productivity indices. Based on the reservoir simulation results, productivity index ratios with permeability and non-darcy factors kept constant at initial effective stress (6,000 psi) are less than 0.05 percent higher than those with stress-dependent permeability and non-darcy factors for a gas rate of 30 MMscf/D. Samples The rock/fracture/proppant system was built by placing CarboHSP high-strength sintered bauxite proppant in a cylinder, bordered above and below by Berea Sandstone. The proppant used in the test was mesh (sieve) with a median particle diameter of 697 micrometer. The cylinder containing the sample has a diameter of 6.14 mm and contains 9 to 10 layers of proppant. The packing was imaged using computed microtomography (personal communication, C. S. Willson, Louisiana State University). Cross-sections of the fracture system under different stress levels are shown in Figure 1. As the loading stress on the fracture system increases, the width of fracture decreases. Some grain sliding and embedment into the sandstone surfaces are observed as loading stress increased. Crushing of grains observed at 20 kpsi. (a) No stress (b) 4,000 psi (c) 12,000 psi (d) 20,000 psi Figure 1. Cross-sections of the Berea sandstone/fracture/proppant system at different stress levels. Problem Formulation Lattice Boltzmann method The Lattice Boltzmann method (LBM) is a simplification of the Boltzmann equation and based on kinetic theory (Sukop and Thorne, 2007). The Boltzmann equation with BGK type collision operator is (1) where, (2) 65

3 in which R is the ideal gas constant, f are the distribution functions in velocity space, f eq are the equilibrium distributions, e is the molecular velocity, τ is the relaxation time, and ρ, u, and T are the macroscopic density, velocity, and temperature, respectively. To convert the above equation to lattice Boltzmann equation, a set of discrete molecular velocity is defined, on which the distribution function is evaluated. The molecular velocity and fluid velocity are normalized by 3RT; thus, the sound speed defined by c s = RT becomes as 1 (Takashi, 1997). The LBM equation with streaming and single velocity relaxation operator collision becomes 3 in which e i are the molecular velocities in direction i, f i are the discrete distribution functions in direction i, and fi eq are the equilibrium distributions in direction i, in which w i are weight factors specific to different directions, (3) (4) In this work, the D 3Q 19 model (three dimensions and 19 directions of fluid movement) is used. Velocity directions of a particle in a typical 3D lattice node are shown in Figure 2. Velocity vectors for this model are described below, (5) Figure 2. Velocity directions of a particle on a typical 3D lattice node. The medium used for the LBM simulations is converted to a periodic domain by adding ten layers of void voxels to each side. A body force approach, which is an alternative to specifying pressure values at the inlet and outlet of the domain, is used. Periodic boundary condition is applied on all the external faces. The body force was not applied on these added layers (Takbiri-Borujeni, 2013; Chukwudozie, 2011). LBM simulations in this study are performed using the Parallel Lattice Boltzmann Solver (PALABOS, 2012) on Louisiana Optical Network Initiative (LONI) resources. The PALABOS (Parallel Lattice Boltzmann Solver) code is used for solving the flow problems in this study ( The bounceback boundary scheme is used to implement the no-flow boundary conditions at the void-solid interfaces. Permeability Permeability is calculated from (6) in which k is the permeability tensor of the porous medium, is average velocity of the fluid in the domain, µ is viscosity of the fluid, and is dynamic pressure gradient of the fluid. Velocity values in each node are computed in all directions using LBM simulations to determine the permeability tensor. Fracture conductivity Fracture conductivity is the product of fracture width left open at the end of the hydraulic fracturing treatment and the permeability of the proppant packs inside the fracture. In practice, width loss due to embedment is difficult to quantify and cannot be observed without disassembly of the proppant 66 Volume 3 - Number 2

4 pack. In this study, it is possible to analyze the width changes by direct visual inspection of the images followed by direct measurement of the width. Because the embedment of the proppant grains into the Berea sandstone was not uniform (more embedment near the container walls because the rock was not cut at a perfect normal angle), average of the proppant pack at each stress level was calculated. Non-Darcy factor Darcy s law stipulates a linear relation between fluid velocity and pressure gradient in porous media, and is valid only at low Reynolds numbers (Re 1). At high fluid velocities, which most commonly occur in regions around the wellbore and inside hydraulic fractures, Darcy s law is no longer valid. Forchheimer (1901) proposed predicting the pressure drops at higher Reynolds numbers using in which β is the non-darcy factor. To compute the non-darcy factor, the Forchheimer equation is rearranged and the apparent permeability (k app) is defined. Apparent permeability is not a property of the porous media because it changes with flow velocity, As shown in Equation 6, at low velocities (Re < 1), apparent permeability and permeability values are approximately equal; as velocity increases, (7) (8) apparent permeability decreases. A plot of inverse of apparent permeability versus inverse of the permeability. is a line with slope of non-darcy factor and intercept of There have been two main types of criteria for determining the onset of the non-darcy flow in the porous media (Ma and Ruth, 1993). The first type is based on the Reynolds number in the porous media. Reynolds number is defined as, where D p is the diameter of the particles, and v is the velocity of the fluid in the porous media. Critical value of the non-darcy flow varies from 1 to 100. The second type is based on the Forchheimer number, which is defined as, where k is the intrinsic permeability of the medium and β is the non-darcy flow parameter. According to this criterion, the critical value varies from to 0.2 (Zeng and Grigg, 2006). Reservoir simulation This study uses a two-dimensional finite-difference single-phase gas reservoir simulator. The reservoir assumed to be horizontal, homogeneous, isotropic, and initially filled with constant pressure of 6,000 psi. The exterior surfaces of the domain are represented by no-flow boundaries. A rectangular hydraulic fracture is placed at the center of the reservoir. The hydraulic fracture extends from both sides of the wellbore and the fracture is assumed to be fully filled with a proppant pack. Due to the symmetry, flow simulations in a quarter of the flow domain are performed. Because hydraulic fracture apertures are usually orders of magnitude smaller than the grid spacing used in the reservoir simulators grid spacing is increased geometrically with distance away from the fracture. In order to do this, a constant multiplier is used for successive grid sizes. Results and discussion Stress-strain relations of the fracture system is plotted by measuring the width of the proppant pack by visual inspection of locations of the interfaces of the proppant grains and the surrounding sandstone samples at each loading stress. Strain is a measure of the material deformation as a result of applied stress. Strain in z-direction,, can be calculated by: where, z is the original height of the proppant pack, and δz are changes in height. Stress-strain plot of the proppant pack is shown in Figure 3. For LBM simulations, voxel regions of the proppant pack away from the Berea sandstone/proppant interface are selected. These regions consist only of proppant grains. Cross sections of the selected regions for LBM simulations at each stress level are shown in Figure 4. In LBM simulations, velocity velocities of the fluid in all nodes are computed. Contour plots of normalized velocity magnitude (velocity magnitude divided by the average velocity magnitude in the medium) in a xy-cross section under 0, 4, 12, and 20 kpsi at small Reynolds numbers (Re < 0.03) are shown in Figure 5. At lower loadings, pores are larger and the cross-sections contain higher velocities. As loading increases, pore spaces become smaller and velocity values decrease. At all the stress levels, the contour plots for higher flow velocities (higher Reynolds numbers) have more regions with elevated velocities compared to the lower velocities. Contour plots of normalized velocity magnitude in xy-cross section under 12 kpsi for high (Re = 18.1 and F o = 3.83E-01) and low (Re = and F o = 5.07E-04) are shown in Figure 6. High pressure gradients (high body force values) cause inertial flows to increase, which causes reductions in k app. (9) (10) (11) 67

5 Figure 3. Proppant pack deformation in response to mechanical loading. (a) No stress (b) 4 kpsi (c) 12 kpsi (d) 20 kpsi Figure 4. Cross section of the selected regions at each stress level. (a) No stress (b) 4 kpsi (c) 12 kpsi (d) 20 kpsi Figure 5. Contour plots of normalized velocity magnitudes for each stress level at Re < (a) Re = (F o = 5.07E-04) (b) Re = 18.1 (F o = 3.83E-01) Figure 6. Normalized velocity magnitude in xy-cross section under 12 kpsi for (a) Re = (F o = 5.07E-04) and (b) Re = 18.1 (F o = 3.83E-01). 68 Volume 3 - Number 2

6 In Figure 7, computed permeabilities are compared with Modified API RP-61 (ISO ) experimental results published by CARBO Ceramics, the manufacturers of the proppant used in the proppant/berea imaging experiments (Palisch et al., 2007). Permeability slightly decreased as stress increased below 12 kpsi compared to reduction in permeability at 20 kpsi. At 20 kpsi, there is 63.0 percent reduction in the z-direction permeability compared to the computed value at zero stress. In At this stress level, due to the crushing of the proppant grains, topology of the medium changes and pore spaces become more compact, and therefore, permeability is lower. As loading stress increase up to 12 kpsi, computed and measured permeability values become closer. At 12 kpsi, computed permeability is close to the vendor reported permeability (difference is almost 4 percent). Figure 7. Computed z-direction permeability values versus loading stress. Conductivity is the product of the permeability and width of the propped fracture. Computed proppant pack conductivity values versus the confining stress are plotted in Figure 8. Similar to permeability results, change in conductivity slightly decreases with increase in stress below 12 kpsi. At 20 kpsi, computed value of conductivity decreases by 60.6 percent compared to the zero stress state. Figure 8. Conductivity variations against loading stress. Due to relatively large pore sizes in the proppant pack (pores, in average, consist of more than 10 voxels), permeability at each stress level is computed without substantial numerical error (finite-size errors and relaxation-time dependence of the no-flow boundaries are not significant). However, computation of non-darcy factor requires applying high-pressure gradients (large body forces) to reach Re > 1 in LBM simulations. This may cause continuity errors that can grow at each time-step and break down the simulation. In simulations on the original images (300 3 voxels with a numerical resolution of 11.8 micrometer), Reynolds numbers as high as 10 are achieved. In order to reach higher ranges of Reynolds number, the numerical resolution of the images are increased by dividing each side of each voxels in the domain by two. This causes the numerical domain to be 8 times larger and the numerical resolution to be 6.9 micrometer. In this case, Reynolds numbers as high as 20 (F 0 = 0.47) are achieved. Apparent permeability values in z-direction are plotted versus Reynolds number for each stress level, as shown in Figure 9. Apparent permeability values decrease as Reynolds number increases (Re > 1). For Re < 1, there are small differences between apparent permeability and permeability and they are essentially the same; however, at Re > 1, apparent permeability can reduce to half of the permeability. Variations of apparent permeability values with Reynolds numbers are used for estimation of non-darcy factors. According to Equation 6, non- Darcy factor can be calculated from the slope of the changes of inverse of apparent permeability versus pseudo-reynolds number. Inverse of apparent z-direction permeability versus are depicted in Figure 10. Changes of non-darcy factor below 12 kpsi are slight compared to that at 20 kpsi, at 69

7 which non-darcy factor increases more than 60 percent compared to no-loading state. Figure 9. Log-log plot of apparent permeability values in z-direction for each loading stress versus Reynolds number. Figure 10. Inverse of apparent z-direction permeability vs. under 20 kpsi with the slope equal to non-darcy factor. In Figure 11, computed z-direction non-darcy factor is compared with experimental results published by the proppant manufacturer. Computed non- Darcy factor slightly increased as stress increased up to 12 kpsi compared to increase in non-darcy factor at 20 kpsi. At 20 kpsi, there is percent increase in z-direction non-darcy factor compared to the computed value at zero stress. In At this stress level, due to the crushing of the proppant grains, pore spaces become more compact, and therefore, inertial flow becomes more significant. Differences between the computed and measured non-darcy factor at 4 and 12 kpsi are 68.0 and 23.0 percent, respectively; as loading stress increase to 12 kpsi, computed and measured non-darcy factors become closer. Figure 11. Computed z-direction non-darcy factors versus stress at each stress level. 70 Volume 3 - Number 2

8 Computed non-darcy factor and permeability for all flow directions are listed in Table 1. At 20 kpsi, some of the proppant grains crushed, z-direction permeability reduced 63 percent and non-darcy factor increased 181 percent compared to their calculated values at zero stress. Computed values of the permeability and non-darcy factor in all directions are approximately equal and off-diagonal terms in permeability and non-darcy factor tensors are negligible compared to the diagonal terms, which shows that they are both isotropic. Permeability (10 3 D) Non-Darcy Factor (10 6 ft -1 ) Table 1. Calculated values of the permeability and non-darcy factor for the proppant pack at each stress. Stress (psi) 0 4,000 12,000 20,000 k xx k yy k zz β xx β yy β zz Comparison of the computed results of permeability and non-darcy factors using LBM simulations with the vendor published results shows that the computed values are less sensitive to the loading stress below a stress level at which proppant grains crush (i.e., 12 kpsi). One of the possible reasons is that the fracture system in this study is close to be an ideal pack (i.e., the proppants are carefully put together and have more compact pore spaces and highest number of point contacts). This eliminates the changes in pore volume and geometry under stress because there is not enough room for the proppant sliding and movement, and therefore, grain rearrangement and sliding are more limited. In experiments, as stress increases, pack becomes more compact and movement of proppant grains becomes harder. As stress increases, LBM results of permeability and non-darcy factor become closer with increasing stress up to 12 kpsi, which shows that after applying stress on the proppant in experiments, grain movement becomes limited and pore structure becomes more similar to that of imaged pack. In the LBM simulations, crushed particles cannot move (because solid movement in not considered in LBM simulations). In practice, crushed particles can shift to the open spaces because of the exerted drag force by the fluid. This can substantially change the topology of the porous media. Better results can be achieved using numerical simulations capable of simulating solid particle movement in the porous media. Reservoir simulations A reservoir model with the parameters listed in the Table 2 is considered. Computed values of permeability and non-darcy factor from LBM simulations are input in the reservoir simulator. Two sets of simulations are run, one with stress-dependent permeability and non-darcy factors and the other without them. In all the reservoir simulations, considered effective stresses are lower than 10,000 psi (that is lower than the stress at which proppant grains crush). Table 2. Reservoir properties used for the reservoir model. Simulation Input Initial reservoir pressure (psia) 4000 Reservoir volume (ft 3 ) Fracture half-length (ft) 500 Fracture width (ft) Porosity (%) 8 Minimum in-situ stress (psi) Formation permeability (md) 1 Biot s constant 1 In the simulations, the well is at the bottom left corner (Figure 12) and the hydraulic fracture is along the bottom of the flow domain, with a half-length of 500 ft. Pressures are relatively low near the well and along the fracture compared to regions away from the wellbore and hydraulic fracture. The nonuniform fluid pressure induces different effective stresses at different locations in the reservoir. σ eff = σ αp, where α is Biot s constant of poroelasticity. The productivity index for low-pressure gas flow is approximately (Lee and Wattenbarger, 1996) in which J g is the productivity index of gas reservoir, Z is the gas compressibility factor, µ is the gas viscosity, q is the gas flow rate in standard conditions, p r is the reservoir average pressure, and p wf is the flowing bottom hole pressure. The initial rapid changes in productivity index are because of transient (12) 71

9 Figure 12: Pressure distribution in the reservoir. Pressure is lower close to the well (bottom left corner) and in the fracture (along the bottom boundary of the reservoir). flow. Later, the profile stabilizes and the flow regime is boundary-dominated (or stabilized flow) and the productivity index is approximately constant. The non-darcy J g is always less than or equal to the productivity index that neglects inertial effects. Results with varying flow rate can be compared using the productivity index ratio, J D. This is the ratio of the productivity index including non-darcy flow, J β, to the pure Darcy index, J K; J d = J β/j K. Increasing the flow rate reduces J D because non-darcy effects increase with increasing velocity (Figure 13). For gas rates of 10, 20 and 30 MMScf/D, J D departs from unity by 2, 4, 7 percent, respectively. Productivity index ratios with permeability and non-darcy factors kept constant at initial effective stress (black dashed line in Figure 13) are ca. 0.02, 0.08, and 0.18 percent higher than those with stress-dependent permeability and non-darcy factors for 10, 20 and 30 MMScf/D, respectively. Figure 13. Productivity index ratio JD = Jβ/Jk for three gas rates. Conclusions For the fracture system, stresses, and rates investigated, grain crushing significantly increases flow resistance of proppant packs at effective stresses greater than ca. 12 kpsi. Velocity profiles of the fluid in the system at each stress level show that the flow pathways become narrower as loading stress increases, which causes reduction of permeability and increase in non-darcy factor. LBM simulation results predict permeability reduction of 63 percent and non-darcy-factor increases of 181 percent at 20 kpsi compared to zero stress state. As loading stress increases to 12 kpsi, computed and measured permeability and non-darcy factor become closer. These predictions were used in a two-dimensional single-phase gas reservoir simulator to assess the effects of gas rates on productivity indices. In all the reservoir simulations, considered effective stresses are lower than 10,000 psi (that is lower than the stress at which proppant grains crush) and therefore, changes of permeability and non-darcy factors with effective stress are not significant. Increasing the flow rate reduces the productivity index ratio because non-darcy effects increase with increasing velocity. Variability of the productivity index with gas flow rate is much greater than with effective stress. For gas rates considered, productivity index ratio departs from unity by up to 7 percent. Productivity index ratios with permeability and non-darcy factors kept constant at initial effective stress are less than 0.2 percent higher than those with stress-dependent permeability and non-darcy factors for 30 MMScf/D. Acknowledgments We thank Prof. Clint Willson (LSU Civil and Environmental Engineering Dept.) for the digital proppant images. Tyagi acknowledges and thanks ExxonMobil Upstream Research for funding parts of this research. We also acknowledge the PALABOS group for the open source code used for 72 Volume 3 - Number 2

10 LBM simulations in this study ( We thank the LSU Center for Computation and Technology (CCT) and the Louisiana Optical Network Initiative (LONI) for providing high performance computing resources. Nomenclature c s D p e e i f f eq f i f eq i F o J g J β J k J D k k app p r p wf q R Re u w i T Z β = Speed of sound = Diameter of the particles = Molecular velocity = Directions in which fluid particles move = Distribution functions in velocity space = Equilibrium distribution functions in velocity space = Discrete distribution functions in i direction in velocity space = Discrete equilibrium distribution functions in i direction = Non-Darcy factor = Productivity index of gas reservoir = Productivity index of gas reservoir including non-darcy flow = Productivity index of gas reservoir excluding non-darcy flow = Productivity index ratio = Permeability = Apparent permeability = Reservoir average pressure = Flowing bottom hole pressure = Gas flow rate in standard conditions = Gas constant = Reynolds number = Fluid velocity = Average velocity of the fluid in the domain = Weight factors specific to i direction = Temperature = Gas compressibility factor = Non-Darcy factor = Strain in z-direction = Pressure gradient = Fluid density ρ µ = Viscosity of the fluid τ = Relaxation time References Barree, R. D., Cox, S. A., Barree, V. L., and MConway, M. W. (2003). Realistic assessment of proppant pack conductivity for material selection. SPE Annual Technical Conference and Exhibition, 5-8 October, Denver, CO. doi: /84306-ms. Chukwudozie, C. (2011). Pore-scale lattice Boltzmann simulations of inertial flows in realistic porous media: a first principle analysis of the Forchheimer relationship. Master s Thesis, Louisiana State University. Cooke, C. E. (1973). Conductivity of Fracture Proppants in Multiple Layers. Journal of Petroleum Technology 25(09), Donaldson, E., Alam, W., and Begum, N. (2014). Hydraulic fracturing explained: evaluation, implementation, and challenges. Elsevier Science & Technology Ferréol, B., and Tothman, D. H. (1995). Lattice Boltzmann simulations of flow through lattice Boltzmann simulations of flow through Fontainebleau sandstone. Transport in Porous Media 20, Forchheimer, P. (1901). Wasserbewegung durch boden. Z. Ver. Deutsch. Ing 45(1782), Fredd, C. N., McConnell, S. B., Boney, C. L., and England, K. W. (2001). Experimental study of fracture conductivity for water-fracturing and conventional fracturing applications. SPE Journal 6(03), Ginzbourg, I., and Adler, P. M. (1994). Boundary flow condition analysis for the three-dimensional lattice Boltzmann model. Journal of Physics II France 4(2), Jin, G., Patzek, T., and Silin, D. (2004). Direct prediction of absolute permeability of unconsolidated and consolidated reservoir rock. SPE Annual Technical Conference and Exhibition, September, Houston, TX. doi: /90084-ms Kaufman, P. B., Anderson, R. W., Parker, M. A., Brannon, H. D., Neves, A. R., Abney, K. L., Cortes, G. W. K. P., Joyce, S. A., Ziegler, M. J., and Penny, G. S. (2007). Introducing new API/ISO procedures for proppant testing. SPE Annual Technical Conference and Exhibition, November, Anaheim, CA. doi: / ms Lee, J., and Wattenbarger, R. A. (1996). Gas Reservoir Engineering. Volume 5 of SPE Textbook Series. Society of Petroleum Engineers, Inc. Ma, H. and Ruth, D. W. (1993). The microscopic analysis of high Forchheimer number flow in porous media. Transport in Porous Media 13, Maier, R., and Bernard, R. (2010). Lattice Boltzmann accuracy in pore-scale flow simulation. Computational Physics 229,

11 Manwart, C., Aaltosalmi U., Koponen, A., Hilfer, R., and Timonen, J. (2002). Lattice Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media. Physical Review E 66(1), Much, M. G., and Penny, G. S. (1987). Long-term performance of proppants under simulated reservoir conditions. Low Permeability Reservoirs Symposium, May, Denver, CO. doi: /16415-ms Palisch, T. T., Duenckel, R. J., Bazan, L. W., Heidt, J. H., & Turk, G. A. (2007). Determining realistic fracture conductivity and understanding its impact on well performance-theory and field examples. SPE Hydraulic Fracturing Technology Conference, January, College Station, TX. doi: / ms Succi, S., Foti, E., and Higuera, F. (1989). Three-dimensional flows in complex geometries with the lattice Boltzmann method. Europhys. Lett. 10, Sukop, M. C., and Thorne, D. T. (2007). Lattice Boltzmann modeling: an introduction for geoscientists and engineers. Springer Publishing Company, Incorporated. Takashi A., Derivation of the Lattice Boltzmann Method by Means of the Discrete Ordinate Method for the Boltzmann Equation, Journal of Computational Physics, Volume 131(1), 1997, Takbiri-Borujeni, A. (2013). Multi-scale modeling of inertial flows through propped fractures. Ph. D. Dissertation, Louisiana State University. Takbiri-Borujeni, A., Lane, N., Thompson, K., and Tyagi, M. (2013). Effects of image resolution and numerical resolution on computed permeability of consolidated packing using LB and FEM pore-scale simulations. Computers & Fluids 88, Zeng, Z., and Grigg, R. (2006). A criterion for non-darcy flow in porous media. Transport in Porous Media 63(1), Biographies Ali Takbiri-Borujeni, PhD is an assistant professor of petroleum and natural gas engineering at West Virginia University. His research interests include image-based porescale modeling, computational fluid dynamics (CFD), reservoir simulation, Enhanced Oil Recovery (EOR), and hydraulic fracturing. His recent research focus is on the modeling of shale gas transport phenomena. He holds a Ph.D degree in petroleum engineering from Louisiana State University and two M.S. degrees in reservoir engineering and reservoir geoscience from Petroleum University of Technology (Iran) and Institute Français du Petrole (France). Mayank Tyagi, PhD is an associate professor in the Craft & Hawkins department of petroleum engineering, Louisiana State University, Baton Rouge. He is also jointly appointed at the Center for Computation & Technology (CCT), LSU. His current research interests include imagebased pore-scale modeling, computational fluid dynamics (CFD), reservoir simulations using high performance computing (HPC) platforms, and quantitative risk assessment (QRA) of offshore operations. He has published over 30 peer-reviewed articles in a various research areas. He graduated in mechanical engineering from Indian Institute of Technology, Kanpur and obtained his Ph.D. in mechanical engineering from Louisiana State University, Baton Rouge. 74 Volume 3 - Number 2