Impact of the loading stress variations on transport properties of granular packs

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1 ARMA Impact of the loading stress variations on transport properties of granular packs Takbiri Borujeni, A. West Virginia University, Morgantown, WV, USA Tyagi, M. Louisiana State University, Baton Rouge, LA, USA Kazemi, M. West Virginia University, Morgantown, WV, USA Copyright 2014 ARMA, American Rock Mechanics Association This paper was prepared for presentation at the 48 th US Rock Mechanics / Geomechanics Symposium held in Minneapolis, MN, USA, 1-4 June This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review of the paper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any position of ARMA, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMA is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 200 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented. ABSTRACT: Hydraulic fracturing is a well stimulation technique that makes the recoveries from the vast unconventional hydrocarbon resources in US economically feasible. Proppants are granular materials that are injected into hydraulic fractures to keep them open following a fracturing treatment. Hence, the proppant selection is of particular significance in petroleum industry. Due to recent advances in imaging technologies and high-performance computing, estimation of the elastic and transport properties of proppant packs at different closure stresses using imaged-based simulations is a credible alternative to direct experiments. In this study, transport properties (permeability and inertial flow par ameter) of a ceramic proppant pack exposed to varying loading stresses are calculated using Lattice Boltzmann (LB) model simulations. The images of this packing sh ows rearrangement of the packing structure, embedding of the grains at the rock wall, and crushing of individual proppants. LB simulation results of this packing show that the permeability and inertial flow parameter are less sensitive to stress variations before crushing of the grains occurs. 1. INTRODUCTION Since its advent in the 1950s, hydraulic fracturing has proven to be a robust technology, lending itself to many different types of reservoirs. It is one of the key methods for extracting unconventional oil and gas resources. Conductivity and non-darcy factors of the propped hydraulic fracture are among the most important design parameters of the hydraulic fracturing. These parameters are functions of proppant types, sizes, and their concentration. The size and mechanical strength of the proppants should be sufficient to allow high fracture conductivity and low non-darcy factor values, and to withstand the closure stress without getting mechanically crushed. Therefore, knowing the mechanical and hydraulic properties of the proppants is of particular importance. Stress-related changes of conductivity and non-darcy factor can be attributed to the alteration of pore space topology resulted from grain slippage and rotation, changes in proppant grain shapes, grain embedment, and grain fracturing. The proppant pack conductivity and non-darcy factors are usually estimated from standardized API conductivity tests of proppants in a linear flow cell at a specified pack concentration, closure stress, and temperature [1]. In 2001, the International Organization for Standardization (ISO) and American Petroleum Institute (API) have organized a new committee to write procedures for measuring the properties of proppants used for hydraulic fracturing. In 2003, a second committee was formed to write procedures on measuring long term conductivity of a proppant pack. These two new procedures will enable researchers to evaluate and to compare proppant characteristics under the described test conditions for use in hydraulic fracturing operations. They investigated impacts of elevated temperatures, fracturing fluid residues, cyclic stress loading, embedment, formation fines, and other factors on the proppant pack conductivity [2].

2 Barree et al. [7] performed a statistical survey of laboratory conductivity measurements on similar proppant samples at similar stress, temperature, and flow conditions. They concluded that small variations in proppant properties (less than 15 to 20% difference in conductivity) may not be statistically significant when applied to performance measurements in the field. According to them, conductivity description for a granular pack must rely on a statistically significant number of observations under similar packing conditions and along similar loading paths. They performed statistical analyses of pack width and permeability for different loadings. They concluded that conductivity variations (ensemble of variations of pack width and permeability) of +/- 20% about a mean are within laboratory accuracy for a given proppant type and size. Stephen et al. [6] described a vast series of crush tests performed on several proppant types and sizes. After the tests, the samples were sieved for a detailed analysis to measure a median particle diameter and the standard deviation of the distribution. At the end, a relationship was observed between the median particle diameters and the fracture conductivity and non-darcy factor. Rivers et al. [8] conducted long term conductivity tests on both coated and uncoated proppants. They have also performed some cyclic tests on uncoated proppants. They measured changes of fracture conductivity which was verified by dynamically measuring fracture width. In this work, a 20/40 mesh proppant sample is placed under varying loading stresses and imaged. Images are then used for estimations of mechanical and hydraulic properties using numerical pore-scale simulations. 2. BACKGROUND Proppant packs (or any granular media) are different from rocks in the sense that in the latter strong coupling forces exist among the solid grains and they could be considered as continuous media, while in the former such coupling forces are negligible. There are several properties of the proppant packs that are of importance in hydraulic fracturing. Among those are permeability, conductivity, and non-darcy factors. There are several approaches for estimations of these properties. Fracture permeability and non-darcy factors can be calculated using core flood experiments, correlations, and numerical image-based flow simulations. Fracture width can also be calculated using loading experiments and numerical estimations using three-dimensional and pseudo-three dimensional fracture models. In this study, permeability and non-darcy factors are calculated at each loading stress using image-based flow simulations on the three-dimensional images of the proppant pack. Width of the propped fracture is calculated by closely examining the locations of the proppant-berea interfaces in the images. In this section, a brief introduction to the numerical method and properties of the proppant packs are presented. 2.1 Lattice Boltzmann model Lattice Boltzmann (LB) method is a relatively new simulation technique to the well-known continuum hydrodynamic equations based on the kinetic theory to simulate various hydrodynamic systems. Owing to its particulate nature and local dynamics, LB has several advantages over other conventional CFD methods, especially in dealing with complex boundaries (such as porous media) and parallelization of the algorithm. Due to its attractive features, LB has many applications in petroleum engineering. LB equation with steaming and single relation collision operator (LBGK) can be written as: eq fi(x,t) fi (x,t) fi(x+ei Δt,t + Δt)= fi(x,t) (1) τ where e i are directions in which fluid particles can move, eq f i are the discrete distribution, and f i are the discrete distribution functions calculated from 2 2 eq (e i. u) (ei. u) u f i (x)= wi ρ(x) (2) cs 2c s 2c s where, w are weight factors, = 1/ 3 i c s is the sound speed in the fluid, u is the velocity of the fluid, and ρ is the density of the fluid. At the pore-solid interfaces, bounce back boundary condition is used. Flow is bodyforce driven and the lateral boundaries are periodic. For detailed discussions about the boundary conditions and simulation set-up refer to [2]. In this study, the PALABOS (Parallel Lattice Boltzmann Solver) code, which is a parallelized LB code for solving flow problems and is available from the web address has been used. All flow simulations were carried out on Louisiana Optical Network Initiative (LONI) resources. 2.2 Permeability Permeability of the samples are calculated using the following equation: u K = μ (3) P where K is the permeability tensor of the porous medium, u is the volume averaged velocity of the fluid inside the porous media, μ is fluid viscosity, and P is the pressure gradient across the medium. In this study, velocity and pressure fields are calculated using LB simulations and then, for each loading stress, Eq. 3 is

3 used to determine the permeability values at different directions. 2.3 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. Factors that can alter fracture conductivity are elevated temperatures, fracturing fluid residues, cyclic stress loading, embedment, and formation fines [2, 3]. In practice, loss of width to embedment is difficult to quantify and cannot be observed without disassembly of the proppant pack. In this study, it is possible to have accurate analyses of the width changes by analyzing the images of the proppant packs at each stress conditions. Fig. 1: Schematic cross-sections of propped fractures at different loading stresses. As the stress on the propped fracture increases, the width of fracture starts to decrease. (a) shows the proppant pack at no loading stress. (b) shows the propped fracture at a stress lower than which proppants start to crush. At this stress, there are some grain rearrangement, sliding, and embedments into the walls of the fracture. (c) shows the propped fracture at or above a stress at which grains start to mechanically fail. 2.4 Non-Darcy flows According to the Darcy s law, there is a linear relationship between fluid velocity and pressure gradients in the porous media at low Reynolds numbers (Re<1). For high flow rates, especially in regions around the wellbore and in hydraulic fractures, this linear relationship is no longer valid. Forchheimer (1901) proposed a relationship for predicting the pressure drops at higher Reynolds numbers. Δp L uμ = K + ρβu u where, β is Fochheimer coefficient or non-darcy factor. For calculations of non-darcy factors, Forchheimer equation is rearranged and a new parameter called apparent permeability, K app, is defined. Apparent permeability is defined the same way as permeability with a difference that apparent permeability is defined for all Reynolds numbers. Unlike permeability, apparent permeability is not a property of the porous media and it changes with flow velocities (as shown in equation (5)), (4) 1 1 = K app K ρu + μ β As can be seen in this equation, when flow velocities are small ( Re<1), apparent permeability and permeability values are close to one another. As flow velocity (and therefore Reynolds number) increases, apparent permeability decreases. According to equation (5), a plot of inverse of apparent permeability against ρu/ μ (pseudo-reynolds number) results a line with a slope of non-darcy factor. 3. SAMPLES Experiments, imaging, and segmentation of the proppant packs at different stress conditions are performed by Prof. Clint Willson s research group at LSU civil and environmental engineering department. Proppants are located at the mid-section of a cylinder and are sandwiched by a different type of porous medium (Berea Sandstone). For calculation of the permeability and non- Darcy factor, smaller regions from the original proppant packs at each stress conditions are selected for flow simulations. These regions are selected such that they are away from the Berea/proppant interfaces and consist merely the proppant grains. For further information about the samples refer to [3]. 4. RESULTS AND DISCUSSIONS The cylinder containing the proppants is of diameter 6.14 mm and 6.48 mm height, loaded with 9 to 10 layers of proppant stacking. Primary causes of the alterations in the width are grain rearrangement, embedment, and grain failure at contact points. The largest variation in width is observed by applying stress to the unstressed packing arrangement. Strain is a measure of the material deformation as a result of applied stress. If stress is exerted in z-direction, the original height of the proppant pack, z, changes by z. Strain in z-direction,, can be calculated by: z z = z z (5) (1) The width changes of the proppant pack were measured by closely examining the locations of the interfaces of the proppant grains and the surrounding Berea samples at each loading stress. Stress-strain plot of the proppant pack is shown in Fig. 2.

4 Reynolds numbers, apparent permeability and (absolute) permeability of the proppant packs are the same. Variations of apparent permeability values with Reynolds numbers are used for estimation of non-darcy factors. Fig. 2. Proppant pack deformation in response to mechanical loading (calculated from digital images). The original height and diameter of the proppant pack were 6.48 and 6.14 mm, respectively. Calculated values of non-darcy factor and permeability are tabulated in the Table 1 for all the flow directions. For the applied stresses less than 82.7 MPa (12 Kpsi ), flow simulation results show 15.3% reduction and 28.4% increase in the calculated values of z-direction permeability and non-darcy factor, respectively, compared to zero stress state. At MPa (20Kpsi), where some of the proppant grains crush, z-direction permeability and non-darcy factor changes are 63.0% reduction and 181.2% increase, respectively, compared to the calculated values at zero stress. In this stress, porespace topology changes due to creation of new narrow flow pathways because of the grain crushing. These new pathways can create local high velocity regions that can be responsible for generating high values of non-darcy factor and low values of the permeability. Table 1: calculated values of permeability and non-darcy factor for the proppant pack at different stresses. As loading stress increases, permeability values decrease and non-darcy factors increase. Fig. 3. Log-log plot of apparent permeability values in z- direction for each loading stress versus Reynolds number. Higher velocities (higher Reynolds numbers) results in reductions in apparent permeability values for each loading stress. At low Reynolds (R e<1) numbers, apparent and (absolute) permeability values are essentially the same. As loading stress increases, permeability values of the proppant pack decrease. By plotting inverse of apparent permeability against ρu/ μ, non-darcy factors can be calculated. Fig. 3 shows apparent permeability values in z-direction versus Reynolds number for each loading stress. As can be seen, as flow velocities increase (Re numbers increase), apparent permeability values decrease. At low Fig. 4. Calculated z-direction permeability values versus loading stress. A comparison of the calculated results with published results in [11] and [12] are performed. Our results show that the changes in the permeability values at stresses lower than 82.7 MPa is 15.3%. At MPa, the calculated value is 37.0% of the calculated permeability in zero stress. This significant decrease is attributed to the changes in the topology of the proppant pack due to grain fracturing. Stress-dependent variations of non-darcy factor and permeability are plotted against published values in Fig. 4 and Fig. 5, respectively. In Fig. 4, calculated permeability values are compared to the published

5 results in [11] and [12]. Reduction of the permeability values at 82.7 MPa is 15.3% compared to the calculated value at zero stress. Calculated permeability values also show small variations with stress below 82.7 MPa compared to the published results. In Fig. 5, a comparison of the calculated non-darcy factors of the proppant pack at different closure stresses against the published values of the Bauxite Ceramic ( BC), Intermediate Density Ceramic ( IDC), and Light Weight Ceramic (LWC) proppants in [6], and proppants in [10] and [12] is performed. Calculated non-darcy factor values increase by 28.4% at 82.7 MPa compared to the values at zero stress. At Mpa, they increase to 281.2% of the calculated values at zero stress state. It seems that grain crushing has a more distinct impact on the calculated values of non-darcy factor in this work compared to the published results. Fig. 5: Calculated non-darcy factors versus stress. Non-Darcy values are compared with the published results in [11] and [6]. The calculated results in this study show higher non-darcy values compared to the plotted published results at loading stresses below 27.6 MPa. At 82.7 MPa, a 28.4% increase in the non-darcy factor is observed compared to the zero stress state. The calculated values at MPa increase to 281.5% of the calculated values at zero stress. Fig. 6. Conductivity variations against loading stress. Based on the calculated results in this study, conductivity values exhibit less variations with the increasing loading stress at stresses below 82.7 MPa (12 Kpsi)) compared to the published results. Fig. 6 shows the calculated proppant pack conductivity values versus the confining stress. The conductivity values are plotted against the published values of IDC and BC proppants in [6], and proppants in [11] and [12]. Conductivity is the product of the z-direction permeability values and width of the propped fracture. At 82.7 MPa, calculated value of conductivity decreases by 21.2% compared to the zero stress state. 60.6% reduction of the conductivity compared to the zero stress state is observed at MPa. 5. CONCLUSIONS & RECOMMENDATIONS Based on the results, following conclusions can be drawn: Analyses of the three-dimensional images of the propped fractures at different loading stresses provide accurate and insightful tools for estimation of elastic and hydraulic properties. Based on the calculated results in this study, permeability, non-darcy-factor, and fracture conductivity values change by 63.0%, 181.2%, and 60.1%, respectively, at MPa compared to zero state stress state. Grain fracturing has a major impact on the calculated hydraulic properties of the propped fractures. In this study, crushed particles cannot move (since our tests are numerical and we did not consider solid movement in our simulations). In practice, crushed particles can shift to the open spaces because of the drag force exerted on them 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.

6 Pore size distributions of the proppant packs at different loading stresses can provide insightful information about the variations of pore space topology as result of compaction. 6. ACKNOWLEDGEMENT The authors are grateful to Prof. Clint Willson (LSU Civil and Environmental Engineering Dept.) for access to the digital proppant images. Author (MT) would like to acknowledge ExxonMobil Upstream Research for funding parts of this research at LSU. We also acknowledge the PALABOS group for the open source code used for LB simulations in this study ( The authors also thank LSU Center for Computation and Technology (CCT) and Louisiana Optical Network Initiative (LONI) for providing the high performance computing resources. REFERENCES 1. Kaufman, P. B. (2007). Introducing New API/IS O Procedures for Proppant Testing. Society of Petroleum Engineers. doi: / ms 2. Takbiri Borujeni, A. (2013). Multi-scale modeling of inertial flows through propped fractures. PhD dissertation, Louisiana State University. 3. Shen Y., Sanematsu P., Bradley S. M., Wilson C., Ali Takbiri Borujeni A., Thompson K. E., Tyagi M. (2012), Non-Darcy flow parameter estimation of proppant packs under varying confining stress using pore-scale flow simulations, AICHE Annual Meeting 4. Hazzard, J.F., R.P. Young, and S.C. Maxwell Micromechanical modeling of cracking and failure in brittle rocks. J. Geophys. Res. 105: 16,683 16, Wu, A., & Sun, Y. (2008). Basic Physical and Mechanical Properties of Granular Media. In Granular Dynamic Theory and Its Applications (pp. 5 29). 6. Stephen, S., & David, M.-T. (2004). Investigating How Proppant Packs Change Under Stress. In Society of Petroleum Engineers. doi: /90562-ms 7. Barree, R. D., Cox, S. A., Barree, V. L., & Conway, M. W. (2003). Realistic Assessment of Proppant Pack Conductivity for Material Selection. In Society of Petroleum Engineers. doi: /84306-ms 8. Rivers, M., Zhu, D., & Hill, A. D. (2012 ). Proppant Fracture Conductivity with High Proppant Loading and High Closure Stress. In Society of Petroleum Engineers. doi: / ms 9. Ye, X., Tonmukayakul, P., Weaver, J. D., & Morris, J. (2012). Experiment and Simulation Study of Proppant Pack Compression. In Society of Petroleum Engineers. doi: / ms /1001_70_C_HSP_ts_Lr.pdf 12. FTSI Proppants Manual for Premier Partially Cured (PPC) 20/40 Resin Coated Proppants 10. Schubarth, S. K., Cobb, S. L., & Jeffrey, R. G. (1997). Understanding Proppant Closure Stress. In Society of Petroleum Engineers. doi: /37489-ms