Displacement Discontinuity Method Research Articles

Fully Coupled Model for Calculating the Effective Acid Penetration Distance during acid fracturing

A new thermal-hydro-mechanical-chemical coupled acid fracturing model is proposed to analyze the effective acid penetration distance during acid fracturing. To improve the computational efficiency of the model, we used a simple three-dimensional displacement discontinuity method. Additionally, a finite-volume method was used to calculate the flow field. Temperature- and chemical-field models were also established based on energy- and material-conservation criteria, respectively. Subsequently, a new full implicit solution was established to solve the problem of rock deformation, stress interference, temperature, and acid–rock reaction coupling. The results show that the greater the injection rate, the greater is the effective acid penetration distance. To ensure that this distance is as large as possible, the injection speed should be maximized in the acid-fracturing process. Similarly, a high acid concentration results in a larger effective acid penetration distance. Therefore, the original concentration of the acid may be moderately increased, which is the simplest and most straightforward way to achieve a high acid penetration distance. If increasing the viscosity does not reduce the transfer rate of H+ and the order of the acid–rock reaction, the effective acid penetration distance is reduced. Therefore, the inter-dependence of multiple fields during construction optimization should be considered. In addition to construction and fluid parameters, reservoir parameters also have a significant effect on the effective acid penetration distance and acid-fracturing process. Reservoir temperature influences the effective acid penetration distance, fluid-temperature distribution in the fracture, and etching width. Finally, a high reservoir temperature results in a rather small effective acid penetration distance.

A thermo-hydromechanical displacement discontinuity method to model fractures in high-pressure, high-temperature environments

Geothermal reservoirs, oil wells, radioactive waste disposals, and deep underground mines deal with high-temperature and high-pressure problems. The thermo-hydromechanical coupling may significantly affect the behavior of the rock mass in these applications. Fractures and joints are the main conduits of thermal and hydraulic transition. The displacement discontinuity method (DDM) is ideally suited to model problems containing fractures. However, the DDM in its original formulation is restricted to elasticity problems. It is formulated in this study to take into account the thermo-hydromechanical effects. A numerical formulation and implementation for the thermo-hydromechanical DDM is derived. The proposed numerical model is validated in three parts. The poroelastic, thermoelastic, and thermo-hydromechanical couplings are each validated by analytical or experimental results. The results showed good agreement between the proposed numerical model and analytical or experimental results over various periods. The validations proved the accuracy and applicability of the proposed thermo-hydromechanical numerical model in a wide range of problems. Furthermore, the thermo-hydromechanical effect on crack opening displacement (COD) is modeled. Numerical simulation showed that the maximum COD due to only thermal effects may be reached after almost a year.

Optimization of horizontal well fracturing in shale gas reservoir based on stimulated reservoir volume

Optimization of multi-stage fracturing horizontal wells is with great significance in shale oil and gas reservoir production to enhance the gas recovery. The stimulated reservoir volume (SRV) is an important factor for fracturing horizontal well design and gas reservoir development. However, very few research has studied horizontal well optimization considering the effect of SRV. This paper proposes a mathematical model for SRV estimation during hydraulic fracturing. The model couples both rock and fluid mechanic effects. The displacement discontinuity method (DDM) is adopted to calculate the formation stress field. Then a natural fracture failure criterion based on formation pressure and stress fields is used to estimate the SRV. Additionally, a stress dependent permeability (SDP) algorithm is applied to estimate permeability change within the SRV region. Based on SRV and SDP models, an automatic optimization framework is established, with the net present value (NPV) employed as the optimization objective function, in which the overall cost and the ultimate income are involved simultaneously. A Modified-PSO algorithm is proposed and adopted to realize the optimization process, which can significantly reduce the calculation time while maintaining the accuracy of the result. In the end, two cases with shale gas reservoirs are presented, and different types of hydraulic fracturing wells are optimized to validate this optimization framework. Parameters including hydraulic fracturing spacing and half-length, well spacing and well length are optimized to achieve the optimum well design. The optimization results are compared and discussed, which show that this approach can contribute to obtaining the optimal shale gas recovery and provide a new insight into hydraulic fracturing well design under the effect of SRV.

Modeling of Borehole Hydraulic Fracture Initiation and Propagation with Pre-existing Cracks Using the Displacement Discontinuity Method

Pre-existing borehole cracks result in hydraulic fracture deflection, which causes sand screen-out in advance and even fracturing failure. Based on a simplified three-dimensional displacement discontinuity method, a model to simulate the initiation and propagation of hydraulic fractures from a borehole with pre-existing cracks was established. It is found that the effects of injection rate and fracturing fluid viscosity on fracture propagation are similar. When the effects are smaller, hydraulic fractures quickly turn perpendicular to the minimum horizontal stress. With injection rate or viscosity increase, the injection pressure at the well bottom increases, and the turning degree and width of hydraulic fractures increase. The length of pre-existing cracks dramatically affects the breakdown pressure. With increasing pre-existing crack length, the breakdown pressure decreases. With increasing horizontal stress contrast, the hydraulic fracture path clearly changes, and the required extension pressure also increases when the injection rate and fracturing fluid viscosity both remain constant.

Analysis of anti-plane interface cracks in one-dimensional hexagonal quasicrystal coating

The displacement discontinuity method is extended to study the fracture behavior of interface cracks in one-dimensional hexagonal quasicrystal coating subjected to anti-plane loading. The Fredholm integral equation of the first kind is established in terms of displacement discontinuities. The fundamental solution for anti-plane displacement discontinuity is derived by the Fourier transform method. The singularity of stress near the crack front is analyzed, and Chebyshev polynomials of the second kind are numerically adopted to solve the integral equations. The displacement discontinuities across crack faces, the stress intensity factors, and the energy release rate are calculated from the coefficients of Chebyshev polynomials. In combination with numerical simulations, a comprehensive study of influencing factors on the fracture behavior is conducted.

A 3-D hydraulic fracture propagation model applied for shale gas reservoirs with multiple bedding planes

Rock layering, a critical factor in determining fracture height growth, is pervasive in Longmaxi shale formation in the southwest of China. This formation has characteristics of large burial depth, low porosity and multiple bedding layers that hamper reaching the target fracture height even after increasing the pumping rate and treatment size. Hence, it becomes much significant to develop a fracture propagation model considering the effect of bedding layers on fracture height growth. This paper introduces a coupled 3-D hydraulic fracture propagation model and investigates the influence of shear displacement discontinuities along bedding planes on fracture height growth. Our model addresses rock deformation and fluid flow. Rock deformation is governed by a fully three-dimensional displacement discontinuity method (3D DDM). The fluid flow model employs a finite difference method (FDM) being able to capture fluid movement along vertical fractures and bedding planes. Additionally, a propagation criteria determines whether the fracture would penetrate bedding planes. In this paper, we selected two different fracture geometries and analyzed profiles of fracture width, pressure and two types of shear displacement discontinuities. From numerical investigations, we found that the maximum width can be obtained at the junction after the vertical fracture penetrated the bedding planes as a result of the decrement of the compressive stress acting on the bedding planes. As the fracture penetrates the bedding planes, a certain amount of fluid would leak into the planes, which leads to fracture height containment. Moreover, the slope utilized for characterizing the correlation between leak-off volume and fracture height, is regarded as a tool to identify the number of BPs that fracture penetrates through. This paper illustrates the potential application of our 3-D fracture propagation model for Longmaxi shale formation with multiple bedding layers. Shear displacements along bedding planes are regarded as a primary mechanism of fracture height containment.

A high-efficiency 3D boundary element method for estimating the stress/displacement field induced by complex fracture networks

The displacement discontinuity method (DDM) is widely used in engineering problems such as hydraulic fracturing stimulation in unconventional reservoir and tunnel excavation simulation. Because the geometry of the artificial/natural fracture is intrinsically three-dimensional and quite complex, the 3D-DDM is more suitable than 2D-DDM to obtain more accurate and reliable calculation results such as the fracture-induced stress/displacement field. However, the analytical solution for 3D-DDM with triangular elements is very complicated and can only be applied to planar fractures and the numerical solution would introduce the singular and hyper-singular integrals in the calculation process. A new method that combines analytical and numerical solutions is proposed in this paper, which is validated by the solution of a penny-shaped fracture and the new Gaussian quadrature formula for standard triangles (GQSTS). Furthermore, we found that the new method is 32% faster than that of the pure numerical solution. The proposed method is then applied to perform sensitivity analyses associated with the fracture opening under a single fracture and multiple hydraulic fractures. Numerical results demonstrate that (1) not the fracture length or height but the aspect ratio is the dominant factor of the rectangular-shaped fracture opening; (2) the fracture inclination angle affects the fracture opening via altering the net pressure; (3) rock elastic properties also have great impact on the fracture opening, which is more sensitive to Young's modulus than Poisson's ratio; (4) both the cluster spacing and the fracture geometry could significantly alter the stress shadowing effect, which can be negligible when the cluster spacing is twice greater than the fracture radius or the short edge; and (5) the fracture opening and the induced stress become more intricate in complex fracture networks, and the potential stress-reorientation regions extend further in the outside directions of the entire network. This study establishes a high-efficiency 3D-DDM with triangular elements for calculating the stress/displacement field induced by complex fractures and may serve as a basis for hydraulic fracturing modeling via incorporating fluid flow and proppant migration.

Study of the cylindrical wave propagation across a single rock joint with nonlinear normal deformation

ABSTRACT Rock joints, usually with nonlinear deformation, widely exist in many natural rock masses, which greatly affect stress wave propagation. In practical engineering, the generated stress wave is taken as a cylindrical wave when an explosion with cylindrical charge or a tunneling explosion occurs. The cylindrical wave propagation in jointed rock mass is a great concern of underground structure safety. In this paper, the time-domain recursive method (TDRM) is extended to investigate cylindrical stress wave propagation through a rock joint with nonlinear normal deformation. Based on the conservation of momentum at the wave fronts and the displacement discontinuity method, a quantitative analysis for the interaction between wave and the joint is carried out and the wave propagation equation is accordingly deduced by considering the attenuation. Verification of the TDRM-based results is conducted by comparison with the simulated results from UDEC code. Finally, parametric studies are carried out, which include the influences of joint stiffness, the maximum allowable closure of joint and the frequency of the incident wave on the transmission and reflection coefficients.

Optimization of Plug-and-Perforate Completions for Balanced Treatment Distribution and Improved Reservoir Contact

Summary Heel-dominated treatment distribution among multiple perforation clusters is frequently observed in plug-and-perforate (plug-and-perf) stages, causing small propped surface areas, suboptimal production, and unexpected fracture hits. A multifracture simulator with a novel wellbore-fluid and proppant-transport model is applied to quantify treatment distribution among multiple perforation clusters in a plug-and-perf operation. A simulation base case is set up on the basis of a field treatment design with four clusters. Simulation results show that the two toe-side clusters screened out early in the treatment and the two heel-side clusters were dominant. The simulated proppant placement is consistent with distributed-acoustic-sensing observations. The impact of different perforating strategies and pumping schedules on final treatment distribution is investigated. Two criteria are defined that quantify the proppant distribution and fracture area: the weighted average (WA) and standard deviation (SD) of the final fluid and proppant distribution, as well as the hydraulic surface area (HSA) and propped surface area (PSA) of the created fractures. An optimal plug-and-perf design is defined as one that minimizes the SD of the treatment distribution among perforation clusters, and maximizes the PSA. Both perforating strategy and pumping schedule are found to affect the final treatment distribution significantly, and uniform treatment distribution is shown to create more PSA. Having fewer perforations per cluster was found to promote uniform fluid and proppant placement. Other helpful strategies include reducing the number of perforations near the heel and using small, lightweight proppant. The stress shadow effect is accounted for using the displacement discontinuity method (DDM) and was found to play a smaller role than perforation friction and proppant inertia in most cases. An automated process is developed to optimize plug-and-perf completion design with multiple decision variables using a genetic algorithm (GA). Thirteen parameters are optimized simultaneously. The optimal design solution creates an almost even treatment distribution and more than doubles the PSA compared with the base case. The multifracture model presented in this paper provides a way to quantify fluid and proppant distribution for any perforating strategy and pumping schedule, and provides more insight into the physics relevant to plug-and-perf treatment distribution. The perforation and pumping schedule recommendations presented in this paper provide directional guidance for the design of fracturing jobs with balanced treatment distribution and large PSA.

Applications of the fast multipole fully coupled poroelastic displacement discontinuity method to hydraulic fracturing problems

In this study, a fast multipole method (FMM) is used to decrease the computational time of a fully-coupled poroelastic hydraulic fracture model with a controllable effect on its accuracy. The hydraulic fracture model is based on the fully-coupled poroelastic formulation of the displacement discontinuity method (DDM) which is a special formulation of the boundary element method (BEM). DDM is a powerful and efficient method for problems involving fractures. However, this method involves the multiplication of a dense matrix with a vector in several places. Thus, the DMM algorithm slows down drastically as the number of elements increases, and even more so with the inclusion of necessary details such as poroelasticity, which makes the solution history-dependent. Rather, FMM is a technique to expedite matrix-vector multiplications, within a controllable error range, by approximating the far-range interactions without forming the matrix explicitly. In doing so, FMM offers a leverage in the computational efficiency by modifying the algorithm of the fully-coupled poroelastic displacement discontinuity method (PDMM) in two places. The first modification is in the time-marching scheme, which accounts for the solution of previous time steps to compute the current time step. The second modification is in the generalized minimal residual method (GMRES), where the unknowns are solved for iteratively.Several examples are provided to show the efficiency of the proposed fast multipole fully-coupled poroelastic displacement discontinuity model (FMPDDM) in problems with large degrees of freedom (in time and space). These examples include hydraulic fracturing of a horizontal well and randomly distributed pressurized fractures at different orientations with respect to principal horizontal stresses. The results of FMPDDM are compared to PDDM in terms of computational time and accuracy, which demonstrates that FMPDDM decreases the computation time by up to a factor of 70 for a case with 20000 piece-wise constant elements and a single time step. The difference between the computational times of FMPDDM and PDDM can be higher for a larger number of elements and time steps. Moreover, the associated absolute value of relative error for a problem of this size is observed to be less than 4% for the hydraulic fracture example, and less than 0.5% for the example of randomly distributed pressurized fractures. Furthermore, the solution of tip displacements using both methods are used to compare the computation of stress intensity factors (SIF) in mode I and II, which are needed for fracture propagation. The calculated displacement discontinuities reveal that the SIF calculation using FMPDDM has a negligible error compared to PDDM. Therefore, FMPDDM will not affect the estimation of the fracture propagation direction. Accordingly, the proposed algorithm may be used for fracture propagation studies to substantially reduce the computational time.

Analysis of elastic−plastic fracture in two-dimensional decagonal quasicrystals using the displacement discontinuity method

The generalized Dugdale model of cracks in two-dimensional decagonal quasicrystals is studied by developing the extended displacement discontinuity boundary element method. Expressions for the phonon and phason stress intensity factors and energy release rate are given in terms of the phonon and phason displacement discontinuities near the crack front. The extended displacement discontinuity boundary element method, with an iterative approach of determining the size of the yielding zone, is adopted to numerically simulate the nonlinear fracture problem. The effects of phonon and phason loadings, as well as the yield stress on the size of yielding zone and energy release rate, are discussed.

Investigation of Rupture and Slip Mechanisms of Hydraulic Fractures in Multiple-Layered Formations

SummaryWeak bedding planes (BPs) that exist in many tight oil formations and shale–gas formations might strongly affect fracture–height growth during hydraulic–fracturing treatment. Few of the hydraulic–fracture–propagation models developed for unconventional reservoirs are capable of quantitatively estimating the fracture–height containment or predicting the fracture geometry under the influence of multiple BPs. In this paper, we introduce a coupled 3D hydraulic–fracture–propagation model considering the effects of BPs. In this model, a fully 3D displacement–discontinuity method (3D DDM) is used to model the rock deformation. The advantage of this approach is that it addresses both the mechanical interaction between hydraulic fractures and weak BPs in 3D space and the physical mechanism of slippage along weak BPs. Fluid flow governed by a finite–difference methodology considers the flow in both vertical fractures and opening BPs. An iterative algorithm is used to couple fluid flow and rock deformation. Comparison between the developed model and the Perkins–Kern–Nordgren (PKN) model showed good agreement.I–shaped fracture geometry and crossing–shaped fracture geometry were analyzed in this paper. From numerical investigations, we found that BPs cannot be opened if the difference between overburden stress and minimum horizontal stress is large and only shear displacements exist along the BPs, which damage the planes and thus greatly amplify their hydraulic conductivity. Moreover, sensitivity studies investigate the impact on fracture propagation of parameters such as pumping rate (PR), fluid viscosity, and Young's modulus (YM). We investigated the fracture width near the junction between a vertical fracture and the BPs, the latter including the tensile opening of BPs and shear–displacement discontinuities (SDDs) along them. SDDs along BPs increase at the beginning and then decrease at a distance from the junction. The width near the junctions, the opening of BPs, and SDDs along the planes are directly proportional to PR. Because viscosity increases, the width at a junction increases as do the SDDs. YM greatly influences the opening of BPs at a junction and the SDDs along the BPs. This model estimates the fracture–width distribution and the SDDs along the BPs near junctions between the fracture tip and BPs and enables the assessment of the PR required to ensure that the fracture width at junctions and along intersected BPs is sufficient for proppant transport.

Three‐dimensional elastic analysis of cracks with the g2 constant displacement discontinuity method

SummaryA new triangular element was created that could be used for the improvement of the accuracy of the constant displacement discontinuity method (CDDM). This element is characterized by three degrees of freedom in the three‐dimensional space as in the classical CDDM approach. The element is based on strain gradient elasticity theory that accounts for the difference of the average value of stress with the local stress at surfaces with large curvature (eg, crack borders, corners, and notches) in elastic bodies. The new element is characterized by a strain gradient term in addition to the two Lamè constants that gives a more representative value of the stresses at the centroid of crack edge elements compared with the classical elasticity solution and thus an accurate stress intensity factor. In this approach, special crack border elements with square‐root radius dependent displacements and numerical integrations are avoided. The extra strain gradient term is calibrated once only on the analytical solution for the penny‐shaped crack. In a verification stage, the accuracy of the computational algorithm for the elliptic and rectangular crack problems is demonstrated. Then, the algorithm, which also accounts for crack closure in compression, is applied for the modeling of crack propagation and crack interaction in uniaxial tension and compression loading. It is illustrated that the numerical predictions are in accordance with experimental evidence pertaining to uniaxial compression of transparent precracked specimens in the lab.

Three-Dimensional Propagation Simulation and Parameter Analysis of Rock Joint with Displacement Discontinuity Method

Displacement discontinuities method (DDM) is convenient for and efficient at dealing with discontinuity problems such as fracture and fault. However, the known method of ease iteration is not available for nonlinear joint surface problems. This paper introduces Barton-Bandis nonlinear joint deformation failure criterion, figures out the propagation model of joint through the maximum energy release rate theory of rock fracture mechanics, and establishes three-dimensional nonlinear displacement discontinuity model for rock joint. This paper gives results of the joint propagation pattern and its distribution law under tension and compression with different parameters and side pressure coefficients via compiled program.

Fundamental solutions and analysis of the interface crack for two-dimensional decagonal quasicrystal bimaterial via the displacement discontinuity method

The extended displacement discontinuity method is developed to analyze an interface line crack in a two-dimensional decagonal quasicrystal bimaterial. According to the general solutions and Fourier transform, fundamental solutions for phonon and phason displacement discontinuities and phonon and phason stresses are obtained. The delta function in the phonon and phason stresses is replaced by the Gaussian distribution function to eliminate the oscillatory singularity near the crack front. The expressions of phonon and phason stress intensity factors are given in terms of phonon and phason displacement discontinuities. The finite element analysis software COMSOL is used to validate the developed method.

A numerical scheme of coupling of fluid flow with three-dimensional fracture propagation

In this paper, we combine a displacement discontinuity method (DDM) for rock deformation and a finite volume method (FVM) for fluid flow in fractures in rock mass to set up a numerical simulation scheme for coupling of hydraulic fracture flow with three-dimensional fracture propagation. Both the DDM and FVM discretise the fractures as the problem domain into small planar parts and same discretisation mesh is used by the two methods. The fluid flow is modelled with the cubic law in each control volume. A sequential coupling iteration procedure is employed to solve the two systems of linear equations. The numerical simulation scheme has been implemented in an existing three-dimensional fracture propagation code FRACOD3D. Both the DDM and FVM have been verified previously by the authors and the sequential iteration procedure is tested in this paper. Then the numerical simulation scheme is employed to simulate an illustration example of hydraulic fracturing and investigate the effects of fractures’ sizes and positions on the fracture propagation configurations.

Deflected trajectory of a single fluid-driven crack under anisotropic in-situ stress

Deflected crack trajectories are detected in hydraulic fracturing. There are numerous physical quantities impacting the deflection of a fluid-driven crack. A 2D model of the elasto-hydrodynamics system is established to predict the crack trajectory and to investigate the coupling effects of anisotropic in-situ stress, fluid viscosity, fluid injection rate and crack inclination on the crack deflection. The displacement discontinuity method (DDM) and the finite difference method (FDM) are used to solve the nonlinear problem. The concept of deflection region is defined to quantify the crack deflection. Results show that the size of the deflection region is dominated by the inclination angle. Increase of the magnitude and the difference of in-situ stresses promotes crack deflection. The combined effects of fluid viscosity and fluid injection rate are divided into three stages: significant prevention stage, significant promotion stage and insignificant prevention stage. This study provides the method for crack trajectory prediction and crack deflection control.

Analysis of stress interference among multiple hydraulic fractures using a fully three-dimensional displacement discontinuity method

Due to the extremely low permeability of shale matrix, a great number of hydraulic fractures are required to enhance the flow capacity of shale reservoirs to obtain economic productivity. The mechanical interactions among closely spaced hydraulic fractures could result in fracture curving, intersection and unbalanced growth. Current hydraulic fracturing models are mainly based on the assumption of 2D or Pseudo-3D, in which the fractures are vertical with constant or equilibrium height. To better analyze and understand the stress interference among multiple hydraulic fractures, a fully three-dimensional (3D) model based on the displacement discontinuity method (DDM) is developed in this paper. The challenges (eq. grid twisting, hyper-singular integrals) in implementing 3D DDM for fracture growth problems are systematically discussed and solved from different aspects. Optimization strategies for the 3D model, including adaptive mesh growth, non-crowding Gaussian points and distance-dependent integrals, are proposed. Using this model, we first compare the strength of stress interactions with different fracture geometries. The simulation results show that the length of the shorter edge of the hydraulic fracture dominates the strength of stress shadowing effect. The stress redistribution due to fracture interference in 3D space is also calculated which delivers a much more complex shape of potentially stress re-orientation region than the 2D results. Then, the simultaneous propagation of multiple hydraulic fractures in different in-situ stress fields is analyzed, highlighting the importance of in-situ stress distribution on fracture geometries and interactions. The limitation on fracture height growth can reduce the mechanical interactions under the constant net pressure assumption. Besides, the detailed introduction of the present stable and efficient 3D DDM-based fracture propagation model can be used as a basis for other investigation purposes.

Numerical Investigation of Effects of Subsequent Parent-Well Injection on Interwell Fracturing Interference Using Reservoir-Geomechanics-Fracturing Modeling

SummaryBecause of interwell interference, the completion and production of infill wells in unconventional reservoirs often change established production profiles for parent wells and lead to infill-well production lower than expected. Parent-well injection has been used in some fields in an attempt to reduce interwell interference. However, mixed responses were received from these attempts, and few modeling studies have been presented to investigate the mechanisms of the mixed responses. This study investigates the effects of subsequent injection in parent wells with legacy production on interwell interference using a data set from Eagle Ford Shale. A numerical-modeling work flow is presented for the characterization of poroelastic behaviors of multiphase-fluid diffusivity and rock deformation using the finite-element method and multifracture propagation using the displacement discontinuity method. It solves for the spatial-temporal evolutions of pore pressure and in-situ stress because of parent-well production and injection and models the fracture propagation during infill-well completion on the basis of updated heterogeneous in-situ stresses. Thus, the approach obtains the interwell fracture network comprising parent-well fractures and fractures from infill-well completion and captures fracture hits, which are necessary for the analysis of the injection effectiveness. Numerical results indicate that subsequent injections in parent wells make infill-well fractures grow more transversely, denoting improved completion qualities of infill wells. Also, the required subsequent injection volume leading to transverse infill-well fractures is positively correlated with the volume of legacy production in parent wells. In addition to subsequent injection volume, locations of perforation clusters along the infill well are another key parameter affecting the associated interwell interference. Results show that it is easier to generate fracture hits after infill-well completion, when perforation-cluster locations along the infill wellbore are identical to those along parent wellbores. In contrast, certain infill-wellbore perforation-cluster locations different from those in parent wellbores guarantee transverse infill-well fractures and avoid fracture hits during/after infill-well completion. On the basis of the numerical results in this specific study, when infill-well perforation cluster locations are properly placed, the volume of parent-well subsequent injection should be at least 76.9% of the total depleted liquid volume during the legacy production of parent wells for subsequent injection to be effective in avoiding fracture hits. This value is on a case-by-case basis and should not be generalized. The contribution of this work lies in its analyses of the mixed performance by parent-well subsequent injection in the reduction of interwell interference using a reservoir-geomechanics/fracturing modeling work flow.

Approximations for fluid pressure and flux of hydraulic flow in three-dimensional fractures

In this paper, we implement a finite volume method for approximating the pressure of fluid flowing within three dimensional fractures and formulate a scheme for computation of the fluid velocity and flux in the fractures. These are needed to couple with an existing fracture propagation code based on a displacement discontinuity method for simulation of hydraulic fracturing processes. Both of the methods discretise the fracture only, not its two surfaces. The pressure and velocity computed with the proposed numerical schemes are shown to qualitatively agree with analytic solutions for several cases of flow in planar annular fractures.