Fluid-driven fracturing is fundamental to geophysical processes (e.g. fluid-induced seismicity) and engineered applications such as geothermal energy extraction. This study investigates the coupled influence of rock heterogeneity and fracture propagation regime on hydraulic fracture growth. Using a fully coupled discrete element-pore network model, we simulate fracture propagation driven by both Newtonian and shear-thinning power-law fluids. Rock heterogeneity is explicitly represented by a Weibull distribution of mechanical properties, allowing fractures to initiate and propagate without predefined paths. Model validation shows excellent agreement with classical Khristianovic–Geertsma–de Klerk (KGD) solutions across viscosity- and toughness-dominated regimes. Our results demonstrate that heterogeneity strongly influences fracture morphology. In highly heterogeneous rocks, viscosity-dominated propagation creates broad process zones with distributed microcracking and high orthogonal roughness. In contrast, toughness-dominated propagation leads to localized stress perturbations, forming tortuous and asymmetric fractures. Shear-thinning fluids reduce the morphological contrast between regimes by decreasing the effective near-tip viscosity, while time-evolving propagation behavior further moderates regime-dependent differences. Ultimately, fracture complexity is governed by the interplay between fluid rheology and material heterogeneity. These findings provide critical insights for optimizing fluid selection and injection parameters to achieve desired fracture complexity and containment in heterogeneous reservoirs.

It is crucial to understand how multiple hydraulic fractures interact with each other to improve effectiveness of fracturing treatments. This study employs a fully coupled DEM-based hydraulic fracture model to investigate the effect of intra- and inter-stage stress shadowing across different propagation regimes under limited entry conditions. Fracture propagation is modeled during injection and shut-in, with the coupled hydro-mechanical scheme validated against existing analytical solutions for radial fractures. The results show that the competitive propagation of multiple fractures is influenced by the fracture propagation regime, stress orientation angle and fracture spacing. Fractures tend to grow uniformly in the viscosity-dominated regime, whereas in the toughness-dominated regime, fractures exhibit petal-like shapes with directional growth preferences. The interaction between fractures during shut-in further complicates the process. Inter-stage stress shadowing promotes secondary propagation of early-stage fractures due to compression stress caused by later fractures, resulting in more significant variations in fracture radii across stages. In contrast, intra-stage stress shadowing causes multiple fractures within a single stage to deviate from symmetrical growth. The stress orientation angle, however, has a relatively minor effect on fracture morphology. As fracture spacing increases, the effect of stress shadowing diminishes, allowing each fracture to behave more like an isolated fracture in its respective propagation regime.

This work presents a unified scaling law for a plane strain hydraulic fracture propagation in layered rocks, where fracturing behavior is influenced by both layer thickness and property contrasts across interfaces, leading to non-self-similar propagation over time. The singular integral balance equation is derived by introducing a kernel function that accounts for the varying elastic modulus and a modified load term that incorporates in-situ stress. The layered distribution of elastic modulus and fracture energy is considered using dynamic tip asymptotics. The governing equations and boundary conditions are then made dimensionless by new characteristic scales of fracture opening, fluid pressure, and fracture length, which are proposed to depict the spatial relationship between the interface and fracture. Solutions are obtained through a decoupled approach to match the fracture morphology and tip boundary conditions. This model captures a novel time-sensitive propagation mode of a hydraulic fracture, that can be dominated by toughness in the tip region and viscosity dissipation at the interface region, especially in multilayer rocks. Consequently, the temporal scale of injection time and the spatial scale of layer thickness are integrated into the characteristic scales. A scaling law is thus proposed to fully consider the variations of elastic modulus, fracture energy, fluid injection rate, injection time, and layer thickness on propagation behavior. The law spans the temporal-spatial parameter space within a general framework that quantifies the evolution of viscosity dissipation induced by interfaces, summarizing four specific cases: the homogeneous model, single-interface model, multi-layer model, and homogenized model. The proposed law provides a universal measure for modeling hydraulic fracturing of layered heterogeneous rocks with arbitrary thickness and propagation stage.

Hot dry rock (HDR) is an important geothermal resource whose development relies on hydraulic fracturing to create high-conductivity fracture networks for efficient geothermal energy extraction in Enhanced Geothermal Systems (EGS). In this study, a coupled Thermo-Hydro-Mechanical (THM) discrete element model was developed to investigate the hydraulic fracturing process in HDR. The model incorporates fluid flow, heat transfer, and mechanical deformation to achieve coupled THM processes. After verification under hydro-mechanical (HM), thermo-mechanical (TM), and thermo-hydro (TH) conditions against analytical benchmarks, the model was adopted to examine the effects of temperature (20-300 degrees C), confining pressure (20-40 MPa), and fluid viscosity (1-100 mPa & sdot;s) on fracture propagation. Higher breakdown pressure is more easily achieved under coupled conditions of low temperature, high confining pressure, and high fluid viscosity, whereas complex fracture networks are more readily generated under high-temperature and low-confinement conditions. Temperature plays a key role by weakening the rock matrix and inducing thermal damage zones near the borehole, which enhances fracture complexity but reduces breakdown pressure. Stronger confining pressure increases breakdown pressure and limits fracture complexity, while increasing stress anisotropy shifts fracture control from thermal-stressdominated to confinement-dominated behavior. Higher-viscosity fluids increase breakdown pressure by delaying fracture propagation and enabling greater energy accumulation. The associated fluid lag reduces tip pressure and inhibits the uniform activation of multiple fractures, thereby limiting fracture complexity and promoting dominant fracture growth. These findings provide numerical support for understanding fracture behavior in high-temperature reservoirs and offer practical guidance for optimizing hydraulic fracturing strategies in EGS development.

Understanding how hydraulic fractures interact with rock heterogeneity across different propagation regimes is essential for interpreting fracture evolution in deep crystalline reservoirs. This study presents an improved mesostructure-resolved hydromechanical DEM model to investigate fracture propagation in granite, explicitly capturing the interaction between realistic mineral grain structures and fluid flow. The numerical model is benchmarked against the KGD analytical solution. The dimensionless toughness Km is used to delineate the transition between dominant energy dissipation mechanisms, distinguishing fracture growth controlled primarily by fluid viscosity from that governed by rock fracture energy. The results reveal distinct, regime-dependent propagation behaviors. In the viscosity-dominated regime, a steep pressure gradient along the fracture provides viscous stabilization, resulting in planar and nearly symmetric growth that effectively suppresses the influence of local mineralogical heterogeneity. Conversely, the toughness-dominated regime promotes unstable fracture propagation, manifested by step-wise advance and asymmetric growth due to the increased sensitivity of the fracture tip to local mineral-scale heterogeneity. Furthermore, thermal treatment intensifies mesoscale heterogeneity through differential thermal expansion among constituent minerals, generating heterogeneous internal stresses and inducing intergranular microcracks that act as preferential pathways and thus enhance fracture network complexity. These findings clarify the coupling between fracture propagation regime and mesoscale heterogeneity and provide a mechanistic basis for evaluating injection conditions in geothermal reservoirs.

Accurately predicting fracturing pressure is critical for optimizing the safety and efficiency of hydraulic fracturing operations, particularly in newly developed blocks where data scarcity poses significant challenges. Traditional machine learning methods require large, high-quality datasets to train algorithms. To address these limitations, this study presents physics-boosted transfer learning frameworks designed to enhance fracturing pressure prediction in data-scarce scenarios. By integrating a gated recurrent unit (GRU) deep learning model with physical modeling principles, three transfer learning frameworks were developed and evaluated, including a pure data-driven framework, a hybrid-modelling framework, and a physics-informed framework. Field data from only three shale gas wells were utilized to train the GRU algorithm-simulating real-field data-scarcity scenarios. Fine-tuning technologies are optimized based on the pure data-driven framework. The physics-informed framework demonstrated superior performance, achieving root mean square errors (RMSE) as low as 2-3 MPa, significantly outperforming both the pure data-driven and hybrid frameworks in terms of accuracy, stability, and adaptability. By bridging the gap between data-driven methods and physical modeling, this new framework offers a robust solution, for improving operational safety and cost-effectiveness in hydraulic fracturing, particularly under data-scarce conditions.

Fluid injection in fractured rock reservoirs can induce fault slip, potentially compromising subsurface integrity and triggering seismic events. Field observations in shale gas development suggest that the relative orientation between the wellbore and nearby faults influences the injection-induced fault reactivation behavior, but the controlling pattern remains unclear. This study develops a site-scale, three-dimensional, distinct-element model calibrated to the Luzhou region in Southwest China. Single-stage hydraulic fracturing is simulated to interact with a preexisting fault under nine intersection angles between the wellbore and the fault. The simulations show a consistent sequence of fluid penetration, stable sliding, and unstable stick–slip. The initial unstable slip is driven by pore-pressure-induced reduction in effective normal stress near the intersection region and by poroelastic stress redistribution during fracture growth. Elevated shear stress develops near the edges of the slipped patch and tends to advance ahead of the overpressure front. The results show a clear dependence of reactivation potential and slip consequences on fault orientation relative to the wellbore. The dependence is broadly consistent with Mohr–Coulomb predictions under the strike-slip stress regime considered here. Moderate intersection angle cases produce larger slipped areas and greater shear displacement. When the intersection angle is below 30°, unstable slip may be accompanied by borehole pressure surges and relatively larger microseismic events. These findings advance the mechanistic understanding of injection-induced fault reactivation and support more targeted prefracturing risk assessment and warning strategies during stimulation.

There are numerous commercial hydraulic fracturing simulators and even a higher number of their academic counterparts. At the end of the day, only a few end up being used regularly and for extended periods of time, and this has a reason. In this presentation, I will share my experience in contributing to both academic and commercial simulators, covering their overall goals, general characteristics, similarities, and differences. Certain aspects lie on a surface, such as the purpose of an academic simulator is to generate good quality results for a few publications, and the purpose of a commercial simulator is to solve an industry problem for many scenarios and as fast as possible. At the same time, there are subtle nuances that are not obvious and are not often discussed, while they can determine the long term outcome. Finally, I will cover an example of different hydraulic fracture front tracking algorithms and how they made contributions to both academic and commercial simulators.

The development of unconventional shale gas is gradually evolving towards multi-layer and multi-well developments with small well spacing to improve resource recovery. However, the inter-well interference mechanism and the procedures for determining an optimal well spacing in multi-layer developments remain unclear. This study establishes a field-scale distinct element numerical modeling based on the data from the Luzhou deep shale gas reservoir, in Southwest China. The hydraulic fracturing process of three staggered wells is hydro- mechanically simulated, followed by an analysis of the interaction mechanism between multiple wells and the effect of well spacing on hydraulic fracturing. The results show that the effect of interference between wells on fracturing effectiveness varies non-linearly with increasing well spacing. Fracture growth is limited when well spacing is too small, resulting in reduced stimulated reservoir volume (SRV). Excessive well spacing is detrimental to the formation of complex fracture networks and results in a waste of reservoir resources between wells. Furthermore, a multi-factor scoring method was proposed to evaluate the effectiveness of hydraulic fracturing. This method considers reservoir stimulation, fracture network quality, and resource utilization to generate comparable scores. The assessment result clearly shows that an optimal well spacing exists for multi-well and multi-layer development and numerical modeling can be used to identify this value. This study describes a possible mechanism for multi-well interactions and provides a methodology for the design of multi-well and multi-layer developments in Luzhou and related shale gas development areas.

Plug and perf hydraulic fracturing is performed with high-pressure injection of fluid and proppant from perforation clusters along a wellbore. During this process, uniform placement of fluid and proppant is important for maximizing economic performance. In prior work, we developed a wellbore-proppant transport simulator, which accounts for a wide range of phenomena, including proppant suspension, proppant settling, perforation erosion, perforation pressure drop, inertial effects, perforation orientation, and random variance, among others. In the present work, we calibrate the simulator to downhole imaging measurements of perforation erosion from wells in the Midland Basin, Montney, and Bakken Shale plays. The simulator uses several empirical coefficients. We identify coefficients that have consistent values in the calibrations to all datasets. On the other hand, a few of the coefficients exhibit variability from dataset to dataset. We show how these parameters can be calibrated on a case-by-case basis prior to using the simulator for design optimization. Based on these case studies, we identified several opportunities to improve the simulator physics—by accounting for perforation ‘inline’ effects, including random variance in erosion coefficient, and increasing the amount of proppant suspension. Comparison across datasets shows that there is not a single consistent trend in heel-side or toe-side erosion bias. Different physical processes have opposing effects on heel/toe-side bias, and depending on the stage design and practical conditions, these processes can have different relative magnitudes. Correspondingly, the optimal perforation design varies from case-to-case, depending on which type of ‘bias’ is observed in the base design. The simulator predicts that measured erosion uniformity should be lower than the proppant or slurry uniformity. This result is supported by observations from a Bakken dataset in this study, where fiber-based slurry allocations yielded a significantly higher uniformity index than downhole imaging-based measurements. The implication is that the ‘uniformity index’ of erosion, observed from downhole imaging, cannot be taken as a direct proxy for the uniformity of proppant or fluid outflow. Finally, the simulator was applied to each field dataset to identify the optimal perforation design. The optimization procedure identified opportunities to improve the cluster-level uniformity index of proppant placement by a range of 0.12–0.19. Because each dataset requires case-specific calibration prior to optimization, there is no single ‘best’ design for all circumstances.

Hydraulic fracturing then fluid circulation in enhanced geothermal system (EGS) reservoirs have been shown to induce seismicity remote from the stimulation-potentially generated by the distal projection of thermoporoelastic stresses. We explore this phenomenon by evaluating stress perturbations resulting from stimulation of a single stage of hydraulic fracturing that is followed by thermal depletion of a prismatic zone adjacent to the hydraulic fracture. We use Coulomb failure stress to assess the effect of resulting stress perturbations on instability on adjacent critically-stressed faults. Results show that hydraulic fracturing in a single stage is capable of creating stress perturbations at distances to 1000 m that reach 10-5-10-4 MPa. At a closer distance, the magnitude of stress perturbations increases even further. The stress perturbation induced by temperature depletion could also reach 10-3-10-2 MPa within 1000 m-much higher than that by hydraulic fracturing. Considering that a critical change in Coulomb failure stress for fault instability is 10-2 MPa, a single stage of hydraulic fracturing and thermal draw-down are capable of reactivating critically-stressed faults at distances within 200 m and 1000 m, respectively. These results have important implications for understanding the distribution and magnitudes of stress perturbations driven by thermoporoelastic effects and the associated seismicity during the simulation and early production of EGS reservoirs. (c) 2025 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/ 4.0/).

We investigate the problem of simultaneous propagation of multiple hydraulic fractures in situations when the effect of buoyancy is important. While such fractures often occur in nature as magmatic dikes, they also bear technological relevance. For instance, some petroleum reservoirs have strong upward stress gradient and multiple hydraulic fractures are often created simultaneously from a horizontal well to promote operational efficiency. More recently, enhanced geothermal systems employed a similar multi-fracture from a horizontal well approach to stimulate more homogeneous hot rock. The stress state in the latter rock formations is often dominated by a constant stress gradient and promotes buoyancy effects. In geothermal applications, there is at least one more additional well that is drilled to establish circulation. Connectivity between the wells is therefore a crucial aspect of the treatment design. For this reason, we aim to investigate hydraulic fracture morphology of multiple fractures, since it is directly related to connectivity between the wells. To address the problem, we utilize a recently constructed parametric space for a single buoyancy-driven hydraulic fracture, and analyze the fracture shapes for the multi-fracture cases in this dimensionless space to ensure coverage of all possible scenarios. We find that hydraulic fractures exhibit instabilities in all of the cases by trying to prevent an overlap between the fractures. As a result, fractures form multiple fracture “heads” or fingers. At the same time, the results noticeably depend on the fracture propagation regime respectively on the location in the parametric space. In particular, horizontal fracture extension, vertical fracture height, aperture, and the number of “heads” vary considerably within the parametric space. Implications of the results to practical applications are discussed and recommendations on instability avoidance are provided.

Hydraulic fracturing involves injecting a high-pressure fluid mixture to create fractures in underground rock formations, thereby enhancing hydrocarbon flow. Recent field observations, such as those from the Hydraulic Fracture Test Site (HFTS) project and tests in the Eagle Ford, have revealed surprising complexity of hydraulic fractures, including the presence of densely distributed fracture swarms. These findings challenge conventional expectations and necessitate a deeper understanding of fracture mechanisms. Existing studies have explored the propagation, connectivity, and implications of multiple fractures, but questions remain about the mechanisms behind the formation of fracture swarms, particularly the distribution of these secondary fractures. Our research introduces an alternative mechanism, proposing that secondary fractures result from a pore-pressure wave in which the pore pressure exceeds the compressive stress. This hypothesis suggests that pore-pressure variations within the rock and fluid exchange between the fracture and rock after shut-in can initiate secondary fractures. Through theoretical modeling and scaling, we have identified the governing dimensionless parameters that determine the number and distribution of secondary fractures. Numerical simulations enabled us to construct the parametric space for these parameters. Finally, we propose a procedure to interpret field observations of fracture swarms. Our approach provides new insights into predicting fracture swarms and using the observed fracture swarms to constrain some parameters that are relevant to hydraulic fracturing.

Abstract During plug and perf completion, perforation pressure drop is used to encourage a uniform distribution of flow between clusters by overcoming stress shadowing, stress variability, and nonuniform breakdown pressure. However, proppant inertia, gravitational settling, and perforation erosion contribute to nonuniformity, even with an aggressive limited-entry design. In prior work, Dontsov (2023) developed a correlation for predicting proppant outflow from the wellbore as a function of slurry velocity, perforation phasing, and other parameters. In the present study, the Dontsov (2023) correlation is integrated into a wellbore dynamics simulator capturing key physical processes that control slurry and proppant outflow from the wellbore, such as erosion, stress shadowing, and near-wellbore tortuosity. The simulator is fast running and incorporated into a tool for Monte Carlo uncertainty quantification and design optimization. First, we run a series of sensitivity analysis simulations to evaluate the effect of key model inputs. The simulations demonstrate processes that can cause heel bias, toe bias, or heel/toe bias in the erosion distribution. Next, we apply the tool to analyze field datasets from the Eagle Ford and the Montney. Downhole imaging of erosion data enables model calibration. Calibration is necessary because differences in casing, cement, and formation properties cause differences in erosion behavior and flow distribution. Parameters controlling the magnitude of erosion and stress shadow are modified to match the trends observed from the downhole imaging. After calibration is performed, the model is applied to maximize the uniformity of proppant placement by optimizing perforation phasing, diameter, count, and cluster spacing.

The purpose of this study is to develop a model for the flow of suspensions consisting of Herschel-Bulkley fluid mixed with spherical particles. In particular, the focus is to investigate the effect of non- Newtonian rheology of the carrying fluid on the flow behavior of a suspension. Two-dimensional steady flow problem in a vertical channel is considered, in which both the pressure gradient and gravity drive the suspension flow. Dependence of the velocity profile and particle concentration across the channel on the fluid rheology parameters and orientation of the pressure gradient is investigated. It is found that the non-uniform particle distribution in the flow across the channel leads to the non-uniform density of the suspension, which causes sinkage of the denser regions and promotes downward migration of the particles even without slip velocity. Particle and suspension fluxes are calculated for various fluid rheologies and pressure gradient orientations. The effect of slip velocity between the phases is added via filtration term that captures fluid flow once particles reach the maximum concentration and stall, and via the settling term that describes gravitational particle settling.

ABSTRACT Uniform proppant distribution between perforation clusters is one of the goals in hydraulic fracturing operations. However, this can be challenging to achieve, even if limited entry is used to acheive a nearly uniform fluid distribution. Two mechanisms contribute to a non-uniform proppant distribution between perforations. The first is particle settling in the wellbore, which becomes especially important towards the toe of the stage where the flow velocity is lower. The second mechanism is related to proppant inertia, whereby some particles are unable to follow the fluid streamlines and miss the perforation. In this paper, a mathematical model is developed to simulate both of these physical phenomena and is calibrated against available experimental and computational data. The model is then combined with an optimization algorithm to investigate perforation designs leading to nearly uniform proppant distribution between perforations. It is found that it is possible to significantly improve proppant placement by varying perforation orientation along the stage. If the same orientation is required for all perforations, then perforations located in the lower part of the wellbore with azimuths between 110° and 120° lead to more uniform proppant distribution, compared to other cases. INTRODUCTION Many studies have addressed the problem of particle distribution between perforations because this is a critical issue for optimization of hydraulic fracturing treatments. One of the first experimental studies (Gruesbeck and Collins, 1982) investigated the problem of slurry flow in a perforated vertical well, as is typical for conventional hydraulic fracturing treatments. It was observed that particle size and viscosity play a significant role on the resultant amount of proppant leaving through the perforation. With the rise of unconventional resource development as well as computational capabilities, Computational Fluid Dynamic (CFD) models have been used to address the same problem of particle turning into a perforation (Wu and Sharma, 2016; Wu, 2018). What is interesting is that in this study authors observed practically no dependence on particle size and fluid viscosity. A model that is employed in this study for the optimization purposes (see Dontsov (2023)) is able to explain such a contradiction by showing that the parameters used in Gruesbeck and Collins (1982) and Wu and Sharma (2016); Wu (2018) lead to dominance of different particle turning mechanisms, which in turn caused different sensitivities.

Hydraulic Fracturing (HF) is the prime technology for enhanced production from shale plays. Due to their laminations at different scales, shales exhibit transverse isotropic (TI) properties. In most cases the lamination planes are nearly horizontal with the symmetry axis being vertical, therefore, shales are referred to as TI vertical (TIV) rocks. This significantly influences the initiation and propagation of the induced fractures during fracturing operation. However, many studies on HF modeling in shales assume isotropic medium to simplify the problem. The anisotropic toughness is a distinct feature of layered formations such as shales, and its modelling is important to estimate the hydraulic fracturing initiation pressure, as well as the fractures geometry. In this work, to address the effect of TIV properties of the rock, we used the anisotropic toughness to develop the analytical models for fracture initiation pressure (FIP) based on data from the Bakken Shale, as a case study. In particular, we used ResFrac for calculations, which is a fully 3D HF and reservoir numerical simulator. Different scenarios of radial fracture for cases of single and multiple fractures were simulated for isotropic and anisotropic toughness conditions. The results showed that the hydraulic fracture initiation and propagation are strongly affected by the toughness anisotropy. An increase in the magnitude of anisotropy in toughness leads to an increase in the FIP. We also observed that the FIP varies with direction and consequently the fracture becomes more elongated in the direction of the minimum toughness and contained in the direction of the maximum toughness. In the case of multiple HFs, the combined effect of the anisotropy and the stress shadow was observed. This effect was stronger in the anisotropic case compared to the isotropic case. Hence, the isotropic approximation for TIV shale rocks can lead to inaccurate prediction of FIP and incorrect fracture morphology that can possibly affect the HF design.

To improve the effectiveness of fracturing treatment, it is vital to understand how multiple fractures interact with each other. This study presents a fully coupled hydraulic fracture model that considers both Newtonian fluids and power-law fluids to investigate the effect of stress shadowing in different propagation regimes, from toughness- to viscosity-dominated, under limited entry conditions. The dimensionless parametric space for a plane-strain fracture driven by power-law fluids is outlined, and numerical analysis is performed in the parametric space to simplify the problem. Moreover, the numerical scheme for the coupled hydromechanical problem is verified by comparing the results to the existing analytic solutions. It is shown that the simultaneous propagation of multiple fractures is significantly influenced by the fracture propagation regime and cluster spacing. Fractures tend to grow uniformly in the viscosity-dominated regime, and the fracture reorientation can be observed if both the cluster spacing and stress anisotropy are small. In contrast, fractures propagating in the toughness-dominated regime tend to propagate in the opposite direction of the adjacent fractures to avoid each other, resulting in minimal overlap between fracture lengths. The influence of fracture propagation regime is consistent for Newtonian and power-law fluids. Additionally, as the cluster spacing increases, the effect of stress shadow decreases, and the behavior of each individual fracture resembles that of an isolated fracture.

This paper analyses the problem of a semi-infinite fluid-driven fracture propagating through multiple stress layers in a permeable elastic medium. Such a problem represents the tip region of a planar hydraulic fracture. When the hydraulic fracture crosses a stress layer, the use of a standard tip asymptotic solution may lead to a considerable reduction of accuracy, even for the simplest case of a height-contained fracture. In this study, we propose three approaches to incorporate the effect of stress layers into the tip asymptote: non-singular integral formulation, toughness-corrected asymptote, and an ordinary differential equation approximation of the non-singular integral formulation mentioned above. As illustrated in the paper, these approaches for stress-corrected asymptotes differ in computational complexity, the complexity of implementation, and the accuracy of the approximation. In addition, the size of the validity region of the stress-corrected asymptote is evaluated, and it is shown to be greatly reduced relative to the case without layers. In order to address the issue, the stress relaxation factor is introduced. This, in turn, allows for enhancing the accuracy of the layer-crossing computation on a relatively coarse mesh to utilize the stress-corrected asymptote in hydraulic fracturing simulators for the purpose of front tracking.

This document provides technical background for StageOpt. This tool helps to better understand fluid and proppant distribution between perforations and clusters within a fracturing stage. Limited entry technique is often used to achieve uniform fluid distribution between clusters. This, however, does not automatically guarantee uniform proppant distribution. In view of this observation, StageOpt allows users to overcome this obstacle and to better design perforation design by focusing on both fluid and proppant distribution between clusters. The algorithm in StageOpt is based primarily on the paper Dontsov (2023), but it also incorporates practically important features, such as the effect of stress shadow from previous stage, perforation erosion, and other phenomena. In this write-up, flow distribution between perforations is described first, and the effects of perforation friction, near-wellbore pressure drop, as well as perforation break down and stress shadow are included. Then, proppant distribution between perforations is discussed as a function of perforation orientation. Finally, perforation erosion is added to the model. The erosion changes perforation diameter as a function of several parameters, including the flow rate through the perforation as well as the amount of proppant that flows through it. The change of perforation size in turn changes magnitude of perforation friction, which then changes the distribution of slurry, that consequently changes proppant flow distribution. Such a coupled system often leads to complex results that are hard to deduce without systematic physical modeling, which is exactly what StageOpt provides.