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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. • The problem of a semi-infinite fracture crossing a stress layer is investigated. • Three approaches to incorporate the effect of stress layers are proposed. • The accuracy and computational complexity of different approaches are compared. • The validity region of the solution is increased by introducing a stress relaxation factor. [ABSTRACT FROM AUTHOR]

This study investigates the relation between two approaches for modeling fracture propagation. The first one is the classical approach, in which fracture propagates once the stress intensity factor exceeds a critical value, called fracture toughness. In the second approach, the fracture propagates once the tensile stress ahead of the fracture tip exceeds a critical value, called tensile strength. The purpose of this study is to examine the relation between the two approaches and to determine a methodology to make them equivalent. To address the goal, propagation of a radially symmetric fracture is first analyzed. A universal relation between the tensile strength and fracture toughness is obtained, which is then verified via a series of numerical examples. It is found that in order to capture the fracture toughness the tensile strength should be varied with respect to the mesh size and other material parameters. The developments are then applied to a three-dimensional distinct element code, which can be used in various applications involving modeling of a jointed and blocky material. An additional challenge with the distinct element code lies in the fact that the use of uniform value of tensile strength does not lead to a spatially uniform apparent fracture toughness. The latter is caused by mesh distortions and orientation of the elements relative to the fracture front. This problem is successfully addressed by introducing a variation of the tensile strength relative to local geometry of the mesh in the vicinity of the fracture front. The obtained result develop a procedure to accurately model fracture toughness in numerical methods that use tensile strength as a fracture propagation criterion. [ABSTRACT FROM AUTHOR]

The capability to simulate a hydraulic fracturing process is an essential tool that can be used to optimize treatment design and increase the efficiency of field operations. In most practical cases, hydraulic fractures propagate in a multi-layered rock formation. As a result, there is a need to incorporate the effect of such heterogeneities in fracturing models to achieve an accurate prediction. To capture the layered structure of rocks, a hydraulic fracture simulator typically requires a fine mesh, which leads to a drastic reduction in computational performance. An alternative is to use more sophisticated models that are capable of providing reasonably accurate predictions even on a relatively coarse mesh. In the case of fracture growth modeling, the pivotal component of the simulation is a fracture front tracking algorithm that accounts for the layered structure of the formation. Consequently, this paper aims to extend the established Implicit Level Set Algorithm (ILSA) to account for the effect of multiple stress layers within the tip asymptote. The enhanced front tracking algorithm involves the stress-corrected asymptote that incorporates the influence of stress layers within the near-tip region. To further increase the validity region of the stress-corrected asymptote, the stress relaxation factor is introduced, and its accuracy is examined. The numerical algorithm is validated against the reference semi-analytical solutions as well as experimental observations. In addition, we investigate the sensitivity of the fracture geometry to mesh size to demonstrate that the front tracking algorithm based on the stress-corrected asymptote retains its accuracy on a coarse mesh. • A numerical algorithm for a planar hydraulic fracture is presented. • Tip asymptotic solution is used to determine the evolution of fracture boundary. • The near-tip logic includes the effects of stress layers. • A level set algorithm is used to track the moving fracture front. • The developments improve the overall accuracy of the algorithm. [ABSTRACT FROM AUTHOR]

This paper presents a model to simulate behavior of particle-laden slurry in a horizontal perforated wellbore with the goal of quantifying fluid and particle distribution between the perforations. There are two primary phenomena that influence the result. The first one is the non-uniform particle distribution within the wellbore's cross-section and how it changes along the flow. The second phenomenon is related to the ability of particles to turn from the wellbore to a perforation. Consequently, the paper considers both of these phenomena independently at first, and then they are combined to address the whole problem of flow in a perforated wellbore. A mathematical model for calculating the particle and velocity profiles within the wellbore is developed. The model is calibrated against available laboratory data for various flow velocities, particle diameters, pipe diameters, and particle volume fractions. It predicts a steady-state solution for the particle and velocity profiles, as well as it captures the transition in time from a given state to the steady-state solution. The key dimensionless parameter that quantifies the latter solution is identified and is called dimensionless gravity. When it is small, the particles are fully suspended and the solution is uniform. At the same time, when the aforementioned parameter is large, then the solution is strongly non-uniform and resembles a flowing bed state. A mathematical model for the problem of particle turning is developed and is calibrated against available experimental and computational data. The key parameter affecting the result is called turning efficiency. When the efficiency is close to one, then most of the particles that follow the fluid streamlines going into the perforation are able enter the hole. At the same time, zero efficiency corresponds to the case of no particles entering the perforation. Solutions for the both sub-problems are combined to develop a model for the perforated wellbore. Results are compared (not calibrated) to a series of laboratory and field scale experiments for perforated wellbores. Comparison with the available computational results is presented as well. In addition, the comparison is presented in view of the parametric space defined by the dimensionless gravity and turning efficiency. Such a description allows to explain seemingly contradictory results observed in different tests and also allows to highlight parameters for which perforation orientation plays a significant role. • A model for particle dynamics in a perforated pipe is developed. • Solution for particle distribution in a turbulent slurry flow is constructed. • Higher particle concentration is predicted at the bottom of the pipe. • The problem of particles missing the perforation is solved. • The developments are validated against numerous experimental observations. [ABSTRACT FROM AUTHOR]

The goal of this study is to investigate the effect of turbulent fluid flow on the propagation of planar hydraulic fractures. Modeling a hydraulic fracture includes solving the elasticity equation that ensures the equilibrium of the rock, the fluid volume balance equation, and the fluid flow equation, which are solved together with a propagation condition. In this paper, the influence of turbulent flow is condensed into a single friction factor that influences the fluid flow equation, i.e. the relationship between the fluid flux and the pressure gradient. To capture all possibilities, an approximation for the friction factor, that captures the laminar, the turbulent, and the transitional flows is utilized in this study. Results for the axisymmetric fracture geometry demonstrate that the solution is dominated by turbulent flow at early times and near the source, while transitions to the laminar regime at larger times and close to the fracture tip. In the situation when turbulence dominates, the fracture is shorter and wider, since there is a strong pressure drop in the vicinity of the source, which causes the local fracture width increase. Results for a planar fracture propagating in a three stress layer geometry demonstrate that the turbulence leads to a more circular fracture that promotes height growth through a high stress zone. [ABSTRACT FROM AUTHOR]

The goal of this study is to compare and evaluate the accuracy of different approaches to incorporating the effect of lateral fracture toughness into reduced models for blade-like and pseudo-3D hydraulic fractures. The following three methods are used for the comparison: (i) a classical model with a plane strain (or local) elasticity assumption and a pressure boundary condition calculated based on energetic considerations, (ii) a classical model with local elasticity and pressure boundary condition originating from “stitching” a radial fracture tip to the rest of the fracture, and (iii) a novel model with non-local elasticity and a boundary condition at the tip that is consistent with the linear elastic fracture mechanics propagation criterion. Predictions of all three approaches are compared to a reference solution calculated using a fully planar hydraulic fracturing simulator. The results indicate that the reduced model with non-local elasticity is able to provide an accurate approximation for a wide range of fracture toughness values. The models that feature the local elasticity assumption are able to provide reasonably accurate results for moderate values of fracture toughness, while they become less accurate for blade-like geometries and significantly less accurate (and in some cases unstable) for the pseudo-3D geometry for large values of the fracture toughness. [ABSTRACT FROM AUTHOR]

The goal of this paper is to develop a more accurate pseudo-3D model for hydraulic fracturing. The primary weaknesses of the classical pseudo-3D model are: (1) its inability to capture viscous height growth and (2) its failure to include lateral fracture toughness. These flaws are addressed respectively by: (1) introducing an apparent fracture toughness in the vertical direction and (2) using an approximate non-local elasticity operator. To evaluate the accuracy and the level of improvement of the model we have developed, the results are compared to the predictions calculated using a recently developed fully planar hydraulic fracturing simulator. [ABSTRACT FROM AUTHOR]

The aim of this study is to develop a model for proppant transport in hydraulic fractures capable of capturing both gravitational settling and tip screen-out effects, while prohibiting the particles from reaching the crack tips by imposing a width restriction based on the particle size. First, the equations that govern the propagation of hydraulic fractures and the proppant transport inside them are formulated. They are based on the solution for the steady flow of a viscous fluid, mixed with spherical particles, in a channel, which is obtained assuming an empirical constitutive model. This proppant transport model is applied to two fracture geometries – Khristianovich–Zheltov–Geertsma–De Klerk (KGD) and pseudo-3D (P3D). Numerical simulations show that the proposed method makes it possible to capture proppant plug formation and growth, as well as the gravitational settling for both geometries. A dimensionless parameter, whose magnitude reflects the intensity of the settling, is introduced for the P3D fracture. [ABSTRACT FROM AUTHOR]

We analyse the influence of fluid yield stress on the propagation of a radial (penny-shaped) hydraulic fracture in a permeable reservoir. In particular, the Herschel–Bulkley rheological model is adopted that includes yield stress and non-linearity of the shear stress. The rock is assumed to be linear elastic, and the fracture is driven by the point source fluid injection with a constant volumetric rate. The fracture propagation condition follows the theory of linear elastic fracture mechanics, and Carter's leak-off law is selected to govern the fluid exchange process between the fracture and formation. The numerical solution of the problem is found using the algorithm based on Gauss–Chebyshev quadrature and Barycentric Lagrange interpolation techniques. We also construct an approximate solution with the help of the global fluid balance equation and the near-tip region asymptote. The latter approximation is computationally efficient, and we estimate its accuracy by comparing the primary crack characteristics such as opening, pressure, and radius with those provided by the full numerical solution. We present examples corresponding to typical field cases and demonstrate that the addition of yield stress can lead to a shorter radius and wider opening compared to the corresponding case with simpler power-law fluid rheology. Further, we quantify the limiting propagation regimes (or vertex solutions) characterised by the dominance of a particular physical phenomenon. Relative to the power-law results, there are two new vertices that are associated with the domination of yield stress: storage-yield-stress and leak-off-yield-stress. To understand the influence of various problem parameters, we utilise the constructed approximate solution to investigate the dimensionless parametric space of the problem, in which the applicability domains of the limiting solutions are quantified. This enables one to quickly determine whether yield stress provides a strong influence for given problem parameters. • Analysis of the impact of fluid yield stress on radial fracture propagation. • Comparison with the power-law rheological model. • Numerical and rapid semi-analytical approximate solutions for the problem. • Construction of the dimensionless parameter space for the problem. • Quantitative identification of parameters for which yield stress is important. [ABSTRACT FROM AUTHOR]

• Coupled problem of a hydraulic fracture (HF) with proppant transport is solved. • Enhanced pseudo-3D model is used for rapid HF modeling. • Saffman-Taylor instability is predicted for proppant schedule with pulses. • Fracture growth is noticeably influenced by the flow instabilities. • Proppant schedule with pulses can lead to longer fractures. This paper presents the coupled model of a hydraulic fracturing and proppant transport. The former is described in terms of enhanced pseudo-3D model that considers height growth across two symmetric stress barriers, while the latter is given by two-dimensional transport model, stemming from the solution of an elliptical equation for fluid pressure and advection equation for the proppant transport. These two sub-modules are solved numerically using implicit time integration in the hydraulic part and explicit time stepping in the transport part. In addition, interpolation is used to couple the two models with different grids. Results of several numerical simulations are presented for different configurations to demonstrate the interplay between these two modules. In particular, the developed coupled scheme allows us to study phenomena associated with complex fluid flow within the fracture, such as for the case of Saffman-Taylor instability. [ABSTRACT FROM AUTHOR]

This study considers the problem of simultaneous propagation of multiple closely spaced hydraulic fractures in homogeneous rocks, under the limited entry condition that ensures uniform flux distribution between the fractures. In particular, the sensitivity of the results to fracture propagation regimes is investigated. There are four primary regimes of hydraulic fracture propagation that are related to interactions between two dissipation mechanisms, related to fluid viscosity and rock toughness, and a fluid storage mechanism (fracturing fluid leak-off), which can occur either inside the fracture or inside the formation. It is shown that the fracture behavior varies dramatically with respect to the fracture propagation regime. In particular, fractures that propagate in the viscosity dominated regime exhibit weak stress shadow interactions, grow predominantly radially, and develop nearly identical fracture geometries. In contrast, fractures that propagate in the toughness regime exhibit strong stress shadow interactions, grow as individual segments avoiding others that are propagating simultaneously, and develop geometrical structures resembling petals in a flower, where the overall geometry is radially uniform, but the individual fractures form non-overlapping segments of the total surface area. Results are presented for the case of Newtonian fluids with and without stress anisotropy, power-law fluids, and consider the case of fracture toughness anisotropy. Finally, the effect of fracture spacing on fracture geometry is investigated for different fracture regimes. [ABSTRACT FROM AUTHOR]