Radial Hydraulic Fracturing Research Articles

Natural vibrations of fluid in a well connected with the reservoir by a system of radial fractures

The problem of natural vibrations of a fluid in a horizontal well with multiple fractures obtained by hydraulic fracturing is considered. A mathematical model of the natural vibrations of fluid in a horizontal oil well connected to the reservoir by a system of radial hydraulic fractures is constructed and the frequency characteristics of the natural vibrations of fluid as functions of the hydraulic fracture and reservoir parameters are determined. Using a numerical analysis of the frequency characteristics of vibrations, the effect of changes in the fracture width, the number of fractures, and the reservoir permeability on the natural frequencies is demonstrated.

Fluid Flow to a Well through a Radial Hydraulic Fracture in a Porous Medium at Constant Flow Rate

Fluid Flow to a Well through a Radial Hydraulic Fracture in a Porous Medium at Constant Flow Rate

Fluid Flow to a Well through a Radial Hydraulic Fracture in a Porous Medium at Constant Flow Rate

The problem of fluid withdrawal from a reservoir into a well or injection of fluid from a well into a reservoir in the presence of a hydraulic fracture perpendicular to the wellbore is considered in the constant flow rate regime. Analytical solutions that describe the evolution of pressure in the fracture at a constant reservoir-to-well fluid flow rate are obtained. Approximate solutions are constructed using the method of successive change of stationary states (SCSS). A comparison of the numerical results of the exact and approximate solutions shows that these solutions are almost identical. The influence of the reservoir features of the stratum and fracture on the evolution of pressure in the fracture and well is analyzed.

Effects of velocity-dependent apparent toughness on the pre- and post-shut-in growth of a hydraulic fracture

We investigate the growth of a plane-strain/radial hydraulic fracture in an infinite impermeable medium driven by a constant injection rate assuming that the apparent toughness scales with the decreasing fracture growth rate in a power-law relation. The viscosity dominated regime always governs the fracture growth at large time for the plane-strain geometry. For a radial hydraulic fracture, we report a transition from early-time fracture growth dominated by viscous fluid flow to large-time propagation dominated by fracture toughness. Such a transition results from an overall increase of the energy dissipation by fracture surface creation. After shut-in, both plane-strain and radial fractures propagate at a lower velocity with decreasing fracture toughness. The fracture growth then depends on the dimensionless toughness at the stop of fluid injection and transitions towards a self-similar pulse-viscosity solution when viscous fluid flow dominates the energy dissipation. The fracture arrests when fulfilling two conditions — the apparent toughness reaching its minimum, and viscous forces being negligible in the fluid flow. The fracture dimension at arrest is independent of the velocity-dependent power-law relation.

Impact of the tangential traction for radial hydraulic fracture

The radial (penny-shaped) model of hydraulic fracture is considered. The tangential traction on the fracture walls is incorporated, including an updated evaluation of the energy release rate (fracture criterion), system asymptotics and the need to account for stagnant zone formation near the injection point. The impact of incorporating the shear stress on the construction of solvers, and the effectiveness of approximating system parameters using the first term of the crack tip asymptotics, is discussed. A full quantitative investigation of the impact of tangential traction on solution is undertaken, utilising an extremely effective adaptive time-space solver.

Three-dimensional buoyant hydraulic fractures: constant release from a point source

Hydraulic fractures propagating at depth are subjected to buoyant forces caused by the density contrast between fluid and solid. This paper is concerned with the analysis of the transition from an initially radial fracture towards an elongated buoyant growth – a critical topic for understanding the extent of vertical hydraulic fractures in the upper Earth crust. Using fully coupled numerical simulations and scaling arguments, we show that a single dimensionless number governs buoyant hydraulic fracture growth, namely the dimensionless viscosity of a radial hydraulic fracture at the time when buoyancy becomes of order 1. It quantifies whether the transition to buoyancy occurs when the growth of the radial hydraulic fracture is either still in the regime dominated by viscous flow dissipation or already in the regime where fracture energy dissipation dominates. A family of fracture shapes emerge at late time from finger-like (toughness regime) to inverted elongated cudgel-like (viscous regime). Three-dimensional toughness-dominated buoyant fractures exhibit a finger-like shape with a constant-volume toughness-dominated head and a viscous tail having a constant uniform horizontal breadth: there is no further horizontal growth past the onset of buoyancy. However, if the transition to buoyancy occurs while in the viscosity-dominated regime, both vertical and horizontal growths continue to match scaling arguments. As soon as the fracture toughness is not strictly zero, horizontal growth stops when the dimensionless horizontal toughness becomes of order 1. The horizontal breadth follows the predicted scaling.

The arrest and recession dynamics of a deflating radial hydraulic fracture in a permeable elastic medium

In this paper we describe the post injection deflation dynamics of a radially symmetric hydraulic fracture in a permeable elastic medium. Depending on the parameters of the problem, the fracture may arrest almost immediately after injection has ceased, or continue to propagate despite the fluid loss to the porous medium. After arrest the fracture continues to deflate while the stress intensity factor decreases to zero, after which it recedes until the fracture finally collapses. In order to establish rigorous numerical solutions to explore this deflation dynamics, we make use of recent research (Peirce and Detournay, 2021) that derived vertex asymptotes for the arrest and recession phases of the deflation process as well as multiscale asymptotic solutions that connect these vertex asymptotes: during arrest, through the arrest–recession transition, and during recession. If only vertex asymptotes are used to capture the arrest and recession, the solution exhibits jump discontinuities through this arrest–recession transition point. We describe how the multiscale asymptotes can be used to obtain a smooth solution valid through the arrest–recession transition. Significantly, after the arrest–recession transition, the jump-induced transients decay and the solutions using vertex asymptotes converge to those that use multiscale asymptotes. Thus unless it is important to obtain a smooth solution through the arrest–recession transition, a practical, and more efficient, approach would be to use an algorithm based solely on vertex asymptotes. We also provide numerical confirmation of the emergence, beyond the arrest–recession transition time, of the dominant balance used in the asymptotic analysis to establish the linear recession asymptote wˆ∼rˆ. We present a novel scaling analysis to establish the characteristic power laws for the arrest time, arrest radius, arrest aperture, and the deflation time, in terms of two new dimensionless parameters, which make it possible to unify the power laws between the zero and finite toughness cases. These power laws show close agreement with those obtained by regression of the numerical results over a range of the dimensionless parameters. Numerical solutions are provided to illustrate the solution landscape in parameter space and the effect that each of the dimensionless parameters has on: the duration of the period of propagation after injection has ceased, the duration of the arrest period, and the time from the initiation of recession to collapse.

A radial hydraulic fracture driven by a Herschel–Bulkley fluid

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.

An Analytical Model for Fracture Initiation of Radial Drilling-Fracturing in Shale Formations

Abstract Radial drilling-fracturing, the combination of radial drilling and hydraulic fracturing, can guide fractures toward the target area and effectively enhance the recovery of the low permeable reservoir. In this paper, based on the stress superposition principle, we establish an analytical model to predict fracture initiation pressure (FIP) and the shale failure mode for radial drilling-fracturing applied in shale formations. In contrast with the former studies, this model can additionally consider the failure from shale beddings and is more applicable in the shale reservoir. The model classifies the shale failure into three modes and, respectively, gives the criterion for each failure mode. Then, a series of sensitivity analyses is conducted by examining effects of various parameters. By analyzing the variation characteristic of the initiation pressures required for three failure modes, the main conclusions are as follows. Firstly, matrix failure and shear failure along bedding tend to take place when the azimuth of radial borehole is moderate. Small and large azimuths are favorable for the occurrence of tensile failure along bedding. Secondly, a high ratio of horizontal in situ stress predisposes shale to generate matrix failure, and bedding tensile failure and bedding shear failure are apt to occur when the ratio of horizontal in situ stress is low. Thirdly, with the increasing intersection angle of the radial borehole wall and bedding plane, the failure mode apt to occur changes from bedding tensile failure to bedding shear failure and then to matrix failure. Fourthly, shale prefers to yield bedding shear failure under a small Biot coefficient and generate the other two failure modes when Biot coefficient is large. Fifthly, permeability coefficient virtually has no influence on the failure mode of shale. The research clarifies the fracture initiation characteristics of radial drilling-fracturing in shale formations and provides a reference for the field application of radial drilling-fracturing.

A fast and reliable methodology to evaluate maximum CO2 storage capacity of depleted coal seams: A case study

Evaluating CO2 storage capacity of formation before implementing CO2 injection is of great importance for CO2 geological storage. Coal seams are promising CO2 geological storage sites due to their high surface area and wide geographical spread. In this study, a fast and reliable methodology is proposed to assess the CO2 storage capacity of coal seams. The methodology provides a model considering gas flow within multiple radial hydraulic fractures (MRHF) connected to depleted well, gas transport in natural fractures, diffusion in coal seam matrix, and adsorption onto matrix surface. Using Stehfest numerical inversion technology and Gauss elimination method, the semi-analytical solution for bottom-hole pressure (BHP) is obtained based on the derived continuous line-source function in coal seams. Following that, a case study regarding the CO2 storage capacity of a coal seam in Wyoming is conducted, which shows that CO2 storage capacity in the coal seam reaches 4.8 × 108 m3. In addition, the sensitive analyses of the reservoir and engineering parameters on CO2 storage capacity are also investigated. The findings confirm that, on the one hand, higher constrained injection pressure (CIP) results in larger CO2 storage capacity. On the other hand, among these key parameters, hydraulic fracture length exhibits the most apparent effects on CO2 storage capacity whereas the impacts of skin factor and stress sensitivity are insignificant. To be more specific, CO2 storage capacity increases almost 7.5 times when the hydraulic fracture length increases 3 times when CIP = 3.5 MPa whereas CO2 storage capacity slightly increases by approximately 4.5% from permeability modulus = 0.0125 to 0.05 for the identical CIP. The methodology provided in this study lays the foundation of highly efficient CO2 geological sequestration in coal seams.

Arrest of a radial hydraulic fracture upon shut-in of the injection

We investigate the propagation and arrest of a radial hydraulic fracture upon the end of the injection. Depending on the regime of propagation at the time of shut-in of the injection, excess elastic energy may be stored in the surrounding medium. Once the injection has stopped, the hydraulic fracture will arrest when the energy release rate falls under the material fracture energy. Fluid leaking-off to the surrounding medium acts as an energy sink such that the available excess energy for fracture growth decreases faster and as a result impacts the arrest (actually controls it in the zero toughness limit). Under the assumption of a homogeneous elastic medium and the Carter’s leak-off model, we show that the post shut-in propagation of the hydraulic fracture depends on the dimensionless toughness Ks and leak-off Cs coefficient at the time of shut-in. Our investigation highlights that for an impermeable rock, the arrest radius is independent of the dimensionless toughness at shut-in. In the limit of a permeable rock with zero fracture toughness, the arrest radius is independent of the dimensionless leak-off coefficient only for Cs<0.25. For larger values of Cs, the radius of arrest reduces with increasing Cs. We delineate the limit above which the arrest is immediate upon shut-in. This limit is given by a critical leak-off coefficient at shut-in Cs,c≈0.53 for the large leak-off/small toughness cases and by the relation Cs,cKs,c≈0.78-0.313·Ks,c for small leak-off/large toughness (where Ks,c is equivalently the critical dimensionless toughness at shut-in). Immediate arrest in the impermeable limit is observed for Ks,c≈2.5. If both (Ks and Cs) are smaller than their critical value for immediate arrest, post shut-in propagation occurs and a self-similar pulse viscosity storage solution emerges. Scaling arguments combined with numerical simulations, show that the propagation post shut-in scales as 1.23Ks-2/5 in the impermeable and small leak-off cases, and as 0.75Cs-2/13 in the zero toughness limit. The growth post shut-in can be significant in impermeable rocks - with a final radius up to twice larger than the radius at shut-in for realistic material and injection parameters.

Simulation of proppant transport and fracture plugging in the framework of a radial hydraulic fracturing model

AbstractHydraulic fracturing technology is widely used in the oil and gas industry. A part of the technology consists in injecting a mixture of proppant and fluid into the fracture. Proppant significantly increases the viscosity of the injected mixture and can cause plugging of the fracture. In this paper we propose a numerical model of hydraulic fracture propagation within the framework of the radial geometry taking into account the proppant transport and possible plugging. The finite difference method and the singularity subtraction technique near the fracture tip are used in the numerical model. Based on the simulation results it was found that depending on the parameters of the rock, fluid, and fluid injection rate, the plugging can be caused by two reasons. A parameter was introduced to separate these two cases. If this parameter is large enough, then the plugging occurs due to reaching the maximum possible concentration of proppant far from the fracture tip. If its value is small, then the plugging is caused by the proppant reaching a narrow part of the fracture near its tip. The numerical experiments give an estimate of the radius of the filled with proppant part of the fracture for various injection rates and leakages into the rock.

A semi-analytical model for the transient pressure behaviors of a multiple fractured well in a coal seam gas reservoir

Multiple radial fractures can be generated in unconventional gas reservoirs after stimulation treatments such as hydraulic refracturing, high-energy gas fracturing, and explosive fracturing. This study presents a semi-analytical model to simulate the bottom-hole pressure (BHP) performances of a fractured well with multiple radial hydraulic fractures (MRHF) in a stress-sensitive coal seam gas reservoir. The adsorption–desorption and diffusion for fluid in reservoirs are considered. The continuous line-source function associated with discretization method in a composite coalbed gas reservoir is applied in Laplace domain. By means of Stehfest numerical inversion algorithm and Gauss elimination, the transient pressure responses in the well bottom are obtained and discussed.The results demonstrate that, from type curves, the main flow regimes for the proposed model in the coalbed gas reservoir are bilinear flow between adjacent radial hydraulic fractures, linear flow between adjacent radial hydraulic fractures, pseudo radial flow in stimulated region, and radial flow in un-stimulated region. Moreover, an obvious “hump” appears between the bilinear flow regime and the linear flow regime in stimulated region, which represents the interference between adjacent radial hydraulic fractures. The impacts of the ratio of permeability in stimulated region to that in un-stimulated region, the radius of stimulated reservoir volume (SRV), the permeability modulus, and the properties of hydraulic fractures on the transient pressure responses in well bottom are analyzed. The findings obtained in this study can pave the way for the quantitative understanding of the transient performances of multiple fractured vertical wells in unconventional reservoirs.

A radial hydraulic fracture with pressure-dependent leak-off

This paper investigates the problem of a radial (or a penny-shaped) hydraulic fracture propagating in a permeable reservoir. In particular, we consider the fluid exchange between the crack and ambient porous media as a pressure-dependent process. In most of the existing models, the fluid exchange process is represented as one-dimensional pressure-independent leak-off described by Carter’s law. We modify this mechanism by including the dependence of the fluid-exchange rate on the fluid pressure inside the fracture. The proposed approach allows the liquid not only to flow out of the fracture but also to leak-in along the region adjacent to the fracture front. A complete model for hydraulic fracturing involves several physical processes that determine the crack propagation, namely, brittle rock failure, elastic equilibrium of rock, viscous fluid flow inside the fracture channel, and fluid exchange. In order to resolve these phenomena near the fracture front in the numerical scheme, we utilise a special asymptotic multi-scale model for the near-tip region, which is used as a propagation condition for the finite fracture. The main aspect of the analysis is a comparison of fracture characteristics such as aperture profile, radius and others calculated via the proposed model with the results of the radial fracture model that assumes standard pressure-independent fluid exchange mechanism governed by Carter’s law. Based on the comparison, we determine parameter ranges, for which the effect of the pressure-dependent fluid exchange mechanism is essential, and, on the other hand, we outline the zones for which Carter’s leak-off model provides accurate results. Finally, by using an approximate solution for the radial fracture with Carter’s leak-off, we obtain estimates of the effect in the entire parametric space, which allows one to evaluate the influence of the pressure-dependent leak-off for any problem parameters.

Fracturing Gels as Analogs to Understand Fracture Behavior in Shale Gas Reservoirs

Hydraulic fracturing is widely used in the exploitation of unconventional reservoirs, such as shale gas and tight gas. However, a full understanding of the activation of natural fractures, prediction of fracture growth, distribution of proppant, and network fracture system effectiveness remain unresolved. The onset of fracturing in the media requires energy and this is due to the buildup of pressure within the rock due to continuous injection of fluid. In other words, when the energy associated with the injection fluid reaches the fracture strength of the rock, the fracture initiates and propagates into the formation. Here, we use gelatin in hydraulic fracturing laboratory tests and compare the results to a modified radial hydraulic fracturing theory. The mechanics of the gelatin, procedures to make a testing gelatin block, and procedures to conduct the test are described. The results show that the fracture evolving behaviours from experiments are well matched by the theory. The results are then scaled up to understand fracture growth behaviour in a tight rock reservoir.

PyFrac: A planar 3D hydraulic fracture simulator

Fluid driven fractures propagate in the upper earth crust either naturally or in response to engineered fluid injections. The quantitative prediction of their evolution is critical in order to better understand their dynamics as well as to optimize their creation. We present an open-source Python implementation of a hydraulic fracture growth simulator based on the implicit level set algorithm originally developed by Peirce & Detournay (2008). This algorithm couples a finite discretization of the fracture with the use of the near tip asymptotic solutions of a steadily propagating semi-infinite hydraulic fracture. This allows to resolve the multi-scale processes governing hydraulic fracture propagation accurately, even on relatively coarse meshes. We present an overview of the mathematical formulation, the numerical scheme and the details of our implementation. A series of problems including a radial hydraulic fracture verification test, the propagation of a height contained hydraulic fracture, the lateral spreading of a magmatic dyke and an example of fracture closure are presented to demonstrate the capabilities, accuracy and robustness of the implemented algorithm. Program summaryProgram title:PyFracCPC Library link to program files:http://dx.doi.org/10.17632/gv7yy9mmwj.1Licensing provisions: GPLv3Programming language: PythonNature of problem: Simulation of the propagation and closure of a planar three-dimensional hydraulic fracture driven by the injection of a Newtonian fluid in a material having heterogeneous fracture toughness under a non-uniform in-situ stress field.Solution method: The fully coupled hydro-mechanical moving boundary problem is solved combining a finite volume scheme for lubrication flow with a boundary element method for elasticity. The algorithm couples a finite scale discretization of the fracture with the near-tip asymptotic solution of a steadily moving hydraulic fracture. The fracture front is tracked via a level set approach using a fast marching method.

The range of influence of the poroelastic effects in terms of dimensionless complexes for the radial hydraulic fracturing model

In our previous work the fully coupled model of the planar hydraulic fracture propagation in a poroelastic medium was developed. In present paper we simplify this model to the case of axisymmetric fracture propagation in order to find the conditions whereby the poroelastic effects are essential. We consider two dimensionless complexes responsible for the transition from 1D to 3D filtration and for the poroelastic influence on the fracture geometry. Analyzing a large set of case studies, we find the critical values for these parameters below which the aforementioned physical effects are not important. Particularly, we demonstrate that 1D pressure diffusion is preserved until the diffusion length scale is 2.4 times smaller than the fracture size. In turn, the diffusion scale plays a central role in the poroelastic stress influence on the fracture propagation. Also, we show that in case of low closure stress the pressure diffusion is dominated not by the filtration process but by the rock deformation with a slight impact on the fracture geometry.We believe that this study will become useful for the appropriate model selection in engineering practice.

Reduced order model for simultaneous growth of multiple closely-spaced radial hydraulic fractures

A new reduced order model (ROM) provides rapid and reasonably accurate prediction of the complex behavior of multiple, simultaneously growing radial hydraulic fractures. The method entails vastly reducing the degrees of freedom typically associated with fully-coupled simulations of this multiple moving boundary problem by coupling together an approximation of the influence of the stress interaction among the fractures (“stress shadow”) with an approximation of fluid flow and elasticity, ensuring preservation of global volume balance, global energy balance, elasticity, and compatibility of the crack opening with the inlet fluid flux. Validating with large scale (“high-fidelity”) simulations shows the ROM solution captures not only the basic suppression of interior hydraulic fractures in a uniformly-spaced array due to the well-known stress shadowing phenomenon, but also complex behaviors arising when the spacing among the hydraulic fractures is non-uniform. The simulator's usefulness is demonstrated through a proof-of-concept optimization whereby non-uniform spacing and stage length are chosen to maximize the fracture surface area and/or the uniformity of growth associated with each stimulation treatment.

Numerical model for a penny-shaped hydraulic fracture driven by laminar/turbulent fluid in an impermeable rock

As hydraulic fracturing at high injection rates with low viscosity fluids grows in popularity, so also there is a growing need to include not only the more common laminar fluid flow, but also the turbulent and transition flow regimes in numerical simulators. One common scenario is embodied in the behavior of a radial (penny-shaped) hydraulic fracture where flow is turbulent near the inlet, laminar near the tip, and in transition somewhere between. The main goal of this paper is to investigate the impact of this transition on hydraulic fracture growth through development and use of a numerical simulator for penny-shaped hydraulic fractures using the so-called drag reduction method to estimate the friction factor inside the crack for all relevant flow regimes. Upon solving this problem numerically for the case of zero toughness, comparing the results with fully laminar and fully turbulent asymptotic solutions shows that the early time behavior of radial hydraulic fractures is predominantly turbulent while large time behavior if predominantly laminar. The time scale associated with this transition determines the relevance of either limiting regime to practical cases, i.e. when the transition takes place in a small fraction of the total treatment time it suffices to approximate growth using the laminar asymptotic solution and when the transition requires are large time compared to the treatment time it suffices to approximate growth using the turbulent asymptotic solution.

Approximate semi-analytical solution for a penny-shaped rough-walled hydraulic fracture driven by turbulent fluid in an impermeable rock

The popularity of high injection rate hydraulic fracturing treatments using low viscosity fluids is driving a need to consider the turbulent and laminar-turbulent transition regimes of fluid flow in hydraulic fracture simulators. The radial model is one of the most important geometries both for benchmarking and as a starter solution for 3D and Planar 3D models. Here we provide a semi-analytical, orthogonal polynomial series solution for a rough-walled radial (penny-shaped) hydraulic fracture driven by a fully turbulent fluid. Embedding the appropriate pressure singularities in a family of orthogonal polynomials used for derivation of the solution leads to very rapid convergence of the series, requiring just two terms for an accurate result. We conclude with an investigation of the occurrence of this limiting regime by comparison with numerical simulations, illustrating that the fully turbulent regime is typically not encountered for the radial geometry, although the present solution remains necessary as a starter solution and benchmark for the numerical simulators that are required to capture the laminar-turbulent transition. By comparison with numerical simulations that consider the laminar-turbulent transition, we find that such an estimate is expected to be sufficient for practical purposes when the inlet opening predicted by the turbulent solution exceeds the inlet opening predicted by the laminar solution.