Self‐similar solution of a plane‐strain fracture driven by a power‐law fluid

This paper analyses the problem of a hydraulically driven fracture, propagating in an impermeable, linear elastic medium. The fracture is driven by injection of an incompressible, viscous fluid with power‐law rheology and behaviour index n?0. The opening of the fracture and the internal fluid pressure are related through the elastic singular integral equation, and the flow of fluid inside the crack is modelled using the lubrication theory. Under the additional assumptions of negligible toughness and no lag between the fluid front and the crack tip, the problem is reduced to self‐similar form. A solution that describes the crack length evolution, the fracture opening, the net fluid pressure and the fluid flow rate inside the crack is presented. This self‐similar solution is obtained by expanding the fracture opening in a series of Gegenbauer polynomials, with the series coefficients calculated using a numerical minimization procedure. The influence of the fluid index n in the crack propagation is also analysed. Copyright © 2002 John Wiley & Sons, Ltd.     

The influence of pumping parameters in fluid‐driven fractures in weak porous formations

In this work, we investigate the main pumping parameters that influence a fluid‐driven fracture in cohesive poroelastic and poroelastoplastic weak formations. These parameters include the fluid viscosity and the injection rate. The first parameter dominates in the mapping of the propagation regimes from toughness to viscosity, whereas the second parameter controls the storage to leak‐off dominated regime through diffusion. The fracture is driven in weak permeable porous formation by injecting an incompressible viscous fluid at the fracture inlet assuming that the fracture propagates under plane strain conditions. Fluid flow in the fracture is modeled by lubrication theory. Pore fluid movement in the porous formation is based on the Darcy law. The coupling follows the Biot theory, whereas the irreversible rock deformation is modeled with the Mohr–Coulomb yield criterion with associative flow rule. Fracture propagation criterion is based on the cohesive zone approach. Leak‐off is also considered. The investigation is performed numerically with the FEM to obtain the fracture opening, length, and propagation pressure versus time. We demonstrate that pumping parameters influence the fracture geometry and fluid pressures in weak formations through the viscous fluid flow and the diffusion process that create back stresses and large plastic zones as the fracture propagates. It is also shown that the product of the propagation velocity and fluid viscosity, µv that appears in the scaling controls the magnitude of the plastic zones and influences the net pressure and fracture geometry. These findings may explain partially the discrepancies in net pressures between field measurements and conventional model predictions for the case of weak porous formation. Copyright © 2014 John Wiley & Sons, Ltd.     

Uniform investigation of hydraulic fracturing propagation regimes in the plane strain model

The hydraulic fracturing propagation regimes in the plane strain model are uniformly investigated using a numerical method based on the finite element method. The regimes range from toughness‐dominated cases to viscosity‐dominated cases, covering zero leak‐off situations and small leak‐off situations. Unlike the asymptotic solutions, the numerical method is independent of the energy dissipation regimes and fluid storage regimes. The numerical method pays no special attention to the fracture tip, and it simulates fracture tip behaviors by increasing the number of functions in a natural and uniform manner. The numerical method is verified by comparing its results with the asymptotic solutions. The effect of the model sizes on the numerical method is discussed along with the robustness of the numerical method. Copyright © 2014 John Wiley & Sons, Ltd.     

Propagation of a hydraulic fracture parallel to a free surface

This paper analyses the plane strain problem of a fracture, driven by injection of an incompressible viscous Newtonian fluid, which propagates parallel to the free surface of an elastic half‐plane. The problem is governed by a hyper‐singular integral equation, which relates crack opening to net pressure according to elasticity, and by the lubrication equations which describe the laminar fluid flow inside the fracture. The challenge in solving this problem results from the changing nature of the elasticity operator with growth of the fracture, and from the existence of a lag zone of a priori unknown length between the crack tip and the fluid front. Scaling of the governing equations indicates that the evolution problem depends in general on two numbers, one which can be interpreted as a dimensionless toughness and the other as a dimensionless confining stress. The numerical method adopted to solve this non‐linear evolution problem combines the displacement discontinuity method and a finite difference scheme on a fixed grid, together with a technique to track both crack and fluid fronts. It is shown that the solution evolves in time between two asymptotic similarity solutions. The small time asymptotic solution corresponding to the solution of a hydraulic fracture in an infinite medium under zero confining stress, and the large time to a solution where the aperture of the fracture is similar to the transverse deflection of a beam clamped at both ends and subjected to a uniformly distributed load. It is shown that the size of the lag decreases (to eventually vanish) with increasing toughness and compressive confining stress. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical modeling of fluid‐driven fractures in cohesive poroelastoplastic continuum

In this article, we investigate the main parameters that influence the propagation of a fluid‐driven fracture in a poroelastoplastic continuum. These parameters include the cohesive zone, the stress anisotropy, and the pore pressure field. The fracture is driven in a permeable porous domain that corresponds to weak formation by pumping of an incompressible viscous fluid at the fracture inlet under plane strain conditions. Rock deformation is modeled with the Mohr–Coulomb yield criterion with associative flow rule. Fluid flow in the fracture is modeled by the lubrication theory. The movement of the pore fluid in the surrounding medium is assumed to obey the Darcy law and is of the same nature as the fracturing fluid. The cohesive zone approach is used as the fracture propagation criterion. The problem is modeled numerically with the finite element method to obtain the solution for the fracture length, the fracture opening, and the propagation pressure as a function of the time and distance from the pumping inlet. It is demonstrated that the plastic yielding that is associated with the rock dilation in an elastoplastic saturated porous continuum is significantly affected by the cohesive zone characteristics, the stress anisotropy, and the pore pressure field. These influences result in larger fracture profiles and propagation pressures due to the larger plastic zones that are developing during the fracture propagation. Furthermore, it is also found that the diffusion process that is a major mechanism in hydraulic fracture operations influences further the obtained results on the fracture dimensions, plastic yielding, and fluid pressures. These findings may explain partially the discrepancies in net pressures between field measurements and conventional model predictions. Copyright © 2012 John Wiley & Sons, Ltd.     

Solution for a plane strain rough‐walled hydraulic fracture driven by turbulent fluid through impermeable rock

The impact of turbulent flow on plane strain fluid‐driven crack propagation is an important but still poorly understood consideration in hydraulic fracture modeling. The changes that hydraulic fracturing has experienced over the past decade, especially in the area of fracturing fluids, have played a major role in the transition of the typical fluid regime from laminar to turbulent flow. Motivated by the increasing preponderance of high‐rate, water‐driven hydraulic fractures with high Reynolds number, we present a semianalytical solution for the propagation of a plane strain hydraulic fracture driven by a turbulent fluid in an impermeable formation. The formulation uses a power law relationship between the Darcy‐Weisbach friction factor and the scale of the fracture roughness, where one specific manifestation of this generalized friction factor is the classical Gauckler‐Manning‐Strickler approximation for turbulent flow in a rough‐walled channel. Conservation of mass, elasticity, and crack propagation are also solved simultaneously. We obtain a semianalytical solution using an orthogonal polynomial series. An approximate closed‐form solution is enabled by a choice of orthogonal polynomials embedding the near‐tip asymptotic behavior and thus giving very rapid convergence; a precise solution is obtained with 2 terms of the series. By comparison with numerical simulations, we show that the transition region between the laminar and turbulent regimes can be relatively small so that full solutions can often be well approximated by either a fully laminar or fully turbulent solution.     

Scaling laws for hydraulic fractures driven by a power-law fluid in homogeneous anisotropic rocks

This study investigates parametric space of solutions for a planar hydraulic fracture propagating in a homogeneous anisotropic rock. It is assumed that the fracture has an elliptical shape and is driven by a power-law fluid. The purpose of this study is to investigate the influence of anisotropy and power-law fluid rheology on the parametric space of solutions. Rock anisotropy is represented by having two values of fracture toughness, one in the vertical direction and another one in the horizontal direction. Similarly, the effect of elastic anisotropy is approximated by using two different effective elastic moduli in the vertical and horizontal directions. In contrast to the isotropic case, for which there are four limiting solutions, the problem for anisotropic rocks features six different limiting cases. These cases represent competition between toughness and viscosity in the vertical and horizontal directions and competition between fluid storage inside the fracture and fluid leak-off into formation. Approximate expressions for the limiting solutions are obtained using global volume balance and tip asymptotic solutions. Despite the developed solutions rely on a series of approximations, they precisely capture all the scaling laws associated with the problem. Zones of applicability of these limiting solutions are calculated, and their dependence on the problem parameters is investigated.     

Compaction-driven fluid flow in viscoelastic rock

Abstract

Compaction driven fluid flow is inherently unstable such that an obstruction to upward fluid flow (i.e. a shock) may induce fluid-filled waves of porosity, propagated by dilational deformation due to an effective pressure gradient within the wave. Viscous porosity waves have attracted attention as a mechanism for melt transport, but are also a mechanism for both the transport and trapping of fluids released by diagenetic and metamorphic reactions. We introduce a mathematical formulation applicable to compaction driven flow for the entire range of rheological behaviors realized in the lithosphere. We then examine three first-order factors that influence the character of fluid flow: (1) thermally activated creep, (2) dependence of bulk viscosity on porosity, and (3) fluid flow in the limit of zero initial connected porosity. For normal geothermal gradients, thermally activated creep stabilizes horizontal waves, a geometry that was thought to be unstable on the basis of constant viscosity models. Implications of this stabilization are that: (1) the vertical length scale for compaction driven flow is generally constrained by the activation energy for viscous deformation rather than the viscous compaction length, and (2) lateral fluid flow in viscous regimes may occur on greater length scales than anticipated from earlier estimates of compaction length scales. In viscous rock, inverted geothermal gradients stabilize vertically elongated waves or vertical channels. Decreasing temperature toward the earth’s surface can induce an abrupt transition from viscous to elastic deformation-propagated fluid flow. Below the transition, fluid flow is accomplished by short wavelength, large amplitude waves; above the transition flow is by high velocity, low amplitude surges. The resulting transient flow patterns vary strongly in space and time. Solitary porosity waves may nucleate in viscous, viscoplastic, and viscoelastic rheologies. The amplitude of these waves is effectively unlimited for physically realistic models with dependence of bulk viscosity on porosity. In the limit of zero initial connected porosity, arguably the only model relevant for melt extraction, travelling waves are only possible in a viscoelastic matrix. Such waves are truly self-propagating in that the fluid and the wave phase velocities are identical; thus, if no chemical processes occur during propagation, the waves have the capacity to transmit geochemical signatures indefinitely. In addition to solitary waves, we find that periodic solutions to the compaction equations are common though previously unrecognized. The transition between the solutions depends on the pore volume carried by the wave and the Darcyian velocity of the background fluid flux. Periodic solutions are possible for all velocities, whereas solitary solutions require large volumes and low velocities. © Elsevier, Paris     

Capturing the two‐way hydromechanical coupling effect on fluid‐driven fracture in a dual‐graph lattice beam model

Fluid‐driven fractures of brittle rock is simulated via a dual‐graph lattice model. The new discrete hydromechanical model incorporates a two‐way coupling mechanism between the discrete element model and the flow network. By adopting an operator‐split algorithm, the coupling model is able to replicate the transient poroelasticity coupling mechanism and the resultant Mandel‐Cryer hydromechanical coupling effect in a discrete mechanics framework. As crack propagation, coalescence and branching are all path‐dependent and irreversible processes, capturing this transient coupling effect is important for capturing the essence of the fluid‐driven fracture in simulations. Injection simulations indicate that the onset and propagation of fractures is highly sensitive to the ratio between the injection rate and the effective permeability. Furthermore, we show that in a permeable rock, the borehole breakdown pressure, the pressure at which fractures start to grow from the borehole, depends on both the given ratio between injection rate and permeability and the Biot coefficient.     

Fully coupled simulation of a hydraulic fracture interacting with natural fractures with a hybrid discrete‐continuum method

A hybrid discrete‐continuum numerical scheme is developed to study the behavior of a hydraulic fracture crossing natural fractures. The fully coupled hybrid scheme utilizes a discrete element model for an inner domain, within which the hydraulic fracture propagates and interacts with natural fractures. The inner domain is embedded in an outer continuum domain that is implemented to extend the length of the hydraulic fracture and to better approximate the boundary effects. The fracture is identified to propagate initially in the viscosity‐dominated regime, and the numerical scheme is calibrated by using the theoretical plane strain hydraulic fracture solution. The simulation results for orthogonal crossing indicate three fundamental crossing scenarios, which occur for various stress ratios and friction coefficients of the natural fracture: (i) no crossing, that is, the hydraulic fracture is arrested by the natural fracture and makes a T‐shape intersection; (ii) offset crossing, that is, the hydraulic fracture crosses the natural fracture with an offset; and (iii) direct crossing, that is, the hydraulic fracture directly crosses the natural fracture without diversion. Each crossing scenario is associated with a distinct net pressure history. Additionally, the effects of strength contrast and stiffness contrast of rock materials and intersection angle between the hydraulic fracture and the natural fracture are also investigated. The simulations also illustrate that the level of fracturing complexity increases as the number and extent of the natural fractures increase. As a result, we can conclude that complex hydraulic fracture propagation patterns occur because of complicated crossing behavior during the stimulation of naturally fractured reservoirs. Copyright © 2017 John Wiley & Sons, Ltd.     

Coupled hydromechanical‐fracture simulations of nonplanar three‐dimensional hydraulic fracture propagation

This paper presents an algorithm and a fully coupled hydromechanical‐fracture formulation for the simulation of three‐dimensional nonplanar hydraulic fracture propagation. The propagation algorithm automatically estimates the magnitude of time steps such that a regularized form of Irwin's criterion is satisfied along the predicted 3‐D fracture front at every fracture propagation step. A generalized finite element method is used for the discretization of elasticity equations governing the deformation of the rock, and a finite element method is adopted for the solution of the fluid flow equation on the basis of Poiseuille's cubic law. Adaptive mesh refinement is used for discretization error control, leading to significantly fewer degrees of freedom than available nonadaptive methods. An efficient computational scheme to handle nonlinear time‐dependent problems with adaptive mesh refinement is presented. Explicit fracture surface representations are used to avoid mapping of 3‐D solutions between generalized finite element method meshes. Examples demonstrating the accuracy, robustness, and computational efficiency of the proposed formulation, regularized Irwin's criterion, and propagation algorithm are presented.     

Three‐dimensional time‐harmonic fundamental solutions for a fluid‐saturated poroelastic half‐space with partially permeable free surface

By virtue of a pair of scalar potentials for the displacement of the solid skeleton and the pore fluid pressure field of a saturated poroelastic medium, an alternative solution method to the Helmholtz decomposition is developed for the wave propagation problems in the framework of Biot's theory. As an application, a comprehensive solution for three‐dimensional response of an isotropic poroelastic half‐space with a partially permeable hydraulic free surface under an arbitrarily distributed time‐harmonic internal force field and fluid sources is developed. The Green's functions for the poroelastic fields, corresponding to point, ring, and disk loads, are reduced to semi‐infinite complex‐valued integrals that can be evaluated numerically by an appropriate quadrature scheme. Analytical and numerical comparisons are made with existing elastic and poroelastic solutions to illustrate the quality and features of the solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Buoyancy-driven propagation of an isolated fluid-filled crack in rock: implication for fluid transport in metamorphism

A new model for upward transport of buoyant fluid released during metamorphism is proposed. The model is fluid transport by buoyancy-driven propagation of isolated fluid-filled cracks. The mechanical behavior of a two-dimensional, isolated, vertical, and fluid-filled crack in impermeable rock is investigated using linear fractire mechanics and fluid dynamics. The results show that steady-state crack propagation which causes long-distance transport of the fluid occurs when the vertical cross-sectional area of the crack exceeds a critical value. Propagation velocity and average thickness of the crack under the steady-state propagation regime are expressed explicitly by the following seven parameters: vertical crack length; rigidity, Poisson's ratio, and fracture toughness of the rock; fluid viscosity; density difference between the rock and the fluid; gravitational acceleration. An isolated H₂O-filled crack of vertical length 100 m, for example, propagates upwards in the crust at 0.3 m/s with the average thickness 0.2 mm when the following likely values are assumed: 0.1 mPa s for the H₂O viscosity; 3 MPa m^1/2 for the fracture toughness of the crustal rock. The application of the obtained results to the transport of H₂O released during metamorphism suggests that the number density of isolated cracks propagating in the crust is very low. Since the propagation velocity is high, our model is suitable particularly for fluid transport through hot quartz-rich rock where fluid-filled cracks have geologically short lifetimes.     

Similarity solutions for wedge-shaped hydraulic fractures driven into a permeable medium by a constant inlet pressure

Similarity solutions are derived for wedge-shaped hydraulic fractures driven by a constant inlet pressure P₀ into a permeable medium under a uniform confining stress σ. These results describe the seepage-dominated regime in which most of the injected fluid is lost into the permeable walls of the fracture; they complement previous results for the capacitance-dominated regime in which seepage is negligible. Fracture propagation velocity is obtained as an analytical function of fracture length, driving pressure, confining stress, material properties and a single separation constant or eigenvalue which is determined numerically. Self-similar profiles of pressure, opening displacement and fluid velocity along the fracture are presented, together with the self-similar isobars of the two-dimensional pressure field within the permeable medium. Comprehensive results are reported for laminar or turbulent flow of a constant-compressibility liquid or an ideal gas driven by overpressures (P₀?σ)/σ ranging from 10^?2 to 10².     

Anisotropic porothermoelastic solution and hydro‐thermal effects on fracture width in hydraulic fracturing

In this study, the effects of the temperature difference between hydraulic fracturing fluid and rock formation on the time‐dependent evolution of fracture width were investigated using a newly derived one‐dimensional anisotropic porothermoelastic analytical solution. The solution is shown to correctly reproduce existing solutions for special cases and corrections for an earlier publication are provided. An analysis of time‐dependent fracture width evolution using Woodford Shale data was also presented. It was found that when the fracturing fluid has the same temperature as the shale formation, the fracture gradually closes back after the initial opening due to the invasion of the fracturing fluid. Practically, in this scenario, proppants should be pumped into the fracture as soon as possible to obtain maximum fracture conductivity. On the other hand, with a fracturing fluid 60 °C colder than the formation, the thermal contraction of the rock dominates the fracture aperture evolution, resulting in a fracture aperture approximately 70% larger than that produced by the hotter fracturing fluid. Consequently, in this case, it is beneficial to delay proppant placement to take advantage of the widening fractures. Finally, it was found that the fracture aperture is directly controlled by the spacing of natural fractures. Therefore, the presence of natural fractures in the shale formation and their spacing influence not only the type of hydraulic fractures created but also what kind and size of proppants should be used to keep them open. Copyright © 2013 John Wiley & Sons, Ltd.     

Strain‐softening analysis of a spherical cavity

This paper studies the excavation of a spherical cavity subjected to hydrostatic initial stresses in the infinite homogeneous and isotropic rock mass with strain‐softening Mohr–Coulomb (M‐C) and Hoek–Brown (H‐B) behaviors. Numerical solutions of the spherical cavity are obtained and the application to determining stress–strain curve of strain‐softening M‐C and H‐B rock mass is studied. A closed‐form solution for the elastic–brittle–plastic medium is introduced first, and then a numerical procedure that simplifies the strain‐softening process into a series of brittle–plastic ones is presented. The approach is validated against the facts that the strain‐softening process evolves into a brittle–plastic one when the softening slope is very steep, whereas it evolves into an elasto‐plastic one when the softening slope approaches zero. Numerical solutions for the prediction of displacements and stresses around the spherical cavity in the strain‐softening M‐C and H‐B rock mass are presented. On the basis of the analysis of the spherical cavity in strain‐softening rock mass, the stress–strain relationship at an infinitesimal cube around the cavity is obtained and discussed with different evolution laws for the strength parameters considered. Copyright © 2011 John Wiley & Sons, Ltd.     

Modeling of dynamic cohesive fracture propagation in porous saturated media

In this paper, a mathematical model is presented for the analysis of dynamic fracture propagation in the saturated porous media. The solid behavior incorporates a discrete cohesive fracture model, coupled with the flow in porous media through the fracture network. The double‐nodded zero‐thickness cohesive interface element is employed for the mixed mode fracture behavior in tension and contact behavior in compression. The crack is automatically detected and propagated perpendicular to the maximum effective stress. The spatial discretization is continuously updated during the crack propagation. Numerical examples from the hydraulic fracturing test and the concrete gravity dam show the capability of the model to simulate dynamic fracture propagation. The comparison is performed between the quasi‐static and fully dynamic solutions, and the performance of two analyses is investigated on the values of crack length and crack mouth opening. Copyright © 2010 John Wiley & Sons, Ltd.     

Validating the ability of the discontinuous deformation analysis method to model normal P‐wave propagation across rock fractures

The effects of fractures on wave propagation problems are increasingly abstracting the attention of scholars and engineers in rock engineering field. This study aims to fully validate the ability of discontinuous deformation analysis (DDA) to model normal P‐wave propagation across rock fractures. The effects of a single fracture and multiple parallel fractures are all tested. The results indicate that DDA can accurately reflect the fracture effects, including the fractures stiffness, the fracture spacing and the fracture number, and the effects of incident wave frequency on one‐dimensional P‐wave propagation problems. Thus, DDA is able to deal well with normal incident P‐wave propagation problems. Copyright © 2017 John Wiley & Sons, Ltd.     

Finite element simulations of 3D planar hydraulic fracture propagation using a coupled hydro-mechanical interface element

Two-dimensional hydraulic fracturing simulations using the cohesive zone model (CZM) can be readily found in the literature; however, to our knowledge, verified 3D cohesive zone modeling is not available. We present the development of a 3D fully coupled hydro-mechanical finite element method (FEM) model (with parallel computation framework) and its application to hydraulic fracturing. A special zero-thickness interface element based on the CZM is developed for modeling fracture propagation and fluid flow. A local traction-separation law with strain softening is used to capture tensile cracking. The model is verified by considering penny-shaped hydraulic fracture and plain strain Kristianovich‑Geertsma‑de Klerk hydraulic fracture (in 3D) in the viscosity- and toughness-dominated regimes. Good agreement between numerical results and analytical solutions has been achieved. The model is used to investigate the influence of rock and fluid properties on hydraulic fracturing. Lower stiffness tip cohesive elements tend to yield a larger elastic deformation around the fracture tips before the tensile strength is reached, generating a larger fracture length and lower fracture pressure compared with higher stiffness elements. It is found that the energy release rate has almost no influence on hydraulic fracturing in the viscosity-dominated regime because the energy spent in creating new fractures is too small when compared with the total input energy. For the toughness-dominated regime, the released energy during fracturing should be accurately captured; relatively large tensile strength should be used in order to match numerical results to the asymptotic analytical solutions. It requires smaller elements when compared with those used in the viscosity-dominated regime.     

High‐grade localized metasomatic alteration of the granitic gneiss surrounding a clinopyroxene‐rich pegmatoid dyke: Söndrum stenhuggeriet,Halmstad, SW Sweden

The Söndrum stone quarry (Halmstad, SW Sweden) exposes a transition from migmatized granitic gneissic country rock into a foliated clinopyroxene‐free granitic gneiss, and then a central pegmatoid dyke dominated by clinopyroxene megacrysts. This transition zone represents a fracture‐controlled, fluid‐alteration zone that developed under conditions of 650–700 °C and 790 MPa. Mineral chemical trends in F, Cl, Fe, Ti, Mn and Y are interpreted as documenting a fluid infiltration event associated with the formation of the pegmatoid dyke. Fluid inclusions from the pegmatoid dyke are CO₂ dominant, whereas in the surrounding country rock they are dominated by H₂O‐NaCl‐CaCl₂. Fluid inclusions from the intermediate foliated clinopyroxene‐free granitic gneiss are a mixture of the two types. The pegmatoid dyke appears to have originated from a high Ca activity, Fe‐Mg enriched, fluid‐rich granitic melt with a CO₂ component, which was emplaced along a tectonic fracture in a regionally migmatized granitic gneiss in the lower crust. This was accompanied by limited partial melting of the surrounding granitic gneiss. The Ca activity of the melt was high enough to allow for the formation of clinopyroxene megacrysts as opposed to orthopyroxene. H₂O‐enriched fluids expelled from the crystallizing pegmatoid dyke, which retained the majority of the CO₂ helping to stabilize the clinopyroxene. The expelled fluids coarsened and chemically affected the surrounding country rock resulting in trends in the mineral and fluid inclusion chemistry seen today in the foliated and regional granitic gneiss.