Fluid-driven Crack Research Articles

A Model for Simulating the Upward Flow of a Viscous Fluid in a Fracture Network

Fluid migration in a fracture network plays an important role in the oil accumulation mechanism and hence is key to oil exploration. In this study, we build a model by combining one-dimensional (1D) Navier–Stokes equations, linear elastic equations, and energy equations, and validate the model by reproducing the thickness profile of a fluid-driven crack measured in an experiment. We employ this model to simulate the upward flow of viscous fluid in a single fracture during hydrocarbon migration. The simulation suggests that the parameters of both the fluid and the surrounding rock matrix, as well as the boundary condition imposed on the fracture outlet, affect the upward flow in the fracture. We then extend our model from the single fracture to the bifurcated fracture and the fracture network by maintaining homogeneous pressure and mass conservation at the connection of the channels. We find that the increase in network complexity leads to an increase in the inlet pressure gradient and inlet speed, and a decrease in the outlet pressure gradient and outlet speed. The effective area where the fluid is driven upward from the inlet to the outlet is offset toward the inlet. More importantly, the main novelty of our model is that it allows us to evaluate the effect of inconsistencies in individual branch parameters, such as matrix stiffness, permeability, temperature, and boundary conditions, on the overall upward flow of viscous fluid. Our results suggest that the heterogeneity enforces the greater impact on the closer branches.

Single fluid-driven crack propagation in analogue rock assisted by chemical environment

During the operation of hydraulic fracturing as used in many geo-energy and geo-environment applications, chemical stimulation is often incorporated for cracking enhancement in low-permeability geological formations for the purpose of an optimization of energy recovery. The mechanism of subcritical crack propagation in a chemically reactive environment is essential for understanding of the involved coupled chemo-mechanical process and a better control of acid-assisted hydraulic fracturing. It has been postulated that the rate of crack propagation under environmental loads is inherited from the chemical processes involved including the reaction and the diffusive transport. However, laboratory explorations focusing on the evolving interplay between the propagation of a fluid-pressurizing individual crack and the environment it is subject to via a variable chemical intensity imposed have been rare. Here we present an experimental investigation on a single tensile crack propagation in alginate hydrogel as an analogue material for brittle rocks, driven by the injection of a chemically reactive fluid kept at constant pressure using a Hele-Shaw cell setup. We show that an intensified chemical environment can accelerate tensile crack propagation in both subcritical crack growth and fracturing regimes, while leading to the Region III fracturing of less brittle characteristics. During the experiment, crack-tip blunting upon injection of reactive solutions was observed, suggesting a competing mechanism between the crack-tip geometry induced toughening and the chemically induced softening within the process zone as the crack advances. Our results provide quantitative insights into how a chemically reactive environment facilitates the growth of a single macroscopic crack of mode I opening in a low-permeability matrix through coupled chemo-mechanical feedback. The imposed chemical environment promotes crack propagation while alleviating the stress concentration at the advancing crack tip, suggesting a more energy-efficient method compared to pure water fracturing. We anticipate our experimental investigation presented here to be a starting point of sound laboratory support for future studies towards a more controllable technique of chemical stimulation in geomaterials as well as complementing the ongoing modeling efforts in reactive chemo-mechanics.

Poroelastic problem of a non-penetrating crack with cohesive contact for fluid-driven fracture

A new class of unilaterally constrained problems for fully coupled poroelastic models stemming from hydraulic fracturing is introduced and studied with respect to its well-posedness. The poroelastic medium contains a fluid-driven crack, which is subjected to non-penetration conditions and cohesion forces between the crack faces. Existence of solution for the governing elliptic–parabolic variational inequality under the unilateral constraint with a small cohesion is established using the incremental approximation based on Rothe’s semi-discretization in time.

Hydraulic fracturing analysis in fluid-saturated porous medium.

This paper addresses fluid-driven crack propagation in a porous medium. Cohesive interface elements are employed to model the behaviour of the crack. To simulate hydraulic fracturing, a fluid pressure degree of freedom is introduced inside the crack, separate from the fluid degrees of freedom in the bulk. Powell-Sabin B-splines, which are based on triangles, are employed to describe the geometry of the domain and to interpolate the field variables: displacements and interstitial fluid pressure. Due to their -continuity, the stress and pressure gradient are smooth throughout the whole domain, enabling a direct assessment of the fracture criterion at the crack tip and ensuring local mass conservation. Due to the use of triangles, crack insertion and remeshing are straightforward and can be done directly in the physical domain. During remeshing a mapping of the state vector (displacement and interstitial fluid pressure) is required. For this, a new methodology is exploited based on a least-square fit with the energy balance and mass conservation as constraints. The accuracy to model free crack propagation is demonstrated by two numerical examples, including crack propagation in a plate with twonotches.

A multi-resolution approach to hydraulic fracture simulation

We present a multi-resolution approach for constructing model-based simulations of hydraulic fracturing, wherein flow through porous media is coupled with fluid-driven fracture. The approach consists of a hybrid scheme that couples a discrete crack representation in a global domain to a phase-field representation in a local subdomain near the crack tip. The multi-resolution approach addresses issues such as the computational expense of accurate hydraulic fracture simulations and the difficulties associated with reconstructing crack apertures from diffuse fracture representations. In the global domain, a coupled system of equations for displacements and pressures is considered. The crack geometry is assumed to be fixed and the displacement field is enriched with discontinuous functions. Around the crack tips in the local subdomains, phase-field sub-problems are instantiated on the fly to propagate fractures in arbitrary, mesh independent directions. The governing equations and fields in the global and local domains are approximated using a combination of finite-volume and finite element discretizations. The efficacy of the method is illustrated through various benchmark problems in hydraulic fracturing, as well as a new study of fluid-driven crack growth around a stiff inclusion.

Thermo-hydro-chemo-mechanical coupling peridynamic model of fractured rock mass and its application in geothermal extraction

In this paper, a thermo-hydro-chemo-mechanical coupling peridynamic model of fractured rock mass is proposed to simulate the fluid-driven crack and the characteristics of water pressure and temperature under thermo-hydro-chemo-mechanical coupling condition. Firstly, the coupling model is employed to simulate the thermo-hydraulic coupling consolidation problem of rock samples, and its accuracy is verified by comparing with the analytical solution and the previous numerical results. Secondly, the coupling model is employed to simulate the problems of fluid-driven crack and chemical corrosion, and the present numerical results are in good agreement with those obtained by the phase field method. Finally, thermo-hydro-chemo-mechanical coupling peridynamic model is used to simulate the process of geothermal extraction, and to investigate water pressure and thermal field distribution.

The energy release rate for non-penetrating crack in poroelastic body by fluid-driven fracture

A new class of constrained variational problems, which describe fluid-driven cracks (that are pressurized fractures created by pumping fracturing fluids), is considered within the nonlinear theory of coupled poroelastic models stated in the incremental form. The two-phase medium is constituted by solid particles and fluid-saturated pores; it contains a crack subjected to non-penetration condition between the opposite crack faces. The inequality-constrained optimization is expressed as a saddle-point problem with respect to the unknown solid phase displacement, pore pressure, and contact force. Applying the Lagrange multiplier approach and the Delfour–Zolésio theorem, the shape derivative for the corresponding Lagrangian function is derived using rigorous asymptotic methods. The resulting formula describes the energy release rate under irreversible crack perturbations, which is useful for application of the Griffith criterion of quasi-static fracture.

An XFEM-based approach for 3D hydraulic fracturing simulation considering crack front segmentation

Hydraulic fracturing is a commonly adopted and effective well stimulation technique in the oil and gas extraction area. Under a mixed I/III loading condition, the crack front segmentation (i.e., the parent crack segments into echelon-shaped daughter cracks) usually occurs, which highly complicates the paths of fluid-driven cracks. This paper presents an efficient numerical model for 3D hydraulic fracturing simulation considering crack front segmentation on basis of the extended finite element method (XFEM). Solutions of the momentum balance equation and the fluid flow equation are simultaneously determined by the Newton-Raphson method along with a reduction technique. In the XFEM framework, a robust local mesh-refinement scheme of the tip-enriched elements is designed to enhance the resolution of the near-front stress field which is crucial for the determination of crack segmentation and propagation behaviors. The locally refined tip-enriched elements are then divided into a series of tetrahedra to perform high-accuracy numerical integration. After verification of the proposed approach, the effects of several critical parameters in hydraulic fracturing treatments are investigated. Results show that crack front segmentation has significant effects on the resulting crack paths and crack aperture distribution. The propagation of hydraulic fractures will be depressed on account of the stress shadow induced by overlapped segments, leading to higher pumping pressure compared to the case without considering front segmentation. The sensitivity analyses indicate that larger elastic modulus of rock formation, larger fluid viscosity, higher fluid pumping rate, and smaller fluid leak-off coefficient can alleviate the influence of crack front segmentation on the pumping pressure. Larger elastic modulus, larger fluid viscosity, higher fluid pumping rate, and greater fluid leak-off coefficient lead to smaller twisting angles of the segments and smaller overlapping ratios.

Fluid-Driven Cracks Tunneling in Cemented Hydrocarbon Wells

Abstract Pressurized fluids may invade into the cement sheath of hydrocarbon wells during hydraulic fracturing and other construction operations in shale gas and oil exploitation. Excessive fluid pressure provides a driving force for small cracks to grow and tunnel through the length of the well, leading to the loss of zonal isolation and severe consequences. This work studies fluid-driven cracks tunneling in the bulk or along the interfaces of the cement sheath. We calculate the energy release rate of the tunneling crack as a function of the width of the tunnel. As long as the maximum energy release rate is below the fracture energy of the cement or the interfaces, no tunnels will form. This failsafe criterion predicts that the interfaces are much more vulnerable to tunneling cracks than the bulk of cement sheath. It is further shown that cement sheath with high Young’s modulus and high Poisson’s ratio can better resist fluid-driven cracks. The influence of casing pressure is also examined. Different loading conditions may have conflicting requirements on the properties of cement sheath. Compromises need to be made in engineering practice.

Asymptotic series solution for plane poroelastic model with non-penetrating crack driven by hydraulic fracture

A new class of coupled poroelastic problems describing fluid-driven cracks (called fractures) subjected to non-penetration conditions between opposite crack faces (fracture walls) is considered in the incremental form. The nonlinear crack problem for a plane isotropic setting in a two-phase medium is expressed in polar coordinates as a variational inequality with respect to the solid phase displacement and the pore pressure. Applying nonlinear methods, the asymptotic theory and Fourier analysis, a semi-analytic solution given as the power series in the sector of angle 2π is proven using rigorous expansions with respect to the distance to the crack-tip. Here no logarithmic terms occur in the asymptotic expansion. Consequently, a square-root singularity for the poroelastic medium with a non-penetrating crack is derived, and the integral formulas for calculating the corresponding stress intensity factors are obtained.

Poroelastic medium with non-penetrating crack driven by hydraulic fracture: Variational inequality and its semidiscretization

A new class of unilateral variational models appearing in the theory of poroelasticity is introduced and studied. A poroelastic medium consists of solid phase and pores saturated with a Newtonian fluid. The medium contains a fluid-driven crack, which is subjected to non-penetration between the opposite crack faces. The fully coupled poroelastic system includes elliptic–parabolic governing equations under the unilateral constraint. Well-posedness of the corresponding variational inequality is established based on the Rothe semi-discretization in time, after subsequent passing time step to zero. The NLCP-formulation of non-penetration conditions is given which is useful for a semi-smooth Newton solution strategy.

Enhanced Pseudo-Three-Dimensional Model for Multiple Hydraulic Fractures

Abstract Modeling the propagation of hydraulic fractures is a complex problem that involves solving the equations that describe the intimate coupling between the cracked and porous solid phase and the fracturing fluid that drives the process according to established crack extension criteria. Most of the existing computational techniques become prohibitively expensive if applied to three-dimensional configurations, especially those involving multiple fractures because their accuracy relies on extremely fine discretizations of the geometry. These include the standard finite element method and standard displacement discontinuity method, which require substantial mesh densities in the vicinity of the crack front to capture the complex nature of the singular fields associated with fluid-driven cracks. Models have recently been proposed that offer sufficient accuracy and a significant increase in computational efficiency. These include the enhanced pseudo-3D model that explicitly incorporates a non-local elasticity relation and known crack-front asymptotic solutions that capture the multiscale near-front behavior. Here, we present an enhanced pseudo-3D model for simulating the propagation of multiple planar and nonplanar hydraulic fractures in a linear elastic medium. The algorithm is built on a fixed mesh with an adaptive tip element, and it incorporates leak-off, as well as stress interaction between the fractures. The model is validated against reference solutions of plane strain, radial, and multi-planar fracture geometries, for various sets of parameters that cover the majority of the problem parametric space.

Hybrid FEM and peridynamic simulation of hydraulic fracture propagation in saturated porous media

This paper presents a hybrid modeling approach for simulating hydraulic fracture propagation in saturated porous media: ordinary state-based peridynamics is used to describe the behavior of the solid phase, including the deformation and crack propagation, while FEM is used to describe the fluid flow and to evaluate the pore pressure. Classical Biot poroelasticity theory is adopted. The proposed approach is first verified by comparing its results with the exact solutions of two examples. Subsequently, a series of pressure- and fluid-driven crack propagation examples are solved and presented. The phenomenon of fluid pressure oscillation is observed in the fluid-driven crack propagation examples, which is consistent with previous experimental and numerical evidences. All the presented examples demonstrate the capability of the proposed approach in solving problems of hydraulic fracture propagation in saturated porous media.

Coupling of meshfree peridynamics with the Finite Volume Method for poroelastic problems

Peridynamics is a non-local theory of continuum mechanics that has been developed primarily for understanding material failure due to different mechanisms, including fluid-driven crack propagation during hydraulic fracturing of subsurface reservoirs. Because of its non-local nature, this theory is computationally expensive. To improve its performance, a scheme was recently proposed for coupling Peridynamics (PD) with less expensive Finite Element Method (FEM) for static equilibrium problems. This scheme has been adapted in the current paper to couple a PD-based poroelastic model with the Finite Volume Method (FVM) for simulating problems in porous media. Coupling is implemented by dividing the computational domain into two subdomains, one of which is discretized and solved with PD and the other with FVM. The formulation is developed for porous flow involving fluid mass balance and is extended for poroelastic problems to include rock momentum balance. The coupled model is verified against the analytical solutions to classical problems. No spurious behavior is observed near the PD-FVM interface region. Improvements in computational performance over the pure PD model are demonstrated. Moreover, due to differences in the sparsity patterns and the magnitudes of PD and FVM transmissibility/Jacobian terms, it is shown that appending the PD equations after all the FV equations in the global matrix has additional computational benefits.

Hydro-mechanical fracture modeling governed by topological derivatives

The process of fluid-driven crack propagation in permeable rocks is investigated by a simple hydro-mechanical model and the concept of topological derivative. Analytical and numerical computations are made to propose a promising model tested and validated on a series of bidimensional benchmark examples. The main guidelines for this model are the use of simple finite elements with a minimal number of user-defined algorithmic parameters.

Numerical investigation of fluid-driven crack propagation and coalescence in granite specimen with two pre-existing flaws

Natural oil-gas-bearing rock formation generally contains a large number of discontinuities, which have a large influence on the cracking process inside the rock under high water pressure condition. Hence, a good understanding of the propagation and coalescence of fluid-driven cracks is important to improve the oil and gas exploitation efficiency. This study numerically investigates the fluid-driven crack propagation process in rock specimen possessing two pre-existing flaws using a fluid coupled discrete element method. The micro-parameters are first calibrated against the mechanical properties of the Lac du Bonnet granite. The numerical specimen models are generated by installing two flaws with different ligament angles, ligament length, and flaw angles. The injection test with a constant rate is then conducted to study the propagation and coalescence patterns in these pre-cracked numerical models. The numerical results show that a relatively large ligament angle is better to accelerate the crack coalescence in the specimen with two paralleled flaws. Numerical model possessing two pre-existing flaws with a small ligament length is inclined to traverse the ligament area. Overall, the results in this study reveal that the pre-existing flaws inside the rock have a significant effect on the fluid-driven cracking process and much attention should be paid to the crack propagation behavior when many discontinuities are associated in the oil-gas-bearing rock formation.

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

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

Experimental exploration of fluid-driven cracks in brittle hydrogels

Hydraulic fracturing is a procedure by which a fracture is initiated and propagates due to pressure (hydraulic loading) applied by a fluid introduced inside the fracture. In this study, we focus on a crack driven by an incompressible Newtonian fluid, injected at a constant rate into an elastic matrix. The injected fluid creates a radial fracture that propagates along a plane. We investigate this type of fracture both theoretically and experimentally. Our experimental apparatus uses a brittle and transparent polyacrylamide hydrogel matrix. Using this medium, we examine the rate of radial crack growth, fracture aperture, shape of the crack tip and internal fluid flow field. Our range of experimental parameters allows us to exhibit two distinct fracturing regimes, and the transition between these, in which the rate of radial crack propagation is dominated by either viscous flow within the fracture or the material toughness. Measurements of the profiles near the crack tip provide additional evidence of the viscosity-dominated and toughness-dominated regimes, and allow us to observe the transition from the viscous to the toughness regime as the crack propagates. Particle image velocimetry measurements show that the flow in the crack is radial, as expected in the viscous regime and in the early stages of the toughness regime. However, at later times in the toughness regime, circulation cells are observed in the flow within the crack that destroy the radial symmetry of the flow field.

Simulate intersecting 3D hydraulic cracks using a hybrid “FE-Meshfree” method

In this work, a discontinuous hybrid “FE-Meshfree” method is developed to simulate three-dimensional (3D) cracking. Then, an algorithm of Local Adaptive Background Sub-element (LABS-element) is developed in the context of hybrid “FE-Meshfree” method to simulate intersecting cracks. Compared to the extended finite element method (XFEM), the present hybrid “FE-Meshfree” method is not required to increase the size of global stiffness matrix, introduce extra unknowns, and construct special enrichment functions. The present algorithms of LABS-element and “FE-Meshfree” method are validated by intensive numerical tests, which achieves the balance between accuracy and flexibility. Additionally, a hydraulic and mechanical (H-M) coupling model is generated, in which the deformation of rocks and the propagation of cracks are solved by the hybrid “FE-Meshfree” method, meanwhile the fluid flow in cracks is solved by a fluid simulator based on the principle of parallel-plate flow model. This H-M coupling model is then used to investigate the propagation of fluid-driven cracks in rock mass with multiple pre-existing cracks, and the observed fluid pressure feedback is potentially useful during shale oil/gas well simulation.

Combined Effect of Pressure and Shear Stress on Penny-Shaped Fluid-Driven Cracks

Penny-shaped fluid-driven cracks are often detected in many fluid–solid interaction problems. We study the combined effect of pressure and shear stress on the crack propagation in an impermeable elastic full space. Boundary integral equations are presented, by using the integral transform method, for a penny-shaped crack under normal and shear stresses. The crack propagation criterion of stress intensity factor is examined with the strain energy release rate. Dominant regimes are obtained by using a scaling analysis. Asymptotic solution of the toughness-dominant regime is derived to show the effect of shear stress on the crack opening, crack length, and pressure distribution. The results indicate that a singular shear stress can dominate the asymptotic property of the stress field near the crack tip, and the stress intensity factor cannot be calculated even though the energy release rate is finite. Shear stress leads to a smaller crack opening, a longer crack, and a slightly larger wellbore pressure. A novel dominant-regime transition between shear stress and pressure is found. Unstable crack propagation occurs in the shear stress-dominant regime. This study may help in understanding crack problems under symmetrical loads and modeling fluid–solid interactions at the crack surfaces.