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 a Python implementation of an open-source hydraulic fracture propagation simulator based on the implicit level set algorithm originally developed by Peirce & Detournay (2008) -- "An implicit level set method for modeling hydraulically driven fractures". Comp. Meth. Appl. Mech. Engng, (33-40):2858--2885. 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 growth accurately, even with 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 benchmark, the propagation of a height contained hydraulic fracture, the lateral spreading of a magmatic dyke and the handling of fracture closure are presented to demonstrate the capabilities, accuracy and robustness of the implemented algorithm

Confined flow of suspensions modeled by a frictional rheology

We investigate in detail the problem of confined pressure-driven laminar flow of neutrally buoyant non-Brownian suspensions using a frictional rheology based on the recent proposal of Boyer et al., 2011. The friction coefficient and solid volume fraction are taken as functions of the dimensionless viscous number I defined as the ratio between the fluid shear stress and the particle normal stress. We clarify the contributions of the contact and hydrodynamic interactions on the evolution of the friction coefficient between the dilute and dense regimes reducing the phenomenological constitutive description to three physical parameters. We also propose an extension of this constitutive law from the flowing regime to the fully jammed state. We obtain an analytical solution of the fully-developed flow in channel and pipe for the frictional suspension rheology. The result can be transposed to dry granular flow upon appropriate redefinition of the dimensionless number I. The predictions are in excellent agreement with available experimental results, when using the values of the constitutive parameters obtained independently from stress-controlled rheological measurements. In particular, the frictional rheology correctly predicts the transition from Poiseuille to plug flow and the associated particles migration with the increase of the entrance solid volume fraction. We numerically solve for the axial development of the flow from the inlet of the channel/pipe toward the fully-developed state. The available experimental data are in good agreement with our predictions. The solution of the axial development of the flow provides a quantitative estimation of the entrance length effect in pipe for suspensions. A analytical expression for development length is shown to encapsulate the numerical solution in the entire range of flow conditions from dilute to dense.Comment: Submitted to J. Fluid Mech. on Dec. 24, 2013, Revised version July 10, 2014, Accepted for publication Sept. 19, 201

Fluid-driven slow slip and earthquake nucleation on a slip-weakening circular fault

We investigate the propagation of fluid-driven fault slip on a slip-weakening frictional interface separating two identical half-spaces of a three-dimensional elastic solid. Our focus is on axisymmetric circular shear ruptures as they capture the most essential aspects of the dynamics of unbounded ruptures in three dimensions. In our model, fluid-driven aseismic slip occurs in two modes: as an interfacial rupture that is unconditionally stable, or as the quasi-static nucleation phase of an otherwise dynamic rupture. Unconditionally stable ruptures progress through four stages. Initially, ruptures are diffusively self-similar and the interface behaves as if it were governed by a constant friction coefficient equal to the static friction value. Slip then accelerates due to frictional weakening while the cohesive zone develops. Once the latter gets properly localized, a finite amount of fracture energy emerges along the interface and the rupture dynamics is governed by an energy balance of the Griffith's type. In this stage, fault slip transition from a large-toughness to a small-toughness regime. Ultimately, self-similarity is recovered and the fault behaves again as having a constant friction coefficient, but this time equal to the dynamic friction value. When slow slip is the result of a frustrated dynamic instability, slip also initiates self-similarly at a constant peak friction coefficient. The maximum aseismic rupture size varies from a critical nucleation radius (shear modulus divided by slip-weakening rate) to infinity near the limit that separates the two modes of aseismic sliding. We provide analytical and numerical solutions for the problem solved over its full dimensionless parameter space. Due to its three-dimensional nature, the model enables quantitative comparisons with field observations as well as preliminary engineering design of hydraulic stimulation operations.Comment: 51 pages, 11 figure

Initiation and Breakdown of an Axisymmetric Hydraulic Fracture Transverse to a Horizontal Wellbore

We investigate the initiation and early-stage propagation of an axi-symmetric hydraulic fracture from a wellbore drilled in the direction of the minimum principal stress in an elastic and impermeable formation. Such a configuration is akin to the case of a horizontal well and a hydraulic fracture transverse to the well axis in an open hole completion. In addition to the effect of the wellbore on the elasticity equation, the effect of the injection system compressibility is also taken into account. The formulation accounts for the strong coupling between the elasticity equation, the flow of the injected fluid within the newly created crack and the fracture propagation condition. Dimensional analysis of the problem reveals that three dimensionless parameters control the entire problem: the ratio of the initial defect length over the wellbore radius, the ratio between the wellbore radius and a length-scale associated with the fluid stored by compressibility in the injection system during the well pressurization, and finally the ratio of the time-scale of transition from viscosity to toughness dominated propagation to the time-scale associated with compressibility effects. A fully coupled numerical solver is presented, and validated against solutions for a radial hydraulic fracture propagating in an infinite medium. The influence of the different parameters on the transition from the near-wellbore to the case of a hydraulic fracture propagating in an infinite medium is fully discussed

Propagation of a plane-strain hydraulic fracture accounting for a rough cohesive zone

The quasi-brittle nature of rocks challenges the basic assumptions of linear hydraulic fracture mechanics (LHFM): linear elastic fracture mechanics and smooth parallel plates lubrication fluid flow. We relax these hypotheses and investigate the growth of a plane-strain hydraulic fracture in an impermeable medium accounting for a rough cohesive zone and a fluid lag. In addition to a dimensionless toughness and the time-scale of coalescence of the fluid and fracture fronts as in the LHFM case, the solution now also depends on the in-situ-to-cohesive stress ratio and the intensity of the flow deviation induced by aperture roughness. The solution is appropriately described by a nucleation time-scale, which delineates the fracture growth into a nucleation phase, an intermediate stage and a late time stage where convergence toward LHFM predictions finally occurs. A highly non-linear hydro-mechanical coupling takes place as the fluid front enters the rough cohesive zone which itself evolves during the nucleation and intermediate stages. This coupling leads to significant additional viscous flow dissipation. As a result, the fracture evolution deviates from LHFM solutions with shorter fracture lengths, larger widths and net pressures. These deviations ultimately decrease at late times as the lag and cohesive zone fractions both become smaller. The deviations increase with larger dimensionless toughness and in-situ-to-cohesive stress ratio, as both further localize viscous dissipation near the fluid front located in the rough cohesive zone. The convergence toward LHFM can occur at very late time for realistic values of in-situ-to-cohesive stress ratio encountered at depth. The impact of a rough cohesive zone appears to be prominent for laboratory experiments and short in-situ injections in quasi-brittle rocks with ultimately a larger energy demand compared to LHFM predictions.Comment: submitted to J. Mech. Phys. So

Time-lapse reconstruction of the fracture front from diffracted waves arrivals in laboratory hydraulic fracture experiments

4D acoustic imaging via an array of 32 sources / 32 receivers is used to monitor hydraulic fracture propagating in a 250~mm cubic specimen under a true-triaxial state of stress. We present a method based on the arrivals of diffracted waves to reconstruct the fracture geometry (and fluid front when distinct from the fracture front). Using Bayesian model selection, we rank different possible fracture geometries (radial, elliptical, tilted or not) and estimate model error. The imaging is repeated every 4 seconds and provide a quantitative measurement of the growth of these low velocity fractures. We test the proposed method on two experiments performed in two different rocks (marble and gabbro) under experimental conditions characteristic respectively of the fluid lag-viscosity (marble) and toughness (gabbro) dominated hydraulic fracture propagation regimes. In both experiments, about 150 to 200 source-receiver combinations exhibit clear diffracted wave arrivals. The results of the inversion indicate a radial geometry evolving slightly into an ellipse towards the end of the experiment when the fractures feel the specimen boundaries. The estimated modelling error with all models is of the order of the wave arrival picking error. Posterior estimates indicate an uncertainty of the order of a millimeter on the fracture front location for a given acquisition sequence. The reconstructed fracture evolution from diffracted waves is shown to be consistent with the analysis of $90^{\circ}$ incidence transmitted waves across the growing fracture.Comment: submitted to Geophys. J. In

Propagation of a Radial Fluid-driven Fracture in Nonlinear Solid --around the tip of a fluid-driven fracture: process zone vs deviated fluid flow

Deviations from conventional hydraulic fracturing simulators’ predictions are sometimes observed in the field and laboratory. This questions the basic assumptions adopted in linear hydraulic fracture mechanics (LHFM): a linear elastic solid and a simplified fluid flow in the fracture. Some rocks encountered in the domain of hydraulic fracturing show a strong non-linear behavior with the existence of a process zone, and fluid flow in small rough apertures also presents a departure from the Poiseuille law. By relating the solid non-linearity to the deviated fluid flow through the fracture roughness, we study the propagation of a radial fluid-driven fracture in quasi-brittle materials. We adopt a linear-softening cohesive zone model to simulate the non-linear behavior in the solid and introduces a friction factor to characterize the deviated fluid flow. We investigate the interplay between the fluid front and the process zone and verify that the solid non-linearity will impact the growth of hydraulic fractures through the fracture roughness by decreasing the permeability of the fracture tip. Fluid driven fracture propagation in quasi-brittle materials shows a similar behavior as in linear elastic solid under the assumption of a zero fluid lag. The cohesive zone develops with time and reaches a stable value, which depends on the fluid viscosity. When considering a non-zero fluid lag, an interplay exists between the fluid front and process zone. Deviated fluid flow pushes the fluid away from the fracture tip and results in a larger fluid lag. We observe a localization of the pressure drop near the tip. Cohesive zone length will either be shortened or enlarged by the deviated lubrication, depending on whether the squeezing or bubbling effect is dominant. However, normal stress drives the fluid front to the tip, leading to a smaller fluid lag and higher net pressure. Cohesive length is shortened as a result of a strengthened suction effect in the lag region. Due to this interplay, more energy dissipation is expected during propagation

Laminar - turbulent transition in the propagation of height contained hydraulic fracture

High injection rate hydraulic fracturing can have Reynolds number as high as 10000. For such high Reynolds numbers, turbulent flow is likely to occur. In this paper, we investigate the effect of turbulence on the propagation of hight contained hydraulic fractures, commonly referred to as PKN fractures. We discuss different scalings for the fracture width, length and pressure under limiting laminar, turbulent smooth and turbulent rough flow regimes. We implement an explicit, central numerical scheme to solve the continuity and friction factor based momentum conservation equations, taking into account the full variation of friction factor with Reynolds number and relative fracture roughness. The scheme is validated against the analytical solution of the PKN model. The results show that the local Reynolds number evolves from a maximum value at the inlet to zero at the tip, with a transition from turbulent to laminar at some point along the fracture length, depending on the value of inlet Reynolds number. Results showing the effect of smooth and rough turbulence on the fracture length and fracture width depending on the Reynolds number are finally presented