Propagation of a plane-strain hydraulic fracture accounting for the presence of a cohesive zone and a fluid lag

We revisit the problem of the propagation of a plane-strain fluid-driven fracture in a quasi-brittle impermeable medium accounting for the presence of a fluid lag. The fracture process zone is simulated using a linear-softening cohesive model while lubrication flow accounts for the possible occurence of a fluid lag. The solution is obtained numerically via a fully implicit scheme based on a boundary element method for the fracture deformation and finite difference for fluid flow. The fluid lag is first automatically captured using the Elrod-Adams lubrication cavitation model during the initiation and early stage of fracture growth. We then switch to an algorithm tracking the fluid front for computational efficiency. Using dimensional analysis, we show that the propagation is governed by a dimensionless toughness and a time scale characterizing the disappearance of the fluid lag, (both similar to the linear elastic fracture mechanics case) and a ratio between the in-situ minimum confining stress and the material tensile strength. The cohesive forces reinforce the suction effect associated with a fluid lag, and leads to the further localization of the fluid pressure drop near the tip. This ultimately results in a slight increase of fracture opening and net pressure.

Hydraulic fracture growth in quasi-brittle materials

Dong Liu

Hydraulic fracturing is a technique often used in the oil and gas industry to enhance the production of wells in low-permeability reservoirs. It consists of fluid injection at a sufficiently large injection rate to create and propagate tensile fractures inside the rock formation. The growth of a hydraulic fracture is a highly non-linear moving boundary hydro-mechanical problem. Many physical processes get tightly involved and influence the fracture growth, such as the elastic deformation of the material, fracture surface creation, viscous fluid flow inside the propagating fracture, and fluid leak-off into the solid principally. The linear hydraulic fracture mechanics (LHFM) theory describes well this coupled process, with adequate comparisons between predictions and observations in the linear elastic medium. However, deviations from LHFM predictions have been reported at the laboratory and field scales in some cases, notably a larger fluid pressure and a shorter fracture length. These deviations are often associated with the non-linear nature of rock fracture deformation. However, the impact of these material non-linearities on the coupled problem of hydraulic fracture propagation has not been fully understood. In this context, this thesis investigates numerically and experimentally the effect of the quasi-brittle nature of rocks on hydraulic fracture growth. We first solve the propagation of a hydraulic fracture using a power-law fracture length-dependent fracture energy to model macroscopically the effect of an enlarging process zone. The use of Gauss-Chebyshev quadrature and barycentric Lagrange interpolation techniques reduces the hydraulic fracture propagation problem to a set of non-linear ordinary differential equations. We find that this increasing apparent toughness leads to a slower fracture growth compared with constant toughness LHFM solutions and a faster transition to the toughness dominated regime. We show that the solution can be approximated by the self-similar constant toughness solutions with the instantaneous toughness value. We then explore the impact of the quasi-brittle nature of rocks by using a cohesive zone model in combination with elastohydrodynamic lubrication to simulate HF growth. We notably account for fluid cavitation at the fracture tip and the effect of fracture roughness on flow. The fracture growth finally converges toward the LHFM predictions but deviate from LHFM predictions when the cohesive zone cannot be neglected. These deviations indicate an increase of energy dissipation. The additional energy dissipation compared with LHFM mainly comes from the viscous fluid flow in the rough fracture process zone and is larger for a larger ratio between the in-situ stress and the material peak cohesive stress. We finally discuss HF experiments performed on two impermeable rocks under different confinement and injection conditions. We notably develop an inversion technique to reconstruct the fracture front geometry using diffracted waves recorded via an array of 32 sources and 32 receivers. We find that transmitted waves attenuate before the arrival of the fracture front estimated from diffracted waves, indicating the existence of a fracture process zone of centimeters extent. This fracture process zone is found to be comparable with the fracture extent, thus implying an important artifact of laboratory HF experiments performed in quasi-brittle rocks on finite-size centimeter-decimeter samples.

EPFL2021

Constitutive behavior and fracture properties of tempered martensitic steels for nuclear applications

Raul Alejandro Bonadé

In this study, the tensile and fracture properties and the microstructure of the reduced-activation tempered martensitic steel Eurofer97 have been investigated. This technical alloy is a 9%Cr steel developed within the European fusion material research program. To a lesser extend, the plastic flow properties of a equiaxed ferritic Fe9%Cr model alloy were also studied for comparison with those of the tempered martensitic structure. The main objectives of this work are described hereafter: to correlate the microstructural features with the plastic flow properties measured by the tensile tests for both the Eurofer97 steel and the model alloy. The correlation established should be reflected in a physically-based model of plastic flow. to study the fracture properties of the Eurofer97 steel in details in the lower ductile-to-brittle transition region. to calculate with finite element modeling the stress fields at the crack tip. This information is further used in conjunction with a local approach model for quasi-cleavage to reconstruct the fracture toughness-temperature curve. Tensile tests were carried out at different imposed nominal strain rate at several temperatures from 77K up to 473K, on both the Eurofer97 steel and the Fe9%Cr model alloy. The temperature dependence of the yield stress was precisely determined. As expected for body-centered cubic (BCC) materials, a strong increase of the yield stress by decreasing temperature, below 200 K, was observed. At higher temperatures, the temperature dependence of the yield stress was found much weaker, being associated with the temperature dependence of the shear modulus. Efforts were made to analyze in details the post-yield behavior (strain-hardening) as a function of temperature. The post-yield behavior was modeled using the Kocks phenomenological model based on the competition of storage and annihilation of dislocations. While this model was originally developed for face-centered cubic (FCC) metals where the rate-controlling mechanism of dislocation motion is the dislocation-dislocation interaction, we used is model in the high temperature domain (T>200K) of BCC materials, to model the strain-hardening evolution of the Eurofer97 steel and the model alloy. The values obtained for the key parameters of the model, namely the dislocation mean free path and annihilation coefficient, were found consistent with the microstructural features. The parameters temperature-dependence observed was also consistent with the physics of two basic mechanisms of dislocation storage and annihilation (dynamical recovery). For the low temperature domain, the strain-hardening model was modified to account for the strong Peierls lattice friction, based on an original idea of Rauch. Transmission electron microscopy observations were done to characterize the undeformed microstructures and their evolution with strain. A clear correlation was established between the stress-dependence of the strain-hardening and the microstructures. Fracture toughness tests on the Eurofer97 steel were performed with 0.2T C(T) and 0.4T C(T) specimens in the lower transition. Five temperatures were selected at which many tests were repeated to determine the amplitude of the inherent scatter of quasi-cleavage. The temperature were chosen in such a way that the measure fracture toughness remain below 150 MPa m1/2, which correspond for the 0.4T C(T) at a M value larger than 70 at the highest temperature. Such a M value for 0.4T C(T) specimens is known to ensure enough constraint. An attempt to analyze the experimental data in the framework of the master-curve approach by following the ASTM E-1921 standard was done. It was clearly demonstrated that the assumed shape of the toughness-temperature curve as described in the ASTM E-1921 standard for the reactor pressure vessel steels is not adequate for the tempered martensitic Eurofer97 steel, which present a particularly steep transition. Our Eurofer97 fracture database was then compared to the existing one on another similar steel, the F82H. Differences in the amplitude of the scatter of both steels was found while the lower bound of the toughness-temperature curve, describing the 1% failure probability was shown to be the same. 2D finite element simulations of the compact tension specimens were performed at various temperatures using the constitutive laws determined previously. The stress field around the crack tip were calculated and used to determine a local criterion of quasi-cleavage. The criterion is defined by the attainment of a critical stress encompassing a critical area. The lower bound of the toughness-temperature curve was then successfully reconstructed by using this local criterion. Finally, the relationship between the critical area and the applied stress intensity factor for the C(T) specimen was shown to follow a power law whose coefficients are dependent on the real dimensions of the specimen. Such a relationship allows scaling the C(T) toughness data from one size to another in case of in-plane constraint loss.

EPFL2006

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

Brice Tanguy Alphonse Lecampion, Dong Liu

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.

2018