Modeling of the propagation of hydraulic fracture using cohesive zone models

Hydraulic fracturing (HF) is widely used in the oil and gas industry to enhance production from tight reservoirs. The process involves the injection of fluid at a given flow rate into a wellbore in order to propagate a fracture in rocks and thus increase their permeability. Linear hydraulic fracture mechanics (LHFM) theories have been developped to predict the fracture propagation, assuming a linear elastic solid and the lubrication fluid flow in the fracture. However, some studies (Chudnovsky et al. 2008; Papanastasiou 1999) have shown deviations from LHFM predictions which indicates an existence of solid non-linearity and a deviation of the Poiseille law. We revisit the problem of a plane-strain hydraulic fracture driven by the injection of a Newtonian fluid in a tight rock. By modelling the quasi-brittle nature of the rock with different cohesive zone models, we study the effect of solid non-linearity on toughness and viscosity-dominated HF regimes. A fixed regular grid is used and an implicit scheme is adopted to solve the increments of the pressure and dislocation for this fully-coupled problem. The elasticity is solved using displacement discontinuity method using piece-wise linear element while the equation of mass conservation is solved by a finite volume method. The numerical algorithm reproduces the same results as LHFM solutions at large time when the cohesive zone is small compared to the fracture length. The shape of the material softening law or cohesive zone model does not influence the results. The accuracy of this method shows a dependence on the mesh refinement in the cohesive zone, which can be costly using a fixed uniform mesh.

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

Mapping of hydraulic fractures from tiltmeter measurements

Brice Tanguy Alphonse Lecampion

In considering the problem of inverse modeling of tiltmeter data for hydraulic fracture mapping, we address the issues of selecting the elastic model to represent the hydraulic fracture and limitations imposed by distance and fracture size on the information that can be recovered about the fracture. A tiltmeter measures, at its location, the changes in the surface inclination in two orthogonal directions. These inclinations are a direct measure of the horizontal gradient of the vertical component of the displacement field. Since advances in instrumentation in the last two decades, this type of apparatus have become extremely precise and can detect inclination changes down to a nanoradian. The simplicity of tiltmeter measurements has attracted interest not only in geophysics, but also in the petroleum industry. The idea of using tiltmeters to monitor hydraulic fractures can be traced back to the paper of Sun te{S} and is now a commercial service offered to the petroleum industry te{W}. However, the modeling and associated inverse problems required to analyze tiltmeter data raise difficult questions. The object(s) (fault, dyke, fracture) responsible for the recorded tilt are often modeled by finite Displacement Discontinuities, also called dislocation models. The validity of this type of model has been extensively discussed te{O,E} and many solutions for different configurations can be found in the literature. We show that it is possible to construct the solution for any type of dislocation model from the fundamental solution for an infinitesimal Displacement Discontinuity tensor. The eigenstrain theory te{M} is used to obtain this fundamental solution from the Green's function for the desired elastic domain (e.g. full or half space). Comparisons with known solutions demonstrate the flexibility of such method. We then focus on the problem of obtaining information about the orientation and size of an opening mode hydraulic fracture from the measured tilt field. One important problem is the identification of all the dimensions of the fracture model (length, width). The ability to obtain these parameters is controlled by limits, expressed in terms of the distance between the measurements and the fracture compared to the size of the fracture itself. The value of this ratio provides a condition that must be met before the fracture length-scales can be resolved. Determination of the fracture orientation is then investigated using a spatial Fourier Transform on the data set. This procedure highlights the requirement on the measurement array needed for a reliable identification: extension, number of tiltmeters, relative angle between the array and the fracture plane.

2003

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

Dong Liu

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.