A new fracture propagation simulator, based on a time-stepped expansion (TLSM) of the 2D linear superposition method (LSM), is applied here to study the growth of hydraulic fractures from the perforations outward. Based on the resultant displacement vector field, TLSM computes the displacement gradient tensor field, and strain tensor field at each time-step. A constitutive equation is used to compute the local stress field and stress concentrations from the strain tensor field solutions. We apply TLSM to a field case comprised of a subset of wells from the Hydraulic Fracture Test Site (Reagan County, Midland Basin). The well spacing and perforation spacing in the fracture treatment stages are used, together with the zipper mode fracture treatment schedule and the typical natural fracture trend, as identified from core data and associated fracture diagnostics. TLSM results provide a more nuanced insight in the possible mode of fracture propagation and observed diffuse response in micro-seismic clouds monitored during fracture treatment. Our TLSM results suggest that real-world fracture propagation is not necessarily constrained to planar fractures, which may be a geometrical constraint following from the initial assumptions in commonly used commercial fracture simulators.


The traditional suite of fracture propagation simulators is based on stress state computations using discretized volumes (e.g., Finite Element Model (FEM)) or discretized boundary segments (e.g., Bounded Element Model (BEM)). Computations are robust but with discretized methods require gridding and meshing (Borst, 2017), which is time-consuming due to elaborate grid refinements which may take hours or days (if not weeks) to run when modeling dynamic crack propagation. A fast, grid-less approach was recently introduced based on static analytical solutions of the stress state around single and multiple pressurized crack(s) (Pham and Weijermars, 2020). The static solution method is adapted into a dynamic solution for fracture propagation by time-stepping (see Section 2). The stress state solutions for static cases (Linear Superposition Method, LSM) can be computed almost instantaneously. Dynamic cases that show the propagation of the individual fractures of the sequential execution of fracture stages can be solved in 10 seconds for each time-step (including compilation of matrix data for post-processing image rendering). For a satisfactory visual resolution, about 200 steps are required in each stage. Modeling the stress interference of three consecutive fracture stages involves 600 (3×200) steps of 10 s, resulting in total Time-stepped Linear Superposition Method (TLSM) run times of about 1 hour and 40 minutes.

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