A pseudo-3D model for hydraulic fracture growth in a layered rock

Abstract

This paper presents a new pseudo-3D (P3D) model for a hydraulic fracture growing in a layered rock with contrasts in both material properties and in situ stresses. In the model, the vertically planar fracture is divided along the lateral direction into cells. Within each cell, the cross-sectional deformation is plane strain, and the fluid pressure is allowed to vary vertically. The cells are discretized by displacement discontinuity elements that are formulated to include the elastic layered effect. The fluid flow in the cell is in two directions. Along the central part, which is of uniform pressure, the fluid flow is lateral, corresponding to the main component of fluid transport. Near the vertical fracture edge of a cell, the flow can be vertical and is generated by the vertical pressure gradient. This part of the cell is called the filling part. When the pressure in the filling part reaches the level equal to that in the central part, the flow direction switches from vertical to lateral. The filling and central parts both contribute to fracture height growth. The proposed P3D problem is solved in a coupled manner that accounts for the two-directional flow and cross-sectional deformation through a two-loop iterative method. In the outer loop, the fluid storage of the central part is updated by satisfying mass conservation in the lateral direction, and in the inner loop, the cross-sectional elastic deformation and the influxes to the filling parts are found by satisfying energy minimization subject to an equality constraint on the central-part volume of a cell. The results of pressure ad fracture width at a given elapsed time are thus obtained. After that, fracture growth in both lateral and vertical directions is controlled by the fracture toughness criterion based on linear elasticity. In describing the P3D model, the governing equations are provided and their dimensionless forms are derived. The numerical algorithm used for solving the P3D problem is also described. Numerical examples are presented, including a constant-height fracture, a radial one and asymmetric fractures in three-layered rocks. Comparisons of our results are made with other published results and good agreements between them are found.