Simulation of penny-shaped hydraulic fracturing in porous media.

Abstract

In another major step, this work derives a pseudo-explicit finite difference scheme (PEFD) to compute the nonlinear, coupled problem of modeling the HF propagation. It is fully implicit in the time marching and is thus stable. But it solves one point at one time, i.e., it is explicit in the solution of the discretized equations. Furthermore, the Newton-Raphson method for a system of nonlinear equations is applied to speed up the convergence. Numerical tests verify its stability and consistency as well as mathematical accuracy. Many other numerical strategies are presented to compute the multiple PCR fracturing events during multiple ISF pumping cycles. Examples are run to verify overall performance of the resultant HF simulator and to show its computational capabilities.

However, this work is not a simple application of a standard numerical method to the general governing equations. Instead, it first proceeds to derive analytical expressions for the early- and late-time asymptotic poroelastic responses of a pressurized fracture. It then builds a composite approximate analytical formula to cover the interim transient poroelastic response between the two asymptotic time regimes. When compared against the numerical computations using commercial FEM software for the full mechanical model, the simplified model is shown to involve less than 10% relative error in the significant part of the poroelastic domain. Therefore, an adequately accurate mathematical accuracy exists.

More examples are computed to exemplify the poroelastic effect. The computations suggest that the poroelastic effect increases the wellbore pressure response and reduces both the fracture aperture and radius. In the computed examples, a maximum of 150% increase in the wellbore pressure is registered as compared to the purely elastic case. The magnitude of the poroelastic effect is linearly proportional to the in-situ minimum stress and pore pressure difference. It is also a positive power function of the formation permeability. The magnitude also increases with the number of pumping cycles.

A Duhamel's theorem-like principle is further derived to extend the foregoing stationary fracture-based simplified 2-D model to a propagating fracture. The only assumption used is that the pore pressure ahead of the fracture tip remains at the in-situ pore pressure level. When the fracture propagation speed is much faster than fluid diffusion rate, this assumption is valid. Several examples are computed. Analyses of the computations show the physical validity of the extension principle and the mathematical accuracy in the limiting cases. Moreover, the computation shows that the 1-D poroelastic model underestimates the poroelastic effect when the fracture propagates a considerable distance.

This dissertation presents a mathematical study and numerical procedure to simulate penny-shaped hydraulic fracture (HF) propagation in porous media. It accounts for the two-dimensional (2-D) pressure-dependent leakoff and poroelastic backstress contributions. It allows for multiple fracture propagation/closure/reopening (PCR) events during multiple injection/shut-in/flow-back (ISF) pumping cycles.