Effects of the nonlinear gradient term on the transient pressure solution for a radial flow system

Abstract

This study presents new analytical pressure solutions for high pressure-gradient flow of a single-phase, slightly compressible fluid under transient conditions. A nonlinear partial differential equation is generally used to describe the pressure behavior during such flows through porous media. The nonlinearity of this equation arises from the presence of the quadratic pressure-gradient term in the diffusion equation. In order to obtain a standard linear diffusion equation so that closed-form analytical solutions can be developed, the original nonlinear equation has traditionally been linearized by assuming the pressure gradient to be small throughout the reservoir at all times. However, during certain operations such as hydraulic-fracturing, high-drawdown flows, slug testing, large-pressure pulse testing, etc., the assumpution of a small pressure gradient may not be justified and the nonlinear pressure-gradient term must be taken into account. Moreover, in an age of increased sophistication of reservoir flow analysis and prediction methods and improved resolution of pressure-measurement devices, the effects of the quadratic pressure-gradient term on the transient pressure behavior must be understood quantitatively. In this paper, analytical dimensionless pressure solutions of the nonlinear diffusion equation are derived by using the Laplace transform. Constant-rate and constant-pressure inner boundary conditions and infinite, closed and constant-pressure outer boundary conditions have been considered. For the constant-rate inner boundary condition, solutions are presented for both injection and production problems by taking into account the presence of wellbore storage. Deviations from existing linear solutions are identified and are related to a dimensionless group, α, which is proportional to the fluid compressibility. It is shown that for constant-rate inner boundary condition, and infinite or constant-pressure outer boundary conditions, the linear pressure solutions are within 5% of the corresponding nonlinear solutions for magnitudes of α less than 0.01 (within the dimensionless time range considered). However, for a closed outer boundary, the wellbore pressure predicted by the linear solution may be significantly smaller than that predicted by the nonlinear solution at large times. Analytical steady- and pseudosteady-state solutions are also presented and compared with the corresponding linear solutions. It has been shown, with examples, that significant errors may be incurred by using the linear pressure solutions in some cases.