Laplace transform deconvolution and its application to perturbation solution of non-linear diffusivity equation

Abstract

The primary objective of this dissertation is to extend the conveniences of deconvolution to non-linear problems of fluid flow in porous media. Unlike conventional approaches, which are based on an approximate linearization of the problem, here the solution of the non-linear problem is linearized by a perturbation approach, which permits term-by-term application of deconvolution. Because the proposed perturbation solution is more conveniently evaluated in the Laplace-transform domain and the standard deconvolution algorithms are in the time-domain, an efficient deconvolution procedure in the Laplace domain is a prerequisite. Therefore, the main objective of the dissertation is divided into two sub-objectives: 1) the analysis of variable-rate production data by deconvolution in the Laplace domain, and 2) the extension of perturbation solution of the nonlinear diffusivity equation governing gas flow in porous media presented by Barreto (2011) into the Laplace domain. For the first research objective, a new algorithm is introduced which uses inverse mirroring at the points of discontinuity and adaptive cubic splines to approximate rate or pressure versus time data. This algorithm accurately transforms sampled data into Laplace space and eliminates the Numerical inversion instabilities at discontinuities or boundary points commonly encountered with the piece-wise linear approximations of the data. The approach does not require modifications of scattered and noisy data or extrapolations of the tabulated data beyond the end values. Practical use of the algorithm presented in this research has applications in a variety of Pressure Transient Analysis (PTA) and Rate Transient Analysis (RTA) problems. A renewed interest in this procedure is inspired from the need to evaluate production performances of wells in unconventional reservoirs. With this approach, we could significantly reduce the complicating effects of rate variations or shut-ins encountered in well-performance data. Moreover, the approach has proven to be successful in dealing with the deconvolution of highly scattered and noisy data. The second objective of this research focuses on the perturbation solution of the nonlinear gas diffusivity equation in Laplace domain. This solution accounts for the nonlinearity caused by the dependency of gas properties (viscosity-compressibility product and gas deviation factor) on pressure. Although pseudo-pressure transformation introduced by Al-Hussainy et al. (1966) linearizes the diffusivity equation for compressible fluids (gas), the pressure dependency of gas properties is not completely removed. Barreto (2011) presented a perturbation-based solution using Green's functions to deal with the remaining non-lineraities of the gas diffusion equation after pseudo-pressure transformation. The presented work is an extension of the work of Barreto (2011) into Laplace domain. The extension of the solution into Laplace domain is an advantage as less effort is required for numerical integration. Moreover, solutions of different well and reservoir geometries in pressure transient analysis are broadly available in Laplace domain. Field application of the solution will involve analysis of gas-rate data after deconvolution.