TDM-Based Semi-Analytical Solutions of the 3D Problem of Oil Production from Shale Reservoirs
Authors: Moridis George J. , Anantraksakul Niwit , Blasingame Thomas A. 
Published: July 2020

Abstract
The primary objective of this study is to develop fast analytical and/or semi-analytical (A/SA) solutions for the problem of liquid flow/production and pressure interference in multi-fractured systems between parallel horizontal wells in ultra-low permeability (ULP) reservoirs. We propose a new A/SA method that reduces the 3D flow equation into either a simple algebraic equation or an ODE in a multi-transformed space, the inversion of which yields solutions at any point in space and time.
In the proposed Transformational Decomposition Method (TDM), a general, fully linearized form of the 3D PDE describing low-compressibility liquid flow through porous and fractured media is subjected first to Laplace transforms to eliminate time, and then to successive Finite Cosine Transforms (FCTs) that eliminate either (a) all three dimensions, yielding a simple algebraic equation or (b) two dimensions, yielding an ODE in space only. Inversion of the solutions of the multi-transformed space equations provides solutions that are analytical in space and semi-analytical in time. The TDM completely eliminates the need for time and space discretization, thus drastically reducing the input data requirements and long execution times of numerical simulations.
The Fortran95 code for the TDM solutions requires limited inputs and is easy to use. Because of the linearity requirements of the Laplace transformation of the underlying PDE, the TDM is only rigorously applicable above the bubblepoint pressure. Using 3D stencils (the minimum repeatable elements in the horizontal well and hydraulically-fractured system) as the basis of our study, solutions over extended production times were obtained for (a) a range of isotropic and anisotropic matrix and fracture properties, (b) constant and time-variable production regimes (rates or bottomhole pressures), (c) combinations of SRV and non-SRV subdomains, (d) variable hydraulic fracture dimensions and (e) inner and boundary (toe and heel) stencils.
The results were compared to analytical solutions (available for simple problems and domain geometries), as well as to numerical solutions from a widely-used, fully-implicit 3D simulator that involves very fine discretization of a 3D domain comprising in excess of 356,000 elements. The TDM solutions were shown to be in excellent agreement with the reference analytical and/or numerical solutions, while requiring a fraction of the memory and of the execution times of the latter because of the elimination of the need for time and space discretization.
The TDM is an entirely new approach to the analysis of low-compressibility liquid flow and pressure interference in hydraulically fractured ULP reservoirs. The TDM solutions have the potential to provide a reliable and fast tool to identify the dominant mechanisms and factors controlling the system behavior and can act as the basis for a rapid initial parameter identification in a history-matching process, for possible further refinement using full numerical modeling below the bubblepoint pressure.