The Partial Transformational Decomposition Method PTDM for the Solution of the Gas Flow Problem in a Hydraulically Fractured Ultra-Low Permeability Reservoir
Authors: Anantraksakul Niwit , Moridis George J. , Blasingame Thomas A. 
Published: July 2020

Abstract
The analysis of gas production from fractured ultra-low permeability (ULP) reservoirs is most often accomplished using numerical simulation, which requires large 3D grids, many inputs and typically long execution times. We propose a new hybrid analytical-numerical method that reduces 3D equation of gas flow into either a simple Ordinary Differential Equation (ODE) in time or a 1D Partial Differential Equation (PDE) in space and time without compromising the strong non-linearity of the gas flow relation, thus vastly decreasing the size of the simulation problem and the execution time.
In the proposed hybrid Partial Transformational Decomposition Method (PTDM), successive Finite Cosine Transforms (FCTs) are applied to the pseudo-pressure-based 3D diffusivity equation of gas flow, leading to the elimination of the corresponding physical dimensions. For production under a constant- or time-variable rate (q) regime, 3 levels of FCTs yield a 1st-order ODE in time. For production under a constant- or time-variable pressure (pwf) regime, 2 levels of FCTs lead to a 1D 2nd-order PDE in space and time. The fully-implicit numerical solutions for the FCT-based equations in the multi-transformed spaces are inverted, providing solutions that are analytical in 2 or 3-space dimensions and account for the non-linearity of gas flow.
The PTDM solution was coded in a FORTRAN95 program that used (a) the Laplace Transform Analytical solution for the q-problem and (b) a Finite Difference Method for the pwf-problem in their respective multi-transformed spaces. Using a 3D stencil (the minimum repeatable element in the horizontal well and hydraulically-fractured system), solutions over an extended production time and a substantial pressure drop were obtained for (a) a range of isotropic and anisotropic matrix and fracture properties, (b) constant and time-variable q and pwf production schemes, and (c) combinations of SRV and non-SRV subdomains. The results were compared to the numerical solutions from a widely-used, fully-implicit 3D simulator that involved a finely-discretized (high-definition) 3D domain involving 220,000 elements.
Of the two versions of PTDM, the PTD-1D was by far the better option and its solutions were shown to be in very good agreement with the full numerical solutions, while requiring a fraction of the memory and orders of magnitude lower execution times because these solutions require discretization of only (a) the time domain and (b) a single axis (instead of three). The PTD-0D method was slower than PTD-1D (but still much faster than the numerical solution) and while its solutions were accurate for t &lt; 6 months, these solutions deteriorated beyond that point.
The PTDM is an entirely new approach to the analysis of gas flow in hydraulically-fractured ULP reservoirs. The PTDM solutions preserve the strong non-linearity of the gas flow equation and are analytical in 2 or 3 spatial dimensions. This being a semi-analytical approach, it needs fewer input data and requires computer storage and computational times that are orders of magnitude smaller than those in conventional (numerical) simulators because its discretization is limited to time and (possibly) a single spatial dimension.