Anantraksakul, Niwit (2020-04). Development and Application of a New Set of TDM-Based Semi-Analytical Solutions to the Problem of Pressure Interference in a Hydraulically-Fractured Reservoir. Master's Thesis.
Thesis
The analysis of production from fractured Ultra-Low Permeability (ULP) reservoirs is routinely conducted using numerical simulation, which requires large 3D grids, a very large number of time-steps, numerous input data, long execution times, and the specialized computational expertise. The main objective of this research is to develop a set of new semi-analytical results that reduce or eliminate the need for spatial and temporal discretization of numerical models and address their shortcomings, thus providing fast and simpler alternatives that can deliver reliable results for the analysis of production and reservoir performance, as well as for history matching. The new set of solutions is based on variants of a hybrid analytical-numerical approach called the Transformational Decomposition Method (TDM) that involves successive applications of Finite Cosine Transforms for the elimination of multiple dimensions in space, and Laplace Transforms for the elimination of the time variable. Application of this method reduces the 3D diffusivity equation of the nearly-incompressible oil flow into either an algebraic equation (referred to as TD-0D) or a simple Ordinary Differential Equation (ODE) in the x-dimension (referred to as TD-1D). The strongly non-linear 3D flow of compressible gas does not allow the application of Laplace transforms, leading to the development and application of the Partial Transformational Decomposition Method (PTDM), which reduces the 3D gas diffusivity equation into either a simple ODE in time (referred to as PTD-0D) or a 1D Partial Differential Equation (PDE) in space and time (referred to as PTD-1D). Both the TDM and PTDM solutions exist in multi-transformed space and/or time environments, from which they are inverted numerically to obtain solutions at any point in space and time through a process that vastly simplifies and decreases the size of the simulation problem, as well as the execution time. Both the TDM and the PTDM solution were coded in a FORTRAN90. Using a 3D stencil (the minimum repeatable element in hydraulically-fractured reservoirs produced by horizontal wells), solutions over an extended production period and covering a substantial pressure drop were obtained for (a) a range of isotropic and anisotropic matrices and fracture properties, (b) constant and time-variable flow rates and bottom-hole pressure regimes, and (c) combinations of Stimulated Reservoir Volume (SRV) and non-SRV subdomains. The results were compared to the solutions from the FTSim code -- a fully implicit 3D simulator -- using a finely-discretized (high-definition) 3D domain. The TDM results of oil flow and production were shown to be in excellent agreement with the FTSim numerical solutions during the entire production periods. The PTDM performance was not uniform across the time spectrum: the FTSim solutions were in excellent agreement with the PTD-1D results at any time during production and with the PTD-0D solutions at early times, but the PTD-0D performance deteriorated at later times.
