Transient Well Index for Numerical Well Test Analysis

Abstract

Abstract Incorporating well test data into integrated reservoir characterizations may require the well test to be modeled in a reservoir simulator that also forward simulates other dynamic data being matched. Care is required to ensure the simulator is not introducing unnecessary numerical artifacts. Standard finite-difference reservoir simulators use the Peaceman well index, which is based on the solution to single-phase, steady-state, incompressible flow. For a pressure transient test the assumptions of steady-state, incompressible flow are not applicable. However instead of modifying the well index it is common to account for this by using highly refined grids around wells and making very careful choices of timestep sizes to model well tests. This work presents a new well index formulation that allows well tests to be simulated accurately in finite-difference simulators using uniform, relatively coarse grids, without the problem of artifact wellbore storage (early time unit slope on the pressure derivative only) that otherwise occurs. The well index model computes the average pressure in the well block directly from the analytical solution for infinite acting radial flow. This approach can also be applied to non-square gridblocks and anisotropic reservoirs. Well tests simulated using finite-difference simulation with the proposed transient well index and the Peaceman well index are compared to analytical well test solutions. The simulated well tests using the proposed transient well index closely follow analytical solutions – even on coarse grids (e.g. 15 cells by 15 cells, 167 feet by 167 feet). When the same simulations were performed using the Peaceman well index, the simulated pressure transients showed significant artifact wellbore storage, especially in low permeability reservoirs. Background Peaceman1 initially proposed the use of a well index in reservoir simulators. His later work2–6 made provision for a variety of reservoir and well geometries. Babu et al.7 presented a well index relationship for horizontal wells and wells at arbitrary locations within grids. Mochizuki8 considered well indices for arbitrarily inclined wells. Chen et al.9,10 developed a pseudo-skin factor to accurately predict well pressure and productivity for wells with various inclinations. Sharpe and Ramesh11 modified the Peaceman well model, restoring its validity for problems with non-uniform local grid refinement. Sharpe and Ramesh also introduced a modified Peaceman well index suitable for modeling problems dominated by vertical flow such as gas and water coning. Ding et al.12 presented a new representation of wells in reservoir simulators which is particularly applicable to non-uniform grids and leads to improvements in the calculation of the productivity index. Wan et al.13 compared horizontal well performance simulated by using a uniform coarse grid, a uniform fine grid, and a non-uniform fine grid using Peaceman's well index model. Wan et al. found that when a coarse grid was used to simulate a partially penetrating horizontal well, the flow rate of the well was under-predicted. None of the works cited (except a brief mention by Peaceman1) specifically address the problem of fully transient flow such as the flow occuring in pressure transient tests. Integrated reservoir characterizations frequently require reservoir simulation models to accurately reproduce dynamic data such as pressure transient tests, production data and long term pressure measurements. When well tests are interpreted in isolation the measured data are compared to analytically generated models to determine the reservoir properties. The catalog of models well test software has available for this process is limited to simple geometries such as square and circular reservoirs.