Transient Well Index for Numerical Well Test Analysis

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.

Modeling the Interfering Effects of Gas Condensate and Geological Heterogeneities on Transient Pressure Response

“…These include the application of a transient productivity index (TPI) (Blanc et al 1999;Archer and Yildiz 2001;Archer 2010), use of a corrected-transmissibility approach (CTA) for the nearwellbore region (Ding and Renard 1994), or, finally, use of a suitable LGR. Some authors (e.g., Blanc et al 1999;Archer and Yildiz 2001) have studied application of the two former techniques for their single-phase 2D homogeneous model. However, to the best of our knowledge, their implementations on the compositional 3D heterogeneous reservoirs under multiphase-flow conditions have not been reported.…”

Transient Behavior of Multilateral Wells in Numerical Models: A Hybrid Analytical-Numerical Approach

“…They also concluded that there was an upper limit to the well-grid coarseness for the conventional well index, beyond which the transient results would loose numerical accuracy, but applying the transient well index in a coarse grid would yield a good match of the analytical solution. In addition to the above studies, the other pertinent references include Goktas and Ertekin (1999), Ding and Jeannin (2001), Archer and Yildiz (2001), Ozkan et al (1997), and Ozkan et al (1989).…”

Coarse Scale Simulation in Tight Gas Reservoirs

“…This error is known as artifact wellbore storage (13) and has been given this name because the well gridblock transient effect causes an error of unit slope of the log-log plot of Δp vs. t. The unit slope numerical error reminds the reservoir engineer of the physical phenomena of wellbore storage that causes mismatching at early time of actual field data; the classical dimensionless well testing type curves. This error may be more common in field models for tight reservoirs that are modelled with wellbore gridblocks that are coarse and which have a high value for diffusivity η.…”

“…Archer and Yildiz (13) pointed out that an early time numerical artifact occurred when uniform coarse areal grids are used to (7) well model equation used in all conventional reservoir simulators. They proposed a new well model formulation to remove the numerical artifact but it still occurred in the level of the pressure derivative.…”