Analytical Solutions for the Radial Flow Equation With Constant-Rate and Constant-Pressure Boundary Conditions in Reservoirs With Pressure-Sensitive Permeability
- Torsten Friedel (Schlumberger Data & Consulting Services) | Hans-Dieter Voigt (Freiberg University)
- Document ID
- Society of Petroleum Engineers
- SPE Rocky Mountain Petroleum Technology Conference, 14-16 April, Denver, Colorado
- Publication Date
- Document Type
- Conference Paper
- 2009. Society of Petroleum Engineers
- 5.2 Reservoir Fluid Dynamics, 5.6.3 Pressure Transient Testing, 5.5 Reservoir Simulation, 4.1.4 Gas Processing, 5.6.4 Drillstem/Well Testing, 5.8.1 Tight Gas, 1.2.2 Geomechanics, 1.2.3 Rock properties, 4.6 Natural Gas, 5.8.3 Coal Seam Gas, 5.8.6 Naturally Fractured Reservoir
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The transient pressure response in stress-sensitive formations is derived by solving analytically the radial flow equation with pressure-dependent rock properties. New solutions are obtained for slightly compressible fluids and real gas flow with constant-rate and constant-pressure wellbore boundary conditions using the Boltzmann transformation. In particular in low-permeability formations, a sensitivity exists to pore pressure changes. Permeability there normally declines with the increase of effective stress. The mathematical model presented here takes into account the reduction in permeability caused by an increase in effective stress for two common parameterized pressure-permeability functions, i.e., a linear and an exponential relationship. The analytical solutions are verified with numerical simulation. They are applicable for well test analysis and prediction of flow rates, as demonstrated in this paper.
Virtually all classical analytical techniques to solve the fluid flow equation are based on the assumption of constant rock permeability. In contradiction to the changes in porosity during production, which are captured in the solutions by introducing a constant compressibility for the porous media, permeability is considered to be independent of the pore pressure. Linearization of the pore volume changes and the assumption of a constant permeability result in a constant diffusivity which has been the basis for a wide range of analytical solutions for the parabolic differential equation in the past.
This constant property assumption is justified for many typical reservoir engineering problems, in which both local pressure and permeability changes are small and negligible. However, this is not strictly applicable to all kinds of reservoirs. Certain reservoir rocks exhibit a stronger sensitivity to changes of stress conditions during the depletion of the reservoirs, in particular, naturally fractured reservoirs and low-permeability reservoirs. As a rule of thumb, the impact of stress on property alteration increases with the tightness of the reservoir rock. This makes it particularly important in tight-gas or coalbed methane reservoirs.
Stress sensitivity impacts both classical well test analysis and performance predictions. In this paper, new analytical solutions are presented for transient radial flow in stress sensitive reservoirs with constant-rate and constant-pressure boundary conditions. The solutions are simple to use and provide a means to derive the transient response of a radial well in pressure-sensitive environment using a typical engineering application such as spreadsheet programs. The solutions are validated against numerical simulation and applied for well test analysis.
Theoretical consideration of stress-dependent mechanisms goes back to Tezaghi's work, which was introduced in reservoir engineering in the early 1950's. The first important publications were from Fatt and Davis (1952) and Dobrynin (1962). In their papers, the authors investigated the relation between effective stress and reservoir permeability. They concluded that permeability declines with increasing effective stress. Dobrynin (1962) also introduced a mathematically derived capillary model to describe the change of permeability as a function of the changes in in-situ stresses.
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