The choice of state variable in the non-linear diffusion equation may be of importance when analytical solutions are desired. The behavior of stress-sensitive reservoirs can best be predicted when the diffusion equation is formulated in terms of a permeability-like variable. Then the resulting equation becomes practically linear and approximate analytical solutions are possible.
We find that the validity of a steady-state skin factor is uncertain for stress-sensitive reservoirs. Different conceptual models are possible. A skin factor based on the equation of steady state flow in a stress-sensitive reservoir is recommended.
The most common linearization techniques, the Cole-Hopf and the Kirkhoff transformation, are equivalent when the permeability pressure relationships is given by an exponential function.
A material interface leads to a discontinuity in the transformed variable. The jump condition is given by powerlaw relationship. Hence application of linearization techniques to layered and commingled reservoirs may be problematic. We find that the technique may be applied when the modulus to permeability is approximately the same in each layer.
Analytical modelling shapes our thoughts and influence numerical simulation. Results from numerical models are verified against approximate analytical solutions. These serve as benchmarks to check the method and also the utility of the spatial and temporal grid. Hence analytical solutions are important tools even today. The simulation model in turn will have an impact on the analytical model since it may be used to check the validity of the simplifying assumptions implied in the latter.
The existence of analytical solutions depends on linearization of the non-linear terms in the diffusion equation. Two major approaches have been developed, the Kirkhoff transformation, Samaniego and Cinco-Ley, or the Cole-Hopf transformation, Pedrosa. The traditional application of the Kirkhoff transformation requires that the permeability is a known and tabulated function of pressure. In this study we investigate the properties of both techniques as seen in the light of permeability as the independent variable.
The major non-linear term for flow in a stress-sensitive reservoir is associated with the pressure dependency of the permeability. Muskat found that the diffusivity equation formulated in terms of density is linear while the corresponding one in terms of pressure is not. The permeability and density appear almost the same way in the diffusion equation. The linearity of the density equation depends on an exponential relationship between density and pressure. If permeability is related to pressure in the same way, the degree of non-linearity may be reduced by using permeability as independent variable. Modelling of stress-sensitive reservoirs depends on the pressure permeability relationship. This could either be assumed or known. Since well testing and core analysis involve different length scales, the well test model ought to be independent of core analysis. Hence it is advantageous to use an assumed relationship between permeability and pressure. This could be in the form of polynomials or exponential functions. The model needs to be calibrated to the reservoir by matching the predicted and observed pressure behavior. This can be achieved by adjusting the free parameters of the rock model. If reliable core analysis exists, the results may be used to check the consistency of the well test interpretation.