Use of Composite Elastic Modulus to Predict Inflow Performance

The proposed methodology is valid for any problem described by a diffusivity equation. As an application of the more general technique, we investigate fluid flow in a stress-sensitive reservoir. In case of production, the permeability, viscosity and fluid density may decrease in the near wellbore region. The reservoir thickness may also decrease. The objective of the present study is to quantify such changes and point out the effect on the inflow performance relationship.

The flow equation for stress-sensitive reservoirs may be highly non-linear. A non-linear variable shows up as a quadratic pressure gradient term in the diffusivity equation, Matthews and Russel (1967) . A logarithmic pressure transform may reduce the effect of the quadratic gradient term. The method involves the assumption of a constant elastic modulus. The equivalent assumption is that the variable may be approximated by an exponential function of pressure. Such variations have some experimental support. Many studies investigate the effect of a single pressure dependent variable. Then, a constant permeability modulus or compressibility is assumed. We propose a composite elastic modulus to investigate the simultaneous effect of an arbitrary number of pressure dependent variables in the transport term. The methodology depends on the assumption that every pressure-dependent variable may be approximated by an exponential function.

We investigate steady state flow, then the linearization of the diffusivity equation in terms of the transformed variable is complete, and analytical solutions are readily available. Solutions in terms of pressure are obtained by the inverse transform. Due to the non-linearities, the reservoir behaves different during production and injection. The pressure sensitivity depends on the value of the composite modulus and may be negligible. We provide equations to estimate the error incurred by neglecting the pressure sensitivity. The proposed methodology may be extended to time-dependent problems, but with reduced accuracy. Perturbations techniques are available to improve the accuracy.

Conclusions:

A composite elastic modulus has been proposed. The effect of non-linear terms may be estimated one by one and in combination.

A transformation based on the modulus will linearize the steady state diffusivity equation and the boundary conditions.

The non-linear pressure solutions may be obtained by the inverse transformation.

The degree of non-linearity may be characterized by the numerical value of the composite elastic modulus. Increasing values leads to decreased performance.

By use of L’Hospitals rule, we find that the conventional reservoir solution (without Stress-sensitivity) is included in the generalized equation as limiting behavior.