Rate- and pressure-transient analysis of unconventional gas and oil reservoirs is a challenge because of complex reservoir characteristics that dictate flow. An important flow regime for analysis of these reservoirs is transient linear flow, which can be associated with linear flow to induced hydraulic fractures or to horizontal wells. One of the complications in the analysis of this flow regime is stress-sensitivity of porosity and permeability. This work aims to provide a method for analyzing transient linear flow in reservoirs with stress-sensitive permeability.
Flow of a compressible fluid in a stress-sensitive formation is governed by a nonlinear second order partial differential equation (PDE) with nonlinearities in both the accumulation and flow differential terms. A version of the Kirchhoff transformation is used to make the accumulation term linear, while a monotonically varying nonlinearity (as a function of a new pseudopressure function introduced in this work) exists in the flow differential term. The transformation, however, does not introduce any nonlinearity to the constant-wellbore pressure-condition, which is the case for constant well flow rate.
An exact solution of the transformed nonlinear PDE is provided for pressure distribution and flow rate calculations. The results are compared with approximate solutions in which fluid and rock properties are considered to be constant and evaluated at a specified pressure, as obtained by the error function (erf) solution. The results show that, at the wellbore, the value of the slope of the square-root of time plot (reciprocal of flow rate vs. square-root of time) can be used to calculate one of the two parameters, permeability modulus or initial permeability. This is the case if the derivative of the Kirshhoff parameter with respect to the Boltzmann variable is known for different values of fluid and rock compressibility, permeability modulus, and pressure drawdown. In this study, the Fujita’s method is used to calculate the derivative for some ranges of the affecting parameters. The results are presented as plots which can be used for analyzing the production data.