A New Rigorous Analytical Solution for a Vertical Fractured Well in Gas Reservoirs

摘要

Significant gas reserves are found in low permeability reservoirs world wide. Economical flow rates are only achieved on these reservoirs by massive hydraulic fractures. Well testing is one of the most used techniques to reservoir management and monitoring, however pressure transient analysis can be a real challenge under these conditions. The gas hydraulic diffusivity equation is commonly linearized by means of pseudo pressure function m(p). The resulting partial differential equation having m(p) as dependent variable remains non-linear because the viscosity-compressibility product that multiplies the partial derivative of m(p) which respect to time varies with m(p). This type of differential equation is named quasi-linear. A wide spread assumption in the well testing literature is to consider that the viscosity-compressibility product remains approximately constant throughout the well test. This assumption is acceptable for small pressure drawdown only but rarely met in tight gas reservoirs. One can find in literature a few attempts to account for the viscositycompressibility product variation analytically; hence, there is still room for improvements in this research topic. This paper considers a vertical fractured well in a homogeneous isotropic infinite gas reservoir produced by constant flow rate. In this work the variation of the viscosity-compressibility product is considered as a non-linear source term. This approach leads to a closed form analytical solution based on Green’s Functions. The solution is presented as an integraldifferential equation which must be evaluated numerically. A multidimensional numerical integration package was used to obtain the pseudo pressure results. This numerical scheme is capable of handling accurately large viscosity-compressibility variations. Results for a constant rate test for a vertical fractured well in an infinite isotropic reservoir show good agreement to a finite difference numerical simulator. It is shown that the behavior of dimensionless pseudo pressure and its log-time derivative is similar to the classical slightly compressible fluid fracture well solution. During the pseudo-radial flow regime the pseudo pressure solution is given by the correspondent liquid solution plus a small negative constant. This constant, however is rate sensitive. The solution technique presented in this paper may be successfully applied to harder gas well testing problems