Line fracture in an anisotropic reservoir with incompressible liquids and compressible gases

Example 2-2. Line fracture in an anisotropic reservoir with incompressible liquids and compressible gases. 

Having demonstrated the power and elegance behind the use of distributed line sources and the use of singular integral equations, we now consider a slightly more complicated example involving incompressible liquids and compressible gases in anisotropic reservoirs under steady-state flow conditions. This second example will illustrate the flexibility of the thin airfoil technique. But it will also reveal the weaknesses inherent in analytical approaches and why a well formulated numerical method is necessary. 

To make progress, we consider constant permeabilities (a log r function is not available for heterogeneous reservoirs). This leads to the simpler equation 

Equations 2-41 to 2-43 resemble Equations 2-7 to 2-9, except that m is nonzero. The scaling in Equations 2-38 and 2-39 is chosen so that x remains between -1 and +1. This requirement is imposed so that existing results can be used without renormalization. This problem, while it involves powers of pressure, is not nonlinear our pressure boundary conditions are easily rewritten in powers of pressure, resulting in a linear Dirichlet problem for p.  

Singular integral equation analysis. A closed-form analytical solution can be obtained. Observe that the standard source solution log r, centered at the origin r = + y ) = 0, solves Equation 2-41 for Similarly, the 

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