Most composite solutions used in modem well test interpretation use the "step-permeability" model where the altered formations are divided into two or more concentric zones, each with constant permeability. Depending on the causes of alteration or variation, this model can deviate significantly from the real condition. A formation can be altered because of stress redistribution, drilling damage, solids co-production, mineral precipitates, and drilling mud invasion, each factor more likely to generate spatially-varying permeabilities rather than constant k zones. Experimental data confirm that a continuous permeability model is more realistic than the step permeability model.
A new semi-analytical solution with smooth radius-dependent permeability has been developed to analyze well test results from such altered formations. The altered formation is approximated as a concentric two-zone composite model: an exterior intact zone that retains the original unaltered permeability, and an interior zone where the permeability values have been altered and change as a function of radius. The permeability in this near-well zone can change continuously to the original permeability or to a different value at the boundary with the exterior zone, where there is a step-change in permeability. The choice of an increase or decrease of permeability with radius within the altered zone depends on the nature of the actual alteration. The permeability can also be simulated as a step-wise function if appropriate parameters are used; in such situations, the model collapses to conventional step-wise composite models. Other parameters such as formation porosity and compressibility of each zone can also be different, but they are assumed constant for this derivation. The new solution is derived in Laplace space, and numerical Laplace inversion is used.