New analytical functions for a skin and storage effect – An insight to decouple fracture half-length and square-root of permeability

Abstract

Several solutions to the double porosity model have been presented. There are solutions for closed outer boundary systems as well as infinite acting boundary conditions with constant rate or constant pressure inner boundary conditions. A common assumption in all of these solutions is that the fracture has a quadrilateral (linear) geometry. Very few solutions exist for cases were the fracture is assumed to have a circular geometry. This is the first work that presents approximate early and late time solutions for flow from fractures that have a circular geometry where the inter porosity transfer function is model with an unsteady state function. The result obtained from this analysis shows that the function behaves in a manner identical to production data from fractured horizontal wells in unconventional reservoirs (including early time data that is usually assumed to be obscured by skin and wellbore/fracture storage effects).The double porosity model was formulated for a fractured horizontal well in which the fractures were assumed to have a circular geometry. The associated mathematical model was then solved with Laplace transforms to obtain a closed form analytical solution in Laplace space. We then derive the approximate early and late time solutions by using special properties of Bessel functions and the Laplace space solution. The solutions were derived for a constant-pressure and constant-rate inner-boundary condition with a no flow outer-boundary condition.The approximate early time solution obtained is identical to the solution obtained when the fracture is assumed to have a quadrilateral geometry. But it has two additional parameters. A detailed sensitivity analysis showed that fracture thickness does not have a significant impact on the producing characteristics. The approximate late-time solution has a production history identical to that observed from field production data. The approximate late-time solution also has a slope of two on a log–log plot of rate verses time; this has not been report before in the literature. A comparison of the approximate late-time solution to traditional linear fracture solutions with skin and storage effect lead to the development of new skin and fracture storage functions. The significance of this result is that the new skin and fracture storage functions suggest that they can be used to decouple the product of fracture half-length and the square-root of permeability from early-time production data.