Application of the finite Hankel transform to a diffusion problem without azimuthal symmetry

The problem treated in this paper concerns calculating the evolution of the pressure in a single-phase, slightly compressible fluid in a porous medium consisting of communicating layers. The fluid is produced through a point sink located on the side of an otherwise sealed cylindrical wellbore. This location of the sink causes the flow around the wellbore to be azimuthally asymmetric.The problem is solved through successive application of Laplace, finite Fourier and finite Hankel transforms. Although apparently straightforward, this approach leads to serious numerical difficulties. The published form of the inversion formula for the finite Hankel transform leads to inaccurate computation for the higher azimuthal modes even with 128 bit arithmetic. An alternative form is developed which enables accurate evaluation of the solution with the more practical 64 bit arithmetic. The technique for two-layer solution presented here can be directly extended to a problem with a larger number of communicating layers. This is the first instance of successful application of the finite Hankel transform to an azimuthally asymmetric diffusion problem.     

Well Test Solutions for Vertically Fractured Injection Wells

The transient behavior of a vertically fractured pressure response due to the presence of an infinite-conductivity vertical fracture is determined by solving the diffusivity equation in elliptical coordinates. The solution is then extended to a composite elliptical system to provide for the different fluid banks present during water injection. The validity of the analytical solutions presented is demonstrated by comparing limiting forms with those available elsewhere in the literature. Computational issues which became evident during the verification stage of our work are also discussed. The solutions have been developed in the Laplace domain to facilitate the addition of fissures and variable rate production (i.e. wellbore storage).A pressure transient test for tracking the advancement of a water front during the early stages of waterflooding is described. We utilize the composite elliptical model developed herein to provide for two distinct regions in which the flow behavior resulting from an induced fracture is elliptical rather than radial. A relationship between the increasing elliptical distance to the waterfront and the resulting change in the apparent (total) skin factor is obtained. Through the analysis of successive falloff tests, this relationship may be used to monitor the advancement of the front provided the cumulative volume of injected water is known. The fluid saturations and the mobilities of the swept and unswept regions are assumed unknown and are obtained from the test analysis.Finally, we present methods for computing the Mathieu functions necessary in solving the diffusivity equation in elliptical coordinates. Mathieu functions are utilized in many applications involving elliptical geometry and we feel the efficient evaluation of these functions is an important contribution of this work.     

Transient Flow of a Compressible Fluid in a Connected Layered Permeable Medium

We develop a semi-analytical model of transient fluid flow in a 2D layered permeable medium with cross-flow between adjacent layers. It is shown that the pressure satisfies a diffusion equation to leading order, even when the non-linear term and gravity are included in the mathematical model. The solution is based on an analytical expression in the transform domain for the fluid pressure in terms of interfacial flux functions; the algorithm to compute the flux functions accepts an arbitrary number of formation layers. We show some benchmark tests that validate the general model; the model is then applied to an example derived from experiments. Numerical experiments confirm the significance of the cross-flow in a particular scaling of the ratio of permeabilities and quantify the influence of the various physical parameters.