Unsteady-State Spherical Flow With Storage and Skin

This paper presents short-time interpretation methods for radial-spherical (or radial-hemispherical) flow in homogeneous and isotropic reservoirs inclusive of wellbore storage, wellbore phase redistribution, and damage skin effects. New dimensionless groups are introduced to facilitate the classic transformation from radial flow in the sphere to linear flow in the rod. Analytical expressions, type curves (in log-log and semilog format), and tabulated solutions are presented, both in terms of pressure and rate, for all flow problems considered. A new empirical equation to estimate the duration of wellbore and near-wellbore effects under spherical flow is also proposed. Introduction The majority of the reported research on unsteady-state flow theory applicable to well testing usually assumes a cylindrical (typically a radial-cylindrical) flow profile because this condition is valid for many test situations. Certain well tests, however, are better modeled by assuming a spherical flow symmetry (e.g., wireline formation testing, vertical interference testing, and perhaps even some tests conducted in wellbores that do not fully penetrate the productive horizon or are selectively penetrate the productive horizon or are selectively completed). Plugged perforations or blockage of a large part of an openhole interval may also promote spherical flow. Numerous solutions are available in the literature for almost every conceivable cylindrical flow problem; unfortunately, the companion spherical problem has not received as much attention, and comparatively few papers have been published on this topic. papers have been published on this topic. The most common inner boundary condition in well test analysis is that of a constant production rate. But with the advent of downhole tools capable of the simultaneous measurement of pressures and flow rates, this idealized inner boundary condition has been refined and more sophisticated models have been proposed. Therefore, similar methods must be developed for spherical flow analysis, especially for short-time interpretations. This general problem has recently been addressed elsewhere. Theory The fundamental linear partial differential equation (PDE) describing fluid flow in an infinite medium characterized by a radial-spherical symmetry is (1) The assumptions incorporated into this diffusion equation are similar to those imposed on the radial-cylindrical diffusivity equation and are discussed at length in Ref. 9. In solving Eq. 1, the classic approach is illustrated by Carslaw and Jaeger (later used by Chatas, and Brigham et al.). According to Carslaw and Jaeger, mapping b=pr will always reduce the problem of radial flow in the sphere (Eq. 1) to an equivalent problem of linear flow in the rod for which general solutions are usually known. (For example, see Ref. 17 for particular solutions in petroleum applications.) Note that in this study, we assumed that the medium is spherically isotropic; hence k in Eq. 1 is the constant spherical permeability. This assumption, however, does not preclude analysis in systems possessing simple anisotropy (i.e., uniform but unequal horizontal and vertical permeability components). In this case, k as used in this paper should be replaced by k, an equivalent or average (but constant) spherical permeability. Chatas presented a suitable expression (his Eq. 10) obtained presented a suitable expression (his Eq. 10) obtained from a volume integral. It is desirable to transform Eq. 1 to a nondimensional form, thereby rendering its applicability universal. The following new, dimensionless groups accomplish this and have the added feature that solutions are obtained directly in terms of the dimensionless pressure drop, PD, not the usual b (or bD) groups. ......................(2) .......................(3) .........................(4) The quantity rsw is an equivalent or pseudospherical wellbore radius used to represent the actual cylindrical sink (or source) of radius rw. SPEJ p. 804