New Exact Spherical Flow Solution with Storage and Skin for Early-Time Interpretation with Applications to Wireline Formation and Early-Evaluation Drillstem Testing

Abstract

Abstract The wireline formation tester, which is frequently operated over short durations, and a new formation testing tool that obtains pressure measurements during the drilling program, have provided the impetus to develop improved pressure transient analysis methods that support formation evaluation during early-to-intermediate test times. Existing methods, for example, apply to late times only, and are approximate in that they do not model the entire timewise pressure response; alternatively, the detailed models that are available for more exact analysis are numerical, chart-oriented, and not amenable to simple physical interpretation. This paper extends recent work that has solved the spherical flow boundary value problem for storage with an exact analytical solution.1,2 Here, the general spherical flow formulation, including both wellbore storage and skin effects, is solved by inverting the complete Laplace transform to produce an exact, closed form, analytical solution. This new solution reduces to previously published early- and late-time solutions in the respective asymptotic limits. However, the new solution permits convenient pressure response to theory matching over the complete time regime (early-, transitional-, and late-time data), rather than conventional late-time, spherical flow matching; it permits critical pressure-transient interpretation using only early-to-intermediate-time data, thereby reducing required rig time. The new solution is derived for the full initial-boundary value problem for transient, compressible, spherical flow using Laplace transforms. Closed-form expressions are obtained for pressure, pressure derivative, and sandface flow rate in terms of complex error functions. These exact solutions are compared to existing spherical solutions obtained by numerical transform inversion, and also to finite element results that include the three-dimensional effects of wellbore geometry. The accuracy and interpretation speed of the new solution permits systematic determination of all flow parameters, to include storage coefficient, skin effect, formation fluid compressibility, permeability, and pore pressure. Its application to a new type of well test tool designed for pressure testing during the drilling program is described. Synthetic data from wireline formation testers and a new drill pipe conveyed pressure test system are analyzed to illustrate the utility and power of the new solution. These examples are used to demonstrate how the new approach can improve real-time regression analysis used in determining formation properties.