Abstract

Conventional methods used to analyze early-time formation transient test data are approximate and do not model the entire timewise pressure response. Alternatively, detailed models used for early-time storage or late-time analysis are numerical and chart-oriented, and not amenable to simple physical interpretation. In this paper, the general spherical flow formulation, with wellbore storage, is solved by inverting the Laplace transform to produce a closed-form analytical solution. This new solution reduces to previously published early- and late-time solutions in the respective asymptotic limits. Importantly, the availability of the new solution permits convenient pressure-response-to-theory-matching using the complete time regime (early-, transitional-, and late-time data), rather than the typical late-time matching to the asymptotic limit of the spherical flow regime. The new solution also allows accurate formation evaluation using early- and intermediate-time data only, thereby reducing rig down-time required for testing.

The new analytical solution is derived for the full initial- boundary value problem for transient, compressible, spherical flow using Laplace transforms. In this paper, the Laplace transform is inverted to obtain a closed-form expression for the pressure field, and additionally, expressions for pressure derivative and sandface flow rate are derived in terms of the complex error function. These solutions are compared to existing numerical solutions obtained by approximate transform inversion, and to finite element model 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, formation fluid compressibility, permeability, and pore pressure. It circumvents limitations inherent in existing methodologies, e.g., where parameters such as compressibility are assumed a priority.

A new type of drill stem well test tool is also described, and the new solution is used to analyze its field test data. Wireline tester log data examples are analyzed to illustrate the utility and power of the new solution. The examples are used to demonstrate how the new solution saves rig time by using convenient, microprocessor-based, real-time regression analysis to determine formation properties.

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