Amplification of Pulse-Testing Theory

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Abstract
Pulse-testing theory as originally outlined is based Pulse-testing theory as originally outlined is based on the standard exponential integral solution to the radial form of the diffusivity equation. This paper presents a new solution to the subject diffusivity presents a new solution to the subject diffusivity equation and outlines how this type of solution can be used to clarify the "tangent construction method" of analyzing pulse data. The new solution is composed of a periodic part and a transient part. The splitting of the total solution into two parts facilitates the extension of pulse-testing to include steady-state conditions The differential equation and auxiliary conditions that define the periodic pulsing problem are .......(1) with the initial condition of ...........(2) and boundary conditions of .............(3) .............(4) where ............(5) and H(t) is the unit step function. The solution to this boundary value problem is usually represented by the summation of appropriate exponential integral terms. A solution that splits the total solution into periodic and transient portions can be obtained as periodic and transient portions can be obtained as follows. Utilizing Laplace transform theory, the solution to Eqs. 1 through 4 is ................(6) where, the real part of the complex integration limits, is taken large enough so that all singularities of the integrand lie to the left of the line ( - i, + i ). Complex variable theory was used to evaluate the integral in Eq. 6; and the final solution, in field units, can be shown to be .........(7) P. 1245

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