A Further Note on Pulse-Test Interpretation

Startzman, R.A., Chevron Oil Field Research Co.

The development of pulse testing, a special type of well interference testing, has resulted in interpretation techniques that require the use of graphical methods. These methods make use of such variables as "time lag" and "pressure response", obtained visually from plots of well pressure vs time. Such methods require that the pulse-rate schedule be a symmetric, rectangular step-function of constant frequency and amplitude.

In actual field practice it may be impossible to maintain a pulse-rate schedule of this type. Furthermore, even if such a schedule could be adhered to, the resulting pressure signal in the responding well might possess a high degree of random noise. Noise caused, perhaps, by a low-resolution pressure instrument can obscure the test results so that time lag and pressure response cannot be determined accurately.

A more general approach that overcomes these difficulties is that of automatically history-matching observed responding well pressures with pressures computed from a mathematical model. A convenient method is to minimize the least-squares objective function

(1)

Any obvious long-term reservoir pressure trends must be removed from the measured pressures in order to arrive at po. An analysis of the residuals, (pt - po), computed at the minimum of psi, will determine if pressure trends have been successfully removed.

Since pt and its first and second derivatives can be determined analytically from the exponential integral solution to the diffusion equation, a useful minimization technique is the Newton-Raphson method. This method is used to solve the following equations for transmissibility, T, and storage, s:

(2)

and

(3)

This interpretation technique was programmed in FORTRAN for the IBM 360/50. The Newton-Raphson method was found to be quite satisfactory and most problems took less than 10 seconds for adequate convergence.

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