Deconvolution Theory

Assuming the reservoir and wellbore properties do not change over time, the governing equation being linear with respect to pressure, and using the principle of superposition the following equation can be derived:

Where Pu is the unit rate pressure response of the reservoir.

If we define:

The superposition equation can be re-written as:

Where z is the variable used in the least squares minimization process to calculate the derivative and Pu such that the total least squares error ETLS is minimized.

The error terms used to determine the minimum error are defined as follows:

Where pm is the measured pressure and pc is the pressure calculated from the preceding superposition equation.

Where qm is the measured rate and qc is calculated by the least squares minimizer. Erate is zero unless the Adjust Rates option has been turned on.

The curvature error (Ecurv), is defined as the amount of curvature on the derivative curve. it is a measure of how smooth the derivative is, so that the smoother the curve the smaller Ecurv. For more detailed equations on this term please refer to SPE papers 71574 and 77688.

The total least squares error (ETLS) is defined as follows:

Where wx  is the weighting parameter for each Ex error term. These values can be adjusted to improve the fit when needed by placing more emphasis on specific error terms.

References

This deconvolution method is based on the work of Thomas Von Schroeter et al. and Michael M. Levitan et al. for more information on the concepts of deconvolution please refer to the following papers: SPE 71574, SPE 77688, SPE 84290, SPE 90680.