Incorporating Constraints Improves Least-Squares Multiwell Deconvolution
Authors: Carpenter Chris 
Published: August 2020

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 195271, “Constrained Least-Squares Multiwell Deconvolution,” by J. Cumming, Durham University; V. Jaffrezic, SPE, Total; and T. Whittle, Imperial College, et al., prepared for the 2019 SPE Western Regional Meeting, San Jose, California, 23-26 April. The paper has not been peer reviewed.
In the complete paper, the authors reduce nonuniqueness and ensure physically feasible results in multiwell deconvolution by incorporating constraints and knowledge to methodology already established in the literature. The paper demonstrates that a combination of constraints on the shapes of the deconvolved derivatives and knowledge of the reservoir results in an improved level of quality and consistency in multiwell deconvolution solutions. The new constrained least-squares multiwell deconvolution approach is illustrated by synthetic examples with known solutions involving up to nine wells.
Introduction
To apply multiwell deconvolution to real data, confidence must exist in the validity of the solution it provides and the consequent interpretation that is inferred. Therefore, for practical multiwell deconvolution of field data, reducing the nonuniqueness in a solution, ensuring the results are physically feasible, and providing the desired consistency and quality of solutions are essential. With these goals in mind, the authors extend existing methodology to incorporate additional constraints and introduce additional knowledge. These constraints allow encoding of information about the behavior of the deconvolved derivatives to penalize and discourage nonphysical solutions and improve solution quality. Additionally, the approach encodes a priori knowledge, for instance that all pressure responses ultimately will show a closed reservoir by imposing a constraint that future derivative behaviors should tend toward a common unit slope. Through a combination of these techniques, an improved level of quality and consistency of deconvolution solutions can be achieved.
Constrained Least-Squares Multiwell Deconvolution
A key issue in any deconvolution, single-well or multiwell, is the lack of a unique solution, which results in sets or ranges of possible response functions that yield pressure matches of indistinguishable quality. When this space of possible solutions is large, then the associated response function could correspond to reservoir models with multiple differing (and potentially contradictory) interpretations. Clearly, reducing the range of possible solutions as much as possible, and assessing and quantifying the uncertainty surrounding those solutions, is critical to provide confidence in the accuracy of conclusions.
One primary reason why the space of potential deconvolution solutions is so large is the choice of parameterization as a sequence of straight-line segments. In single-well deconvolution, such concerns motivated the introduction of the smoothness component to the error function in order to penalize potential deconvolution solutions with unfeasibly rough or oscillatory response functions. A second source of nonuniqueness in deconvolution is the error and uncertainty associated with the measured test data. The smoothness constraint also is particularly useful in addressing this problem.