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.

The nonuniqueness problem is compounded in the multiwell scenario, where the additive nature of the multiwell superposition means that, in addition to the difficulties in identifying single response functions, a potentially infinite number of ways exist by which the collection of responses and interferences can be combined to reproduce the same set of observed pressures. Thus, the challenge for multiwell deconvolution is that, while smoothness is still a beneficial constraint to apply to deconvolved response functions, it is insufficient alone to ensure that the collection of deconvolved responses and interference effects correspond to functions that have feasible physical behaviors, both independently and relative to one another. Therefore, to improve and ensure the consistency and reliability of the results of a multiwell deconvolution, the authors introduce and apply further constraints that encode knowledge about the range of possible physical behaviors of the response functions. These constraints, expressed in terms of the deconvolution parameters, will disfavor a wider range of nonphysical results, penalizing such solutions, pushing them out of the space of potential deconvolution solutions, and thereby improving the quality of deconvolutions. The complete paper provides a detailed discussion of constraints within least-squares devolution and both simple and complex constraints for the response function, including involved equations.

## Synthetic Examples

The four synthetic examples provided in the complete paper include a three-well example, a five-well example, a three-well example with noninterference, and a nine-well example. Because of space limitations, the first and last of these synthetic examples are included in this summary.

**Three-Well Example. **The first example is a three-well problem **(Fig. 1). **The data comprise 904 pressure observations spanning 2,000 hours, with a production history of three overlapping flow periods, one for each well. This is shown in the left panel of Fig. 1; the true response functions are shown in the center panel, and the map in the right panel.

The data were deconvolved both with and without constraints under otherwise identical conditions. The relative weights of the error-function components were chosen to be 1,000 for the pressure match and 0.001 for the response curvature for both multiwell deconvolutions. The low value for the curvature term is appropriate for a synthetic problem in which the data are not contaminated by observational error and other noises. Each response function comprised 30 nodes, with the first node of main effect responses fixed at 10^{–3} hours and 10 hours for interference responses. For the constrained multiwell deconvolution, crossing constraints and future convergence to a closed system were applied, extrapolating forward 1.5 log cycles and imposing convergence constraints only after the test duration. Reciprocity of interference was assumed throughout.

Multiwell deconvolutions took between 3 and 5 minutes. In both constrained and unconstrained cases, excellent pressure matches to the history were obtained, with pressure errors generally within 0.1 psi of the observed values. The deconvolved derivatives are also very similar; however, the authors note that the unconstrained multiwell deconvolution fails to find a solution with a common late-time convergence. Instead, some of the response functions diverge from the expected late-time behavior. Multiwell deconvolution with constraints yielded a set of response functions that was a closer match to the true response functions.

In both cases, the algorithm makes substantial initial improvements before reaching a plateau where no further improvements are possible and convergence is declared. In the case with constraints, convergence is slower, though this is to be expected because of the increased complexity of the error function.

**Nine-Well Example.** The authors consider a synthetic nine-well problem derived from a real 53-well reservoir. The data comprise 10,629 pressure observations over all nine wells and a complex rate history of 2,532 flow periods. The problem includes two pseudowells to represent the entire injection and production of the 46 wells elsewhere in the reservoir. This approach ensures that the cumulative production is preserved, and thus deconvolution should honor the material balance for the entire field.

Because all wells are active and in communication with one another, assuming reciprocal interference results in a problem with 45 unique response functions. Deconvolution was performed both with and without the standard constraints using a pressure weight of 1,000 and a curvature weight of 0.1 while specifying the end of wellbore storage at 0.1 hour and the earliest interference arrival time at 10 hours. A sequence of deconvolutions was performed, each initialized from the output of the previous deconvolution, with each deconvolution taking between 4 and 6 hours to complete because of the scale and complexity of the problem.

The results of the unconstrained deconvolution are generally poor, though, evidently, even without constraints, the algorithm has been able to identify the gross shapes and features of many of the main well responses. Interference effects, however, are less well-distinguished and identified. The most-obvious deficiency is the lack of late-time or future convergence, which results in a somewhat chaotic late-time picture. Additionally, the authors note problems with the early-time behaviors of interferences. Nonetheless, for a problem of this size and complexity, the set of response functions returned yields a fair match to the pressure data, giving a mean-square pressure match error of 7.18 psi (a relative error of less than 1%).

The imposition of constraints, however, provides the additional information required to overcome many of these problems. The constraints applied were positivity of the initial slopes of the responses and convergence to a closed system beyond the end of the test. The match between the deconvolved responses and the solutions is improved considerably, with a mean-square error of 0.013 using constraints compared with 0.542 without. The deconvolved responses are now very close to the solutions throughout, and the sequencing and arrival of interference effects are much improved. Some minor deviation exists for the lower half of the interference effects; however, this is attributable to the practical sizes of interferences at early times being almost undetectable and, in practice, indistinguishable from observational errors. The pressure match is now improved vastly, with the estimated pressures being generally within ±0.5 psi of the observed pressure history (a relative error of approximately 0.01%) and a reduction in the mean-square pressure error by a factor of 41.7.

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.