Assessing the Non-Uniqueness of the Well Test Interpretation Model Using Deconvolution

Abstract

Uncertainty in well test analysis results from errors in pressure and rate measurements, from uncertainties in basic well and reservoir parameters; from the quality of the match with the interpretation model; and from the non-uniqueness of the interpretation model. These various uncertainties, except the non-uniqueness of the interpretation model, were examined in SPE 113888. It was concluded that the permeability-product kh is generally known within 15%; the permeability k, within 20% (because of the uncertainty on the thickness h); and the skin effect S, within 8% for high S values and within ±0.5 for low S values. Distances (half-fracture lengths, horizontal well lengths, and distances to reservoir boundaries) are usually known within 25%. The issue of non-uniqueness of the interpretation model is more complex: not only may there be a multitude of possible models for any one derivative response (the usual inverse problem), but there may be also a multitude of derivative responses, due to the uncertainty inherent in the observed data. This paper presents a methodology for assessing the derivative response uncertainty using deconvolution. It is shown that the uncertainty depends mainly on the error bounds for initial pressure and flow rates, which yield a range of possible shapes for the deconvolved pressure derivative and therefore different possible interpretation models. In some cases, the non-uniqueness of deconvolution can be reduced using knowledge of the expected model response, for instance from geology or seismic. In the absence of differentiating information, however, alternative interpretation models have to be considered, which may lead to completely different development options. The methodology is illustrated with three field examples.

Multiple Well Deconvolution

“…First, we observe that the estimated interference effects are not as steep as the true interference introducing a disagreement in the early stages of the interference derivatives. However, since such a disagreement coincides with very small values of the derivative, the practical impact is negligible and the fact that we recovered such a precise pressure match despite this disagreement highlights the non-uniqueness present in our solution (Cumming et al, 2013). A more appropriate choice for the initial node of the interference response may address this.…”

“…As discussed in Cumming et al (2013), this problem is both non-linear and over-determined, and as such a unique solution cannot be found. Instead, we determine a statistical estimate of the response function which best matches the data via nonlinear least squares methods.…”

“…Consequently, in a multiple well problem with ) wells, there will be a total of ) A distinct response functions to be identified, each of which expresses the effect of production from any one well on the pressure at any other. Identification of such a large collection of response functions is a challenging problem, and one which is made more complex by the non-uniqueness of the solution and noisy data (Cumming et al, 2013),. The complexity can be reduced by assuming reciprocity and treating the interference effects between pairs of wells as symmetric:…”

“…Identifying good choices for these hyperparameters is a challenging problem, and various methods have been proposed -see for example Cumming et al (2013), Levitan (2006), Pimonov et al (2009), von Schroeter et al (2004. In single-well deconvolution, the purpose of these hyperparameters is to balance the contributions of the four components of the error function in such a way that leads to good quality deconvolutions.…”

“…We applied Monte Carlo (Fishman, 1999) uncertainty analysis to this data set following the method of Cumming et al (2013) using uncertainties as specified therein -namely a pressure resolution of ±0.1psi and a stability of up to 0.1psi per month, an error of up to 10% of the value of rate measurements, and a ±20psi uncertainty on the initial pressure. Pressure accuracy was not explicitly considered due to its effects behaving effectively as an error on initial pressure.…”