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Implications of Aggregating and Smoothing Daily Production Data on Estimates of the Transition Time Between Flow Regimes in Horizontal Hydraulically Fractured Bakken Oil Wells

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Abstract

The level to which data are aggregated or smoothed can impact analytical and predictive modeling results. This paper discusses findings regarding such impacts on estimating change points in production flow regimes of horizontal hydraulically fractured shale oil wells producing from the middle member of the Bakken Formation. Change points that signal transitions in flow regimes are important because they subsequently affect estimates of ultimate recovery from wells producing from shale plays. Extending our earlier work, we employ two different statistical approaches, Bacon–Watts Bayesian regression and nonlinear constrained least squares regression, and a designed computational experiment to estimate the time of transition from the transient to the boundary-dominated flow regime for 14 different wells using daily production data rather than aggregated monthly data, as previously considered. The daily data were also smoothed to reduce noise. Computational experiments suggest that both statistical approaches can lead to plausible estimates of the transition point under different data aggregation or smoothing regimes, but that daily data are likely too granular to produce credible estimates. Although the expected value of transition points using smoothed daily data and monthly disaggregated data are generally comparable, the confidence intervals bounding the estimates based on smoothed daily data are generally wider. Our results not only inform the operational practices of oil producers engaged in economic evaluation of their shale resources and additional play development activities, but also the activities of petroleum research groups, government agencies, and financial organizations seeking to improve the trustworthiness of resource projections.

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Appendix

Appendix

This appendix summarizes the nonlinear least squares and Bacon–Watts Bayesian regression algorithms used to estimate a single transition (change point) between flow regimes in horizontal hydraulically fractured Bakken shale oil wells (see Attanasi et al. (2019) for additional details). We assume the change point marks a discrete transition between the two flow regimes.

Using the nonlinear approach, an estimate of t*, the time at which the transition occurs, is obtained using

$$ y_{i} = \left\{ {\begin{array}{*{20}l} {\beta_{0} + \beta_{1} x_{i } + e_{i} , } \hfill & {x_{i} \le t^{*} } \hfill \\ {\beta_{0} + \beta_{1} x_{i} + \beta_{2} \left( {x_{i} - t^{*} } \right) + e_{i} ,} \hfill & {x_{i} > t^{*} } \hfill \\ \end{array} } \right., $$
(1)

where yi = log10 (1/production), xi = log10 (time), and where production is typically given in barrels per day and time is typically given in days (see Fig. 1). t* = log10 (transition time from transient flow to boundary dominated flow) and the ei are assumed to be independent, having additive errors with mean zero, constant variance, and finite absolute moments. Additional restrictions must be imposed on β1 and β2 that constrain their values to roughly conform to the production regime represented in Fig. 1. One possible set of restrictions could be to constrain the values of β1 and β2 to an interval such as 0.45 to 0.55. The imposition of such constraints requires a nonlinear algorithm to estimate the parameter values, including the estimate of t*. The solution to the resulting nonlinear regression depends on the autocorrelation structure of the production series so that the variance–covariance matrix is invertible.

The Bayesian regression approach is based on procedures developed by Bacon and Watts (1971) to estimate the statistical distribution of the uncertain transition (change point), t*. Carlin et al. (1992) facilitated implementation of the Bayesian approach by using a Gibbs sampler to compute parameter posterior density functions, thus avoiding the need for high-dimension integration. The Gibbs sampler is a Markov chain Monte Carlo (MCMC) method for calculating a sequence of observations which are approximated from a specified multivariate distribution and can be used to approximate a joint distribution, a marginal distribution of one of the variables, or the distribution of a subset of variables (Casella and George 1992).

Following Spiegelhalter et al. (1996), the change point problem in this case is specified as

$$ y_{i} \,\sim\,\left\{ {\begin{array}{*{20}c} {{\text{Normal}}\left( {\theta_{0} + \theta_{1} \left( {x_{i} - t^{*} } \right), \sigma^{2} } \right), } & {x_{i} < t^{*} } \\ {{\text{Normal}}\left( {\theta_{0} + \theta_{2} \left( {x_{i} - t^{*} } \right),\sigma^{2} } \right),} & {x_{i} \ge t^{*} } \\ \end{array} } \right.. $$
(2)

For the present investigation, the following priors were used for illustrative purposes: t* ~ Uniform (a, b), θ1 ~ Uniform (0.45, 0.55), θ2 ~ Uniform (0.90, 1.10), θ0 ~ Normal (0, 100), and σ2 ~ Uniform (0, 1,000).

The yi, xi, and t* are defined as in Eq. 1, whereas distributions for t* and θi are defined in Eq. 2. The prior density function on the change point (t*) assumes a uniform distribution where the endpoints, a and b, are expressed in log10(time), and where time is typically specified in days. For the current illustration the original values of a and b are taken to be 100 and 1,000 days, respectively. Setting the lower bound at 100 days mimics industry practice of disregarding the “settling out” period during which production is generally very erratic. Setting the upper bound at 1,000 days permits all wells to be treated consistently since they have different lengths of production life, thereby minimizing computational complexity. If one has point estimates of the appropriate reservoir parameters plus information about well spacing, then more formal estimates of the parameters for the prior distribution for t* can be obtained (Kabir et al. 2011; Ran 2016). Implementation of the Bacon–Watts Bayesian regression algorithm summarized here, along with the associated computations, follows the approach of SAS Institute (2016).

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Coburn, T.C., Attanasi, E.D. Implications of Aggregating and Smoothing Daily Production Data on Estimates of the Transition Time Between Flow Regimes in Horizontal Hydraulically Fractured Bakken Oil Wells. Math Geosci 53, 1261–1292 (2021). https://doi.org/10.1007/s11004-020-09909-7

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