Approximate Bayesian Computation for Probabilistic Decline-Curve Analysis in Unconventional Reservoirs

Abstract

Summary In this work, we developed a methodology that integrates decline-curve-analysis (DCA) models with an approximate Bayesian probabilistic method that is based on rejection sampling to quantify the uncertainty associated with DCA models. This methodology does not require the estimation of the likelihood, which simplifies the Bayesian inference greatly. In approximate Bayesian computation (ABC) with rejection sampling, the posterior distribution is approximated by substituting different values of the decline-equation parameters into the Arps’ DCA model and generating a large number of production-profile realizations. Summary statistics (mean, standard deviation) between the simulated and observed production data are then compared. On the basis of some optimum threshold value, the rejection-sampling technique is applied to either accept or discard the simulated production data. The resulting distribution is used to approximate the posterior. The 10th, 50th, and the 90th percentiles of the accepted data sets provide the P90, P50, and P10 estimates for reserves, respectively. History matching was performed to test the proposed Bayesian model and to check how well the simulation results match with the observed production data. Chosen for analysis were 57 gas wells from Johnson County (Barnett Shale) and 21 oil wells from Karnes County (Eagle Ford Shale) with production history of 90 and 45 months, respectively. Best-fit (deterministic) curves were computed with the least-squares regression method. Only two-thirds of available history was used for modeling, and the remaining one-third of the history was used for validating the methodology. The P50 history match trended well with the best-fit curve for majority of wells. The ABC P90–P50–P10 average cumulative production interval for wells in Johnson County was 1,263–1,410–1,528 MMcf, whereas the true average cumulative production per well was 1,425 MMcf. Similarly, for Karnes County, the ABC P90–P50–P10 average interval per well was 170,000/184,000/204,000 STB, whereas the true average cumulative production per well was 183,000 STB. This implies that the ABC bounds bracket the true reserve well. Approximately 42 and 98% of wells’ true cumulative production at the end of hindcast were greater than their P50 and P90 estimates, respectively. This implies that the P50 and P90 estimates were quite accurate even with short production history (approximately 2 years). The P10 estimates were less accurate but still acceptable with only 4% of wells’ true production higher than their P10 estimate. Therefore, estimates from the ABC methodology are well-calibrated. The proposed ABC methodology combined with rejection sampling provides a procedure that not only produces probabilistic forecasts but also quantifies reserves uncertainty in shale plays quickly and consistently. The ABC methodology can be coupled with any other deterministic DCA model.