# Uncertainty Quantification in the Eagle Ford using a Bayesian Framework

**Author:** Isaac Zhukovsky, Texas A&M University

Once an oil or gas well has been drilled, one of the first questions usually asked is “What volume of hydrocarbons can we expect this well to produce?” To answer this question, the oil and gas industry can choose from a menu of many different methods. These methods vary widely in accuracy and complexity; one of the most common methods employed is called decline curve analysis (DCA). DCA is the process of fitting an equation to production data and using this fitted equation to predict future well performance. There are many different available decline curve equations, some empirical and others derived from physical laws and theories.

Once a decline curve equation has been chosen, it can be applied deterministically or probabilistically. Deterministic analysis is a way of quantifying risk using single point estimates. The engineer picks values for the “best case scenario”, “most likely scenario” and “worst case scenario.” The most likely scenario is usually determined by the fitted decline curve, while the engineer typically modifies the decline curve based on his experience and knowledge of the area and reservoir to come up with the other two scenarios.

Probabilistic, or stochastic, analysis is a way of quantifying risk using probability distributions of input parameters instead of discrete parameter values. These distributions are sampled many times, resulting in a large number of different combinations of the parameters. These combinations can be ranked in terms of their probability of occurrence and used to create a probability distribution of the potential outcomes. One way of performing probabilistic analysis involves applying Bayes Rule. Bayes’ rule is a method for updating your prior knowledge, represented by probability distributions, given the acquisition of new observed data. When Bayes’ Rule is applied, the engineer usually ends up with a narrower parameter distribution, reflecting the knowledge gained by incorporating the observed data. In Bayesian decline curve analysis, the observed data is the production data from the well of interest.

Deterministic analysis is usually appropriate in mature, conventional fields where well behavior and performance are already well understood. Probabilistic analysis is typically more appropriate earlier in well life or in green fields where there is still a large amount of uncertainty regarding well performance. By implementing Bayesian DCA, the engineer can quantify the uncertainty and get a better idea of the range of possible well performance.

Traditional decline curve analysis is usually a deterministic process and quantifies uncertainty in a qualitative manner. Additionally, the equation most commonly used for decline curve analysis in the oil and gas industry, Arps’ (1945) hyperbolic rate/time equation, was developed under several assumptions that do not hold true for shale wells. There are several problems inherent with this approach. An important parameter of Arps’ equation, the b factor can decrease over time, leading to over prediction based on early time forecasts. The deterministic nature of the analysis focuses on one “most likely” prediction, with high and low cases based on the judgment of the engineer. All of these aspects lead to a situation where there is a significant chance for surprises in the future, complicating corporate planning processes.

Therefore, a Bayesian decline curve methodology is proposed using a novel empirical decline curve equation (Zhang et al., 2015) to better quantify uncertainty in shale wells. The equation attempts to accommodate both early time steep decline behavior and later time shallower decline behavior with a smooth transition in between. Coupled with a Bayesian decline process, the particular decline behavior of shale wells can be modeled in a probabilistic manner.

In Bayesian decline curve analysis, the decline curve parameters are assumed to be random variables instead of parameters with “best fit” values. Using the Metropolis algorithm, a Markov Chain Monte Carlo simulation is performed to obtain the posterior distribution of variables of interest, principally well expected ultimate recovery. By performing the simulation on many wells with only the first 12 months of production known, the calibration of the method can be checked on an areal basis by measuring the rate of coverage of “true reserves.”

Preliminary hindcasting results on an areal basis with more than 100 Eagle Ford shale oil wells have demonstrated promising results, with a coverage rate of true reserves approaching 80% for an 80% confidence interval (P90-P10). This method offers many benefits. Principally, it quantitatively assesses uncertainty, generates accurate results for shale wells as the decline curve equation was empirically designed for shale wells. Furthermore, it generates replicable results for given wells regardless of the engineer and offers a fast calculation time of 5-10 seconds per well in the data set.

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