Coupling Probabilisitic Methods and Finite Difference Simulation: Three Case Histories
Authors: Purvis Dwayne C. , Strickland Richard F. , Alexander Richard A. , Quinn M. Anthony 
Published: December 2007

Coupling Probabilisitic Methods and Finite Difference Simulation: Three Case Histories Dwayne C. Purvis, SPE, Cawley, Gillespie and Associates, Inc. (CG&A); Richard F. Strickland, SPE, CG&A; Richard A. Alexander, SPE, CG&A; and M. Anthony Quinn, SPE, CG&A

Abstract
Advances in computing power have made possible the combination of probabilistic methods with the conventional practice of finite difference simulation. This paper presents and demonstrates a methodology for probabilistic finite difference simulation to determine and examine a range of potential production rate profiles and ultimate recoveries by simulating hundreds of scenarios.
Due to the high degree of uncertainty in the three case histories discussed, finite difference simulation coupled with probabilistic methods offered the best tool to predict production profiles given a wide variety of assumptions about reservoir character and producing conditions. A Monte Carlo simulation was run in each case to create several hundred possible combinations of the uncertain variables. These variables became inputs to the same number of finite difference simulations. The resulting forecasts were ranked and P10, P50 and P90 values were determined depending on the needs of those using the forecasts.
When probabilistic simulation is used to predict a range of results, care should be taken when combining the forecasts to properly honor their history-dependent nature. Discounting techniques can be applied to the forecasts before ranking in order to make them useful for economic decisions. In the one case in which a comparison was made, it was also observed that, depending on the skewness of the input distributions, the median forecast was similar to the deterministic, "most likely" forecast.

Introduction
Monte Carlo analysis has been frequently applied in the petroleum industry as it permits a quantitative analysis of the associated risks. In traditional Monte Carlo applications the calculations are rather simple, such as a summation to obtain total cost or multiplication to obtain oil-in-place. Distributions of the calculated values are analyzed to ascertain the total and combined uncertainty implied by the input values.
Advances in computing power have made it feasible to extend Monte Carlo techniques to complex calculations such as finite difference simulation. The coupling of these powerful tools allows the modeler to quantify the uncertainty in the generated rate forecasts. In the three examples presented, the uncertainty in the production forecast was particularly important to the development decision, making them good candidates for a probabilistic approach.
All three studies involved large uncertainties in reservoir parameters due to limited information. A single simulation forecast was not deemed adequate because it could not describe the associated uncertainty. In each case, an appropriate finite difference model was created. Probability distributions were developed for the major variables that controlled the production profile and ultimate recovery. The most important parameters controlling the production profile varied between the cases. Among the variables considered were porosity, water saturation, net thickness, gas-water contact, permeability and areal extent.
Monte Carlo techniques were used to create several hundred combinations of uncertain reservoir parameters. Each combination was then simulated producing a unique production profile. The resulting, simulation-based production forecasts can be interpreted for technical risk. Such characterization of technical risk is becoming critically important to the increasingly sophisticated strategies used in the petroleum industry.

Statement of Theory and Definitions
Traditional Monte Carlo applications address uncertainty of data inputs to simple calculations such as oil-in-place. Finite difference simulation, on the other hand, has been used to assess uncertainty on a larger scale, such as that between qualitatively different reservoir characterizations. P. 289