The goal of this project was to develop an efficient and accurate approach to quantify uncertainty for flow in randomly heterogeneous reservoirs. 

Program
This project was funded through DOE's Natural Gas and Oil Technology Partnership Program. The program establishes alliances that combine the resources and experience of the nation's petroleum industry with the capabilities of the national laboratories to expedite research, development, and demonstration of advanced technologies for improved natural gas and oil recovery.

Los Alamos National Laboratory
Los Alamos, NM

University of Oklahoma
Norman, OK

ChevronTexaco
San Ramon, CA

Accurate modeling of reservoir flows requires a detailed spatial description of reservoir properties such as permeability and porosity. However, oil and gas reservoirs exhibit a high degree of spatial variability in medium properties, which are usually measured only at a few locations. This leads to uncertainty about the values of reservoir properties and in turn to uncertainty in predicting flow in such reservoirs. The common tool in quantifying this uncertainty in industry has been the Monte Carlo method, which is computationally expensive. Researchers originally proposed to develop a direct method based on the moment approach. During the course of this study, they developed a completely new approach, a moment-equation method based on KLME, which is much more efficient and accurate than the Monte Carlo method and the moment-equation method originally proposed.

Project Results
Researchers developed a completely new approach to quantify uncertainties associated with resevoir performance simulations-a moment-equation method based on Kahunen-Loeve decomposition (KLME), which is much more efficient and accurate than the moment-equation method they originally proposed. The KLME approach makes it possible to evaluate higher-order terms that are needed for highly heterogeneous reservoirs. The KLME method enables the simulation of flow and transport in large-scale heterogeneous reservoirs and may revolutionize the way of quantifying uncertainty in reservoir performance.

Benefits
Performance predictions obtained from reservoir flow simulations are the primary tool for managing existing operations and planning future developments. The study provides a general, computationally efficient and accurate tool for quantifying uncertainties for flow in heterogeneous porous media. It couuld have a significant impact on a wide range of fields, including energy security (nuclear waste repository sciences), contaminant transport (DOE's site cleanup), oil/gas recovery (energy security), and geological carbon sequestration (global warming).

Project Summary
Project researchers have:

• Developed a moment approach based on KLME for solving stochastic flow in randomly heterogeneous reservoirs.
• Extended the KLME method to a number of different flow conditions, such as multiphase flow.

Because of heterogeneity in reservoir properties such as permeability, the flow equations are treated as stochastic partial differential equations. Researchers first decomposed the log permeability using the Karhunen-Loéve (KL) decomposition, which is of mean square convergence. They then wrote the pressure head as an infinite series whose terms h(n) represent the head contribution at the nth order in terms Y, the standard deviation of log permeability, and derived a set of recursive equations for h(n). This assumes that h(n) can be expressed as infinite series in terms of the products of n Gaussian random variables. In short, using the new approach, with a much lower computational cost, the project performers are able to evaluate the mean pressure head up to fourth order in Y and the pressure head variance to the third order in Y2.

Researchers compared the results from the KLME approach against those from Monte Carlo simulations and the first-order moment method originally proposed. It is evident that the KLME approach, with higher-order corrections, is superior to the first-order approximations and is computationally more efficient than both the Monte Carlo simulations and the first-order moment method. The KLME approach has been extended to simulate water flow in unsaturated soils (Yang et al.), in conditional fields (Lu and Zhang), and two-phase (water and oil phases) flow problems (Chen et al.).

(October 2005)
This method is being combined with a Kalman filter to efficiently characterize geological reservoirs and incorporated into MODFLOW, the most popular groundwater flow simulator.

Publications (partial list)
Lu, Z., and Zhang, D., Stochastic studies of well capture zones in bounded heterogeneous media, Water Resour. Res., 39(4), 1100, doi:10.1029/2001WR 001633, 2003.

Lu, Z., and Zhang, D., Solute spreading in nonstationary flows in bounded, heterogeneous unsaturated-saturated media, Water Resour. Res., 39(3), 1049, doi:10. 1029/2001WR000908, 2003.

Li, L., Tchelepi, H.A., and Zhang, D., Perturbation-based moment equation approach for flow in heterogeneous porous media: applicability range and analysis of high-order terms, Journal of Comput. Physics, v.188 n.1, p.296-317, 2003.

Lu, Z., and Zhang, D., Conditional simulations of flow in randomly heterogeneous porous media using a KL-based moment-equation approach, Advances in Water Resources, 27(9), 859-874, 2004.

$500,000

$600,000 (55% of total)

NETL - Dan Gurney (dan.gurney@netl.doe.gov or 918-699-2063)
LANL - Zhiming Lu (zhiming@lanl.gov or 505-665-2126)