In this article we use a direct approach to quantify the uncertainty in flow performance predictions due to uncertainty in the reservoir description. We solve moment equations derived from a stochastic mathematical statement of immiscible nonlinear two-phase flow in heterogeneous reservoirs. Our stochastic approach is different from the Monte Carlo approach. In the Monte Carlo approach, the prediction uncertainty is obtained through a statistical post-processing of flow simulations, one for each of a large number of equiprobable realizations of the reservoir description.

We treat permeability as a random space function. In turn, saturation and flow velocity are random fields. We operate in a Lagrangian framework to deal with the transport problem. That is, we transform to a coordinate system attached to streamlines (time, travel time, and transverse displacements). We retain the normal Eulerian (space and time) framework for the total velocity, which we take to be dominated by the heterogeneity of the reservoir. We derive and solve expressions for the first (mean) and second (variance) moments of the quantities of interest.

We demonstrate the applicability of our approach to complex flow geometry. Closed outer boundaries and converging/diverging flows due to the presence of sources/sinks require special mathematical and numerical treatments. General expressions for the moments of total velocity, travel time, transverse displacement, water saturation, production rate, and cumulative recovery are presented and analyzed. A detailed comparison of the moment solution approach with high-resolution Monte Carlo simulations for a variety of two-dimensional problems is presented. We also discuss the advantages and limits of the applicability of the moment equation approach relative to the Monte Carlo approach.


Reservoir management decisions are often based on predictions of future performance obtained from numerical flow simulations. Quantifying the predictive reliability of reservoir performance forecasts is necessary for risk management. The reliability of flow predictions depends on the quality of the information used, and then it depends on the ability of the numerical simulations to describe the physics of the flow accurately.

Accurate modeling of the physics that govern complex multiphase reservoir flows requires a detailed spatial description of reservoir properties such as permeability and porosity. However, only limited reservoir characterization information of varying quality from different sources is usually available. Thus, a major element of risk is due to incomplete knowledge of the reservoir description, which leads to uncertainty in the flow responses obtained for the displacement processes under investigation. In this article, we use a direct approach to quantify the uncertainty in flow performance predictions due to uncertainty in the reservoir description.

A stochastic framework provides a formal way for generating detailed models of reservoir description that honor available hard and soft data. Stochastic, or probabilistic, models of reservoir characterization are common practice in the oil industry. However, deterministic mathematical formulations of multiphase multicomponent flow processes continue to serve as the basis for numerical reservoir flow simulators. Subsurface hydrologists, on the other hand, describe the flow and transport of contaminants in heterogeneous aquifers using stochastic formulations and solution methods. The stochastic perturbation-based methods of subsurface hydrology have focused on single-phase and unsaturated (air-water) flows. These methods are not directly applicable to the nonlinear multiphase displacement processes of interest in oil reservoirs.

We must address a number of fundamental challenges before stochastic formulations can be applied to reservoir flows. These challenges span a wide range of issues. They include issues related to nonlinear multiphase flow, complex reservoir geometry and formation heterogeneity, boundary conditions, and the presence of wells. Important considerations related to efficiency, range of applicability, and effectiveness for real-world problems must also be eventually addressed.

Dagan and Cvetkovic1,2 applied a Lagrangian perturbation theory for single-phase and two-phase flows in random velocity fields in the absence of capillary pressure and gravity. Assuming that the mean flow was uniform and that the total velocity was time independent in an unbounded domain, they derived analytical expressions for mean oil production and cumulative recovery. Zhang and Tchelepi3 extended the work of Dagan and Cvetkovic.1,2 They derived expressions for the mean and standard deviation of water saturation for nonlinear two-phase flow using a Lagrangian, moment-based approach. They focused on issues related to the nonlinear character of immiscible two-phase flow, and they presented analytical solutions for simple one- and two-dimensional nonlinear problems that demonstrate the theory and discuss its implications. Zhang and Tchelepi's work3 is also limited to uniform mean flow in unbounded domains.

In this article, we extend the Lagrangian, statistical moment approach to (1) flow in bounded domains and (2) complex flow patterns due to the presence of wells (sources and sinks). These extensions are a prerequisite for solving real-world reservoir problems in a stochastic framework. Although we only tackle the problem of immiscible two-phase flow in this article, this Buckley-Leverett-type problem possesses many of the necessary features that a stochastic moment-based approach for reservoir flows must handle accurately and efficiently.

The class of problem we deal with here is not amenable to analytical solutions. This is because the assumptions of statistical stationarity and uniformity of mean flow are not valid when the flow is driven by wells and the domain is bounded. Thus, although the general theoretical framework described by Zhang and Tchelepi3 is applicable, the moment expressions for the cases considered here are quite different and require special numerical treatment.

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