Conditional Stochastic Moment Equations for Uncertainty Analysis of Flow in Heterogeneous Reservoirs
- L. Li (ChevronTexaco EPTC) | H.A. Tchelepi (Stanford U.)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2003
- Document Type
- Journal Paper
- 392 - 400
- 2003. Society of Petroleum Engineers
- 4.5 Offshore Facilities and Subsea Systems, 4.3.4 Scale, 5.5 Reservoir Simulation, 4.1.2 Separation and Treating, 5.3.2 Multiphase Flow, 4.1.5 Processing Equipment, 5.6.3 Deterministic Methods, 5.1 Reservoir Characterisation, 5.3.1 Flow in Porous Media, 5.1.5 Geologic Modeling
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We assume that permeability is a random space function defined by its mean and covariance. The stochastic nature of the permeability description leads to uncertainty in flow-related quantities such as pressure, saturation, and production rate. We extended our statistical moment equation (SME) approach to accommodate conditioning. The conditional statistical moment equations (CSME) framework is a direct approach for quantifying the uncertainty in flow performance caused by uncertainty in the reservoir description. It is quite different from Monte Carlo Simulation (MCS). In MCS, the performance 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 developed a CSME computational tool for flow in heterogeneous domains, which we use here to analyze the behavior of the second statistical moment of pressure and velocity in the presence of permeability measurements. We study the effect of both the number and spatial arrangement of available measurements on the computed variances of pressure and velocity. This numerical CSME tool allows us to quantify the value of existing and future information, and that helps in the evaluation of existing projects and in steering future developments. We present several examples that demonstrate how to choose the best sampling locations to obtain maximum reduction in prediction uncertainty. We compare our results with high-resolution MCS.
Properties of natural formations, such as permeability, rarely display uniformity or smoothness. Instead, they tend to show significant variability and complex patterns of spatial correlation. In practical settings, predictions of reservoir performance are usually made with very limited information about the actual details of the reservoir properties. Having measurements at a few locations about properties that display such variability and complexity of spatial correlation means that significant uncertainty accompanies our best attempts to provide detailed description of important quantities such as porosity and permeability. This uncertainty in the reservoir description leads to uncertainty in predictions of flow performance. There is a great deal of effort in our industry not only to make predictions of future reservoir performance, but also to quantify the uncertainty associated with such forecasts.
To deal with the problem of characterizing reservoir properties that exhibit such spatial complexity and variability with the help of only limited data, a probabilistic framework is commonly used. That is, we treat reservoir properties (e.g., porosity and permeability) as random space functions. Here, we use a direct method for quantifying prediction uncertainty. Equations governing the statistical moments of flow-related quantities (pressure and velocity) are derived from a stochastic statement of flow in heterogeneous porous media. In our analysis, we assume that the only source of uncertainty is our limited knowledge of the permeability field. We extend the SME method1-3 to conditional permeability fields. That is, in addition to honoring statistical permeability information (mean, variance, correlation structure), we incorporate available permeability measurements in the domain.
Given a state of knowledge, both deterministic and probabilistic, about the permeability field, our CSME framework can be used to make predictions and quantify the uncertainty associated with such predictions for quantities of interest, such as the pressure and velocity fields. Moreover, the CSME approach can be used to design data collection strategies that lead to maximum reduction in prediction uncertainty for the parameter of interest. That is, CSME provides a framework for analyzing the value of information, both present and future. We illustrate the utility of our CSME method for the flow problem (i.e., pressure and velocity) in a simple setting, namely the quarter-five-spot configuration. Although we confine our attention to the flow problem, the transport problem (i.e., the saturation equation) is both interesting and quite challenging. However, we defer such an analysis to a later time.
The idea is to use conditional stochastic simulation in order to quantify the impact of available measurements on the quality of predictions of flow behavior in heterogeneous reservoirs.
There are two categories of conditional simulation. The first deals with the forward problem, in which we explore the effect of having measurements of the input random variables (e.g., permeability) on the statistical moments of the dependent random variables (e.g., pressure and velocity). The second involves the inverse problem, where observations of the dependent variables are used to deduce information about the statistical moments of the independent variables. Several investigators have addressed both scenarios of conditional flow simulation in heterogeneous domains.4-10 However, most of these investigations come with various restrictions, such as unbounded domains, uniform mean flow, and low input permeability variance. The conditional simulation for the inverse problem is a topic of great importance in many field applications. In this paper, however, we focus on forward conditional simulation.
Dagan4 used a first-order perturbation method with the conditional covariance of log-permeability (lnK) as input. He derived analytical forms of the conditional mean and variances of pressure. He analyzed the influence of conditional probability of the input variables on the statistical structure of the dependent variables. The main effect of conditioning is to reduce the variance (i.e., the uncertainty) of these variables. In Dagan's work, the effect is particularly important in the case of flow near wells. However, Dagan's work was restricted to the case of uniform mean flow in an unbounded domain, and the solutions were valid for small input variance.
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