Stochastic Formulation for Uncertainty Analysis of Two-Phase Flow in Heterogeneous Reservoirs

Abstract

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 postprocessing 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 displacements͒. We retain the normal Eulerian ͑space and time͒ 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 ͑mean͒ and second ͑vari-ance͒ 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.

A stochastic mixed finite element heterogeneous multiscale method for flow in porous media

“…, 4 and n @E denotes the unit outer normal of the coarse element boundary @E. According to the definition of RT 0 basis function in Eqs. (33) and (34), this boundary condition applies a uniform flow with magnitude w c i ðn k Þ from one side to the opposite side while keeping no-flow conditions on the other two sides. The example of w 1 (n 1 ) is shown in Fig.…”

A Lagrangian, stochastic modeling framework for multi-phase flow in porous media

“…(8), where phase particles move with the velocity given by Eq. (9). Here, for illustration purpose, it is assumed that the large scale (Darcy scale) fluxes F i are governed by Darcy's law and read…”

“…Extension of the particle-tracking approach to more complex geometry [7] and reactive flows in highly heterogeneous formations [8] appeared later. A hybrid Eulerian-Lagrangian method, where particle tracking is employed to represent the transport, was developed and used to model unstable first-contact miscible (two-component, single-phase flow) displacements in the presence of density and viscosity differences [9]. In these particle tracking methods, each particle represents a physical mass.…”

Numerical methods for multiscale transport equations and application to two-phase porous media flow

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