Summary This paper describes a hybrid finite volume method, designed to simulate multiphase flow in a field-scale naturally fractured reservoir. Lee et al. (WRR 37:443-455, 2001) developed a hierarchical approach in which the permeability contribution from short fractures is derived in an analytical expression that from medium fractures is numerically solved using a boundary element method. The long fractures are modeled explicitly as major fluid conduits. Reservoirs with well-developed natural fractures include many complex fracture networks that cannot be easily modeled by simple long fracture formulation and/or homogenized single continuity model. We thus propose a numerically efficient hybrid method in which small and medium fractures are modeled by effective permeability, and large fractures are modeled by discrete fracture networks. A simple, systematic way is devised to calculate transport parameters between fracture networks and discretized, homogenized media. An efficient numerical algorithm is also devised to solve the dual system of fracture network and finite volume grid. Black oil formulation is implemented in the simulator to demonstrate practical applications of this hybrid finite volume method. Introduction Many reservoirs are highly fractured due to the complex tectonic movement and sedimentation process the formation has experienced. The permeability of a fracture is usually much larger than that of the rock matrix; as a result, the fluid will flow mostly through the fracture network, if the fractures are connected. This implies that the fracture connectivities and their distribution will determine fluid transport in a naturally fractured reservoir (Long and Witherspoon 1985). Because of statistically complex distribution of geological heterogeneity and multiple length and time scales in natural porous media, three approaches (Smith and Schwartz 1993) are commonly used in describing fluid flow and solute transport in naturally fractured formations:discrete fracture models;continuum models using effective properties for discrete grids; andhybrid models that combine discrete, large features and equivalent continuum. Currently, most reservoir simulators use dual continuum formulations (i.e., dual porosity/permeability) for naturally fractured reservoirs in which matrix blocks are divided by very regular fracture patterns (Kazemi et al. 1976, Van Golf-Racht 1982). Part of the primary input into these simulation models is the permeability of the fracture system assigned to the individual grid-blocks. This value can only be reasonably calculated if the fracture systems are regular and well connected. Field characterization studies have shown, however, that fracture systems are very irregular, often disconnected, and occur in swarms (Laubach 1991, Lorenz and Hill 1991, Narr et al. 2003). Most naturally fractured reservoirs include fractures of multiple- length scales. The effective grid-block permeability calculated by the boundary element method becomes expensive as the number of fractures increases. The calculated effective properties for grid-blocks also underestimates the properties for long fractures whose length scale is much larger than the grid-block size. Lee et al. (2001) proposed a hierarchical method to model fluid flow in a reservoir with multiple-length scaled fractures. They assumed that short fractures are randomly distributed and contribute to increasing the effective matrix permeability. An asymptotic solution representing the permeability contribution from short fractures was derived. With the short fracture contribution to permeability, the effective matrix permeability can be expressed in a general tensor form. Thus, they also developed a boundary element method for Darcy's equation with tensor permeability. For medium-length fractures in a grid-block, a coupled system of Poisson equations with tensor permeability was solved numerically using a boundary element method. The grid-block effective permeabilities were used with a finite difference simulator to compute flow through the fracture system. The simulator was enhanced to use a control-volume finite difference formulation (Lee et al. 1998, 2002) for general tensor permeability input (i.e., 9-point stencil for 2-D and 27-point stencil for 3-D). In addition, long fractures were explicitly modeled by using the transport index between fracture and matrix in a gridblock. In this paper we adopt their transport index concept and extend the hierarchical method:to include networks of long fractures;to model fracture as a two-dimensional plane; andto allow fractures to intersect with well bore. This generalization allows us to model a more realistic and complex fracture network that can be found in naturally fractured reservoirs. To demonstrate this new method for practical reservoir applications, we furthermore implement a black oil formulation in the simulator. We explicitly model long fractures as a two-dimensional plane that can penetrate several layers. The method, presented in this paper, allows a general description of fracture orientation in space. For simplicity of computational implementation however, both the medium-length and long fractures considered in this paper are assumed to be perpendicular to bedding boundaries. In addition, we derive a source/sink term to model the flux between matrix and long fracture networks. This source/sink allows for coupling multiphase flow equations in long fractures and matrix. The paper is organized as follows. In Section 2 black oil formulation is briefly summarized and the transport equations for three phase flow are presented. The fracture characterization and hierarchical modeling approach, based on fracture length, are discussed in Section 3. In Section 4 we review homogenization of short and medium fractures, which is part of our hierarchical approach to modeling flow in porous media with multiple length-scale fractures. In Section 5 we discuss a discrete network model of long fractures. We also derive transfer indices between fracture and effective matrix blocks. In Section 6 we present numerical examples for tracer transport in a model with simple fracture network, interactions of fractures and wells, and black oil production in a reservoir with a complex fracture network system. Finally, the summary of our main results and conclusion follows.

Molecular dynamics simulations of ether-derivatized imidazolium-based room-temperature ionic liquids (EDI-RTILs), [C(5)O(2)mim][TFSI] and [C(5)O(2)mim][BF(4)], have been performed and compared with simulations of alkyl-derivatized analogues (ADI-RTILs). Simulations yield RTIL densities, self-diffusion coefficients and viscosity in excellent agreement with experimental data. Simulations reveal that structure in the EDI-RTILs, quantified by the extent of nanoscale segregation of tails as well as cation-ion and cation-cation correlations, is reduced compared to that observed in the ADI-RTILs. Significant correlation between ether tail oxygen atoms and imidazolium ring hydrogen atoms was observed in the EDI-RTILs. This correlation is primarily intramolecular in origin but has a significant intermolecular component. Competition of ether oxygen atoms with oxygen atoms of TFSI(-) or fluorine atoms of BF(4)(-) for coordination of the ring hydrogen atoms was found to reduce the extent of cation-anion correlation in the EDI-RTILs compared to the ADI-RTILs. The reduction in intermolecular correlation, particularly tail-tail segregation, as well as weakening of cation-anion specific interactions due to the ether tail, may account for the faster dynamics observed in the EDI-RTILs compared to ADI-RTILs.

No abstract

The use of a probabilistic framework for dynamic data integration (history matching) has become accepted practice. In this framework, one constructs an ensemble of reservoir models, in which each realization honors the available (static and dynamic) information. The variations in the flow performance across the ensemble provide an assessment of the prediction uncertainty due to incomplete knowledge of the reservoir properties (e.g., permeability distribution). Methods based on Monte Carlo simulation (MCS) are widely used because of the generality and simplicity of MCS. As a black-box approach, only pre-and post-processing of conventional flow simulations are needed. To achieve reasonable accuracy in estimating the statistical moments of flow-performance predictions, however, large numbers of realizations are usually necessary. Here, we use a different, and direct, approach for model calibration and uncertainty quantification. Specifically, we describe a statistical-moment-equations (SMEs) framework for both the forward and inverse problems associated with immiscible two-phase flow. In the SME method, the equations governing the statistical moments of the quantities of interest (e.g., pressure and saturation) are derived and solved directly. We assume that statistical information and a few measurements are available for the permeability field. As for the dynamic properties, we assume that measurements of pressure, saturation, and flow rate are available at a few locations and at several times. For the forward problem, the flow (pressure and total-velocity) SMEs are solved on a regular grid, while a streamline-based strategy is used to solve the transport SMEs. We use a kriging-based inversion algorithm, in which the first two statistical moments of permeability are conditioned directly using the available dynamic data. We analyze the behaviors of the saturation moments and their evolution as they are conditioned on measurements, in both space and time. Moreover, we discuss the relationship between the widely used MCS-based Kalman-filter approach and our SME inversion scheme.

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.