Multiscale Mixed-Finite-Element Modeling of Coupled Wellbore/Near-Well Flow

An accurate well modeling capability is important for both production and reservoir engineering calculations. Ideally, the models used for these two applications should be related in a logical manner. Multiscale methods allow varying degrees of resolution and can therefore provide a natural linkage between production and reservoir models. In this work, we present a coupled wellbore-reservoir flow model that is based on a multiscale mixed finite element formulation for reservoir flow linked to a drift-flux wellbore flow representation. The model is able to capture efficiently the effects of near-well heterogeneity in the reservoir and phase holdup and pressure variation in the wellbore. The formulation presented here is for oil-water systems. The basic reservoir-wellbore linkage is described and validated through comparison to results from an existing simulator. The multiscale methodology is then applied to a heterogeneous reservoir model. Both vertical and deviated wells are considered. Comparisons of the multiscale solution to the fully resolved (fine-scale) solution demonstrate the high degree of accuracy of the method, for both reservoir and wellbore quantities, as well as its efficiency. Introduction The accurate modeling of near-well flow effects is essential for large-scale reservoir simulation and detailed production engineering calculations. Near-well models should ideally include wellbore and reservoir flow effects and should be suitable both as standalone well production applications and as modules in reservoir simulators. Multiscale finite element methods are in concept well-suited for this type of modeling, as they allow varying resolution and provide a systematic procedure for coarsening and refining. Thus they can provide the basis for a modeling framework that maintains consistency between reservoir and production engineering models. To our knowledge, however, multiscale methods have not yet been applied for this problem. In this work, we develop a multiscale mixed finite element method (MsMFEM) for modeling coupled wellbore and nearwell flow. The MsMFEM solves the pressure equation (in the reservoir domain) on a coarse grid, but captures fine-scale effects through basis functions determined from numerical solutions of local single-phase flow problems on the underlying fine-scale geological grid. We fully resolve the well trajectory on the fine scale using a flexible grid, which is close to radial around the well but logically Cartesian away from the well. The flow within the wellbore is represented using a drift-flux model. This model captures the slip between phases and provides the in-situ phase fractions (holdup), which provide a means for computing the pressure profile within the wellbore. Pressure variation within the wellbore is important in many settings as it impacts the local inflow from the reservoir into the well (in the case of a production well). Multiscale methods for reservoir simulation have been introduced as an alternative to standard upscaling as a tool for more fully incorporating fine-scale features at low computational cost. Among the relevant approaches are the multiscale finite element methods (e.g., Hou and Wu, 1997), the multiscale finite volume method (Jenny et al., 2003) and the multiscale mixed finite element method (Chen and Hou, 2003). The variational multiscale approach of Arbogast (2000), also formulated within a mixed finite element context, represents another related methodology. Here we consider a version of the multiscale mixed finite element method (MsMFEM) introduced by Aarnes (2004) and extended in later publications (Aarnes et al., 2006a, 2006b). Well representations have been incorporated into multiscale formulations by, for example, Chen and Yue (2003) and Wolfsteiner et al. (2006), though these formulations did not include wellbore flow effects. As far as we are aware, multiscale formulations have not been applied previously to detailed near-well models that include both reservoir and wellbore flow effects. Wellbore flow modeling within the context of reservoir simulation is often accomplished using drift-flux (or related) models. These models are well suited for use in reservoir simulation as they are simple, continuous and differentiable. The drift-flux representation (Zuber and Findlay, 1965) includes two basic parameters – the profile parameter C0 and the drift velocity Vd. These quantities depend on system parameters such as phase flow rates, well inclination and fluid properties. The general drift-flux model applied here is based on the formulation originally introduced by Holmes (1977) and Holmes et al. (1998). In recent work (Shi et al., 2005a), the model parameters for two-phase flow (gas-liquid and oilSPE 106179 Multiscale Mixed Finite Element Modeling of Coupled Wellbore/Near-Well Flow S. Krogstad, SINTEF ICT, and L. J. Durlofsky, SPE, Stanford University 2 S. Krogstad and L.J. Durlofsky SPE 106179 water) were optimized based on large-scale experiments involving a 15 cm diameter inclinable flow loop (see Oddie et al., 2003, for details of the experiments). In the current work, we apply the drift-flux model described in Shi et al. (2005a). We note that the model was subsequently extended to threephase wellbore flow (Shi et al., 2005b), though we do not consider flows of this type here. This paper proceeds as follows. After introducing the model equations, we briefly present three mixed finite element approximation techniques relevant for this paper: the standard Raviart-Thomas approach, mimetic finite differences, and the MsMFEM. The drift-flux model for computation of wellbore in-situ phase fractions and pressure drop is then presented, followed by a description of its coupling to the MsMFEM. Next, we validate the mimetic finite difference and MsMFEM implementations through comparison to an existing simulator. MsMFEM solutions are then compared to reference solutions for a heterogeneous model computed directly on the fine grid and close agreement is observed. We present results involving vertical and inclined wells, where slip between phases and pressure drop due to wellbore friction occur. Numerical Procedures Governing Equations. We consider model equations for twophase immiscible flow in porous media. Although included in the numerical experiments of this paper, we here neglect compressibility and gravity for ease of exposition. Letting Ω denote the domain of interest, the pressure equation reads: