Streamline Tracing on Curvilinear Structured and Unstructured Grids

Summary

The streamline method has proven to be extremely successful for reservoir simulation.¹ For a Cartesian grid, the standard five-point (seven-point in 3D) discrete pressure equation is employed within the IMPES streamline method. The accuracy of the velocity field plays a critical role in the success of the technique, which relies on precise determination of streamline paths.

Modeling complex geology and geometry requires the use of sophisticated grids, and thus the ultimate power of the streamline method will only be realized when the method is extended to general curvilinear grids.

In this paper, a generalization of the Cartesian tracing algorithm is developed for general tensor fields on structured (corner-point) grids and unstructured grids. The extension of the method requires the identification of appropriate volumes and fluxes over which tracing equivalent to the Pollock Cartesian method² can be performed. Essential components of the method include the use of flux-continuous finite-volume schemes for the different grid types coupled with a novel flux recovery technique and a generalization of streamline tracing to polygons.

Introduction

The streamline method has become increasingly popular in recent years, because it allows a fast evaluation of reservoir performance, and because it has been extended to accommodate realistic flow physics (gravity, compressibility, and three-phase flow). The nature of the method permits a semianalytical 1D treatment along streamlines for fluid transport that minimizes dispersivity effects even when using numerically computed saturation profiles. Accurate resolution of the streamlines permits large timesteps to be taken when solving the pressure equation and results in at least an order of magnitude speed-up for displacement-type problems. Consequently, streamline simulators can routinely run problems that go from a few thousand blocks for compositional models up to a million blocks or more for simpler flow problems^1,3-5 (e.g., immiscible displacements). See Batycky et al.,¹ Thiele et al.,³ King and Datta-Gupta,⁴ and Lolomari et al.⁵ for details.

An effective method for streamline tracing, developed by Pollock, ² involves the use of a piece-wise linear approximation of velocity with respect to each gridblock followed by exact integration for position. This method provides a significant improvement when compared with the original Runge-Kutta streamline tracing technique used by Shafer.⁶ Tracing via the Pollock method is now used in all 3D streamline simulators. However, the method is restricted to Cartesian grid blocks.

Flow simulations of reservoirs with complex geometries and geological features generally require the use of flexible structured and unstructured grids in order to resolve important features such as faults and deviated wells. Corner-point geometry⁷ (CPG) (or nonorthogonal grids) are widely used in reservoir simulation. These grids are structured (logically Cartesian) and are comprised of quadrilateral cells. Unstructured grids are generally comprised of triangular and/or quadrilateral cells (in 2D) and are generally based on a more complex data structure than the standard logically Cartesian grid indices.⁸ Most applications of the streamline method with CPG grids have only been performed with standard diagonal tensor pressure equation approximations using transmissibility modifiers for geometry effects.^9,10 However, nonorthogonal CPG grids generally give rise to full-tensor coefficients, and a consistent approximation of the pressure equation must be used in order to avoid O(1) discretization errors.^11,12

The focus of this paper is on the development of the streamline method for curvilinear CPG grids and unstructured triangular grids. Essential components of the method include the use of consistent flux-continuous finite-volume schemes for curvilinear and unstructured grids, coupled with a novel flux recovery technique and a generalization of Pollock streamline tracing on appropriate volumes. The tracing algorithm is extended for application with a consistent cell-centered finite-volume approximation on curvilinear (corner-point) quadrilateral grids, the standard triangular control- volume finite-element method (CVFE) scheme,¹³ in which permeability is piecewise constant over each cell, and the pointdistributed triangular CVFE scheme,¹⁴ in which permeability is piecewise constant over each control volume. This is achieved in part by extending the flux-continuous recovery techniques of Cordes and Kinzelbach¹⁵ to subquadrilateral tracing.

The flow equations considered are summarized in the next section. A summary of the finite volume methods employed is given in the section by that name. Following that section, novel streamline tracing methods for structured and unstructured grids are presented. The results presented in the final section demonstrate the importance of accurate streamline tracing when using both nonorthogonal grids and unstructured grids. The effectiveness of the new streamline tracing methods is also demonstrated for problems involving full-tensor permeability fields.

Flow Equations

For simplicity, incompressible flow is assumed, and dispersivity, compressibility, capillary, and gravity effects are neglected. However, the methods presented here can still be applied when more complex physics are incorporated in the simulation. Furthermore, all the examples presented will be 2D, although extensions of the methods to three dimensions will be discussed. The conservation equations are written as

• Equation 1

where the total velocity is defined via Darcy's law

• Equation 2

In this paper the permeability tensor K(x,y) can be full with nonzero off-diagonal coefficients because of the frame of reference of the local coordinate system being nonaligned with the principal axes of the tensor and/or upscaling with cross-flow effects included. Curvilinear and unstructured grids can provide additional full-tensor effects.¹⁶ For incompressible flow, away from sources, the velocity field satisfies

• Equation 3