In this paper we consider porous media flow without capillary effects. We present a streamline method which includes gravity effects by operator splitting. The flow equations are treated by an IMPES method, where the pressure equation is solved by a (standard) finite element method. The saturation equation is solved by utilizing a front tracking method along streamlines of the pressure field. The effects of gravity are accounted for in a separate correction step. This is the first time streamlines are combined with gravity for three-dimensional (3D) simulations, and the method proves favourable compared to standard splitting methods based on fractional steps. By our splitting we can take advantage of very accurate and efficient 1D methods. The ideas have been implemented and tested in a full field simulator. In that context, both accuracy and CPU efficiency have tested favourably.

A new 3-D three phase compositional reservoir simulator based on extension of the streamline method has been developed. This paper will focus on the new methods developed for compositional streamline simulation as well as the advantages and disadvantages of this strategy compared with more traditional approaches. Comparisons with a commercially available finite difference simulator will both validate the method and illustrate the cases in which this method is useful to the reservoir engineer. Introduction Streamline methods have been used as a tool for numerical approximation of the mathematical model for fluid flow since the 1800's (Helmholtz28 and later Muskat45) and have been applied in reservoir engineering since the 1950's and 1960's19,29–31,21,57. The reason behind using the approach has been both the needs for solving the governing equations accurately and achieving reasonable computational efficiency. Streamline methods continued to be explored through the 1970's by LeBlanc and Caudle37, Martin et al40,41 and Pitts et al50 and 1980's by Lake et al36,70, Cox11, Bratvedt et al8 and Wingard et al71, but the focus of reservoir simulation was on developing finite difference simulators. In the 1990's streamline methods have emerged as an alternative to finite difference simulation for large, heterogeneous models that are difficult for traditional simulators to model adequately. These efforts are described in numerous papers notably by Renard56, Batycky et al1–4, Peddibhotla et al48,49, Thiele et al64–66, Ingebrigtsen33, Ponting52 and in an overview paper by King and Datta-Gupta35. The application of the method has been described by numerous other authors10,12–17,26,34,53–55,69. Use of the streamline simulator used for the work in this paper has also been extensively described24,39,42,47,58,61,62,68. Several similar approaches such as the method of characteristics18,27,38,43,44, particle tracking20,63 and front-tracking25,7 have also been used in reservoir simulation. Conventional finite difference methods suffer from two drawbacks, numerical smearing and loss of computational efficiency for models with a large numbers of grid cells. Large models (105 –106 cells) are routinely generated in order to accurately represent geologically heterogeneous, multi-well problems.. Finite difference methods based on an IMPES approach suffer from the time step length limiting CFL condition, so as the number of cells increases the maximum time step length get shorter for a given model. For a large number of cells the shortness of the time step can render the total CPU time for a simulation impractical. Fully implicit finite difference simulators can take longer time steps but require the inversion of a much larger matrix than the IMPES approach. This is an even larger issue with compositional simulation where a large number of components will make the matrix very large. Also the non-linearity of the governing equations might require a limitation on the time step length again making very large models impractical to run. This can be improved by using an adaptive implicit aproach73. Conventional streamline methods are based on an IMPES method. In these methods the pressure is solved implicitly and then streamlines are computed based on this pressure solution. In this way the 3D domain is decomposed into many one-dimensional streamlines along which fluid flow calculations are done. This method assumes that the pressure is constant throughout the movement of fluids. One weakness with this concept is the lack of connection between the changing pressure field and the movement of fluids. This can cause instabilities and limitations on the time step length.

This paper presents a new numerical method for solving saturation equations without stability problems and without smearing saturation fronts. A reservoir simulator based on this numerical method is under development. A set of test problems is used to compare the simulation results of the new simulator with those of an existing flnite-difference simulator (FDS).

Advances in reservoir characterization and modeling have given the industry improved ability to build detailed geological models of petroleum reservoirs. These models are characterized by complex shapes and structures with discontinuous material properties that span many orders of magnitude. Models that represent fractures explicitly as volumetric objects pose a particular challenge to standard simulation technology with regard to accuracy and computational efficiency. We present a new simulation approach based on streamlines in combination with a new multiscale mimetic pressure solver with improved capabilities for complex fractured reservoirs. The multiscale solver approximates the flux as a linear combination of numerically computed basis functions defined over a coarsened simulation grid consisting of collections of cells from the geological model. Here, we use a mimetic multipoint flux approximation to compute the multiscale basis functions. This method has limited sensitivity to grid distortions. The multiscale technology is very robust with respect to fine-scale models containing geological objects such as fractures and fracture corridors. The methodology is very flexible in the choice of the coarse grids introduced to reduce the computational cost of each pressure solve. This can have a large impact on iterative modeling workflows.

Two methods for calculation of three-phase compressible flow in a porous media using streamlines are presented. For simplicity, gravity and capillary effects are neglected. Introduction Various aspects of streamline computations have been reported in a number of recent papers. A review of the technique was given by King et.al.[7]. As many other authors [5][9][13], he emphasis on incompressible two-phase flow. Lately, however there have been a few contributions to the field addressing compressible flow [3][10]. In this paper, we consider three-phase compressible flow. The reason why compressibility has been neglected by most authors in the field, is that it represent a strong coupling between the pressure and the saturation equations. In a streamline method, the pressure is calculated first and defines the streamlines. The saturations are then propagated along those streamlines. To obtain a stable solution using an explicit finite difference method (FDM), there is a strong limitation on the time step length. Thus, it is crucial for the efficiency of the streamline method to establish a time stepping sequence where only a small number of pressure updates is needed compared to the number of time steps required by the saturation solver. While this has been done successfully for years for incompressible flow [4], it is still a challenge to obtain both accuracy and efficiency for compressible flow. We present two different approaches to handle the couplings between pressure and saturation. First, we describe a sequential IMPES type method, where we do additional steps for reducing the mass discrepancy error. Then an implicit method for both pressure and saturation along the streamlines is presented. The results are compared to the solutions obtained using an existing black-oil simulator. The Governing Equations Consider three-phase compressible flow in a porous media. For simplicity, the effects of gravity are neglected. Gravity can be accounted for using operator splitting [5] and introducing solvers for three-phase flow along gravity lines. This work is in progress, but will not be the issue of this paper. Also, capillary forces are neglected, since physical effects transverse to streamlines complicates the streamline approach. We will study a black-oil model, with three phases and three components. We will allow gas to dissolve in the oil phase while we assume the water and gas phase to consist of only water and gas respectively. The component conservation equations now read [11] ∂∂t(ϕSwbw)​∇⋅(bwfwv→t)=qw∂∂t(ϕSobo)​∇⋅(bofov→t)=qo∂∂t(ϕ(Sgbg​RsSobo))​∇⋅(bgfgv→t​Rsbofov→t)=qg(1) for water, oil and gas respectively. By summing up the component conservation equations, we obtain the pressure equation ct∂P∂t​∇⋅v→t=Q−b→⋅∇P(2) where v→t=λt∇P(3) is the Darcy velocity. Refer to the nomenclature for an explanation on notation.