Summary We present a new streamline-based simulator applicable to field scale flow. The method is three dimensional (3D) and accounts for changing well conditions that result from infill drilling and well conversions, heterogeneity, mobility effects, and gravity effects. The key feature of the simulator is that fluid transport occurs on a streamline grid rather than between the discrete gridblocks on which the pressure field is solved. The streamline grid dynamically changes as the mobility field and boundary conditions dictate. A general numerical solver moves the fluids forward in space and time along each streamline. Multiphase gravity effects are accounted for by an operator-splitting technique that also requires a numerical solver. Because fluid transport is decoupled from the underlying grid, the method is computationally efficient and very large timesteps can be taken without loss in solution accuracy. We present results of the streamline-based simulator applied to tracer, waterflooding, and first-contact miscible (FCM) displacements in two and three dimensions. Where possible, comparisons with conventional methods indicate that the streamline model minimizes numerical diffusion and is up to two orders of magnitude faster. We also demonstrate the efficiency of the method on a field-scale, million-gridblock, 36-well waterflood that includes a pattern-modification plan to improve oil recovery. Last, we present results of the method applied to the House Mountain waterflood in Canada. Introduction The use of streamlines and streamtubes to model convective displacements in heterogeneous media has been presented many times since the early works of Muskat,1–3 Fay and Prats,4 and Higgins and Leighton.5–7 Important contributions to the field were also made by Parsons,8 Martin and Wegner,9 Bommer and Schechter,10 Lake et al.,11 Emanuel et al.,12 and Hewett and Behrens.13,14 Streamline methods have recently resurfaced as a viable alternative to traditional finite-difference methods for large, heterogeneous, multiwell, multiphase simulations.15–27 The efficiency of the method has made it an ideal tool for ranking equiprobable reservoir images28; rapid assessment of production strategies, such as infill drilling and gas injection29; computing upscaled component flux properties for compositional simulation30; and integration with production data for reservoir characterization31. The method has also allowed for solution of fine-scale models [on the order of 106 gridblocks] on standard computer resources, thus reducing the need for significant upscaling. In this paper, we present advances on our previous work where we mapped analytical solutions along streamlines.19,22 Although the streamline paths were updated periodically to account for changing mobility fields, the method could not account for changing well conditions or gravity - two key phenomena that must be modeled in general field-scale simulations. We account for these mechanisms by mapping one-dimensional (1D) numerical solutions along streamlines, as first proposed by Bommer and Schechter.10 In doing so, nonuniform initial conditions that appear along recalculated streamline paths, resulting from changing well and mobility conditions, can be moved forward in space and time correctly. Streamline paths are updated, and the transport process repeated. The grid on which the pressure field is solved is effectively decoupled from the streamline grid used to transport fluids. There is no longer a global grid Courant-Friedrichs-Lewy (CFL) condition to restrict timestep size. Furthermore, grid-orientation and numerical-diffusion effects are minimized. Finally, operator splitting is used to account for gravity in multiphase flow.32,33 After moving fluids convectively along streamlines, fluids are then moved vertically along 1D gravity lines. Bratvedt et al.24 presented a similar operator-splitting technique in the context of their front-tracking method. Our application of streamlines to field-scale reservoir simulation is a combination of four existing ideas:3D streamlines,34updating the streamline paths to account for changing mobility field and well conditions,9,15,19numerical solutions along streamlines,10 andincluding gravity effects in multiphase flow by use of operator splitting.23,24,32,33 Using streamlines and gravity lines decouples the 3D transport problem into multiple 1D problems and leads to a very fast and accurate method applicable to a wide range of field conditions. Streamline Method In this section we outline the streamline method. The Appendix gives a detailed discussion on how to trace the streamlines. Governing Implicit-Pressure/Explicit-Saturation (IMPES) Equations. The streamline method is an IMPES method. Ignoring capillary and dispersion effects, the governing equation for pressure p, for incompressible, multiphase flow is given by where D=a depth below datum. Total mobility, ?t, and total gravity mobility, ?g, are defined as where krj=relative permeability of Phase j, µj=phase viscosity, ?j=phase density, g =gravity acceleration constant, and np=number of phases present. We also require a material-balance equation for each Phase j35: The total velocity, ut, is derived from the 3D solution to the pressure field (Eq. 1) and application of Darcy's law. The phase fractional flow is given by and the phase velocity resulting from gravity effects is given by Eqs. 1 and 3 form the IMPES set of equations in the formulation of the streamline simulator. We confine our discussion to the solution of these equations for two-phase flow.
We present a fast technique for modeling convective displacements that are dominated by large-scale reservoir heterogeneities. The novel feature of our method is the integration of any 10 solution with periodically updated streamtubes or streamlines to simulate displacements in two and three dimensions. We use streamtubes in two dimensions and streamlines in three dimensions. We construct approximate solutions using two to five orders of magnitude less computation time than by conventional simulation. Our approach allows us to decouple the physics describing the displacement from the size of the grid used to model the reservoir geology. In addition, because I D solutions are often analytical, or can be obtained numerically using higher-order methods, the resulting 20 and 3D solutions are free of numerical diffusion.We present examples in two and three dimensions for a variety of displacement mechanisms and compare our results with conventional finite-difference solutions showing excellent agreement. Because of its speed, our method becomes particularly effective for estimating the uncertainty in forecasting reservoir performance resulting from the uncertainty in the description of the reservoir geology.
Summary This paper describes a novel approach to predict injection- and production-well rate targets for improved management of waterfloods. The methodology centers on the unique ability of streamlines to define dynamic well allocation factors (WAFs) between injection and production wells. Streamlines allow well allocation factors to be broken down additionally into phase rates at either end of each injector/producer pair. Armed with these unique data, it is possible to define the injection efficiency (IE) for each injector and for injector/producer pairs in a simulation model. The IE quantifies how much oil can be recovered at a producing well for every unit of water injected by an offset injector connected to it. Because WAFs are derived directly from streamlines, the data reflect all the complexities impacting the dynamic behavior of the reservoir model, including the spatial permeability and porosity distributions, fault locations, the underlying computational grid, relative permeability data, pressure/volume/temperature (PVT) properties, and most importantly, historical well rates. The possibility to define IEs through streamline simulation stands in contrast to the ad hoc definition of geometric WAFs and simple surveillance methods used by many practicing reservoir engineers today. Once IEs are known, improved waterflood management can be implemented by reallocating injection water from low-efficiency to high-efficiency injectors. Even in the case in which water cannot be reallocated because of local surface-facility constraints, knowing IEs on an injector/producer pair allows the setting of target rates to maintain oil production while reducing water production. We demonstrate this methodology by first introducing the concept of IEs, then use a small reservoir as an example application. Introduction Local areas of water cycling and poor sweep exist as a flood matures. Current flood management is restricted to surveillance methods or workflows centered on finite-difference (FD) simulation, where areas of bypassed oil are identified and then rate changes, producer/injector conversions, or infill-drilling scenarios are tested. However, identifying and testing improved management scenarios in this way can be laborious, particularly for waterfloods with a large number of wells and/or a relatively high-resolution numerical grid. For mature fields that have potential for improved production without introducing new wells or producer/injector conversions, the main goal is to manage well rates so as to reduce cycling of the injected fluid while maintaining or even increasing oil production. Reservoir engineers have no easy or automated way to identify injection patterns, well-pair connections, or areas of inefficiency beyond simple standard fixed-pattern surveillance techniques (Baker 1997; Baker 1998; Batycky et al. 2005). Such methods are approximate at best owing to the need to define geometric allocation factors and fixed patterns, which suffer from "out-of-pattern" flow. These limitations are removed through streamline-based surveillance models (Batycky et al. 2005). By adding a transport step along streamlines, streamline simulation (3DSL 2006) can additionally identify how much oil production results from an associated injector, quantifying the efficiency down to an individual injector/producer pair. It is this crucial piece of information—the efficiency of an injector/producer pair—that allows an improved estimation of future target rates, leading to improved reservoir flood management.
This paper presents the extension of the streamline approach to full-field, three-dimensional (3D) compositional simulation. The streamline technique decomposes a heterogeneous 3D domain into a number of one-dimensional (1D) streamlines along which all fluid flow calculations are done. Streamlines represent a natural, dynamically changing grid for modeling fluid flow. We use a 1D compositional finite-difference simulator to move components numerically along streamlines, and then map the 1D solutions back onto an underlying Cartesian grid to obtain a full 3D compositional solution at a new time level. Because of the natural decomposition of the 3D domain into a number of 1D problems, the streamline approach offers substantial computational efficiency and minimizes numerical diffusion compared to traditional finite-difference methods. We compare our three and four component solutions with solutions from two finite difference codes, UTCOMP and Eclipse 300 (E300). These examples show that our streamline solutions are in agreement with the finite-difference solutions, are able to minimize the impact of numerical diffusion, are faster by orders of magnitude. Numerical diffusion in finite-difference formulations can interact with reservoir heterogeneity to substantially mitigate mobility differences and lead to optimistic recovery predictions. We demonstrate the efficiency and usefulness of the streamline-based simulator on a 518,400 gridblock, 3D, heterogeneous, 36-well problem for a condensing-vaporizing gas drive with four components. We can simulate this problem on an average-size workstation in three CPU days. It takes approximately the same amount of time to simulate the upscaled 28,800 gridblock version of the problem using finite-differences. We conclude with a qualitative discussion explaining the near-linear scaling of the streamline approach with the number of gridblocks and the cubic and higher scaling exhibited by one of the finite-difference codes. Introduction The use of streamlines and streamtubes to model convective displacements in heterogeneous media has been presented repeatedly since the early work by Muskat, Fay and Prats, and Higgins and Leighton. Important subsequent contributions are due to Parsons, Martin and Wegner, Bommer and Schechter, Lake et al., Mathews et al., Emanuel et al., Renard, and Hewett and Behrens. Recently, streamline methods have received renewed attention by several groups as a viable alternative to traditional finite-difference (FD) methods for large, heterogeneous, multiwell, multiphase simulations, which are particularly difficult for FD simulators to model adequately. Large speed-up factors compared to traditional FD solutions, minimization of numerical diffusion and grid orientation effects, and the inherent simplicity of the approach offer unique opportunities for integration with modern reservoir characterization methods. Examples include ranking of equiprobable earth models, estimation of the uncertainty in production forecasts due to the uncertainty in the geological description, rapid assessment of production strategies such as infill drilling patterns and miscible gas injection. In addition, streamlines may offer an attractive alternative to well-known problems with upscaling of absolute and pseudorelative permeabilities by allowing larger geological models and requiring upscaling across a smaller range of scales. Our streamline approach for reservoir simulation hinges on two important extensions to past streamline/streamtube methods:the use of true 3D streamlines andand numerical solutions of the transport equations along periodically changing streamlines. With these extensions we have been able to simulate realistic fluid flow in detailed, heterogeneous, 3D reservoir models much more efficiently than FD methods. We emphasize that reservoir simulation using streamlines is not a minor modification of current FD approaches, but instead represents a significant shift in methodology. P. 471^
We present a generalized streamline method to model flow in porous media, including the effects of gravity and dispersion. We first describe the theory and discuss the approximations of the method, and then compare the predictions using the streamline technique against two-dimensional numerical simulations of incompressible miscible flow. In the cases we studied, the streamline method predicts recovery with an average error of at most 5%, where the principal flow directions are governed by the pattern of permeability, and for gravity numbers less than or equal to 1 and mobility ratios of 10 or less. The streamline method does not suffer from numerical dispersion and is more than 100 times faster than conventional simulation.
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