Marco R. Thiele

Simulating flow in heterogeneous systems using streamtubes and streamlines

The authors present a fast technique for modeling convective displacements that are dominated by large-scale reservoir heterogeneities. The novel feature of their method is the integration of any 1D solution with periodically updated streamtubes or streamlines to simulate displacements in two and three dimensions. They use streamtubes in two dimensions and streamlines in three dimensions. They construct approximate solutions using two to five orders of magnitude less computation time than by conventional simulation. Their approach allows them to decouple the physics describing the displacement from the size of the grid used to model analytical, or can be obtained numerically using higher-order methods, the resulting 2D and 3D solutions are free of numerical diffusion. The authors present examples in two and three dimensions for a variety of displacement mechanisms and compare their results with conventional finite-difference solutions showing excellent agreement. Because of its speed, their method becomes particularly effective for estimating the uncertainty in forecasting reservoir performance resulting from the uncertainty in the description of the reservoir geology.

A Streamline-Based 3D Field-Scale Compositional Reservoir Simulator

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 finitedifferences. 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, 43–45 Fay and Prats, 25 and Higgins and Leighton. 31–33 Important subsequent contributions are due to Parsons, 50 Martin and Wegner, 39 Bommer and Schechter, 9 Lakeet al.,37 Mathewset al.,41 Emanuelet al.,20–22 Renard, 57 and Hewett and Behrens. 30 Recently, streamline methods have received renewed attention by several groups as a viable alternative to traditional finitedifference (FD) methods for large, heterogeneous, multiwell, multiphase simulations, which are particularly difficult for FD simulators to model adequately. 3–6, 8, 10, 11, 51, 52, 59, 64–66Large 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. 3, 19, 22, 51, 67In 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. 2, 15 Our streamline approach for reservoir simulation hinges on two important extensions to past streamline/streamtube methods: (1) the use of true 3D streamlines 55 and (2) and numerical solutions of the transport equations along periodically changing streamlines. 9, 57 With these extensions we have been able to simulate realistic fluid flow in detailed, heterogeneous, 3D reservoir models much more efficiently than FD methods.3, 4, 6, 64 We emphasize that reservoir simulation using streamlines is not a minor modification of current FD approaches, but instead represents a significant shift in methodology. By transporting fluids along periodically changing streamlines, the streamline 2 3D Field Scale Compositional Simulation Using Streamlines SPE 38889 approach is equivalent to a dynamically adapting grid that is decoupled from the underlying, static, grid used to describe the reservoir geology. The 1D nature of a streamline is translated into a 1D transport problem that can be solved easily and efficiently. This is considerably different from FD methods that use the same grid to solve both for pressure and for saturation/composition, and are forced to move fluids only along grid directions. The implementation of our approach uses five key ideas: Heterogeneity and Well Locations: Streamlines represent the natural grid to capture high and low flow regions due to well placements and reservoir heterogeneity.20, 30, 39, 45, 50 Muskat first used streamline to estimate reservoir drainage volumes resulting from well placements. The use of streamlines to capture the impact of heterogeneity is more recent and is due to the work by Lakeet al37 and by Emanuel and co-workers. 20–22, 30, 41 3D Streamlines: Tracing 3D streamlines using a time-offlight (TOF) approach as presented by Pollock 55 and Datta-Gupta and King. 17 Using a TOF formulation along streamlines is significantly easier than using a volumetric formulation along streamtubes, particularly for true 3D flow. It is straightforward to show that a volumetric coordinate along a streamtube is equivalent to a TOF coordinate along a streamline. 64 1D Numerical Solutions: Moving solutions forward in time numerically along 1D streamlines as proposed by Bommer and Schechter. 9 By using a numerical FD solution along streamlines extends the approach to general initial conditions, changing boundary conditions, and any type of displacement or recovery mechanism which can be formulated in one dimension. Updating Streamlines: Capturing the changing total velocity field due to problem nonlinearities as well as time-varying boundary conditions (wells coming online as well as shutting-in) by periodically recalculating the streamlines. 3–6, 40, 57, 64–66 Because streamlines represent the natural flow grid along which fluids want to move, far fewer streamline updates (global pressure solves) are required to move the fluids forward in time compared to FD methods. This results in a significant computational efficiency. Combining numerical 1D solutions (point 3 above) with periodically updated 3D streamlines is the centerpiece of our approach. It allows to capture nonlinear flow mechanism in 3D while retaining the speed advantages that made approximate 2D streamtube methods attractive in the past. Operator-Splitting: Using operator-splitting to capture flow mechanisms that are not aligned with the total velocity field, such as gravity. 4, 11, 28 The inability of 2D streamtube methods in the past to account for gravity effects has been a long-standing criticism. By using operator splitting and 3D streamlines it is possible to account for any mechanism that is not aligned with the total velocity field, such as transverse diffusion, capillary crossflow, and gravity. Operator splitting is a well established mathematical concept and has been used in the past to solve convective-diffusive transport problems, 24 transport problems on multiple grids, 56 and transport problems with gravity. 4, 11, 29 Compositional Displacements In this paper we consider the extension of the streamline approach to compositional displacements. We have described the application of the method to immiscible and miscible displacements in our previous work, 3–5, 8, 64, 65, 67and the reader is referred to those publications for additional details. Modeling mass transfer effects correctly becomes imperative when designing injection processes that will enhance the recovery of oil compared to traditional waterflooding methods. For example, under the right physical conditions gas or solvent injection can lead to very efficient recovery processes, as capillary forces that cause entrapment in immiscible displacements are reduced leading to much lower residual oil saturations. If the injected solvent is completely miscible, i.e. only a single phase is formed over the range of pressures and temperatures present in the reservoir, then the displacement is said to be first-contact-miscible (FCM) and the residual oil saturation in a swept zone is theoretically zero. In many cases however, miscibility is achieved in-situ after the solvent has contacted the oil repeatedly and mass transfer occurs between the injected solvent and the oil in place. These cases are referred to as multicontact miscibility (MCM). If miscibility is never achieved for the range of pressures and compositions in a particular displacement, then the displacement is referred to as being immiscible, although mass transfer still occurs between the flowing phases. For a general overview on the subject the reader is referred to Stalkup’s monograph. 61 Although the residual oil saturations that can be achieved by gas injection processes can be very low, there is an offsetting effect caused by the injected fluids usually being significantly more mobile and of lower density than the resident oil. Higher mobility can lead to channeling and viscous fingering. 63 A significant density contrast can lead to gravity override. These effects have been observed experimentally at the core scale by several investigators 12, 13, 26, 60and have also been confirmed numerically. Experimental work has also underscored the importance of crossflow mechanism due to capillary forces and concentration gradients, but is remains unclear to what extent these mechanisms act at the scale of typical field-scale reservoir simulation gridblock and how they affect the flow between gridblocks at that scale. It is generally agreed though, that successful mode

A generalized streamline method to predict reservoir flow

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.

Reservoir monitoring; a multidisciplinary feasibility study

There is a recognized need to combine the skills of geoscientists and engineers to build quantitative reservoir models that incorporate all available reservoir data. These integrated models are critical for forecasting, monitoring, and optimizing reservoir performance because they will enable more accurate flow simulation studies, identification of permeability flow paths and barriers, mapping of bypassed oil, and monitoring of pressure and saturation fronts.

Quantifying uncertainty in reservoir performance using streamtubes

We present the use of a streamtube approach to study the uncertainty in reservoir performance resulting from a stochastic description of the flow domain. The streamtube technique is an efficient numerical method which is particularly effective for modeling convective displacements that are dominated by large-scale heterogeneities. Stable, numerical-diffusion-free solutions can be obtained in a fraction of the time taken by conventional finite difference simulators, thereby allowing a statistical approach to reservoir simulation and forecasting.

Predicting Multicomponent, Multiphase Flow in Heterogeneous Systems using Streamtubes

We present a fast and accurate numerical technique for modeling convective displacements which are dominated by large-scale heterogeneities. Streamtubes map the direction of flow at any time during the displacement and are recalculated as the fluid mobility distribution changes. Two-dimensional solutions are then constructed by mapping one-dimensional solutions along each streamtube. Large timesteps may be taken, producing stable, diffusion free solutions in a fraction of the time taken by conventional finite difference/finite element simulators.

How Much is an Oil Price Forecast Worth in Reservoir Management

Reservoir management has lately received significant attention in the oil industry because of the sizable potential increment in profit associated with optimized strategies, together with the increased reliability of new reservoir flow simulation techniques. Long-term (more than five years) management is especially challenging partially because of the difficulties brought by the uncertainty associated with forecasting oil production and price. In this work, we first propose a risk measure of a given production strategy with respect to uncertainty in the oil price. This measure is interpreted as the value of information associated with an uncertain oil price. Decision makers can leverage value of information of oil price, for example, to qualify potential risks associated with the lack of knowledge of the market, to compare an exploitation strategy with the optimal strategy in the utopian situation where future oil price is known, and to evaluate possible capital investments for a better forecast of oil price. Then, we present a numerical approach using reservoir flow simulation to estimate efficiently this risk measure. The computational cost of this numerical method does not increase with the number of possible oil price scenarios considered, and this is a desirable feature when simulation is time-demanding. To the best of our knowledge, value of information of oil price has not been addressed using complex reservoir flow simulation. The approach is validated on a synthetic case and two real field waterflooding scenarios: a relatively small field with eight wells, and a larger field with 174 wells. In both cases, we analyze 10,000 oil price scenarios, quantify the risk of waterflooding production strategies obtained through optimization, and estimate the monetary value associated of oil price forecasts with different degrees of market information and knowledge.

Reservoir Management Using Two-stage Optimization with Streamline Simulation

Waterflooding is a common secondary oil recovery process. Performance of waterfloods in mature fields with a significant number of wells can be improved with minimal infrastructure investment by optimizing injection/production rates of individual wells. However, a major bottleneck in the optimization framework is the large number of reservoir flow simulations often required. In this work we propose a new method based on streamline-derived information that significantly reduces these computational costs in addition to making use of the computational efficiency of streamline simulation itself. We seek to maximize the long-term net present value of a waterflood by determining optimal individual well rates, given an expected albeit uncertain oil price and a total fluid injection volume. We approach the optimization problem by decomposing it into two stages which can be implemented in a computationally efficient manner. The two-stage streamline-based optimization approach can be an effective technique when applied to reservoirs with a large number of wells in need of an efficient waterflooding strategy over a 5 to 15 year period.

A Generalized Streamline Method to Predict Reservoir Flow

We present a generalized streamline method that is able to model flow in porous media-including gravity, compositional effects and dispersion in three dimensions. First we describe the theory and discuss the approximations of the method. Then we compare the predictions using the streamline technique against high resolution numerical simulation.