Timestep Selection During Streamline Simulation Through Transverse Flux Correction
- Ichiro Osako (Texas A&M U.) | Akhil Datta-Gupta (Texas A&M U.) | Michael J. King (BP plc America)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2004
- Document Type
- Journal Paper
- 450 - 464
- 2004. Society of Petroleum Engineers
- 4.1.5 Processing Equipment, 5.6.5 Tracers, 5.1 Reservoir Characterisation, 4.3.4 Scale, 5.4.2 Gas Injection Methods, 5.5.7 Streamline Simulation, 4.1.2 Separation and Treating, 5.1.5 Geologic Modeling, 5.5 Reservoir Simulation, 5.3.1 Flow in Porous Media, 5.4.1 Waterflooding, 5.3.2 Multiphase Flow, 3.3.6 Integrated Modeling, 5.2 Reservoir Fluid Dynamics
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Streamline simulators have received increased attention because of their ability to effectively handle multimillion-cell detailed geologic models and large simulation models. The efficiency of streamline simulation has relied primarily on simulators' ability to take large timesteps with fewer pressure solutions within an IMPES formulation. However, unlike conventional finite-difference simulators, no clear guidelines are currently available for the choice of timestep for pressure and velocity updates. This has remained largely an uncontrolled approximation, either managed by engineering judgment, or by potentially time-consuming timestep size sensitivity studies early in a project. This is clearly related to the lack of understanding of numerical stability and to the lack of error estimates during the solution.
We propose a novel approach for timestep selection during streamline simulation based on three elements. First, we reformulate the equations to be solved by a streamline simulator to include all of the 3D flux terms - both aligned with and transverse to the flow directions. These transverse flux terms are totally neglected within the existing streamline simulation formulations. Second, we propose a simple grid-based corrector algorithm to update the saturation to account for the transverse flux. Third, and most importantly, we provide a discrete Courant-Fredrich-Levy (CFL) formulation for the corrector step that leads to a mechanism to ensure numerical stability through the choice of a stable timestep for pressure updates. This discrete CFL formulation now provides us with the same tools for timestep control as are available within conventional reservoir simulators.
We demonstrate the validity and utility of our approach using a series of numerical experiments in homogeneous and heterogeneous 1/4 five-spot patterns at various mobility ratios. For these numerical experiments, we pay particular attention to favorable mobility ratio displacements, as they are known to be challenging to streamline simulation. Our results clearly demonstrate the role of the transverse flux and our proposed CFL formulation on the accuracy of the solution and on the appropriate choice of timestep across a range of mobility ratios. The proposed approach eliminates much of the subjectivity associated with streamline simulation, and provides a basis for automatic control of pressure timestep within full-field streamline applications.
Compared to conventional finite-difference simulation, 3D streamline simulation has moved from a research topic to a commercial product rather recently. Finite-difference simulation made this transition in the 1960s, while commercial streamline simulators have only been available since the 1990s.1 Technically, the oil industry literature on streamtubes dates back to the 1930s, but the 3D streamline approaches are much more recent.2-8 Many of the features that we take for granted in finite-difference simulation are still missing from streamline approaches. This is not unreasonable; the technology is much younger.
Our work attempts to resolve one such gap. We provide an analysis of the numerical stability of the streamline formulation, a prerequisite to the choice of a stable timestep for pressure updates during streamline simulation. The stability analysis is based upon a discrete CFL formulation. It provides us with the same tools for timestep control as are available within conventional reservoir simulators. We expect that this formulation will allow stream- line simulators to provide as robust answers as we take for granted with finite-difference calculations, while retaining the speed and performance characteristics that make streamline simulators of value today.
What controls the timestep size in a flow simulator? Besides numerical stability, clearly there are also external factors, such as reservoir development and management.7 As we develop and manage a field, we are introducing new wells and changing our well rates, potentially on a daily basis. If these changes in boundary conditions are significant, then the timestep size for our flow simulation must honor them. Of course, when screening geologic models, or when developing long-term depletion plans for reservoirs, the frequency of reservoir management activities may not be a severe limitation. In that case, timestep size is then primarily controlled by the requirement for numerical stability.
A numerical stability analysis for streamline simulation is the primary focus of this paper. The development we supply will also explain the ability of streamline simulators to take large timesteps. The results are in complete agreement with our physical intuition, and justify much of the current industry practice. However, it will also show us when timesteps for pressure updates are too large. Paradoxically, favorable mobility ratio waterflood, one of the most common secondary recovery processes, turns out to be a difficult calculation for streamline simulators. Calculations of unstable displacements and miscible viscous fingering are much easier in comparison. As we continue to advance the breadth and complexity of mechanisms that are included within a streamline simulator - for example, compositional processes - the requirement for a stability analysis and the choice of appropriate timestep clearly becomes of paramount importance.
The outline of our paper is as follows. We will start the exposition with a minimal review of streamline simulation, only being explicit about those aspects required for the stability analysis. We begin with a discussion of the streamline time of flight formulation as it provides us with a clear means of distinguishing between longitudinal and transverse flux. We then move to the discussion of the transverse flux, and of unsteady-state effects in general during streamline simulation. Specifically, we provide a new formulation to quantify the errors introduced by the neglect of transverse flux. We also introduce a discrete CFL9 number based on the transverse flux and demonstrate its ability to provide an effective timestep control for streamline simulation. Finally, we return to questions of numerical accuracy, and the saturation corrections introduced by the inclusion of transverse flux in our formulation. Our proposed new formulation, the application of the discrete CFL number, and the discussion of numerical accuracy provide the bulk of the paper. We have chosen to work with simple 2D waterflood models (homogeneous and heterogeneous 1/4 five-spot patterns with quadratic relative permeabilities, at various mobility ratios) to clearly demonstrate the stability mechanisms. Although full-field applications with adaptive timestep controls are certainly achievable, they are beyond the scope of this paper.
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