High-Throughput TVD-Based Simulation of Tracer Flow
- Robert C. Wattenbarger | Khalid Aziz | F.M. Orr Jr.
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- September 1997
- Document Type
- Journal Paper
- 254 - 267
- 1997. Society of Petroleum Engineers
- 5.4.9 Miscible Methods, 4.1.5 Processing Equipment, 5.2 Reservoir Fluid Dynamics, 5.6.5 Tracers, 5.5 Reservoir Simulation, 5.4.2 Gas Injection Methods, 4.1.2 Separation and Treating, 4.3.4 Scale, 5.3.2 Multiphase Flow
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High-throughput, total-variation-diminishing based, numerical techniques for simulating tracer flow are developed. High-throughput (HT) timestepping enables explicit differencing techniques to use timesteps that are larger than allowed by normal stability constraints. Total-variation-diminishing (TVD) based techniques are a means of controlling the solution oscillations associated with high-order differencing schemes. The concept of ‘upstream stability' is introduced to help combine HT timestepping with TVD-based differencing. Additionally, an improved HT timestepping algorithm for solving physically dispersive flow problems is developed.
Numerical experiments indicate that HT timestepping is most beneficial for solving highly convective flows that have both a physical Peclet number and a diffusion-only numerical Peclet number greater than one. Experiments comparing standard and alternating direction TVD-based techniques with both the Superbee (SB) and third-order based (L3) limiters indicate that the TVD-SB scheme is the most accurate for purely convective problems. For physically dispersive problems, the TVD-SB scheme may be too compressive and the TVD-L3 is recommended. The alternating direction TVD scheme is the least sensitive to timestepping and the most sensitive to grid-orientation.
The tracer flow model is a fairly simple one. It does, however, adequately describe a variety of real-world problems. More importantly, tracer flow simulators are relatively fast, accurate and have low storage requirements. This efficiency is due to the reduced-physics of the tracer flow equations 1,2 and to the exact decoupling of the concentration and potential solutions. For many problems, potential need be solved for only once. In this paper, we focus on the concentration step of an implicit-potential explicit-concentration solution procedure to the tracer flow equations.
Typically, tracer transport is dominated by convection, rather than dispersion. For convection-dominated flow, the numerical front spreading associated with low-order techniques may overwhelm the computed solution. This numerical spreading may be reduced, but not eliminated, by high-order differencing.3,4 Unfortunately, high-order differencing may introduce nonphysical oscillations to the solution. A common approach to eliminating these oscillations is flux limiting. Flux limiting adjusts the high-order component of the convective interblock flux to satisfy a non-oscillation condition.5,6,7 One popular group of flux-limiting schemes is the total variation diminishing (TVD) schemes. TVD schemes ensure that the total variation of the computed solution to a one-dimensional (1D) pure-convection problem does not increase in time. TVD schemes, which are based on an essentially 1D concept, have been extended to multidimensions in a variety of ways.
Explicit formulations, which are typically more accurate and require less work per timestep than implicit formulations, are often used to implement flux-limiting schemes. The main disadvantage of explicit methods is that their timestep size is limited by stability conditions. For fine-grid convection-dominated problems, stability constraints are usually most severe for high-throughput blocks near wells. Away from wells, timestep constraints are normally mild enough for explicit techniques to be attractive. This contrast in timestep constraints has led to the use of adaptive-implicit techniques that employ a combination of implicit and explicit differencing.8
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