This paper demonstrates the application of a higher-order Godunov method and adaptive mesh-refinement to a three-phase, seven-component, micellar/polymer (MP) model and use in the simulation of an MP flood designed for North Sea conditions.
Conventional one-point upstream weighting with globally fine mesh is too expensive for obtaining a reasonable level of accuracy for field-scale simulations. The use of higher-order Godunov method with adaptive mesh-refinement not only results in significant reduction in computational times but also reveals more numerical details of the displacement process due to higher- order accuracy. Comparisons are also made between the first- and second-order Godunov methods under field-scale design conditions with and without adaptive mesh-refinement.
Introduction
To speed up computations in reservoir simulation, the recent emphasis has been on developing streamline models, parallel codes, and the use of higher-order methods. The stream, line models are computationally fast and can be used for simulating entire reservoirs. However, they cannot adequately model the complex chemistry and physics typical of an enhanced oil recovery process. The higher-order methods provide more accuracy than lower-order methods for a given computational cost; however, a reasonable accuracy may still not be obtained using fixed uniform grids in practical cases. By employing adaptive mesh-refinement (AMR), it is possible to concentrate numerical effort near fluid fronts or wells where the highest resolution is required. The simulations employing AMR cost less to achieve a desired level of resolution than uniform fine grid calculations and thus, more simulations can be performed within a given computational budget. We envision a new generation of reservoir simulators that would combine higher-order methods with AMR and parallel computing and make the process of reservoir simulation more efficient and accurate than ever.
The application of Godunov methods is relatively new to reservoir simulation. Recent applications to problems like waterflooding, black-oil model, miscible flooding, and polymer flooding have been very successful. The commonly used one- point upstream weighting and the first-order Godunov method are the same in the absence of gravity in one dimension. With the gravity included, these methods are very different because the former looks only at the flow within phases to determine a numerical flux, while the latter constructs an approximation to the local chromatography to determine the flux.
On a regular mesh, comparisons with conventional singlepoint upstream weighting for polymer flooding problems where analytic solutions are available, show that the higher-order method can achieve the same accuracy with about one-tenth the number of gridblocks. In combination with AMR, the performance of the higher-order method improves even more. This is because single-point upstream weighting results in smeared fronts that require more refined mesh around the displacement fronts for a given accuracy. The overhead cost of AMR, i.e., time not spent on solving flow equations, is less than about 10% of the total simulation time and guarantees the desired accuracy throughout the problem domain.
MODEL DESCRIPTION
This is a seven-component, three-phase, three-dimensional, finite-difference model developed to simulate MP flooding and surfactant remediation of aquifers. The seven components include water, oil, surfactant, alcohol, polymer, anions, and cations, the first four of which are the volume occupying components.
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