Adaptive-Implicit Method in Thermal Simulation (includes associated papers 23528 and 23571 )
- Viera Oballa (Computer Modelling Group) | Dennis A. Coombe (Computer Modelling Group) | W. Lloyd Buchanan (Computer Modelling Group)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- November 1990
- Document Type
- Journal Paper
- 549 - 554
- 1990. Society of Petroleum Engineers
- 7.4.4 Energy Policy and Regulation, 7.1.8 Asset Integrity, 5.1.1 Exploration, Development, Structural Geology, 5.1.5 Geologic Modeling, 4.1.5 Processing Equipment, 4.1.2 Separation and Treating, 1.2.3 Rock properties, 4.3.4 Scale, 5.5 Reservoir Simulation, 5.4.6 Thermal Methods, 5.2.1 Phase Behavior and PVT Measurements
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Summary. An adaptive-implicit (AIM) for thermal simulation that provides different degrees of implicitness in various gridblocks is discussed. A special treatment of flow terms during appearance or disappearance of any particular phase is critical. This treatment creates a more stable Jacobian matrix, especially in cases where an implicit block is downstream of an explicit neighbor. The maximum eigenvalue of a local amplification matrix which is related to a dimensionless front velocity, determines the switching between implicit and explicit treatment. The switching, which is automatic and problem-independent, is superior to classical threshold switching based on the magnitude of changes of key reservoir variables. Results from the simulation of several thermal recovery processes are presented. The saving in computer time over a fiery implicit method ranges from 34 to 62 %. The maximum savings can be achieved in the simulation of large field-scale problems in which the fronts are moving fairly slowly.
In early reservoir simulation models, the flow terms in the difference equations were handled explicitly. This posed some restrictions on timestep size because of numerical instability, especially in cases with extensive changes of key reservoir variables over a timestep. As reservoir simulation technology progressed, an emphasis was placed on implicit time-differencing methods. This approach reduced numerical instability, but larger time-truncation errors were introduced. In addition, computational cost and storage requirements are higher for the implicit method. Usually, only a small portion of a reservoir experiences rapid changes in pressure, saturation, and temperature and must be treated implicitly. Thomas and Thurnau proposed an AIM that provides different degrees of implicitness in various gridblocks. The switching between explicit and implicit treatments was based on pressure and saturation changes during an iteration. These changes were compared with specified threshold values. Several authors subsequently implemented this idea of varied implicitness in black-oil, compositional, and thermal simulators with some dissimilarities in the handling of the explicit variables and in the switching between explicit and implicit modes. More particularly, in a steamflood application of the adaptive-implicit technique, Tan compared the performance of an AIM (similar to that of Thomas and Thurnau) and an inexact adaptive Newtonian method (AIN) with different kinds of switching criteria on the problems of the Fourth SPE Comparative Solution Project. The implementation of AIM was unsuccessful [CPU time was higher than for fiery implicit method (FIM) or non-convergence]. With the AIN method, savings from 5 to 28% were achieved. Fung et al. described a new switching criterion based on the numerical stability of a local amplification matrix. It is problem-independent and provides the possibility of backward switching (implicit to explicit), which is not feasible with the classical threshold switching criterion. The implementation of an AIM in a multicomponent, multiphase thermal simulator is discussed in this paper. The switching criterion is similar to that of Fung et al. Results are presented for both steam and combustion problems.
The basic equations used in thermal simulation consist of mass- and energy-conservation equations. The mass conservation is expressed in terms of components; the energy equation is the total energy balance. The equations can be written as follows. Mass balance for Component ic in np phases:
Total energy balance:
In addition to the conservation equations, a saturation or gas mole fraction constraint equation exists that may be solved simultaneously with the reservoir flow equations. In an implicit block, all terms in the conservation equations are evaluated at the latest timestep and iteration level. Derivatives with respect to all primary variables are included in the Jacobian matrix. Treatment of explicit blocks is more complicated. The terms on the right side of the mass- and energy-balance equations (accumulation, reaction, injection, production, heat loss, and external heaters) are fully implicit. Flow and thermal conductivity terms are on the n level (previous timestep)during timesteps without phase change; fiery implicit pressure derivatives are retained. As soon as a phase appears or disappears in the reservoir, the flow and conductivity terms are updated at that Newton iteration and kept on this iteration level for the rest of the timestep. However, the block is still handled explicitly; i.e., only derivatives with respect to pressure are included in the Jacobian matrix. This update is crucial to thermal problems because it creates a more stable Jacobian matrix, especially in cases where an implicit block is downstream of an explicit neighbor. The Jacobian matrix is built columnwise into a packed array and solved by the adaptive implicit matrix solver (AIMSOL). AIMSOL is a general sparse solver that uses incomplete Gaussian elimination as a preconditioning step to ORTHOMIN acceleration. The incomplete factorization is carried out in a block equation sense, implying that the symbolic factorization need be computed only when the block nonzero structure of the Jacobian matrix changes(e.g., well changes) and is independent of the implicitness level.
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