Implementation of an Improved Adaptive-Implicit Method In a Thermal Compositional Simulator
- T.B. Tan (D&S Petroleum Consulting Group Ltd.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- November 1988
- Document Type
- Journal Paper
- 1,123 - 1,128
- 1988. Society of Petroleum Engineers
- 7.1.8 Asset Integrity, 5.5 Reservoir Simulation, 5.1.5 Geologic Modeling, 4.1.2 Separation and Treating, 5.4.6 Thermal Methods, 4.1.5 Processing Equipment
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A multicomponent thermal simulator with an adaptive-implicit-method (AIM) formulation/inexact-adaptive-Newton (IAN) method is presented. The final coefficient matrix retains the original banded structure so that conventional iterative methods can be used. Various methods for selection of the eliminated unknowns are tested. AIM/IAN method has a lower work count per Newtonian iteration than fully implicit methods, but a wrong choice of unknowns will result in excessive Newtonian iterations.
For the problems tested, the residual-error method described in the paper for selecting implicit unknowns, together with the IAN method, had an improvement of up to 28% of the CPU time over the fully implicit method.
The formulation of reservoir simulation models has progressed from implicit pressure, explicit saturation (IMPES) to semi-implicit to sequential implicit to fully implicit. Associated with the increased stability of the fully implicit method, however, are the increased storage requirements and longer computing times. In most solution methods, the work count increases with the cube of the number of equations for each cell to be solved, so any reduction of the number of equations will result in reduced computational time.
The AIM was originally proposed in an excellent paper by Thomas and Thurnau.1 The AIM reduces the number of equations by changing the degree of implicitness required on a block-by-block basis. The selection of the implicit/explicit variables was based on which variables changed most rapidly. In the published formulation, the resultant equations were reduced to an explicit partition and an implicit partition that were solved separately. As a result of the reduction process the matrix coefficients became arbitrarily structured so that conventional iterative solution methods could not be used.
A significant improvement to the AIM was presented by Bertiger and Kelsey.2 They modified the AIM matrix by zeroing out the derivatives of a declared explicit variable only if they appear in the off-diagonal blocks of the coefficient matrix. The resultant matrix was then reduced as in the normal AIM. This method was called the IAN method. The authors mentioned increased material-balance error and difficulty in the solution method at the time of publication.
A variant of the above is the dynamic implicit method proposed by Vinsome.3 If the flow across a gridblock face is less than a suggested 0.02 of the minimum of the two adjacent block PV s, then all but the pressure column of the corresponding off-diagonal block of the matrix is zeroed out. This method does not lead to a reduction in the resultant number of equations; rather, savings in time are achieved in building the Jacobian and in a specialized solution method that can recognize the off-diagonal blocks in the matrix coefficients that have been partially zeroed out.
This paper discusses the implementation of the adaptive-implicit concept in a multicomponent, three-dimensional (3D), multiphase thermal simulator. In multicomponent models, variable substitution as presented by Coats4 is often used to eliminate constraint equations. This complicates the implementation of the adaptive-implicit concept. This paper shows how a resultant matrix can be obtained that has the same structure as the original matrix, allowing conventional iterative methods to be used. Both AIM and IAN have been implemented in the simulator, and we compare the relative performance of the two methods. Various schemes for the selection of the implicit/explicit variables and their effect on the overall performance of the model are presented. The three problems used in the fourth SPE comparative solution project for steam models5 were chosen as the basis for comparison of the various selection methods.
The simulation model is a comprehensive 3D, multiphase multicomponent thermal model capable of simulating steam and combustion recovery processes. The general formulation of the conservation equations is presented, but specific attention is focused on the adaptive-implicit feature.
Any component can exist in any phase, and assuming that there are np phases, nr reactions, and nc components, the component mass balances and energy balance have the familar form described below.
|File Size||4 MB||Number of Pages||6|