We describe a multistage parallel linear solver framework developed as part of the Intersect (IX) next-generation reservoir simulation project. The object-oriented framework allows for wide flexibility in the number of stages, methods and preconditioners. Here, we describe the specific components of a two-stage CPR[1] (Constraint Pressure Residual) scheme designed for large-scale parallel, structured and unstructured linear systems. We developed a highly efficient in-house Parallel Algebraic Multigrid (PAMG) solver as the first stage preconditioner. For the second stage, we use a parallel ILU-type scheme. This new and powerful combination of CPR and PAMG was the result of detailed analysis of the linear system of equations associated with reservoir simulation. Using several difficult reservoir simulation problems, we demonstrate the robustness and excellent parallel scalability of the IX linear solver. For the field case studies, the IX linear solver with CPR and PAMG is at least five times faster than an established and widely used industrial linear solver. The performance advantage of the IX linear solver over traditional reservoir simulation linear solvers increases with both problem size and the number of processors. Introduction Different types of grid may be used for reservoir flow simulation to model geometrically complex, highly detailed models and/or deviated or multi-lateral wells[2]. Grid types are often labeled based on their structure. Examples of simulation grids include:structured Cartesian,structured stratigraphic,multi-block stratigraphic,PEBI (Perpendicular Bisector), andgenerally unstructured. Hybrid grids that combine various types can also be used. It is now widely recognized that complete flexibility in representing complex and highly detailed simulation models can be achieved using generally unstructured grids[3].In recent years, significant efforts have focused on building multi-purpose reservoir flow simulators that can deal with geometrically complex and highly detailed structured and unstructured reservoir models[4,5,6]. These relatively large-scale efforts are being pursued because, for nearly three decades, the reservoir simulation community has focused on building robust and efficient reservoir simulators for structured grid problems. Today, the ability to routinely simulate a wide spectrum of practical black-oil problems on (effectively) structured models with O(105) gridblocks is widespread. However, the performance of traditional reservoir simulators typically deteriorates significantly with problem size and the number of processors. This is because the algorithms and software implementations were not designed for scalable, parallel computation. A scalable algorithm is one whose computational complexity (i.e., the number of operations to reach solution) is proportional to the number of unknowns; moreover, the algorithm should also have a convergence rate that is independent of problem size or the number of processors. In numerical solution algorithms, there is often a tradeoff between convergence rate and degree of parallelism. As a result, to obtain a useful measure of parallel efficiency, the best scalar (uniprocessor) algorithm should be used as reference. Scalable methods are needed because the size of problems of interest continues to grow quite significantly, and we want to avoid methods with computational complexities of O(Na) with an athat is (much) larger than unity.
TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractThe current trend in industry is to use finer and finer grid with larger and larger number of components. This requires the development of faster simulation techniques than those currently available in commercial simulators.The fully implicit (FIM) model, the implicit pressure and explicit saturations (IMPES) model and the adaptive implicit (AIM) model are the traditional approaches. The IMPSAT (implicit pressure and saturations and explicit component mole fractions) model has been proposed in the literatures, but so far it is not considered a viable approach. This may be a result of lack of suitable stability analysis and improper implementation. Our analysis shows that the IMPSAT model is significantly more stable than the IMPES model, and in many cases substantially less expensive than the FIM model. The IMPSAT model becomes particularly attractive as the number of components becomes large. IMPSAT can also be used in an adaptive implicit approach to form various IMPSAT based adaptive implicit (IMPSAT-AIM) models.A new General Purpose Research Simulator (GPRS) using a General Formulation Approach has been developed and tested. The General Formulation Approach used in GPRS can be used to obtain any type of model. We can easily form a model by selecting any reasonable set of primary variables and any level of implicitness from IMPES to FIM. One important subset of models that have been developed from this new approach is the IMPSAT model and the IMPSAT based AIM models.The performance of different compositional models has been evaluated using a wide range of problems. We found that the IMPSAT model and the IMPSAT-AIM models have very good performance, and they are more efficient than the traditional models (FIM, IMPES and AIM). The IMPSAT model is generally over 50% faster than the IMPES model due to its improved stability. For problems that are not very hard, the IMPSAT model is cheaper to run than the FIM model since it reduces the number of unknowns that have to be solved implicitly. The IMPSAT-AIM models are more flexible, more stable and with appropriate linear solvers have the potential to be less expensive in terms of CPU time than the traditional AIM model.
The basic reproductive number R 0 of a discrete SIR epidemic model is defined and the dynamical behavior of the model is studied. It is proved that the disease free equilibrium is globally asymptotically stable if R 0 < 1, and the persistence of the model is obtained when R 0 > 1. The main attention is paid to the global stability of the endemic equilibrium. Sufficient conditions for the global stability of the endemic equilibrium are established by using the comparison principle. Numerical simulations are done to show our theoretical results and to demonstrate the complicated dynamics of the model.
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