Implicit versus explicit modeling

Schimel J. (2001) Biogeochemical models implicit versus explicit microbiology. In Global Biogeochemical Cylces in the Climate System (eds. E. D. Schulze, M. Heimann, S. Harrison, E. A. Holland, J. Eloyd, 1. C. Prentice, and D. Schimel). Academic Press, pp. 177-183. 

Biogeochemical Models Implicit versus Explicit Microbiology... 

Various viscosity models have implicitly included the effects of gelation on the chemoviscosity, and these were reviewed in Table 4.2 incorporating gelation-conversion and glass-transition-temperature effects implicitly in the cure effects on chemoviscosity. Explicit models for the expression of gel time versus temperature and time are sparse, with empirical measurements mainly being used. 

In many books, radial flow theory is studied superficially and dismissed after cursory derivation of the log r pressure solution. Here we will consider single-phase radial flow in detail. We will examine analytical formulations that are possible in various physical limits, for different types of liquids and gases, and develop efficient models for time and cost-effective solutions. Steady-state flows of constant density liquids and compressible gases can be solved analytically, and these are considered first. In Examples 6-1 to 6-3, different formulations are presented, solved, and discussed the results are useful in formation evaluation and drilling applications. Then, we introduce finite difference methods for steady and transient flows in a natural, informal, hands-on way, and combine the resulting algorithms with analytical results to provide the foundation for a powerful write it yourself radial flow simulator. Concepts such as explicit versus implicit schemes, von Neumann stability, and truncation error are discussed in a self-contained exposition. 

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