# Numerical models in Petroleum Reservoirs

Numerical models are becoming increasingly popular in well test analysis, mainly because the address problems far beyond the reach of analytical and semi-analytical models. The two main areas of usage of numerical models in petroleum reservoirs are nonlinearities, such as multiphase or non-Darcy flow, and complex reservoir or well geometries. Numerical

models can also be used to replace rate by pressure constraints when the well flowing pressure goes below a certain point, hence avoiding the embarrassing negative pressures often generated by analytical models.

The first attempts at numerical well testing were done ad hoc across the industry by engineers using standard reservoir simulators with local grid refinement. In the early 1990’s the first industrial project involved pre-conditioning of an industry standard simulator using PEBI gridding. Since then, several technical groups have been working on numerical projects dedicated to transient analysis.

In recent years, improvements in automatic unstructured grids and the use of faster computers have allowed such models to be generated in a time that is acceptable to the end user. The change has been dramatic, the time required to calculate the solution has decreased from days to hours, then to minutes, and now, for linear diffusion problems, to seconds. Using gradient methods even nonlinear regression is possible, and improved well-to-cell models allow simulation on a logarithmic time scale with little or no numerical side effects.

Last, but not least, automatic gridding methods allow such models to be used without the need for the user to have a strong background in simulation.

The main goal of numerical models is to address complex, boundary configurations, but this part of the work is actually easily done by any simulator. The problem is to also address what is easily done by analytical models, for example, the early time response and the logarithmic sampling of the time scale. This requires, one way or the other, to get more grid cells close to the well, and this has been done using three possible means:

• Local grid refinement of Cartesian grids.
• Unstructured gridding.
• Finite elements.

When the diffusion problem to model is linear, taking the same assumption as in an analytical solution, the process only requires one interaction of the linear solver for each time step. The solution is very fast and the principle of superposition can be applied. In this case, the numerical model acts as a “super-analytical-model” which can address geometries far beyond those of an analytical model.

When the problem in nonlinear the numerical module is used more like a standard simulator, with “just” grid geometry adapted to a logarithmic time scale. The nonlinear solver is used, iterating on the linear solver.

A numerical model can also be used to change the well constraint in time. For each well, a minimum pressure is set, below which the simulator changes mode and simulates the well production for this minimum pressure.