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2.G
Other means to solve and model a diffusion problem
The three models presented so far were solved analytically. Analytical models are fast and
accurate. Unfortunately a lot of problems are too complicated to be directly solved analytically.
This section presents the main tools used in the industry to provide such solutions.
2.G.1
Superposition in space of analytical models (image wells)
It is possible but relatively complex and CPU intensive to model linear boundaries and radial
diffusion at the same time. However, for some geometries, linear boundaries can be replaced
by a superposition of interferences from virtual wells. This is called the method of image wells,
which in turn requires the notion of superposition in space.
The principle application is detailed in the chapter ‘Analytical models - 13.D - Superposition in
space’.
Fig. 2.G.1 – 3D representation of the pressure profile
with an image well
Note: in case of multiple faults, if the number of image wells is limited the sum of the image
solutions constitute the exact analytical model. Conversely, when the sum is infinite the
resulting solution could be classified as a semi-analytical solution (developed in ‘Analytical
models - 13.D - Superposition in space’).
This is correct, but still solutions using image wells are an important category of models that
we wanted to single out.
2.G.2
Boundary elements
A particular case of approximate analytical solutions are boundary elements. The principle is to
honor the boundary condition by adding to the infinite solution an integral function on the
boundary that will ‘compensate’ for the infinite solution to honor this boundary condition.
This solution has the advantage of focusing on the boundary only, and solves a 3-D problem
with a 2-D solution, or a 2-D problem with a 1-D solution.
However there are two major shortcomings: (1) each block delimited by a boundary must be
homogeneous, and (2) it involves the inversion of a densely populated matrix, and this can be
very computer intensive.
Because of these two shortcomings, numerical models are currently preferred and more
developed in the industry.