Preprint E2/2005
A simple inverse solver for the filtration function in flow of water with particles in porous media

D. Marchesin | Alvarez, A.C.

**Keywords: **
Inverse problem | Parameter estimation | Formation damage | Deep bed filtration | Convection-reaction equations

Models for deep bed filtration in the injection of sea water with solid inclusions
depend on an empirical filtration function that represents the rate of particle retention in the
porous rock. This function must be calculated indirectly from experimental measurements
of other quantities. In this work, we determine the filtration function from the retained
particle distribution history measured in laboratory tests by means of Xrays. The practical
petroleum engineering purpose is to predict injectivity loss in the porous rock around wells.
The recovery procedure employs optimization to estimate filtration function parameters,
as well as a fast solver for the direct problem, based on integrating ordinary differential
equations. We utilize a simple model of Hertzig et al. because the implementations of the
direct and inverse problem solvers are easier, without causing noticeable loss of accuracy, as
compared to more detailed models.
In this report, the mathematical foundation of the method and its numerical implemen
tation are discussed in detail. Good numerical results for synthetic data are shown.