Preprint A3/2005
The inverse problem of determining the permeability reduction in flow of water with particles in porous media

Alvarez, A.C. | Hime, G. | Marchesin, D.

**Keywords: **
Formation damage | Deep bed filtration | Inverse problem | Tikhonov regularization

Most of the oil in the world is produced by injecting water in some wells and
recovering oil in other wells. In offshore fields sea water containing organic and mineral
inclusions is injected. This practice curtails the well's injectivity because the particles
suspended in the fluid are trapped while passing through the porous rock.
In this work, we study the deep filtration during the injection of water containing
solid particles to predict the loss of injectivity in wells. The mathematical model for the
filtration process is characterized by the filtration and the permeability reduction func�
tions which describe properties of the porous medium where the flow occurs. We develop
a recovery method for determining the permeability reduction function indirectly from
the pressure drop along the core, assuming that the filtration function has been deter�
mined previously by a separate procedure. For this recovery of permeability, we derive
an integral equation of Volterra type for the rock formation damage function k(\sigma) and
we discuss conditions for well�posedness of the operator equation. Finally, we describe
a numerical implementation to calculate k(\sigma) within an appropriate subset of feasible
solutions. The classical Tikhonov�-Phillips regularization is used to reduce the ill�posed
Volterra equation of first kind to a well�posed problem.