The inverse problem of determining the filtration function and permeability reduction in flow of water with particles in porous media
P.G. Bedrikovetsky | Alvarez, A.C. | Hime, G. | Marchesin, D.
Deep bed filtration | Suspension transport | Porous media | Inverse problem | Tikhonov regularization | Formation damage | System of convection-reaction equations
Deep bed filtration of particle suspensions in porous media occurs during water injection into oil reservoirs, drilling fluid invasion of reservoir production zones, fines migration in oil fields, bacteria, viruses or contaminant transport in groundwater, industrial filtering, etc. The basic features of the process are particle capture by the porous medium and consequent permeability reduction. Models for deep bed filtration contain two coefficients that represent rock and fluid properties: the filtration function, which is the fraction of captured particles per unit of particle path length, and formation damage function, which is the ratio between reduced and initial permeabilities. The coefficients cannot be measured directly in the laboratory or in the field; therefore, they must be calculated indirectly by solving inverse problems. The practical petroleum and environmental engineering purpose is to predict injectivity loss and particle penetration depth around wells Reliable prediction requires precise knowledge of these two coefficients. In this work we determine these coefficients from pressure drop and effluent concentration histories, measured in one-dimensional laboratory experiments. The filtration function is recovered by optimizing a nonlinear functional with box constraints. The permeability reduction is recovered likewise, taking into account the filtration function already found. The recovery method consists of optimizing Tikhonov's functionals in appropriate subdomains. In both cases, the functionals are derived from least square formulations of the deviation between experimental data and quantities predicted by the model.