The inverse problem of determining the filtration function and permeability reduction in flow of water with particles in porous media

Abstract

Abstract. 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, industrial filtering, bacteria, viruses or contaminants transport in groundwater, 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 quantities that represent rock and fluid properties: the filtration function, which is the fraction of particles captured per unit particle path length, and formation damage function, which is the ratio between reduced and initial permeabilities. These quantities 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 quantities from pressure drop and effluent concentration histories measured in one-dimensional laboratory experiments. The recovery method consists of optimizing deviation functionals in appropriate subdomains; if necessary, a Tikhonov regularization term is added to the functional. The filtration function is recovered by optimizing a non-linear functional with box constraints; this functional involves the effluent concentration history. The permeability reduction is recovered likewise, taking into account the filtration function already found, and the functional involves the pressure drop history. In both cases, the functionals are derived from least square formulations of the deviation between experimental data and quantities predicted by the model.

A new profile control design based on quantitative identification of steam breakthrough channel in heavy oil reservoirs

“…Such design may not be appropriate because the motion and distribution of the blocking agent among complete breakthrough channels and incomplete breakthrough channels is a dynamic balance process. This phenomenon is especially obvious in the injection process of particle blocking agent (Alvarez et al 2007;Wang et al 2012b). So if the blocking volume between the injector and the incomplete steam breakthrough producer is ignored, the designed blocking agent will partly flow to the incomplete breakthrough well.…”

Laboratory-based mathematical modelling of graded proppant injection in CBM reservoirs

“…The problem of deep bed filtration with the filtration coefficient and drift delay factor as functions of the strained concentration allows for exact solution (Bedrikovetsky, 2008;Vjazmina et al, 2007). It yields well-posed inverse problem (see Alvarez et al, 2006Alvarez et al, , 2007. However, more information from the fine-migration test is required for unique determination of the functions λ ⁎ = λ(σ s ⁎ ) and α ⁎ = α(σ s ⁎ ).…”

Numerical modeling of formation damage by two-phase particulate transport processes during CO2 injection in deep heterogeneous porous media

“…The determination of the coefficient of filtration, which is critical for a realistic prediction of the injectivity decline, is described in much more detail in the literature [78,62,89,25]. Alvarez et al [3] has developed an inverse computational procedure to deduce the filtration coefficient and formation damage as functionals and not constant coefficients. They suggest their recovery from measurements of the time series of suspended particle effluent concentration and of pressure drop along the core involving a double inverse problem.…”