A fast inverse solver for the filtration function for flow of water with particles in porous media

Abstract

Models for deep bed filtration in the injection of seawater with solid inclusions depend on an empirical filtration function that represents the rate of particle retention. This function must be calculated indirectly from experimental measurements of other quantities. The practical petroleum engineering purpose is to predict injectivity loss in the porous rock around wells. In this work, we determine the filtration function from the effluent particle concentration history measured in laboratory tests knowing the inlet particle concentration. The recovery procedure is based on solving a functional equation derived from the model equations. Well-posedness of the numerical procedure is discussed. Numerical results are shown.

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

“…For the latter times of the first two series, in the late time ranges where c e ≈ c i , a good match was only obtained using f (σ) = θσ 4 . We remark that the method proposed in [3] does not work for the zero filtration rate situation observed in the first three data sets, as the quantity actually recovered is 1/λ(σ) rather than λ(σ). The remaining three do not approach this limit.…”

“…First, we recover the filtration function using either the method presented below or the methods presented in [3] and [4]. This first inverse problem determines the filtration function from the outlet concentration, i.e., the kinetic particle capture rate is calculated from the particle concentration history.…”

“…The well-posedness of the boundary/initial value problem (2.1)-(2.2) with boundary and initial data (2.4)-(2.5) was established in [3] and [4], where U was taken as a constant and it was assumed that the filtration function λ(σ) was C 1 , i.e., it had one continuous derivative, and that…”

“…Differently from [3], where λ(σ) was strictly positive, we assume that λ(σ) is a nonnegative piecewise C 1 function that may vanish in its domain 0 < σ < φ, i.e., λ(σ) > 0 for 0 ≤ σ < σ 0 and λ(σ) = 0 for σ 0 ≤ σ < φ. The following Lemma, proved in [4], allows this assumption.…”

“…In [3] and [4] the well-posedness of the direct problem was proved. To solve the inverse problems for the filtration function λ(σ) and the formation damage function k(σ), we need to solve equations (2.1)-(2.5) and (2.7) many times as part of an iterative optimization procedure.…”