Models for deep bed filtration during 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. This phenomenon is studied in laboratory injection tests, where the effluent particle concentration is measured over time.
We presented in a previous work 1 a method for determining the filtration function from these measurements. In this work, we improve on this method introducing a data preprocessing technique which makes the algorithm robust, and present numerical results with data which could not be treated before. The main purpose of this paper is to present this complete method for the solution of the inverse problem of determining the filtration function in the classical deep bed filtration theory from injected and effluent particle concentration measurements.
In offshore fields, it is common practice to (re-)inject produced water and sea water. However, the injection of poor quality water in a well curtails its injectivity because the particles suspended in the fluid are trapped while passing through the porous rock. This is due to particle retention in the pores, or deep bed filtration. The particle capture by the rock is defined by so-called filtration coefficient which is the probability for particle to be captured per unit path length. This coefficient is particular to each porous medium, and if is defined as dependent on the suspended and retained particle concentrations, it is called filtration function. The form of the filtration function determines the particle retention profiles during sea- and produced water injection, during drilling fluid invasion and during sand production in gravel packs. It also determines the propagation of suspension concentration waves during the disposal of produced waters in aquifers.
The classical model for this phenomenon consists of equations expressing the particle mass conservation and the particle retention process 2,3,4. They form a quasi-linear hyperbolic system of equations containing the empirical filtration function ?(s).
The filtration function cannot be measured directly, but the importance of predicting formation damage has motivated the development of laboratory methodologies for its determination. One of them is coreflood with measurements of break-through concentration: the results of such experiments are widely available in the literature. The current work aims at determining the filtration function from the breakthrough curve.
Methods for determining the filtration function from the effluent concentration history at the core outlet ce (t) of such experiments were first presented in 5,6, for constant filtration ?. A recovery method for the general case was presented in 1,7, under the assumption that the injected particle concentration is constant. A general method that relaxes this last assumption was presented in 8: it proved to be extremely accurate, but oversensitive to the input data, particularly to experimental artifacts and to transient behaviour in the early time measurements. A recent result of Alvarez et alii 9 relates this sensitivity to the lack of analyticity of the ce (t) function. We recall that a real analytic function is defined as a function that can be locally represented by a convergent power series.
Based on this new result, in this work we regularize the concentration history by analytic approximations; combined with the previous work 8 this provides a robust recovery method. We show numerical results for data published in the engineering literature, confirming the broader applicability of this improved method. The method can be used to predict injectivity decline due to injection of water with solid particles. It provides smoother filtration functions than the previous method, and is applicable to data that could not be treated before.