ISSN 0005-1179, Automation and Remote Control, 2017, Vol. 78, No. 8, pp. 1404–1416.
Pleiades Publishing, Ltd., 2017.
Original Russian Text
K.R. Aida-zade, A.B. Rahimov, 2017, published in Avtomatika i Telemekhanika, 2017, No. 8, pp. 60–75.
Identiﬁcation of Piecewise Constant Filtration Parameters
and Boundaries of Their Constancy Domains
K. R. Aida-zade
and A. B. Rahimov
Baku State University, Baku, Azerbaijan
Institute of Control Systems, Azerbaijan National Academy of Sciences, Baku, Azerbaijan
Aix Marseille Universit´e, CNRS, Centrale Marseille, Institut Fresnel, Marseille, France
Received June 8, 2015
Abstract—Consideration was given to the numerical solution of the problem of parametric
identiﬁcation of the processes obeying the parabolic equations using an example of the processes
of underground oil ﬁltration. The identiﬁed parameters belong to the given functional classes
such as the piecewise constant and piecewise linear functions. In the problem, needed is not
only to determine the values of the coeﬃcients, but also to identify the constancy boundaries
of the coeﬃcients. For numerical solution of the problem, an approach was suggested based
on reduction of the initial problem to that of ﬁnite-dimensional optimization with a special
structure of constraints. Obtained were the formulas for the gradient of the objective functional
in the discretized problem allowing one to apply the eﬃcient methods of ﬁrst-order optimization.
The results of numerical experiments on the model problems were presented.
Keywords: partial diﬀerential equations, parametric identiﬁcation, inverse problem, mathemat-
ical model of ﬁltration, constancy domain, functional gradient, object with distributed param-
eters, methods of optimization, numerical methods.
The problems of parametric identiﬁcation in the distributed systems represent a topical subject of
research. The models of numerous processes of practical importance are describable by distributed-
parameter systems. Development of eﬃcient algorithms for identiﬁcation of the parameters of such
systems is important both in theoretical and practical terms. Determination of the characteristics
of oil pools arising both at the development and operation of oil ﬁelds may be mentioned as an
important example of the problem of parametric identiﬁcation of the distributed systems. The
present paper examines the problem of determining the coeﬃcients involved in the permeability
formula and the domains of constancy of such coeﬃcients. It is assumed that the pool is horizontal
and single phase, oil is poorly compressible, and oil ﬁltration can be regarded as a two-dimensional
process [1, 2]. These assumptions to some extent simplify the mathematical description of the
process, but for the parametric identiﬁcation approach proposed in what follows these assumptions
are not points of principle. For such processes, they enable one to explain vividly the essence of
the approach and methodology of generation of formulas without speciﬁc involved calculations and
notation. We note that such problems were considered in numerous publications such as [3–7]. It
was assumed in [3–5] that the identiﬁed permeability is piecewise constant and the domains of its
constancy are known. The problem of determination of the domains of piecewise constancy of the
coeﬃcients of an elliptical equation was considered in , and a numerical algorithm based on the
methods of ﬁnite elements was proposed. The problems of optimal allocation of the oil pool wells
and control of their ﬂow rates were studied in [9–11]. The problems of identiﬁcation of the domains