Efficient Eigenvalue Numerical Solutions For
Time Dependent Linear Systems
Author(s)
R. C. Callarotti& E. J. Paez
Abstract
We discuss the new application of an efficient numerical method for the
calculation of the response of linear multidimensional systems: incompressible
fluid flow, heat transfer, electromagnetic problems, and electrochemical systems.
The method avoids finite difference time iterations that can diverge for large (Δt)
intervals. The method determines the poles and zeros for the different systems
allowing the direct calculation of the time response of the different systems from
trivial evaluation of inverse Laplace transforms. The new application is
concerned with finding the temperature distribution inside a cylindrical extra
heavy oil reservoir (API ~ 1) excited by microwave (5 GHz) power. The power
is applied at the surface of the earth, transmitted via a circular waveguide
operating in the TE01 mode (length 1000 meters), inserted at the center of the
reservoir. Our numerical procedure is based on: (i) spatially discretizing the
system, (ii) deriving an equivalent circuit for each volume element, (iii) applying
a time Laplace transformation and obtaining an admittance matrix, (iv) writing
the expressions for the Laplace transforms of the desired variables according to
Cramer’s determinant rule, (v) interpreting the determinants as generalized
eigenvalue problems, (vi) obtaining the proper eigenvalues of the system (poles
and zeros), and (vii) finding the time-dependent solutions by ‘trivial’ inverse
Laplace transformations.
In this paper we compare this solution with standard finite difference
solutions. We discuss the advantages derived from the application of our proper
eigenvalue approach which provides an analytical solution for the calculation of
reservoir heating at ANY desired time. From a control theory point of view this
Keywords
eigenvalues, numerical procedures, heat conduction, microwave petroleum excitation, heavy oil production
Related Book
Other papers in this volume
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