La Houille Blanche
Number 8, Décembre 1969
|Page(s)||861 - 868|
|Published online||23 March 2010|
Restitution automatique des perméabilités d'une nappe le problème inverse et la déconvolution
Arrondissement minéralogique de Bordeaux
2 Ecole des Mines, Paris
The data interpretation problem frequently arises in the following terms : A physical system of unknown characteristics has the effect of transforming an input phenonmenon s into an output quantity S. For example, where constant rates of flow are pumped from a number of wells, the piezometric level of the tapped aquifer is H (x, y, z) when equilibrium is attained. Similarly, the initial concentration of a tracer penetrating into a porous tube is c (t) and, due to diffusion (for example), the concentration of the liquid emerging from the tube is C (t). The following questions then arise : In the first case what must the transmissibility distribution T be to satisfy div T grad H = Q And in the second case, what "operator" p (t) expresses the response to an instantaneous injection of a unit mass of tracer U, so that C (t) = S t 0 ( p) (x) c (t-x) dx Both problems have the same structure in that data on the operator are required, i.e. on the physical system, which is represented either by a description of what it consists of (T') or by its response to an elementary demand (p). The method described starts out from the observation that current iterative calculation methods (especially Fourier series) tend to be unstable, typically due to the physically absurd fact that the local mean of the values of several points is very much smaller than the local variation amplitude. The proposed new method involves an iteration procedure in which the values of the unknown quantity are modified in decreasing steps. In addition, any punctual modification can only result from an overall modification to the whole of the supporting domain for the unknown quantity. The proposed method has been applied to deconvolution, automatic restoration of permeability (the reverse problem) and the solution of elliptical equations of the type encountered in hydrodynamics (direct problem).
© Société Hydrotechnique de France, 1969
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