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Sur la reponse transitoire des formations resistivesNormal access

Author: J. Loeb
Journal name: Geophysical Prospecting
Issue: Vol 12, No 3, September 1964 pp. 325 - 332
DOI: 10.1111/j.1365-2478.1964.tb01908.x
Organisations: Wiley
Language: English
Info: Article, PDF ( 372.14Kb )

Summary:
This paper deals with transient response of resistive formations to pulse sources. Computation of solutions of equations such as:

is linked to computation of solutions of equations such as:

It can be used in the case where several media exist, the conductivities of which are σ1 σ2.

The method consists in establishing a correspondence between the case to be dealt with and a fictitious region of space with the same interfaces, in which the propagation of a disturbance obeys Equation (2).

Velocities C1 C2. are related to conductivities by the condition

At the interfaces, the distribution of incident energy between transmitted and reflected energy is related to the ratio of characteristic impedances, i.e.

in the real case (ω is the angular frequency) and C1/C2 in the fictitious case.

These ratios are the same because of Equation (3).

Time variation g(θ) of solution of (1) in a given point is computed from time variation of solution of (2) at the same point by means of the transformation

(in which τ is the actual time and T the time unit).

Equation (4) originates from a formula of symbolic calculus, that allows the Laplace transform of F(Vp) to be calculated from the known Laplace transform of F(p). This correspondence is the one that allows passage from in (1) to in (2)

Equation (4) becomes considerably simpler when the response to the unit impulse consists (as happens most often) of the sum of a finite or infinite number of pulses The results is then a sum of functions such as:

Hence, this procedure appears as an extension to systems that obey Equation (1) of the image theory very often used when Equation (2) is applied.


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