Transcript
Page 1: Electrical Prospecting using Partial Differential Equation

Presented by,Pradeep Kumar Somasundaram

MTH5230

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AbstractThe process of finding the minerals under the earth’s crustusing the earthed electrodes. The current from the batteryconducted through the earth and the field of constantcurrent created on the surface of the earth are mapped. Byusing Linear Partial Differential Equations the potentials aredetermined and with help of Bessel functions and method ofseparation of variable the prospecting is found in differentmedium and found that electrolytic tank measurementsreplace the direct measurements.

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Instrument

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Introduction

Underground minerals, surface potentials

Homogeneous medium satisfies Laplace equation𝛻2 𝑉 = 0 βˆ’βˆ’βˆ’β†’ (1)

πœ•V

πœ•r|z=0 = 0 βˆ’βˆ’βˆ’β†’ 2

Considering a point electrode at point A

Potential of the field

V =IΟ±

2Ο€Rβˆ’βˆ’βˆ’β†’ (3)

where,

R is the distance of the potential field from the source point AΟ± is the specific resistance of the mediumI is the intensity of the current

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Potential field Potentials differ for an infinite medium

β€’---------β€’--r--β€’--r--β€’

A M O N

V M βˆ’ V N =πœ•V

πœ•rβˆ†r βˆ’βˆ’βˆ’βˆ’β†’ 4

V M βˆ’ V(N)

βˆ†rβ‰…

πœ•V

πœ•rβ‰…

IΟ±

2Ο€r2βˆ’βˆ’βˆ’βˆ’β†’ 5

where,

r is the distance between the point O to the points M and N.

O is the mid-point of the receiving circuit from the feeding electrode.

I is the current intensity of the feeding circuit which is known value.

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Homogeneous resistance Two layers

homogeneous resistance βˆ’ Ο±0homogeneous resistance -- Ο±1thickness l

The resistance can be represented as Ο± z = Ο±0 where 0 ≀ z < lΟ±1 where l < z

r<<l the impedance will be Ο±k = Ο±0 r>>l the impedance will be Ο±k = Ο±1 Conditions of continuity

V0 |z=l = V1 |z=l βˆ’βˆ’βˆ’βˆ’β†’ 6

1πœ•V0

Ο±0πœ•r|z=l =

1πœ•V1

Ο±1πœ•r|z=l βˆ’βˆ’βˆ’βˆ’β†’ 7

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Cylindrical symmetry

πœ•2V

πœ•r2+

1

r

πœ•V

πœ•r+

πœ•2V

πœ•z2= 0 βˆ’βˆ’βˆ’βˆ’β†’ 8

eΒ±Ξ»zJ0 Ξ»r βˆ’βˆ’βˆ’βˆ’β†’ 9

where, J0 is the Bessel function of the zero order

Ξ» is the separation parameter. The solutions will be of

V0 r, z =Ο±0I 1

2Ο€ (z2+r2)+ 0

∞(A0e

βˆ’Ξ»z + B0eΞ»z )J0 Ξ»r dΞ» βˆ’βˆ’βˆ’βˆ’β†’ 10

V1 r, z = 0∞(A1e

βˆ’Ξ»z + B1eΞ»z )J0 Ξ»r dΞ» βˆ’βˆ’βˆ’βˆ’β†’ 11

Find A0, B0, A1, B1 which are the functions of Ξ»

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Special functionsFor arbitrary r, A0 = B0

For V1 the condition of the bounded nature as z∞; B1 = 0

V1 r, z = 0∞(A1e

βˆ’Ξ»z )J0 Ξ»r dΞ»

Formula found in the boundary value problem by the equations of special functions

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(z2+r2)= 0

∞J0 Ξ»r eβˆ’Ξ»z dΞ»

π‘ž =Ο±0I

2Ο€

(12)

(13)

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By substituting the known values

By using the equations (6) and (7)

Derivation

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Solving equations (A) and (B)

Finding the values

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Contd.

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since |k|<1

The equation of V0 can be written as

Assuming z=0 we obtain the distribution of the potential on the earth’s surface by solving the problem using the method of images.

Distribution of the potential

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Change of variables

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The limit of the nth term of the sum will be equal to Kn, from which it follows that

To prove the impedance at infinity

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Conclusion Different conductivity profiles the impedances are also different.

πœŒπ‘˜ π‘Ÿ1 β‰  πœŒπ‘˜(π‘Ÿ2)

Defects are determined by the presence of cavity under the surface.

The cavity of the surface can be measured by placing a metallic piece between the poles of a magnet and the magnetic field on the surface.

Electrolytic tank.

Replaces effectively the direct measurements of temperature, magnetic and other fields.

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