LONG, HORIZONTAL CYLINDRICAL ORE BODY AT ARBITRARY DEPTH IN THE FIELD OF TWO LINEAR CURRENT ELECTRODES*
Authors: PARASNIS D. S. 
Published: April 2006

The electric potential distribution when an infinitely long, horizontal cylinder is embedded in an otherwise homogeneous earth with a flat surface, and the current is supplied by two infinitely long electrodes on the surface, parallel to the axis of the cylinder, can be readily found by solving Laplace's equation in bipolar coordinates. The expression for the apparent resistivity (in the sense of Schlumberger) can be obtained after differentiating the expression for the surface potential. This, like the expression for the potentials themselves, involves infinite series in the general case. For the particular case of a cylinder with zero resistivity the apparent resistivity (ρa) can, however, be expressed in terms of the Jacobian elliptic functions cs (2Kμ/π) and the trigonometrical functions cot u where K is the complete elliptic integral of the first kind. The value of K depends upon the radius‐to‐depth ratio of the cylinder. For a cylinder of infinite resistivity the logarithmic derivative of the theta function replaces cs (2Kμ/π). Excellent tabulations exist from which the cs and the cotangent functions can be easily computed.

A set of 36 apparent resistivity curves have been computed for the particular case of a perfectly conducting horizontal cylinder midway between the electrodes (but at arbitrary depth), from which a number of interesting conclusions emerge.

It is found that the resistivity anomaly (departure of apparent resistivity from its normalized value 1) vertically above the cylinder axis increases with the electrode separation (2L) but attains a maximum value only when the electrodes are at infinity, that is, when the normal electric field is homogeneous. The optimum electrode separation for detecting a cylinder at any depth whatever would therefore appear to be infinity.

However, the resistivity anomaly at infinity, which will be called the asymptotic anomaly Δρa(o), in contrast to the central anomaly Δρa(o) directly above the cylinder, is zero when the electrodes are at infinity and increases as they approach each other. The asymptotic anomaly attains a maximum value when 2L is vanishingly small, that is, when the current source is a (linear) electric dipole. Its value is then exactly the same as that of the central anomaly for infinite electrode separation. The optimum electrode separation if the asymptotic anomaly were to be measured would be zero.

It is interesting to note that a cylinder of finite radius‐to‐depth ratio (not equal to zero) alters the electric field even at infinity by a finite amount.

In general for any 2L, with the single exception of , where h is the depth to the cylinder axis and a is the cylinder radius. In this case the two are equal.

Between the central and the asymptotic resistivities for any electrode separation, there exists the relation where m1is the complementary modulus of the elliptic functions. Since K and m1are not independent, the product is a function of the ratio a/h only. The value of the central resistivity for infinite electrode separation is also given by and this fact could be used for finding the a/h ‐ratio of a horizontal cylinder by Schlumberger electric drilling above it.

It is also found that the variation of the resistivity anomaly of a horizontal cylinder, in a homogeneous field, with change in depth or radius, follows very nearly an inverse 2nd power law, as for the magnetic anomaly of a cylinder in a homogeneous magnetic field.

The paper discusses computations involving elliptic functions in some detail because such computations do not seem to be reported in geophysical literature.