resistivity and IP modelling

8/13/2019 resistivity and IP modelling

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Resistivity and IP modelling of an anisotropic bodylocated in an isotropic environment1

L. Eskola2

and H. Hongisto2

Abstract

A solution based on Tabarovskiis coupled pair of surface integral equations is given for

the potential of a direct current flowing in an electrically anisotropic body and within

the enclosing isotropic surroundings. The sources of the secondary potential exterior

and interior to the body are fictitious surface charge distributions. The equations are

solved numerically using point matching with pulse functions as subsectional basis

functions. The model used in the applications is a long prism, excited by long line

current electrodes aligned parallel to the strike. The strike length is set at a length

sufficient to guarantee 2D behaviour of the model.

Comparisons of computation results indicate that for the models, electrode arrays

and numerical procedures applied, the solutions based on fictitious surface sources

converge faster and behave more regularly than those based on real surface charges.

When compared with previously published integral equation solutions, the present

solution seems to be relatively efficient, even in the case of purely isotropic models. The

model experiments also showed that at moderate resistivity contrasts, the anomaly

shapes are strongly dependent on the directions of the principal axes of the body

resistivity. However, when the external resistivity is more than 100 times that of the

geometric mean of the principal resistivities in the body, with the principal resistivities

differing from each other by at most one order of magnitude, the contribution of the

anisotropy to the anomaly diminishes as a result of electrical saturation.

Introduction

The behaviour of conductivity and IP anomalies of an isotropic polarizable conductor

within an isotropic environment as a function of model parameters is well known from

numerical modelling results. However, mineralized formations often have a laminated

structure that makes them electrically anisotropic. The bulk resistivity and polariz-

ability of an anisotropic body depend on the direction of consideration in relation to the

principal axes of resistivity and polarizability, so that the number of electrical

parameters controlling the anomalies is three times that of isotropic bodies. A current

flowing transversely to the lamination alternately encounters highly conductive mineral

1997 European Association of Geoscientists & Engineers 127

Geophysical Prospecting, 1997, 45, 127139

1 Received September 1995, revision accepted March 1996.2

Geological Survey of Finland, FIN-02150 Espoo, Finland.


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laminae and purely conductive barren layers. The transverse polarizability of the body

may be strong due to the cumulative effect of surface polarizable lamellar interfaces,

whereas the bulk conductivity is not necessarily very high. Conversely, strong bulk

conductivity may exist parallel to the laminae, due to the high longitudinal

conductance of individual laminae, while the bulk polarizability may remain low,since the polarizable interfaces are predominantly orientated parallel to the current

flow.

Numerous methods have been published for solving electromagnetic and direct

current problems of isotropic models, based on both differential and integral equation

techniques. Due to the subject of this work we are mainly concerned with surface

integral equations applied to d.c. problems. Daniels (1977) applied surface integral

equations with real surface charges as secondary sources for modelling resistivity and

IP anomalies of 3D bodies with buried electrodes. Soininen (1985) used this technique

to model resistivity and IP anomalies of inclined prisms in a two-layered earth. Le

Masne and Poirmeur (1988) considered the behaviour of resistivity and IP anomalies

of 3D bodies for a hole-to-surface method. Eloranta (1986a) based his resistivity and

mise-a-la-masse modelling on surface integral equations with fictitious double sources.Xu, Gao and Zhao (1988) applied this technique to 3D terrain modelling associated

with resistivity surveys. Doherty (1988) gave a coupled set of surface integral

equations for modelling electromagnetic anomalies. The secondary sources for the

electric and magnetic fields were represented by tangential electric and magnetic

sources distributed on the surface of the anomalous body.

Integral equation modelling of anisotropic anomalous bodies has received less

attention in the published literature. This is partly due to the fact that formulation of an

integral equation solution for an anisotropic body is conceptually difficult. For

example, a surface charge generated on an anisotropic body is associated with volume

charges in the body (Eskola 1988), which renders the formulation of surface integral

equations using real physical surface charges as secondary sources inappropriate for

modelling anisotropic bodies. The differences between the real and fictitious surfacesources of anisotropic models are also considered by Eskola (1992, pp. 8393).

Eloranta (1988) modelled mise-a-la-masse anomalies of a perfect conductor in an

anisotropic earth. Xiong et al. (1986) calculated IP and electromagnetic anomalies of

an isotropic 3D body in a uniaxially anisotropic two-layer earth using a volume integral

equation.

Based on the work of Tabarovskii (1977), we present a coupled pair of surface

integral equations for a model consisting of an anisotropic body located in an isotropic

half-space. Model results are computed for some very simple truncated 2D models.

The convergence of the numerical solution as a function of boundary element density

is tested by comparing results obtained for some isotropic models using the equations

given in this work and those obtained with the equations given by Soininen (1985).

Some models are also computed in order to consider the effect of anisotropy onapparent resistivity and IP anomalies. The 2D model is used in the present applications

for computational ease and accuracy. In particular this enables testing of the

128 L. Eskola and H. Hongisto

1997 European Association of Geoscientists & Engineers, Geophysical Prospecting, 45, 127139


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convergence of numerical solutions up to quite high boundary element densities along

the cross-sectional line of the prism surface.

Integral equation solution

Consider an electrically anisotropic body Vi located in an isotropic environment Ve.

The subscripts i and e refer to the interior and exterior regions, respectively. In body Vi,

the resistivity is represented by the tensor with components ra,rb,rcin the directions

of the principal axes a, b, c. In Ve, the body is bounded by isotropic resistivity re.

In Ve, the electric potential Uesatisfies Poissons equation

2

Ue reIdR Rp; 1

where d is the Dirac delta function, R is the calculation point and I is the current

emitted into the earth at point Rp.

In Vi, the potentialUi satisfies Laplaces equation

1 r1riUi 0; 2

wherer i is the geometric mean of the principal resistivities, given by

ri rarbrc1=3

: 3

Ue and Ui are required to be regular at infinity. When Ve is a half-space, the normal

component of current density is zero on the earthair interface, i.e.

r1e Ue 1 n 0: 4

Boundary conditions also require that the potential and the normal component of

current density are continuous across boundary Sbetween Vi and Ve, so that

Ue Ui 5

Resistivity and IP modelling 129


Figure 1.The truncated 2D models used in the examples. Line dipole-dipole array with varying

transmitterreceiver separation. The arrows in the small squares show the direction of layering;

empty squares denote isotropy.


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and

r1e Ue 1 n r1

Ui 1 n: 6

Following the procedure given by Tabarovskii (1977), we take Ue and Ui as being

generated by fictitious sourcesje andji distributed on surface S, thus

Ue Up re

S

GejedS0 7



Figure 2. Apparent resistivity and frequency effect of isotropic model A1 as the solution of (a)

fictitious sources and (b) real sources. Line dipole-dipole array with n 15, re=ri 10,

FE 0:2. Boundary element density varies.


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and

Ui ri

S

Giji dS0; 8

whereUp is the primary potential. By analogy with (1) and (2), Greens functions GeandGi are made to satisfy the equations

2

Ge dR R0 9



Figure 2. Continued.


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and

1 r1riGi dR R0: 10

In accordance with (4), the normal derivative of Ge must be zero on the earthair

interface, which gives the solution

Ge 1=4pjR R0j 1=4pjR R00j; 11

whereR00is the image of point R0in the earthair interface. The exterior region Vecan

also be more complicated, and correspondingly Greens functionGe then also becomes

more complicated. For example, ifVeis a layered space, then the solution of (9) must

satisfy continuity of the potential and the normal component of current density across

each layer boundary.

In Vi, Greens function Gi satisfies the regularity condition at infinity but no other

boundary conditions, which yields the solution to (10) in the form

Gi 1=4pr=riR R0 1 R R01=2

: 12

In the system of principal axes a, b, c, (12) takes the form

Gi 1=4paaa02

bbb02

gcc02

1=2

; 13

wherea ra=ri, b rb=ri, g rc=ri.



Figure 3. Apparent resistivity and frequency effect of isotropic model A1 as the solution of (a)

fictitious sources and (b) real sources. Line dipole-dipole array with n 15, re=ri 100,

FE 0:2.


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In order to solve for the source densities jeandji, we form two integral equations by

substituting the integral formulae (7) and (8) in the boundary conditions (5) and (6).

After moving the calculation point R on to the surface S and removing the singular

points R R0 from the integrals, we obtain the following two coupled surface integral

equations:

Up re

S

Geje dS0 ri

S

Giji dS0 14

and

r1e Up 1 n

S

Ge 1 njedS0 je=2 ri

S

r1

Gi 1 nji dS0 ji=2: 15

The integrals in (15) are considered as the principal values. After solving the set of

equations (14) and (15) forje andji, the potentialsUeandUican be calculated at any

arbitrary point within the regions of definition Ve and Vi using (7) and (8).

Model experimentsIntegral equations (14) and (15) are solved numerically using the method of

subsections with pulse functions as basis functions. In more practical terms, the





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surfaceSis divided into boundary elements, for each of which the source densities jeand ji are assumed to be constant. Requiring that (14) and (15) be satisfied in the

centre of each boundary element, the unknown source densities can be solved by a

linear set of algebraic equations. A detailed treatment concerning the principles of

numerical solution of integral equations is given, for example, by Harrington (1968).

The IP anomaly is modelled as the apparent frequency effect FEAaccording to the

formula

FEA rArk1FEk=rArk 1; 16

where rA is the apparent resistivity, which is a function of the resistivity in the

environment and of the k components of resistivity, rk, and frequency effect, FEk, in

the body, with k a, b, cdenoting the principal axes of anisotropy.

The test model (Fig. 1) used in numerical examples is a truncated 2D prism with a

square-shaped cross-section, and the principal axes of resistivity in the prism are

aligned parallel to the prism edges. The electrode array is a line dipole-dipole array

moving perpendicular to the strike of the body, with length and orientation of theelectrodes coinciding with those of the prism. The strike length used in computations is

200 times the side length of the cross-sectional square, which was shown by test



Figure 4. Apparent resistivity and frequency effect of model A2. Line dipole-dipole array with

n 15. re=rk 22, re=r 2:2, re=ri 10, and re=rk 217, re=r 22, re=ri 100.

FEk 0:2 for all k a,b,c.


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computations to guarantee the two-dimensional behaviour of the model up to a

resistivity contrast of 100.

Numerical results obtained for some isotropic models using (14) and (15) (referred

to below as the solution of the fictitious sources) are compared with the results obtainedusing the equations given by Soininen (1985) (referred to as the solution of the real

sources). The body surface is divided homogeneously into boundary elements having

the length of the strike. The element densities are chosen so that the solutions to be

compared with each other have equal numbers of unknowns in discretized integral

equations. In the case of fictitious sources, the number of unknowns is twice the

number of boundary elements.

Figure 2 and 3 show resistivity and IP anomalies of isotropic model A1 (see Fig. 1)

atn values 1 5 (spacing between the inner transmitter and receiver electrodes in dipole

lengths) and resistivity contrast 10 and 100, computed using 20, 40 and 400 boundary

elements for the fictitious sources, and 40, 80 and 800 boundary elements for the real

sources. The computations were performed using a 100 MHz 486 PC with 20 MB

memory. For both source types, the computation time of a series of five curves with thehighest number of boundary elements was 45 minutes. The results corresponding to

eachn value are represented as separate curves.



Figure 5. Apparent resistivity and frequency effect of model A3. Line dipole-dipole array with

n 15. Resistivities andFEk as in Fig. 4.


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The anomaly curves obtained from the fictitious sources show faster convergence

with growing element density than those obtained from the real sources. The solutions

of the fictitious sources obtained using 400 elements also show a reasonable reciprocity

of the anomaly curves. In general, the element density needed to achieve satisfactory

convergence of the results for the applied 2D models turns out to be rather high.Considerable difficulties in obtaining satisfactory convergence occur in the case of real

sources, as shown by Figs 2b and 3b, in which acceptable results are found only for the

resistivity anomaly at a resistivity contrast of 10 and when computed using the highest

number, 800, of boundary elements. This is particularly true when the current earthing

is close to the body.

The effect of resistivity contrast on the anomalies is best considered by comparing

the curves of Figs 2a and 3a when solved using 400 elements. A central high develops in

the apparent resistivity curves with increasing n, which is a feature reflected in the IP

anomaly as a central low. The IP anomaly intensity decreases strongly with increase of

the resistivity contrast from 10 to 100, which is an expression of electrical saturation of

the model, and is in qualitative agreement with the results obtained for 3D models

(Soininen 1985; Eloranta 1986b). In a saturated state it is not possible to get muchmore current inside the body, and therefore the IP effect is weak. However, the

intensity of the IP anomaly is still significant at a resistivity contrast of 100 (Fig. 3a),



Figure 6. Apparent resistivity and frequency effect of (a) model B1 and (b) model B2. Line

dipole-dipole array with n 15. Resistivities and FEk as in Fig. 4.


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which may indicate that the saturation in 2D models develops at higher resistivity

contrasts than in 3D models.

Figure 4 shows resistivity and IP anomalies of the horizontally layered model A2 at

resistivity contrasts re=ri of 10 and 100, computed using 400 elements, while Fig. 5

shows the corresponding anomalies of the vertically layered model A3. r i is now thegeometric mean of the principal resistivities in the prism. The anomalies of the

corresponding isotropic models are shown by the lowest curves in Figs 2a and 3a. The

isotropic case represents an intermediate form between the two layered cases, both as a

model and also with respect to their anomalies. At a resistivity contrast of 10, the

resistivity and IP anomalies of the horizontally and vertically layered models differ

considerably from one another. At a contrast of 100, however, the resistivity anomalies

of the models with mutually perpendicular layering, and also the anomaly of the

isotropic model with the same resistivity contrast, are very similar to each other

with respect to both shape and intensity. The shapes of IP anomalies of these three

models at resistivity contrast 100 differ from each other but considerably less than

at resistivity contrast 10. The decrease in IP anomaly intensity and the increase in

similarity of anomalies associated with different anisotropy following the increase inresistivity contrast from 10 to 100 is a consequence of electrical saturation in the

model.





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Figure 6a shows resistivity and IP anomalies for the isotropic model B1 and Fig. 6b

shows the anomalies for the anisotropic model B2, both at resistivity contrasts of 10

and 100. At a resistivity contrast of 10, the anomalies of the isotropic and anisotropic

bodies again differ considerably from one another, principally in the strong asymmetry

caused by anisotropy. The deepest minimum in the resistivity anomaly is located abovethe upper right-hand face of the prism, which coincides with the lower principal

resistivity of the prism. Compared to the resistivity anomalies, the asymmetry of the IP

anomalies shows complementary behaviour. At resistivity contrast 100, the resistivity

anomalies of the isotropic and anisotropic bodies are almost similar, and the difference

between corresponding IP anomalies appears as a weak asymmetry in the anomaly of

the anisotropic model, which is again due to the effect of electrical saturation.

Summary

We have derived a coupled pair of integral equations for fictitious surface sources for

modelling resistivity and IP anomalies of an anisotropic body located in an isotropic

environment. The solution was originally formulated by Tabarovskii (1977) for solvingdirect current problems for a model consisting of an anisotropic body located in an

anisotropic full-space. The computer code was written at this first stage for a truncated

2D model, in which the anomalous body is represented by a long rectangular prism

with the principal axes of resistivity being aligned parallel to the prism edges. The

numerical solution was made using point matching with pulse functions as the

subsectional basis functions. The electrode configuration applied was a line dipole-

dipole array.

Comparison runs for isotropic models indicate that for the models, electrode arrays

and numerical procedures applied in the present experiments, the solutions based on

fictitious sources converge faster and behave more regularly than the solutions based

on real surface charge distributions. Although originally formulated for anisotropic

problems, the technique seems to compare favourably in efficiency with other integralequation formulations for solving direct current problems, including the case of

isotropic models.

At moderate resistivity contrasts between the model and its environment, the

anomaly shapes are strongly influenced by the direction of the principal axes of

anisotropy. However, when resistivity in the environment exceeds 100 times the

geometric mean of the principal resistivities in the body, and when at the same time the

contrast between the principal resistivities is not very high (in the applications in this

workr=rk 10, the influence of anisotropy on the anomaly weakens as a result of

electrical saturation taking place in the model.

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electrodes. Geophysics 42, 10061019.




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