High electrical conductivity in a model lower crust with unconnected, conductive, seismically reflective layers

Geophys. J . Inf. (1992) 108, 895-905

High electrical conductivity in a model lower crust with unconnected, conductive, seismically reflective layers

A. M. Merzer'" and Simon L. Klemperer2t ' State of Israel, Armament Development Authority, Electromagnetics Division, POB 2250(87), Haifa 3 1021, Israel

British Institutions' Reflection Profiling Syndicate, Bullard Laboratories, Cambridge University, Madingley Road, Cambridge CB3 OEZ, UK

Accepted 1991 October 1. Received 1991 October 1; in original form 1990 May 21

S U M M A R Y In this paper we derive the electrical conductivity for a model lower crust containing unconnected, highly conductive lamellae within a highly resistive matrix. Lateral overlap, with small vertical separation, of lamellae of the dimensions imaged by seismic reflection profiling (a few hundred metres thick and a few kilometres across) could increase lower-crustal conductivity from the low values predicted by laboratory measurements on dry rocks to the high values observed in field experiments. The model does not depend on the cause of high conductivity within the lamellae. However, lamellation of the lower crust may provide a way of lithologically trapping saline water in permeable, conductive lamellae within an impermeable, non-conductive matrix, and so resolve the apparent contradiction between the low crustal permeabilities required for maintenance of high pore pressure over geological time periods and the high degree of pore interconnection required for the high observed conductivity. The permeable lamellae and impermeable matrix would be of very different lithologies, as implied by the high amplitudes of the lower-crustal reflections. For a typical example the model gives resistivities that compare favourably with the modified Archie's Law. The model can also give anisotropic resistivity effects, which are quantitatively compatible with results from field experiments.

Key words: continental crust, electrical conductivity, lower crust, seismic reflection profiling.

1 INTRODUCTION

Electromagnetic and magnetotelluric experiments have shown that continental lower crust almost always has electrical conductivity orders of magnitude higher than predicted by laboratory measurements on dry rocks at lower- crustal temperatures and pressures (Shankland & Ander 1983: Haak & Hutton 1986). Average conductivity values are 20 to 30 Bm for Phanerozoic lower crust, but lo4 Bm for dry rocks (Hyndman & Shearer 1989).

Possible causes of high crustal conductivity, away from areas of elevated geothermal gradient and crustal magma

* Part of this work was carried out while on sabbatical leave at the Material Mechanics Laboratory, Mechanical Engineering Depart- ment, Technion-Israel Institute of Technology, Haifa, Israel.

t Now at: Department of Geophysics, Mitchell Building, Stanford University, Stanford, CA 94305-2215, USA.

chambers, are the existence in the crust of saline water or of minerals such as metals, metal sulphides and oxides, ferrous-ferric silicates and graphite (e.g. Parkhomenko 1982). These minerals are common accessories but are rarely dominant, so that they are implausible explanations for the worldwide occurrence of lower-crustal conductive layers. The one possible exception is graphite, which has recently been reported to be present as a thin grain- boundary film in some plutonic rocks (Frost et al. 1989). However there is no good reason for any of these minerals to be concentrated into the lower crust (Haak & Hutton 1986), which is normally much more conductive than the upper crust. Though specific mineral assemblages may be an adequate explanation of certain crustal conductors, the most plausible explanation for the common occurrence of lower-crustal conductive zones is the presence of a few tenths of a per cent to a few per cent of saline fluids throughout the lower crust.

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896 A . M. Merzer and S . L. Klemperer

If saline fluids are present in the lower crust, crustal conductivity must depend strongly on their detailed distribution within the much more resistive rock matrix. In the lower crust at temperatures above 350" to 400 "C textural equilibrium between fluids and host rock is rapidly achieved on a geological time-scale (Watson & Brenan 1987). The fluid distribution under this circumstance is controlled by the local porosity and the local dihedral angle. For small porosity, if the dihedral angle is greater than 60°, fluid is distributed in isolated pores at grain corners, and the rock has very low permeability. If the dihedral angle is less than a", fluid coats all grain edges and forms an interconnected pattern through the whole rock, and the rock is highly permeable.

A problem for interpreting lower-crustal conductivity as due to saline water has been that if the fluid forms a connected network, and so is conductive [resistivity of typical saline water at lower-crustal temperatures is about 0.02Qm (Hyndman & Shearer 1989)], the lower crust is permeable, and the fluid cannot be stored for geological time periods (Bailey 1990; Warner 1991). If the fluid is trapped impermeably at grain corners, it is not connected and so is not conductive (the crustal resistivity is close to that of the dry rock, about 104Qm). Thus we have the paradox that the low permeability required for maintenance of pore pressure is incompatible with the poor interconnection required for the observed high conductivity. Jones (1987) noted a potential resolution of this paradox in the generally poor depth resolution of magnetotelluric measurements, and proposed that the highly conductive zone is not distributed through the lower crust, but is restricted to a thin layer of extremely high porosity and extremely high

S W A T 3

2.0 KM

conductivity in the middle crust. A middle-crustal high-porosity layer might contain fluids trapped there by an impermeable layer due to mineral precipitation from cooling fluids (Jones 1987) or due to the brittle-ductile transition and associated change from equilibrium to non-equilibrium (fracture-dominated) fluid-rock geometries (Bailey 1990; Warner 1991). Hyndman & Shearer (1989) suggest as alternative resolutions of the paradox the possibilities that lower-crustal grain sizes are much smaller than typically observed for high-grade rocks, thus reducing permeability; that fluid conductivity in the lower crust is much higher than the conductivity of sea-water at elevated temperatures, thus reducing the required porosity; or that an as yet unknown physical bonding mechanism holds the fluid in the rock without reducing its conductivity.

In this paper we suggest that the paradox can be resolved by considering the macroscopic geometries in which the saline fluids-or other conductors-may be contained; and we calculate the conductivity of a model crust, in which saline fluid is confined to isolated, permeable, porous regions within a resistive matrix. Theoretical support for the model is presented, and order-of-magnitude calculations are made. Finally geological aspects of the model are considered including anisotropy effects.

2 SEISMIC REFLECTION IMAGES OF THE LOWER CRUST

Not only is the lower continental crust normally more conductive than the upper crust, but it is also commonly highly reflective (Fig. 1) (e.g. Leven et a/. 1990; Matthews & Smith 1987). Correlations between lower-crustal zones of

10 0

Figure 1. Typical BIRPS (British Institutions' Reflection Profiling Syndicate) seismic section to show the character of lower-crustal reflections (fig. 1 of Reston 1987). Section extends vertically from c. 5 s two-way traveltime (c. 15 km depth) to c. 11 s (c. 35 km). The Moho reflection, base of the lower crust, is the deepest bright reflector at 9.8 s on the left of the reflection section, 9.2 s on the right-hand side.



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Electrical conductivity of the lower crust 897

2 0

krn

I I I 1 ~~~ - - 10 2 0 krn 30 40

(b) 7

8

9

10

5

TWT

Figure 2. (a) Idealized model of lower-crustal reflectors as an overlapping layer structure. (b) Corresponding synthetic seismogram (fig. 3a of Reston 1987). Vertical scale is in two-way traveltime.

high conductivity and of high reflectivity have been made in North America (e.g. Klemperer et af. 1985) and in Europe (e.g. Cazes et af. 1986), and several authors have proposed a genetic link between these observations (e.g. Matthews 1986; Gough 1986; Hyndman & Shearer 1989). Deep seismic profiles that image a reflective lower crust typically show individual reflections that are 1 to 10 km in lateral extent (e.g. Reston 1987; Wever, Trappe & Meissner 1987). Even the shortest reflections that are imaged probably represent reflectors that are at least 3 to 4 km in diameter, the size of the first Fresnel zone in the lower crust (e.g. Matthews 1986), but their images have been disrupted by spatial interference with other reflections from the densely reflective lower crust (Reston 1987). Fig. 2, after Reston (1987), shows a schematic model for the origin of the bright lower-crustal reflections as a spatial interference pattern arising from closely packed lenses within a matrix of different density and seismic velocity. It is this geometric model of lower-crustal reflectors which motivates our development of a model for lower-crustal conductive zones which contain closely packed, highly conductive lenses within a highly resistive matrix.

3 THE MODEL

Based upon the results of deep seismic reflection profiling over continental crust outlined above (Figs 1 and 2), we consider a zone in which isolated regions contain a few per cent of free water (Fig. 3). [For convenience we use the term ‘free water’, although any geometrically equivalent distribution of other good conductors, for example, unbroken graphite films (Frost et al. 1989), will give

-- -- --- -- Free f Water

Figure 3. Zone of regions containing 1 to 3 per cent free water.

A - - - - __ ___ - - B - - - - - - - - -

c - - - - __ - - - Q - - __ ___ - - - - -

Figure 4. Idealization of configuration of Fig. 3, showing four identical layers A to D. Each layer contains separated regions. Each such separated region contains a few per cent water.

identical results.] This zone of free-water regions (Fig. 3) can be idealized to the configuration of Fig. 4. In this configuration all the regions are identical and have been ordered into layers A, B, C, D etc. The regions in alternate layers overlap each other, with those in one layer lying mid-way under those in the layer above.

The configuration of the currents generated by a magnetotelluric electric field E can be shown by considering two adjacent layers e.g. B and C (Fig. 5 ) . The current from region 1 in layer B can flow to its neighbouring region 3 in two ways. It can flow directly, or it can flow indirectly

3 B - - - I - - - s f

-E Figure 5. Current configuration between layers B and C. E = local ambient magnetotelluric field.



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through region 2 in layer C as depicted in Fig. 5. This alternative path will be preferable if the overlaps of 2 on 1 and 3 are large. This is because large overlaps will lower the resistance between 1 and 2 (and similarly between 2 and 3) to relatively small values compared to the resistance along the direct path from 1 to 3.

The above configuration applies between layers B and C. However similar configurations can be constructed between all the other pairs of layers (A, B), (C, D), etc.

3.1 Zone resistivity

A quantitative estimate of the overall zone resitivity can be made by considering regions 1 , 2 and 3 from Fig. 5 in greater detail (Fig. 6).

The resistance between 1 and 2 will be the resistance R across the block ABCD in Fig. 6. This resistance is

P M t R = - nLw

where p,,, is the matrix resistivity, L is the length of each region, t is the separation between regions 1 and 2, n is the relative overlap between regions 1 and 2, and w is the width of each region (into paper).

The resistance between regions 2 and 3 will have the same value assuming identical parameter values. Thus the total resistance (RT) from 1 to 3 through 2 will be

2pMt R , = 2 R = - nLw

assuming that the resistances within the free-water regions themselves are small (see later).

Using R, the overall zone resistivity ( p ' ) can be estimated. RT gives the resistance for current flowing in a volume unit O103P3P1 outlined in Fig. 6, where O,, 0, are point centres of regions 1 and 3 respectively and the line Pl P3 bisects region 2.

If the zone were to be considered uniform with current flowing parallel to the ambient magnetotelluric electric field E (from left to right), the resistance in this volume unit would be

R v = P' l ~ l ~ 3 l / ~ l ~ l ~ l l w )

= p ' ( L + s ) / [ ( g + Owl (3)

where g is the thickness of each region containing free water, and s is the separation between adjoining regions in each layer (Fig. 4) given by

s = ( 1 - 2 n ) L . (3a)

Y d X R..laMly P r

z

Figure 6. Regions 1, 2 and 3 of Fig. 5 in detail. Each conductive region is of length L, thickness g, and width w. It is separated horizontally by a distance s and vertically by a distance r from other conductive regions. The volume unit O,O,P,P, is outlined at its corners.

Equating Rv to R , one obtains

which after substitution for s from equation (3a) gives

(4)

Since the whole zone can be divided up into such identical volume units, the resistivity p' in (4) becomes in effect the overall resistivity for the whole zone.

3.2 Finite resistance of free-water regions

The above analysis assumes that the resistance of the free-water regions is small compared to R , in (2). If this is not the case, the additional resistance can be included approximately by dividing ABCD and EFGH into small circuits (Fig. 7), and applying Kirchhoff's laws using cyclic currents (see Appendix A). After the appropriate analysis, the overall zone resistivity p' (equation 4) becomes

where

- ( Z g r Making the approximation

coth ( n a L ) = l / n a L

one gets

(7)

The first term is as (4), while the additional second term is due to the finite resistivity of the free-water regions.

Even though (7) strictly applies only for values of p c and p,,, that satisfy approximation (6), it effectively can be applied to the whole range of values of pc and p,,, with reasonable accuracy. This is because the first term in (7) is usually significant only when the required approximation (6) applies. When it does not apply, the first term usually becomes small compared to the second term. A typical example (Table lb) shows that (5) and (7) give compatible results to within 10 per cent accuracy.

3.3 Overlap @)-range of values

In the above analysis there are upper and lower limits to the values of n. The upper limit occurs when the direct resistance between adjoining regions on the same layer (e.g. between 1 and 3 in Fig. 6) becomes less than the indirect resistance i.e. 1 to 3 through 2. This occurs when the ratio of region thickness (g) to separation [s = ( 1 - 2n)L.I between regions 1 and 3 becomes equal to the ratio of overlap ( n L ) to separation ( t ) between regions ( 1 ) and ( 2 ) , i.e.



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nL

Figure 7. Electric circuit equivalent of volume unit 0 , 0 s P 3 P , (Fig. 6). The overlap regions ABCD and EFGH are divided into identical small circuits. Parameters of typical small circuit of length 61: 6r = pc & / ( w g / 2 ) , 6 R = p,,.,r/(w 6x). In addition: r, = pc(s/2) / (wg/2) .

The lower limit occurs when the overlap nL equals approximately the separation t, i.e.

nL = t. @a)

For such a small overlap the simple basic resistance formula (equation 1) starts losing accuracy.

Since L / t and L/g are expected to be large [see order-of-magnitude values in Table l(a) deduced from deep seismic reflection data] a reasonable range of values for n can frequently be obtained. For instance, the values of g, t and L in Table l(a) give a range

0.017 < n < 0.4997.

In Table l(b) we calculate the overall lower-crustal resistivity using equations (5) and (7) for the range

0.1 < n < 0.45.

3.4 Wavelength effects

The analysis in the model has been made under the assumption that the field is pseudo-static. This condition holds for magnetotelluric field wavelengths much larger than the dimensions of the free-water regions.

A typical frequency at which the effect of the lower- crustal low-resistivity zone is observed is (e.g. fig. 2 in Jones 1987)

f = 0.01 Hz. (9)

Even for the extreme case of a small resistivity p’ = 10 Qm one gets a wavelength

A = 2 ( ~ ~ p ’ / p , f ) ” ~ = 100 km, (10)

which is an order-of-magnitude larger than the conductive regions of a few kilometres length suggested by deep reflection profiling.

Furthermore even if the wavelength were to approach the size of the free-water region, its most probable effect would be to lower the overall resistivity because of capacitative admittance across the overlaps.

4 THEORETICAL SUPPORT FOR MODEL

As an indication of the correctness of the ‘zig-zag’ current model, an analytic formulation (see Appendix B for details) can be obtained for a special case of Fig. 4. This is (Fig. 8) when:

(i) the conductive free-water regions are perfectly conducting; and

Potential along layer B V ‘: 7b-~ I

- +cV = - - -Ex

I I . I I I

B

Figure 8. Perfectly conducting regions with infinitesimal separation s (for simplification only layers B and C are shown). The potential distributions (see Appendix B) along the layers are shown at the top and bottom of the figure.

(ii) the separation between regions on the same layer (s in Fig. 6) becomes infinitesimal without contact being made.

The current pattern (Fig. 9) computed from the formulation shows the postulated ‘zig-zag’ configuration between each pair of layers. Outside the layers the zig-zag effect is seen to decrease slowly with distance, and to become enveloped in the ambient current resulting from the ambient electric field E.

5 EXTENSION TO THREE DIMENSIONS

The model outlined above is 2-D: the conductive lamellae have an arbitrary extension in the third dimension. However, we can easily develop our model to describe for example square lamellae as in Fig. 10, in which alternate layers are displaced to give overlap, and the centre of each square lamella lies under the centre of the space between four lamellae in the layer above. The resistivity in the horizontal plane in this model, p&,, is that calculated for the 2-D model (Fig. 4) in equations ( 5 ) and (7), p ‘ , slightly modified. Consider one cycle of lamellae between lines AA’



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900 A . M . Merzer and S. L. Klemperer

Figure 9. Current pattern for special case of Fig. 4. Vector arrows with open heads (+) indicate magnitude and direction of field. Very large currents are indicated by a vector arrow with closed heads ( --D ) of length equal to an arbitrarily fixed cut-off magnitude.

OY

2 I - X

A' B'

S

L

A B Figure 10. Extension of Fig. 4 to three dimensions. Two layers of square lamellae are shown, by solid lines for an upper layer and broken lines for a lower layer.

and BB' (Fig. 10). Current flow only occurs over a length L - s as compared to the spacing between AA' and BB' ( = L +s) over which current flow occurs in the 2-D model. Hence,

p;D = p ' ( L + s ) / ( L - s) = p'(1- n y n , (11)

where p' may be obtained from equations (5) or (7). For intermediate degrees of overlap (n - 0.25) the resistivity of the 3-D model is increased by a factor of about three, compared to the 2-D model.

6 ORDER-OF-MAGNITUDE VALUES

A structure with parameter values defined in Table l(a) from typical deep reflection profiles (e.g. Fig. 1) is used as a basis for order-of-magnitude calculations. The resistivity of the free-water regions is taken as 10 a m after Jones (1987) and could represent a rock containing from 1 to 3 per cent seawater depending on the equilibrium pore geometry (Hyndman & Shearer 1989). The resistivity of the matrix is taken as 104Bm appropriate for dry rocks (Hyndman & Shearer 1989). The dimensions of the free-water regions are

the appropriate dimensions of typical reflectors observed within the lower crust (e.g. Reston 1987; Figs 1 and 2).

Values for the 2-D model (Table lb) and for the 3-D model (Table lc) are calculated for three values of relative overlap n (Figs 4 and lo), varying from small overlap (n = 0.1) to nearly complete overlap (n = 0.45). These values are compared with values calculated from other published models for the conductivity of two-phase systems:

(i) the Hashin-Shtrikman (H-S) bounds [equations (4) and (5) in Waff (1974)l; and

(ii) the modified Archie's Law [equation (2) in Hermance ( 1979)].

Though we are technically misusing the Hashin-Shtrikman bounds which strictly apply to a randomly isotropic material, the values we calculate provide a useful comparison between our model and other published conductivity models. The values from our model are seen to lie between the H-S bounds, and are equal to the modified Archie's Law values to within a 30 per cent deviation (Fig. 11).

In addition the accurate values (from equation 5) and the approximate values (from equation 7) are seen to have a 10 per cent maximum difference indicating that the approximate equation is relatively accurate.

More important is that the absolute values obtained for the average zone resistivity p' from our simple model, whether 2-D or 3-D, are in the range 10' to 102Rm, the same range as typical values reported for the resistivity of lower continental crust [e.g. Hyndman & Shearer 1989; whereas Haak & Hutton (1986) regard 10' to lo3 Qm as typical]. Thus our model, which has as its main assumption that seismic reflectors are conductors equivalent to typical rocks containing 1 to 3 per cent free water, has the merit of satisfying the experimental results of both reflection and electromagnetic surveys. Note that although the water-rich lamellae require a porosity of 1 to 3 per cent, the mean lower-crustal porosity in our model is less by a factor of about two ( n = 0.45) to about four (n = 0.1).



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Table 1. Order-of-magnitude values. (a) Parameter values. (b) Comparison with Hashin-Shtrikman bounds and with modified Archie's Law of order-of-magnitude values for the overall zone resistivity p' for various relative overlaps (n) in a 2-D model with infinite lateral extent of the lamellae. (c) Comparison with Hashin-Shtrikman bounds and with modified Archie's Law of order-of-magnitude values for the overall zone resistivity pAD for various relative overlaps (n) in a 3-D model with square lamellae.

Parameter

Free water region length

Free water region thickness

Separation between layers

Free water region resistivity

Matrix resistivity

Overall resistivity

Relative Values (Qm)

overlap accur. approx.

n (Fig. 4) (rqn 5) (eqn 7)

0. I 100 I00

0.25 65 63

0.45 5 1 46

Relaive Values (Qm)

overlap accur. approx.

n (Fig. 10) (eqns 5 (eqns 7

and I I ) and II)

0.1 900 900

0.25 200 190

0.45 63 56

H-S lower

resistivity

bound (Qm)

49

40

28

H-S lower

resistivity

bound (Qm)

95

63

32

Symbol Value

L 6 km

R 100 m

I IN1 m

PC 10 Qm

Pu 10'Qm

P ' , P i 0 To be calculated

H-S upper Modified

resistivity Archie's

hound (Qm) law (Qm)

4700 I30

4oOo 90

ul00 47

H-S upper Modified

resistivity Archie's

bound (Qm) law (Qm)

6500 910

54M 160

3300 49

I" I I - - - - & H-SUPPERBOUND

- E c: -

- CALCULATED VALUES

0

H-S LOWER BOUND

to' 0.1 0 25 0 45

RELATIVE OVERLAP (n)

11. Comparison with Hashin-Shtrikman bounds and with modified Archie's Law of order-of-magnitude values for the overall zone resistivity piD as a function of relative overlap (n) in a 3-D model with square lamellae.

7 FORMATION A N D PRESERVATION OF CONDUCTOR GEOMETRIES

Although the principal aim of this paper has been to calculate the conductivity of a fixed geometry of conductive

but non-connected lamellae, we should at least speculate how such a geometry might be achieved, and consider whether such a geometry would be stable over geological time periods.

If the conductive lamellae are fluid-rich how did the water migrate through the surrounding impermeable regions into the permeable regions, o r if the conductive lamellae are graphitic why might graphite be concentrated in particular reflectors? A possible solution arises from the recognition that the crustal reflectivity cannot be due exclusively to fluid or graphite content of rocks, but must imply a considerable lithological distinction between the reflectors and the country rock. If for example the permeable conductive reflectors are mafic sills intruded from the mantle, they might have carried juvenile fluids with them from the mantle, subsequently exsolved as water or, if carbon dioxide-rich, precipitating graphite during cooling. Alternatively, if the conductive reflectors are tectonically emplaced paragneisses, they might have carried biogenic carbon or connate water down with them from the upper crust. A common petrological objection to free water in the lower crust is the preservation of high-grade mineralogies and the lack of pervasive retrogression (e.g. Yardley 1986); and a common geochemi- cal objection to the presence of a pervasive H,O or COz (required to deposit graphite) fluid phase is the preservation of premetamorphic stable-isotope heterogeneities in carbon and oxygen over distances less than a metre (e.g. Baker 1990). Our model suggests that it may be possible to preserve anhydrous mineralogies and isotope heterogeneities in impermeable parts of the crust while sufficient fluid is present in permeable regions to give rise to the high observed lower-crustal conductivity.

Once graphite is deposited it will remain physically stable. In contrast water will always be physically unstable due to the tendency of the solid matrix to compact and expel the low-density fluid upwards (McKenzie 1987). If sufficiently fast, upward fluid movement will not only change the fluid geometry and hence the lower-crustal conductivity, but will also lead to fluid concentrations which will eventually be of sufficiently high porosity to render any rock permeable (since even rocks with dihedral angles greater than 60" are only impermeable a t low porosities) (Cheadle 1989). For the lOOm thick fluid-rich layers considered in our model, the characteristic emptying time (time to reduce the amount of fluid in each layer by a factor e) may be calculated by the expression and parameter values given by Bailey (1990), and ranges from c. 1 Gyr at 600 "C to c. 1 Myr at 850 "C. Though our calculations can only be approximate because the emptying time is proportional to the matrix viscosity which is poorly known, it seems that thin layers of enhanced porosity may be physically stable for geologically long times in all but the hottest crusts, in which high lower-crustal conductivity is probably due to the presence of silicate melts rather than (or as well as) t o saline fluids or graphite.

8 EFFECTS OF VARIABLE CONDUCTOR/REFLECTOR GEOMETRIES

We have shown that typical reflector geometries, if also applicable to conductors, can explain the observed resistivity. We next speculatively consider the way crustal reflectivity may vary in different tectonic settings or in different



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azimuths (anisotropy), and show how such variation would be observed in measured resistivities, if our model is applicable.

It is widely believed that prominent reflections from the lower crust are most commonly observed in areas that have undergone Phanerozoic tectonism, while reflection sections from stable cratons are often poorly reflective or are dominated by diffractions (e.g. Wever et af. 1987; Nelson 1991). Though this is not ubiquitously true (for counter- examples see e.g. Klemperer et af. 1985; BABEL Working Group 1990), an observation of diffractions rather than reflections suggests heterogeneities of density and seismic velocity with a length-scale smaller than a Fresnel zone (about 3 to 4 km in the lower crust). Wever et al. (1987) measured reflection lengths for many profiles and suggested that the average length of reflections decreases with increasing tectonothermal age. If we relate the length of the reflectors to the length L of the conductive regions in our model, then diffractive, often cratonic, crust should be more resistive than Phanerozoic crust. If, in our model, the length of the conductors is reduced, say from the 6 km used in Table 1 (derived from reflection profiles across Phanerozoic crust such as Fig. 1) to only 1 km (this value must be less than the Fresnel zone of 3 to 4 km in order to give rise to diffractions from Precambrian crust), then the resistivity will increase by a factor of about 20. This agrees well with the observation that cratons have generally higher resistivities than Phanerozoic crust: Hyndman & Shearer (1989) suggest lower-crustal mean resistivities for Precambrian shield areas of about 500 Qm and for Phanerozoic sites of 20-30 Qm, a ratio of about 20, in agreement with our model.

It has been observed that reflection profiles collected in orthogonal directions may have different appearances, suggesting that lower-crustal reflectors are more elongate or more densely packed in one particular azimuth (Reston 1987; Wever et af. 1987). Either an increase in reflector/conductor length ( L ) , as noted above, or an increase in reflector/conductor overlap ( n ) in a particular direction would lead to a decreased resistivity in that direction. In the extreme case, in which there is zero overlap of conductors ( n = 0) in one direction, the resistivity in that direction can be taken to be the Hashin-Shtrikman upper resistivity bound; whereas the (lower) resistivity in the orthogonal direction can be calculated, as before from equations (5) and (11). For the example geometry listed in Table l (a ) and the calculated resistivities in Table l (b) , the ratio of these two resistivities ranges from about 45 to 65. Electrical anisotropy is only rarely reported in studies of the deep lithosphere, possibly because experimental data are not normally sufficiently precise t o justify inversion for parameters beyond those required for isotropic solutions. However, in an example from southwestern Sweden, Rasmussen (1988) reports an electrically anisotropic lower crust for which the orthogonal resistivities differ by a factor of 60, compatible with our type of model.

The more significant anisotropy on reflection profiles is between horizontal and vertical directions. Our model suggests that the vertical conductivity of the lower crust will normally be close to the Hashin-Shtrikman upper resistivity bound, one to two orders of magnitude greater than the maximum horizontal resistivity. Unfortunately, vertical resistivity is not measured in conventional magnetotelluric

A . M . Merzer and S . L. Klemperer

experiments, and so there are no experimental data to confirm this prediction. A measurement of vertical conductivity of the lower crust, though difficult to achieve, would be an important test of our model because the spatial distribution of conductivity inferred from the geometry of seismic reflectors is an essential part of our model. In contrast, in most models for lower-crustal conductivity the conductive phase is presumed to be distributed homogeneo- usly, e.g. graphitic grain-boundary films, rather than concentrated in horizontal lamellae, and so the conductivity in these models should be approximately isotropic in all directions.

9 DISCUSSION A N D CONCLUSIONS

We have presented a model to explain high lower-crustal conductivity using regions containing some free water, which are not interconnected, so that the high conductivity is due to the mutual overlapping of the conductive regions. Although interconnection would increase the conductivity still more, if the conductors are fluids, interconnection must be limited in order to avoid the fluids leaking out of the crust. The model gives order-of-magnitude values compatible with experimentally derived conductivities, and also compatible with predictions of the modified Archie’s law of Hermance (1979). We have assumed that there is a genetic relation between the commonly observed strong reflectivity and high conductivity of the lower crust. The best test of this assumption would be to measure the vertical conductivity of the lower crust to compare with existing measurements of the horizontal conductivity. We suggest that lower-crustal reflective bodies may also be good conductors, a possibility most easily satisfied if lower-crustal reflectors contain free water at near-lithostatic pressure. The trapping of the free water in the reflectors, but not in their matrix, would imply that the reflectors and their surrounding matrix are different lithologies, so that if free water is present it contributes to seismic reflectivity rather than being the only cause. (Indeed, if free water were present only in the lithologies with higher seismic velocity and density, it could tend to reduce the observed reflectivity). The presence of free water will always lower the average velocity of the lower crust (e.g. Hyndman & Klemperer 1989), so that our model provides one possible explanation of the geophysical dichotomy between the commonly more resistive, more diffractive, higher velocity cratonic lower crust and the typically more conductive, more reflective, lower velocity Phanerozoic lower crust.

Our model assumes a homogenous pattern of free-water region layering. In the future it may be worthwhile to consider how far inhomogeneities in a real structure could affect the results obtained, when average parameter values for that structure are used in the model. However qualitative reasoning suggests that effects should be minor. Both highly resistive heterogenities, which will be circumvented by current flows, and highly conductive heterogeneities, which will be inevitably in series with regions of only average conductivity, will be relatively unnoticeable. In addition a scaled-down model experiment would also help to give further insight into the phenomenon.



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ACKNOWLEDGMENTS

AMM thanks Micheal Kaye for very many discussions and for reviewing the typescript for me. I thank Mark Goldmann of the Geophysical Institute at Holon for discussions on magnetotelluric aspects of the paper. Finally I thank Barry Greenberg at the Technion, Haifa and Moshe Schechter, Oren Hartal and all my other colleagues at work, who helped and encouraged me in preparing this paper. SLK was supported by a Royal Society University Research Fellowship. Mike Cheadle and an anonymous reviewer provided critical reviews. Cambridge Earth Sciences Contribution number 1917.

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9475-9484.

APPENDIX A

Inclusion of finite resistance of free-water regions into the overall zone resistivity formula

As mentioned in the main body of the article, the additional finite resistance of the free-water regions can be included approximately by dividing ABCD and EFGH (Fig. 7) into small circuits, and applying Kirchhoffs laws using cyclic currents (Bleaney & Bleaney, 1965, pp. 68-71). In this appendix we give more details of this procedure. In particular we calculate the resistance across ABCD from A to c.

We consider ABCD in isolation (Fig. Al ) , where A and D are joined together via a voltage source V. The configuration is divided into N cyclic currents i , , i,, . . . , i,,, for the small circuits within ABCD, and I for the large circuit VAC. Calculation of the ratio V / I will give the required resistance from A to C.

The potential around the heavily outlined circuit VAUWCV is calculated, where UW is a cross-over at an arbitrary point x = x ' . The potential calculation around the circuit gives

V = l M S r + I m=M+1 2 i m S r - m = l i m S r l + ( i M + L - i M ) S R , N M

(Al)

where M = x ' / S x . Substituting for Sr and SR (see caption of Fig. A l ) and

converting to continuous differential and integral terms, we



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904 A. M . Merzer and S. L. Klemperer

U

-x 0 X' nL

Figure Al. Configuration for estimation of resistance across ABCD from A to C. 6 r = p c 6 x / ( w g / 2 ) , 6R = p M t / ( w ax), and I , i,, i,, . . . , i, = cyclic currents.

obtain

(A21

where i ( x ) is the continuous form of i,, i,, . . . , i,. Differentiating with respect to x' gives

-=---+--=

since V is independent of the exact position of UW. Equation (A3) is a second-order differential equation for

i ( x ) . The boundary conditions for its solution are that the currents across AD and BC are infinitesimally small tending to zero in the limit.

('43) dV 2Ip, 4p&) d2i(X)PMt dr' wg wg h2 w

0,

At AD the current = i(0) = 0 (A4a)

which gives i(nL) = I (A4b)

At BC the current = I - i(nL) = 0

Solving (A3) using equations (A4a) and (A4b) we obtain

1 sinh (u(2.x - nL) i(x)=- 1+

2 '[ sinhnnL

where 112

(A5a) (also 5a)

Substituting equation (A5) into equation (A2), we get

I = & ( n L + - c o t h n a L 2 I wg ff

This is the required resistance between A and C (Fig. Al). Combining it with the two resistances r, in series (Fig. 7), we obtain the equivalent of equation (1) for the now more-complex configuration. Using equation (3) and reasoning similar to that used to obtain equation (4), we obtain the required overall resistivity equation (5).

The simplified formula equation (7) can also be obtained directly by representing ABCD (and EFGH) as one single big circuit (Fig. A2) instead of several small ones as in Fig. Al . Here r,, is the series resistance in the free-water region from A to B: and the two R,'s in parallel (=R,/2) give the resistance R (equation 1) between regions 1 and 2. Simple parallel and series combination of resistances gives equation (7).

APPENDIX B

The analytic solution

In this appendix the current-field pattern for the special case (Fig. 8) mentioned in the main text is calculated. This is done by calculating the electric field pattern, which is identical to the current pattern.

The layers in Figs 4 and 8 are assumed to be infinitely long. In this situation the electric potential along each layer can be assumed to be

V = - E X (B1)

divided into steps (Fig. 8), where E is the ambient electric field. Each step corresponds to a region of perfect conductivity, which therefore has to have a constant potential.

Using these potential distributions along the layers as boundary conditions, it is possible to obtain the expression for the potential [V(x,y)] between layers (say between B and C in Fig. 8). Since the field is quasi-static, V has to satisfy the 2-D Laplace equation

v2v = 0. (B2)

A basic 2-D solution for this equation [see p. 54 in Panofsky & Phillips (1964)l is

V = sin qx[A exp (-qy) + B exp (qy)] (B3)

nL

Figure A2. Simpler equivalent circuit. R, = p,r / (wnL/2) , r, = p c n l / ( w g / 2 ) .



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where q, A, and B are arbitrary constants. Based on this solution one can express V as an analytic series.

m

V = - E x + 2 amsin m-1

exp (-mY) + (-I), exp ( m y ) exp (mC) + (- 1)"' exp ( - m G ) '

X

where

Y = 2n(y - t/2)/L, G = 2n(t/2)/L, a,,, = EL(-l)"+'/mn.

At y = O [the boundary condition at layer C (Fig. 8)] equation (B4) becomes

2mrn m

v = -EX + a, s i n ( 1 ) . m-1

The first term is the ambient electric potential, while the second term is a Fourier series .giving the desired step configuration along layer C (Fig. 8).

At y = t [the boundary condition at layer B (Fig. 8) ] equation (B4) becomes

2nxm m

v = - E ~ + m = l 2 a,(-l)msin(T)

= -EX + 2 m a, sin [ 2nm(xL- L/2)]. m-1

Here the second term is the same Fourier series as at layer C (equation B5) displaced by L/2. It gives the desired step configuration along layer B, which is that of layer C displaced by L/2 (Fig. 8).

Thus (B4) is the solution for the potential in the space between layers B and C. From (B4) the electric fields can be obtained by suitable differentiation:

E, = -av/ax, E~ = -av /ay . (B7) Equation (B4) applies also to the spaces between layers A and B, and between layers C and D. Above A and below D equation (B4) also applies, if t is allowed to approach infinity. In this case the second term of (B4) decreases exponentially with distance from the layers until only the first term (i.e. the ambient electric field E) remains.

Even though this solution was obtained by inspection, it is in fact the unique solution by the theorem of uniqueness (Bleaney & Bleaney 1965, appendix B).



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High electrical conductivity in a model lower crust with unconnected, conductive, seismically reflective layers

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