424B-Fall2014.pdf

Geophysics 424 – September 2014

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B : Resistivity of rocks and minerals B1 : Basic physics of electrical current flow B1.1 Simple resistor in circuit Ohm’s Law states that for a resistor, the resistance (in ohms), R is defined as

IVR = where V = voltage (volts); I = current flow (amps)

B1.2 Electric current flow in a finite volume Ohm’s Law as written above describes a resistor, which has no dimensions. In considering the flow of electric current in the Earth, we must consider the flow of electric current in a finite volume. Consider a cylinder of length L and cross section A that carries a current I

Current density = A

J I=

Resistance of cylinder, AL

AR ρ

=∝L

where ρ is the electrical resistivity of the material (ohm-m). This is the resistance per unit volume and is an inherent property of the material.

LRA

=ρ

If we were to examine two cylinders made of the same material, but with different dimensions, they would have the same electrical resistivity, but different electrical resistances. Often it is more convenient to discuss the conductivity (σ) which is measured in Siemens per metre.

σ = 1/ ρ


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B1.3 Electric current flow across a slab of material Consider an electric current (I) flowing through a slab of material with resistivity, ρ and cross-sectional area, A

Applying Ohms Law

IVR =

IV

Ax ∆=

∆ρ

Rearranging gives AI

xV ρ

=∆∆

Taking limits ρρ J===AIE

dxdV

Thus Ohms Law for a continuous medium can be written as EJ σ= where E is the electric field strength (Volts per m)


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B1.4 Charge carriers Electric current will flow through a medium as charge carriers move under an applied electric field (E). How is the resistivity (ρ) related to the number and type of charge carrier? Consider current flow through a cylinder of length L and area A.

n = number of charge carriers per unit volume q = the charge on each carrier Consider one of the charge carriers. It will accelerate under the applied electric field until it strikes an atom or another charge carrier. This will reduce the velocity, and it will then start to accelerate again. This process will repeat itself, but over a long time it will move through the material with an average velocity, v

The ease with which the charge carrier can move is described by the mobility, μ, which is defined as the drift velocity per unit electric field = v/E In a time Δt, the electric charges will move a distance Δx = vΔt. This corresponds to a volume of charge carriers = AvΔt The total charge leaving the cylinder is thus Δq = nqAvΔt

By definition, the current nqAvnqAvq=

∆∆

=∆∆

=t

tt

I

Thus current density, EnqnqvA

J µ===I

By comparison with Ohms Law, we see that σµ

ρ 11==

nq


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Thus a material will have a low electrical resistivity (high conductivity) if it has many, highly mobile, charge carriers. If several types of charge carriers are present, then the contribution from each charge carrier must be summed. B2 : Conduction mechanisms in mineral, brine and molten rock Several conduction mechanisms are possible in typical Earth materials. A list of some minerals and typical resistivity values is given on Telford, page 285. B2.1 Electronic conduction Electronic conduction occurs in pure metals. The charge carriers are electrons that exist as a gas between ions and can move very easily through the metal. Thus both μ and n are high giving a very low resistivity (~ 1.6 10-8 Ωm)

http://www.kentchemistry.com/links/bonding/metallic.htm B2.2 Semiconductors Semi conduction occurs in minerals such as sulphides. Here the charge carriers are electrons, ions or holes. Compared to metals, the μ and n are lower, and thus the resistivity is higher (typically 10-3 to 10-5 Ωm). This type of conduction occurs in igneous rocks and usually shows a temperature dependence of the form

kTE

e∝ρ

where T is the temperature in K, E is an activation energy and k is the Boltzmann constant.


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Most resistive material is diamond, with carbon atoms covalently bonded in linked tetrahedra. There are no naturally occurring charge carriers. Huge forces would be required to rip atoms from the lattice and create charge carriers. Thus both μ and n are very low, giving ρ ~ 1010 Ωm

Note that if the carbon atoms are rearranged, they form graphite (right). The 3-fold co-valent bonding leaves free electrons that can easily move along the sheets of linked hexagons. The electrons cannot move across the sheets, giving a very low resistivity parallel to the sheets. However the resistivity is higher in a direction orthogonal the sheets. This is an example of electrical anisotropy (more later!) B2.3 Ionic conduction in brines

In liquids the ions can freely move. As the salinity of a brine increases, the resistivity decreases as more charge carriers become available. Empirical studies (Block, 2001) show that the resistivity of brines (ρw ) in the Alberta Basin varies as: ρw = 4.5 TDS

-0.85 where TDS is the amount of total dissolved solids in g/litre.


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Seawater has a salinity of around 30 g/litre, and this can also be expressed as 30,000 ppm

The resistivity of a brine varies with temperature, as shown by the figure below from Ussher et al (2000). Note that:

• At constant salinity, resistivity decreases with temperature in the range 0 – 300°C. This is because the ions become more mobile as their thermal energy increases and bonding with the polar water molecules decreases.

• Above a temperature of 300°C, the resistivity of the brine increases. This is analogous to the effect seen in most metals where increasing temperature increases resistivity. This is because the increased thermal energy of ions and molecules (vibration) impedes the motion of ions through the solution.

• Geothermal fluids can have a salinity significantly higher than seawater. This can cause corrosion problems in geothermal plants.

• This temperature variation of brine resistivity is important in deep hydrocarbon reservoirs and geothermal exploration.

For more details about the resistivity of aqueous fluids, see my recent review article (Unsworth and Rondenay, 2013). This book chapter can be downloaded at : http://www.ualberta.ca/~unsworth/recent_papers.html

http://www.ualberta.ca/~unsworth/recent_papers.html


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B2.4 Ionic conduction in magma When a rock melts, ions can easily move through the liquid and carry electric current. In a silicic melt, the sodium ions are the most important charge carriers since they are smaller and more mobile than the larger potassium ions. In a silicic melt, the anions are the silica tetrahedra ( SiO4

4- ) and they have a low mobility owing to their size. Thus they do not contribute much to melt conductivity. Typical melts have an electrical conductivity in the range 0.1 – 10 S/m. The melt conductivity depends primarily on temperature, sodium content, water content and silica content. A compilation of laboratory experiments is described by Pommier and Le Trong (2011) and can be found on the SIGMELTS webpage. This can be used to calculate the resistivity of a melt of any composition. SIGMELTS : http://www.calcul.isto.cnrs-orleans.fr/sigmelts/ B3 : Resistivity of two phase systems : Archie’s Law

A typical sedimentary rock consists of mineral grains (white) and pores (green). The fraction of the rock occupied by the fluid is called the porosity (Φ). This is a number in the range 0 to 1. Possible pore fluids include water, oil, air, natural gas and melt. Just one of these fluids can present – as can a mixture.

If the pore fluid is saline water, then the resistivity of the pore fluid (ρw) will be much lower than the resistivity of the mineral grains (ρr). In this scenario, Archie (1942) discovered an empirical relationship for the resistivity of a completely saturated whole rock (ρo) is given by

m

w

o F −== φρρ

where F is called the formation factor.


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Note that the resistivity of the rock grains (ρr) does not appear in Archies’s Law. This is because it is assumed to be so high, that no conduction occurs through the rock grains. In cases where there is significant conduction in the grains, Archie’s Law is not valid. For example, in a partially molten rock, the fluid can be melt. At high temperatures there will be thermally activated conduction through the grains. A modified Archie’s Law, or other models (e.g. modified brick layer model) should be used. On a log-log plot of ρo as a function of Φ, a straight line should result with slope –m. This exponent m termed the cementation factor. Typical values include: 1.8-2.0 for consolidated sandstones to 1.3 for unconsolidated sands. The graph below is taken from Archie (1942) from Nacatoch sand from Lousiana. What is the value of m for this set of samples? What is the difference between permeability and porosity? Are they correlated?

The following plots show theoretical results when ρw = 1 Ωm


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Example of using Archie’s law

Magnetotelluric exploration of the San Andreas Fault at Parkfield in California has revealed that the fault is characterized by a wedge of low-resistivity rock. This has been interpreted a zone of breccia, termed the damaged zone (Unsworth et al, 1997). In this area the groundwater is very saline with ρw = 0.26 Ωm. What porosity is required to explain a bulk resistivity of 3 Ωm (orange zone)? Consider the possible values of m, and assume the rock is saturated.

Note that this calculation assumes that no clay is present. As we will see later, clay minerals provide an alternate conduction mechanism. Thus estimates of porosity that ignore the presence of layer will give an upper limit to the possible porosity.

Physical interpretation of the cementation factor, m

Note that the elongated pores will connect to form an interconnected electrical network at a lower porosity than the spherical pores. Is the permeability of the two cases different?


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Partial saturation The analysis above assumes that all the pore space is filled with the low resistivity water. Archie’s study was motivated by the application to hydrocarbon reservoirs, and thus partial saturation must be considered. Typically oil and gas in the pore spaces will be resistive. Additional experiments by Archie showed that the bulk resistivity (ρ) is given by

n

o

S −=ρρ

where S is the saturation, defined as the fraction of pore space filled with fluid and n is (another) empirical constant.

What value of n is implied in the figure on the left that shows S (vertical axis) as a function of ρ/ρo (horizontal axis)? Figure 3, Archie (1942)

Combining the above equations gives the general form of Archie’s Law

mnwS −−= φρρ

Archie’s Law can also be extended to include the case where some conduction occurs through the solid phase (Ussher et al, 2000; Glover et al, 2000). This is important when studying partially molten rocks, since the temperature is high enough for some conduction to occur through the unmolten grains (minerals with highest melting points) B4 : Influence of fluid geometry on bulk resistivity To emphasize how fluid distribution controls the bulk resistivity, consider the two rock samples that both have Φ =0.1 that comprises a set of cracks. In the first example the rock (1000 Ωm) and fluid (0.3 Ωm) form a series electrical circuit. In the second case, the rock and fluid form a parallel circuit. What is the overall electrical resistance of each case? Use the space next to each figure to calculate the resistance of the cube (1 m x 1 m x 1 m).


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This illustrates that the anisotropy of rock can be an important factor in determining the overall resistivity. The resistivity can also be defined for intermediate values of fluid fraction (porosity) and details will be investigated on Assignment 1 in Geophysics 424. The results are sketched below for the case of ρrock = 1000 Ωm and ρf = 0.3 Ωm.


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Note that at low porosity (low fluid fraction) Archie’s Law for m =2 predicts a resistivity greater than 1000 Ωm. This is clearly impossible, and shows that Archie’s Law can be invalid at low porosity, with poorly connected pore fluids. With limited fluid present, conduction through the mineral grains will be significant. We can also define the Hashin-Shtrikman bounds that are closely related to the above example. These are (approximately) the maximum (HS +) and minimum (HS -) possible resistivities for a given amount of fluid in a rock. Review question : What factors can cause Archie’s Law to be invalid?


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B5 : Clay minerals

Clay has two meanings. It can refer to certain sheet silicate minerals, and also to sediment that has particles smaller than 1/256 mm. These definitions are not independent as weathering of clay minerals usually forms small particles. If a rock contains clay minerals, then an extra conduction pathway is possible via the electrical double layer that forms at the interface of the clay mineral and the water.

This effectively allows ions to move through the system with a lower effective viscosity than in the liquid phase.

• Some cations are held tightly right next to the surface • Others held more loosely further away • Anion exclusion zone next to surface • Soluble cations are reached only when concentrations of cations and anions

are equal http://faculty.plattsburgh.edu/robert.fuller/370%20Files/Week5Ion%20Exchange/Electricdoublelayer.htm Waxman and Smits (1968) developed an equation for the resistivity due to conduction through both the liquid and the double layer

)1(w

vBQ

F

ρ

ρ+

=

where B is the equivalent conductance of the ions in solution and F is the formation factor and

φρ

φ mv CECQ )1( −=

with ρm the matrix grain density and CEC the cation exchange capacity of the clay. Thus the term BQv is a measure of how much the clay contributes to conduction.

http://faculty.plattsburgh.edu/robert.fuller/370%20Files/Week5Ion%20Exchange/Electricdoublelayer.htm


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Values of CEC vary from one type of clay to another. For example: smectite, CEC = 120; illite, CEC = 20. Application to geothermal exploration In geothermal exploration, a clay cap is often found above the geothermal reservoir. The clay minerals are formed as the hot water causes alteration of the rock. Smectite has a low resistivity, owing to the high value of CEC. At temperatures above 200° C the most stable mineral is illite. This has a higher resistivity owing to the reduced CEC. The clay cap has a low permeability and traps the high temperature water in the geothermal reservoir below. The figure below from Ussher et al., (2000) shows how this change in resistivity can be measured at a geothermal field in Indonesia. The electrical resistivity was measured by both (a) a well log shown in the left column and (b) a surface based magnetotelluric survey shown in the middle column. Note that both methods show a low resistivity layer that is coincident with the smectite zone (right column).

The depth to the base of the smectite layer can thus be determined by measuring electrical resistivity. This in turn can be used to infer the depth of the 200°C isotherm and often the top of the geothermal reservoir. This exploration strategy is effective once a well has been completed and logged. Extrapolation away from the well can be done with surface based geophysics, and is much cheaper than drilling multiple wells.

More about clay minerals and double layers at: http://www.cartage.org.lb/en/themes/sciences/Chemistry/Electrochemis/Electrochemical/ElectricalDouble/ElectricalDouble.htm http://www.zeta-meter.com/5min.pdf

http://www.cartage.org.lb/en/themes/sciences/Chemistry/Electrochemis/Electrochemical/ElectricalDouble/ElectricalDouble.htm

http://www.cartage.org.lb/en/themes/sciences/Chemistry/Electrochemis/Electrochemical/ElectricalDouble/ElectricalDouble.htm

http://www.zeta-meter.com/5min.pdf


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B6 : Other causes of high electrical conductivity (low electrical resistivity) In the upper 10 km, aqueous pore fluids dominate the resistivity of most rocks. However, other materials, even in small amounts, can dramatically lower the resistivity of a rock. These include: Graphite films: Proposed as an explanation of why the lower crust often has a high conductivity (Frost et al., 2000). This requires interconnection of the graphite films, which may be difficult to maintain over geological time scales. See the debate by Yardley and Valley (1997) and Wannamaker (2000). These papers refer to lot of prior literature on an important research question. The alternative explanation of the lower crustal, low resistivity is aqueous fluids. Iron oxides and metallic sulphides: Usually localized in a discrete body, and allows detection of these deposits with airborne EM methods (see later in this class).

Partial melting: Rocks are complex mineral assemblages, so they don’t melt at one temperature. In a partial melt, molten rock collects along the boundaries of mineral grains that melt at higher temperature. Typical resistivities of partial melts are shown on the right and taken from Shankland and Waff (1977). These values are for dry rocks. Wet rocks melt at lower temperatures and the water generally raises the conductivity.

The wetting angle (θ) determines the shape of the melt pockets and is controlled by the surface energies.


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Consider a drop of liquid on a glass surface (S). If the value of θ is large (A and B) the liquid stays in a drop. If θ is smaller (C) the drop spreads out and wets the surface. Figures above taken from http://en.wikipedia.org/wiki/Wetting where more details can be found. Similar effects occur when melt is in contact with mineral grains in a partial melt. If the wetting angle is relatively small (< 60°), the melt will be distributed in tubes along the grain boundaries. If θ > 60° then the melt will remain in localized pockets.

High wetting angle (5% melt) Low wetting angle (5% melt)

Laboratory experiments (suggest that partial melts seem to interconnect at low melt fractions, with a relatively low wetting angle (see figure on left from ten Grotenhuis et al., 2005). Thus Archie’s Law can be used to calculate the overall resistivity with m = 1.3 (ten Grotenhuis et al., 2005) or alternatively the HS + bound can be used (Li et al., 2003).

Partial melting is believed to be widespread in the asthenosphere and beneath geothermal fields and volcanoes. Partial melting may also occur in the crust tectonically active areas (Tibet, Altiplano) where MT surveys reveal a zone of low resistivity in the crust.

The magnetotelluric (MT) surveys image a layer around 3 Ωm. Recent laboratory studies have confirmed that these resistivities are consistent with a wet, granitic melt (Gaillard et al, 2004). Tectonic processes and erosion extrude these rocks in the Himalaya, some 10-12 million years later.

http://en.wikipedia.org/wiki/Wetting


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Beaumont et al., 2001

Gaillard et al., (2004) Diffusion of hydrogen : At the high temperatures found in the upper mantle, water molecules dissociate into ions. The hydrogen ions (H+) can act as charge carriers and lower the resistivity from the high values expected for dry olivine. Experiments by different research groups gives very different estimates of the change in olivine resistivity for a given concentration of H+ (Karato, 1990, Wang et al., 2006, Yoshino et al., 2009). These differences are significant, and support very different interpretations of upper mantle conductivity (e.g. Karato (1990) proposes that only H+ is needed, while Yoshino et al., (2006) propose that partial melt is needed. The anisotropy of olivine may cause different diffusion rates in different directions and thus produce electrical anisotropy that is anisotropic (Simpson and Tommasi, 2005). Grain boundary effects : Some conduction mechanisms occur along grain boundaries. Thus a change in grain size could enhance the conductivity. Could this explain upper mantle conductivity variations? See ten Grotenhuis et al., (2004) for a discussion.


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B7 : Summary of resistivity of rocks

Factors that will DECREASE the resistivity of a rock

(a) Add more pore fluid (b) Increase the salinity of the pore fluid - more ions to conduct electricity (c) Fracture rock to create extra pathways for current flow (d) Add clay minerals (e) Keep fluid content constant, but improve interconnection between pores

Factors that will INCREASE the resistivity of a rock

(a) Remove pore fluid (b) Lower salinity of pore fluid (c) Compaction - less pathways for electric current flow (d) Lithification - block pores by deposition of minerals (e) Keep fluid content constant, but decrease connection between pores


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Crust and mantle

• Crust/mantle definition is based on chemical properties of rocks • Crust / mantle separated by the Moho • Thickness of continental crust varies from 20 – 90 km

Lithosphere-asthenosphere

• Lithosphere-asthenosphere definition based on mechanical properties • Separated by lithosphere-asthenosphere boundary (LAB) • Depth range of LAB is 150 – 250 km

Causes of high / low resistivity by layer

(a) Sedimentary basins : Low resistivity due to sedimentary rocks with significant porosity and saline pore fluids. Resistivity rises with depth due to compaction, and reduction of porosity.

(b) Upper crust: High resistivity due to low porosity crystalline rocks (igneous and metamorphic).

(c) Lower crust : Low resistivity. Fluids or graphite? (d) Upper mantle : Intermediate resistivity (e) Asthenosphere : Low resistivity, few percent partial melt.


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B8 : References Archie GE, The electrical resistivity log as an aid in determining some reservoir characteristics,

Trans. Am. Inst. Min. Metall. Pet. Eng., 146, 54-62, 1942. Beaumont C, RA Jamieson, MH Nguyen and B Lee, Himalayan Tectonics explained by extrusion

of a low-viscosity crustal channel coupled to focused surface denudation, Nature, 414, 738-742, 2001.

Block D, 2001, Water Resistivity Atlas of Western Canada Abstract, paper presented at Rock the Foundation Convention of Canadian Society of Petroleum Geologists, June 18-22, Calgary.

Frost RB, WS Fyfe, K Tazaki and T Chan, Grain-boundary graphite in rocks and implications for high electrical conductivity in the lower crust, Nature, 340, 134-136, 2000.

Gaillard F, B Scaillet, M Pichavant, Evidence for present-day leucogranite pluton growth in Tibet, Geology, 32, 801-4, 2004.

Glover P, MJ Hole and J Pous, A modified Archie’s Law for two conducting phases, Earth and Planetary Letters, 180, 369-383, 2000.

Karato S, The role of hydrogen in the electrical conductivity of the upper mantle, Nature, 347-348, 1990.

Li S, MJ Unsworth, JR Booker, W Wei, H Tan and AG Jones, Partial melt or aqueous fluids in the Tibetan crust: constraints from INDETH magnetotelluric data, Geophys. J. Int., 153, 289-304, 2003.

Pommier A, E Le Trong, SIGMELTS : A web portal for electrical conductivity calculations in geosciences, Computers and Geosciences, 37, 1450-1459, 2011.

Shankland TJ and HS Waff, Partial melting and Electrical conductivity anomalies in the upper mantle, J. Geophys. Res., 82, 5409-5417, 1977.

Simpson F, and A Tommasi, Hydrogen diffusivity and electircal anisotropy of a peridotite mantle, GJI, 160, 1092-1102, 2005.

ten Grotenhuis SM, MR Drury, CJ Spiers, CJ Peach, Melt distribution in olivine rocks based on electrical conductivity measurements, J. Geophys. Research, 110,B12201, 2005.

ten Grotenhuis SM, MR Drury, CJ Peach, CJ Spiers, Electrical properties of fine-grained olivine: Evidence for grain boundary transport, JGR, 109, B06203, doi:10.1029/2003JB002799, 2004.

Unsworth MJ, PE Malin, GD Egbert and JR Booker, Internal Structure of the San Andreas Fault Zone at Parkfield, California, Geology, 25, 359-362, 1997.

Unsworth MJ, S Rondenay, Mapping the distribution of fluids in the crust and lithospheric mantle utilizing geophysical methods, Chapter 13 in Metasomatism and Metamorphism: The Role of Fluids in Crustal and Upper Mantle Processes, 535-598, edited by DE Harlov and H Austrheim, Springer-Verlag Berlin Heidelberg, Lectures in Earth Sciences, doi 10.1007/978-3-642-28394-9_13, 2013.

Ussher G, C Harvey, R Johnstone, E Anderson, Understanding the resistivities observed in Geothermal systems, Proceedings World Geothermal Congress, Kyushu, Japan, 2000.

Wang D, M Mookherjee, Y Xu, S Karato, The effect of water on the electrical conductivity of olivine, Nature, 977-980, 2006.

Wannamaker PE (2000), Comment on “The petrologic case for a dry lower crust” by Bruce W. D. Yardley and John W. Valley, J. Geophys. Res., 105(B3), 6057–6064.

Waxman MH, LJ M. Smits, Electrical Conductivities in Oil-Bearing Shaly Sands, Society of Petroleum Engineers Journal, 243, 107-122, 1968.

Yardley B and J Valley, The petrologic case for a dry lower crust, J.Geophys. Res., 102, 12,173, 1997.

Yoshino T, T Matsuzaki, A Shatskiy, T Katsura, The effect of water on the electrical conductivity of olivine aggregates and its implications for the electrical structure of the upper mantle, Earth and Planetary Science Letters, 288, 291-300, 2009.

Yoshino T, T Matsuzaki, S Yamashita, T Katsura, Hydrous olivine unable to account for conductivity anomaly at the top of the asthenosphere, Nature, 443, 973-976, 2006.

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