Structure of lecture

Simple resistor in circuit Ohm’s Law states that for a resistor, the resistance (in ohms), R is defined as V = voltage (volts); I = current flow (amps) 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

RESISTIVITY

1 Basic physics of electric current flow

2 Resistivity of rocks

3 Equation on resistivity surveying

4 Summary of resistivity methods: case histories

5 Conclusions

Structure of lecture

1 Basic physics of electric current flow

 Simple resistor in circuit

Ohm’s Law states that for a resistor, the resistance (in ohms), R is defined as

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

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

Electric current flow in a finite volume

 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

1 Basic physics of electric current flow

 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.

 = 1/ 

 Resistivity is the physical property which determines the aptitude of this material to be opposed to the passage of the electrical current

1 Basic physics of electric current flow

Electric current flow across a slab of material

Consider an electric current (I) flowing through a slab of material

=>

1 Basic physics of electric current flow

Electronic conductibility

 The current flows by displacement

of electrons Known as electronic

conductibility or metallic because

it is a similar conductibility to that

of metals This solid conductibility

is really significant only for certain

massive mineral deposits.

1 Basic physics of electric current flow

Electronic conductibility

 The current is carried by ions The

electrical resistivity of rocks

bearing water is controlled mainly

by the water which they contain.

2 Resistivity of rocks

Electronic conductibility

The resistivity of a rock will depend:

 on the resistivity of the natural pore water and consequently

the quantity of dissolved salts in the electrolyte

2 Resistivity of rocks

Archie´s Law

 Pure materials are rarely found in the Earth and most rocks are a mixture of two or more phases (solid, liquid or gas) Thus to calculate the overall electrical resistivity of a rock, we must consider the individual resistivity and then compute the overall electrical resistivity Consider a sandstone saturated with salt water The grains are quartzite and have a high resistivity (>

1000 ohm-m).

 In contrast, the pore fluid is conductive (~1 ohm-m).

 To compute the overall electrical resistivity, we must consider current flow through each phase However, given the much higher resistivity of the grains, most current will flow through the water, with ions as the charge carriers.

Archie´s Law

ρ resistivity of the rock

ρ w resistivity of the fluid (water)

Φ porosity

S saturation in water

a factor which depends of the lithology (varies between 0.6 and 2)

m cementation factor (depends of the pores shape, of the

compaction and varies between 1.3 for unconsolidated sands to 2.2for cemented limestone

n about 2 for majority of the formations with normal porosities

containing water between 20 and 100 %

2 Resistivity of rocks

Formation factor F

 = wa -mS-n

 For sand and sandstones: F ≈ 0.62/2.15

 For well cemented rocks: F ≈ 1/2

2 Resistivity of rocks

Physical interpretation of the cementation factor, m

Note that the elongated pores

interconnected electricalnetwork at a lower porositythan the spherical pores Isthe permeability of the twocases different?

The above discussion showsthat the resistivity of a fluidsaturated rock depends on the

amount of fluid and it’s distribution (degree of

2 Resistivity of rocks

To emphasize this point, consider the two limiting cases of the fluid distribution

A – Fluid in cracks parallel to electric current flow

The sample has 10% porosity This

fluid geometry represents a parallel

effectively bypass the resistive rock grains and travel through the sample entirely in the conductive liquid.

What is the overall resistivity of the cube?

2 Resistivity of rocks

To emphasize this point, consider the two limiting cases of the fluid distribution

B – Fluid in cracks perpendicular to the electric current flow

The sample has 10% porosity This

geometry represents a series circuit,

and electric current cannot effectively bypass the resistive rock grains It

conductive liquid and resistive rock grains.

2 Resistivity of rocks

Result of influence of fluid geometry on bulk resistivity

2 Resistivity of rocks

Resistivity of rocks and minerals

Air, gas or oil: infinite or very high resistivity!

Liquid materials from landfills are generally conductive (<10 ohm.m)

2 Resistivity of rocks

Effect of clay and graphite

 Clay has a high ionic exchange capacity, therefore the conductivity of the pore fluid largely increases

Archie´s Law is not valid if clay is present!

 Graphite, often associated with pyrite, makes the resistivity decrease

 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 (higher mobility) than in the liquid phase

2 Resistivity of rocks

Effect of clay and graphite

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.

2 Resistivity of rocks

Effect of clay and graphite

Waxman and Smits (1968) developed an equation for theresistivity due to conduction through both the liquid and the doublelayer

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

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

Values of CEC vary from one type of clay to another Forexample: smectite, CEC = 120; illite, CEC = 20

2 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) Lithofication - block pores by deposition of minerals

(e) Keep fluid content constant, but decrease connection between pores

2 Resistivity of rocks

The conductivity of a rock increases if…

 The quantity of water increases

 The salinity increases (quantity of ions)

 The quantity of clay increases

 The temperature increases

The low resistivity phase dominates the overall resistivity

Overall resistivity is very sensitive to the geometry(distribution of the fluid)

2 Resistivity of rocks

More time to fill in fractures and pore space

Abundant fractures and/or pore space

3 Equation on resistivity surveying

3.1 Natural potentials and currents

 Electrical investigations of natural electrical properties are based onthe measurement of the voltage between a pair of electrodesimplanted in the ground Natural differences in potential occur inrelation to subsurface bodies that create their own electric fields.The bodies act like simple voltaic cells; their potential arises from

electrochemical action Natural currents (called telluric currents)

flow in the crust and mantle of the Earth

 In studying natural potentials and currents the scientist has nocontrol over the source of the signal This restricts theinterpretation, which is mostly only qualitative The naturalmethods are not as useful as controlled induction methods, such asresistivity and electromagnetic techniques, but they are inexpensive

Self-potential (spontaneous potential)

 A potential that originates spontaneously in the ground is called a

Some self-potentials are due to man-made disturbances of the

environment, such as buried electrical cables, drainage pipes orwaste disposal sites They are important in the study ofenvironmental problems Other self-potentials are natural effectsdue to mechanical or electrochemical action In every case thegroundwater plays a key role by acting as an electrolyte

3 Equation on resistivity surveying

Self-potential (spontaneous potential)

Some self-potentials have a mechanical origin When an electrolyte

is forced to flow through a narrow pipe, a potential difference(voltage) may arise between the ends of the pipe Its amplitudedepends on the electrical resistivity and viscosity of the electrolyte,and on the pressure difference that causes the flow The voltage is

due to differences in the electrokinetic or streaming potential,

which in turn is influenced by the interaction between the liquid

and the surface of the solid (an effect called the zeta-potential) The

voltage can be positive or negative and may amount to somehundreds of millivolts This type of self-potential can be observed

in conjunction with seepage of water from dams, or the flow ofgroundwater through different lithological units

3 Equation on resistivity surveying

Self-potential (spontaneous potential)

Most self-potentials have an electrochemical origin For example, if

the ionic concentration in an electrolyte varies with location, theions tend to diffuse through the electrolyte so as to equalize the

concentration The diffusion is driven by an electric diffusion

potential, which depends on the temperature as well as the

difference in ionic concentration When a metallic electrode isinserted in the ground, the metal reacts electrochemically with theelectrolyte (i.e., groundwater), causing a contact potential If twoidentical electrodes are inserted in the ground, variations inconcentration of the electrolyte cause different electrochemicalreactions at each electrode A potential difference arises, called the

Nernst potential The combined diffusion and Nernst potentials are

called the electrochemical self-potential.

3 Equation on resistivity surveying

Self-potential (spontaneous potential)

A schematic model of the origin of the self-potential anomaly of an ore body The mechanism

3 Equation on resistivity surveying

SP surveying

 The equipment needed for an SP survey is very simple It consists

of a sensitive high-impedance digital voltmeter to measure thenatural potential difference between two electrodes implanted in theground Simple metal stakes are inadequate as electrodes.Electrochemical reactions take place between the metal andmoisture in the ground, causing the build-up of spurious charges onthe electrodes, which can falsify or obscure the small natural self-potentials

3 Equation on resistivity surveying

SP surveying A Gradient method (fixed electrode spacing)

B Total field method (fixed base)

Two field methods are in

common use The field

techniques of measuring

self-potential by (a) the

gradient method and (b)

the total field method The

SP surveying A Gradient method (fixed electrode spacing)

The gradient method employs a

fixed separation between the

electrodes, of the order of 10 m.

The potential difference is

measured between the electrodes,

then the pair is moved forward

along the survey line until the

trailing electrode occupies the

location previously occupied by

the leading electrode The total

potential at a measurement

station relative to a starting point

outside the study area is found by

summing the incremental

potential differences.

3 Equation on resistivity surveying

This gives rise to a small error in each measurement; these add up to a cumulative error in the total potential Cumulative error

is the most serious disadvantage of the fixed electrode configuration A practical advantage of the technique is that only a short length of connecting wire must be moved along with the electrodes.

SP surveying B Total field method (fixed base)

The total field method utilizes

a fixed electrode at a base

station outside the area of

measuring electrode With

this method the total potential

is measured directly at each

station

3 Equation on resistivity surveying

The total field method results in smaller cumulative error than thegradient method It allows more flexibility in placing the mobileelectrode and usually gives data of better quality Hence, the totalfield method is usually preferred except in difficult terrain

SP surveying

Hypothetical contour lines of a

negative self-potential anomaly over

an ore body; the asymmetry of the

anomaly along the profile AB

suggests that the ore body dips

toward A.

3 Equation on resistivity surveying

3.2 Resistivity surveying

 The large contrast in resistivity between ore bodies and their hostrocks is exploited in electrical resistivity prospecting, especially forminerals that occur as good conductors Representative examplesare the sulfide ores of iron, copper and nickel Electrical resistivity

environmental applications For example, due to the good electricalconductivity of groundwater the resistivity of a sedimentary rock ismuch lower when it is waterlogged than in the dry state

3 Equation on resistivity surveying

Potential of a single current electrode

 In the lab, the electrical resistivity of a rock sample can bemeasured by placing flat electrode plates on each side of arectangular sample In this geometry the electric current flow isparallel and the simple equation derived can be used to compute theresistivity, ρ from the measured resistance (R)

 However this approach is not practical for the measuring the Earth,since we cannot inject current from large plates Thus we mustconsider the electric current flow from a simple electrode (metalspike)

3 Equation on resistivity surveying

Potential of a single current electrode

 Consider an electric current, I, flowing from an electrode The airhas a very high electrical resistivity, so all current flows in theEarth From symmetry arguments, the current spreads outuniformly in all directions Now consider a shell of rock, with

radius, r, and thickness dr The voltage (potential) drop across the

shell is ΔV

3 Equation on resistivity surveying

Potential of a single current electrode

 The resistance of the hemispherical shell,

 Rearranging and taking limits gives

 To compute the potential, V, apply the boundary condition that V =

0 when r = ∞ and integrate to give:

3 Equation on resistivity surveying