Borehole Geophysics Electrical measurements, electromagnetic and ...

Borehole Geophysics

Electrical measurements, electromagnetic and magnetic logging Electrical logging was really the beginning of the entire well-logging business, when the two Schlumberger brothers detected in 1929 that measuring the electrical resistivity can be useful for the detection of hydrocarbons. Since 1929, logging has taken it’s own path and nowadays there are many different types of applications based on electrical resistivity, ranging from hydrocarbon detection to hydro-geology and base-metal exploration.

Much of the theory presented in this lecture is part of EPSC-320 (Elementary Earth Physics) and EPSC-435 (Applied Geophysics).

EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])

Slide S6-1

Borehole Geophysics Motivation Electrical resistivity of rocks in the upper portion of the Earth’s crust varies with several factors and measurements of resistivity can reveal much about the nature of the rocks under investigation: (1) Water content: natural water is much more conductive that most rockforming minerals (2) Salinity: natural water is conductive in proportion to the concentration of salts in the water (3) Temperature: water conductivity increases with temperature, thus a rock at a depth of 1 km can be twice as conductive as the same rock at the surface. (4) Conductive minerals: Electric conduction in sulfide and oxide minerals can dominate the measurements if they are abundant in sufficient concentration. (5) Clays: Clays can augment the ionic conduction of pore-water. (6) Geologic strike of formation: Many sediments and metamorphic rocks are anisotropic; they have a lower resistivity along the bedding plane than perpendicular to it. EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])

Slide S6-2

Borehole Geophysics Some basics … reminder from EPSC-435 Definition of current (I): Charge (q) per time (t): I=q/t

(EQ 6.1)

Definition of current density (J): current (I) divided by sectional area (A): J=I/A

Burger et al., 2006

Ohm’s Law: I = U / R, with U as voltage, and R as resistance. (EQ 6.3)

(EQ 6.2) Diagram illustrating current density J in wires with different crosssectional area. Current flow is represented by arrows

Burger et al., 2006 EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])

Slide S6-3

Borehole Geophysics Some basics … reminder from EPSC-435

Various geologic materials can be expected to have different resistances to current flow. A complication is that resistance depends not only on the material itself but also its dimensions.

Burger et al., 2006

Diagram illustrating two wires with different cross-sectional area A and length l.

Consider the diagram to the left. Although both wires are made of the same material, their resistance is different. The resistance (R) depend on length (L) and cross-sectional area (A) as well as a fundamental material property, referred to as resistivity (ρ) and can be described as follows: R=ρ⋅L/A

(EQ 6.4)

Resistivity units are ohm-meter (Ω⋅m). EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])

Slide S6-4

Borehole Geophysics The description of electric (and magnetic) fields is very similar to what we learned in EPSC-320 about gravity.

We define an electric field E as the force F imposed on a static electric charge q (letters in bold identify vector-quantities): E=F/q

(EQ 6.5)

The units of E are volts per meter (V/m). The field can be expressed as the gradient of a potential Ψ: E = - grad Ψ

(EQ 6.6)

And we define a potential difference V between two points P1 and P2 in a field E as the work done moving a unit charge between those two points in the field:

K K V = − ∫ E ⋅ ds P2

P1

(EQ 6.7)

[note the bold letter was replaced with the vector symbol in equation 6.7] EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])

Slide S6-5

Borehole Geophysics

The current density J, given in amperes per square meter (A/m2) is directly proportional to E:

J=CE=E/R

(EQ 6.8)

Where C is the conductivity of the material, expressed in Siemens per meter (S/m). The resistivity of a material R is 1/C, in units of ohmmeter.


Slide S6-6

Borehole Geophysics Electric properties are linked to magnetic properties Three distinct vectors are used to describe magnetic fields: B

magnetic field

H

magnetic field strength

M

magnetic polarization

[see next slide for alternative names and units and definitions] If a charge q is moving with velocity v, the total force acting on it is:

F = qE + q v × B

(EQ 6.9)

The vector v×B is orthogonal to both v and B. The product of q with this vector v×B is called Lorentz force.


Slide S6-7

Borehole Geophysics

Moving electric charges of velocity V represent a current. Those currents generate a magnetic field, which depends on the velocity of the moving charges as all as on the strength of the electric field they represent:

K K 1 K B =V × 2 E c

(EQ 6.10)

Faraday's law of induction states that a magnetic field changing in time creates a proportional electromotive force:

K K ∂B ∇× E = − ∂t

(EQ 6.11)

Right-hand rule: A current (I) produces a secondary magnetic field (B). Use your right hand and stretch the thump into the direction of the current. Your remaining fingers point into the direction of the circular magnetic field.


Slide S6-8

Borehole Geophysics

Symbol

Meaning

SI Unit of Measure

electric field

volt per meter or, equivalently, newton per coulomb

magnetic field also called the auxiliary field

ampere per meter

electric displacement field also called the electric flux density

coulomb per square meter

magnetic flux density also called the magnetic induction also called the magnetic field

tesla, or equivalently, weber per square meter

free electric charge density

coulomb per cubic meter

free current density, not including polarization or magnetization currents bound in a material

ampere per square meter

differential vector element of surface area A, with infinitesimally small magnitude and direction normal to surface S

square meters

differential element of volume V enclosed by surface S

cubic meters

differential vector element of path length tangential to contour C enclosing surface S

meters


Slide S6-9

Borehole Geophysics Maxwell’s equations

Name

Differential form

Integral form

Gauss's law:

Gauss' law for magnetism (absence of magnetic monopoles):

Faraday's law of induction:

Ampère's law (with Maxwell's extension):

(EQ 6.12) EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])

Slide S6-10

Borehole Geophysics

Resistivity of natural water Resistivity of natural water varies strongly with rainwater being the most resistive (least conductive) and brine being the most conductive (least resistive). The resistivity of pure water without any other ions but H+ and OH- is 2.8 × 105 ohm-meter at 17.6ºC. The six most common ions in natural waters are Na+, Ca+, Mg2+, Cl-, HCO3-, and SO42-. For oceanographic purposes, the relationship between density, resistivity (conductivity), pressure and temperature is defined by the UNESCO equation of state of seawater (Foffonoff, 1985). The equations are rather complex and for simple application it is the easiest to use one of the many online calculators, such as this one: http://fermi.jhuapl.edu/denscalc.html


Slide S6-11

Borehole Geophysics

Electric current in aqueous solutions is carried by the ions of the dissociated salts migrating through the fluid in response to an applied electric field. The current density J due to a single ionic species with ni ions per cubic centimeter is given by: Ji = ni zi evi

(EQ 6.13)

Where z denotes the valence (numbers of electrons per ion), e is the electronic charge per electron and v is the drift velocity of the ion. It is customary to express the ionic density n in terms of concentration c (moles per cubic centimeter) by using: n = c Fa / e

(EQ 6.14)

Where Fa here stands for the Faraday constant (96500 Coulomb / mol). Using the definition of conductivity C from before, we get: Ci = ci F zi vi EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])

(EQ 6.15) Slide S6-12

Effect of salinity and temperature on conductivity of seawater. Note that conductivity increases with T and salinity!

Conductivity (mS/cm)

Borehole Geophysics

http://www.saddleback.edu/faculty/thuntley/ms20labs/MS20salinityLAB.pdf

Salinity (ppt) EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])

Slide S6-13

Borehole Geophysics

Resistivity of minerals In contrast to ionic conduction in fluid-filled pore space, electric current in the rock matrix is carried by electrons. In most rocks electrical conduction in the matrix is negligible because silicates and carbonates are insulators, having resistivities of > 107 ohm-m.

Source: Shuey, 1975, from Hearst et al., 2000

Very broad range of resistivities for any given mineral. Wide range is result from mineral impurities, structural defects and anisotropy.


Slide S6-14

Borehole Geophysics Resistivity of minerals and sulfide bearing rocks

Examples of resistivities obtained from sulfide-bearing rock samples (open pit copper mine). Each data point denoted by letter for various samples represents several cubic meter of rock. Note the extreme spread of data across several orders of magnitude for the same type of rock from a single mine.

Source: Nelson and van Voorhis, 1983, from Hearst et al., 2000


Slide S6-15

Borehole Geophysics

Archie’s empirical relationship Electrical properties of rocks are controlled by the pore-fluids because the resistivity of the silicate minerals is extremely high compared with that of the fluids. Silicates: > 107 ohm-m Salt-water at 20 deg C (35 ppt) : ~ 0.2 ohm-m

The most influential study of rock resistivity was Archie’s (1942) examination of sandstone cores from the Gulf Coast region. Archie established that the resistivity of a core sample (R0) fully saturated with brine is proportional to the brine resistivity Rw: R 0 = F Rw

(EQ 6.16)

with F being called the formation factor. EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])

Slide S6-16

Borehole Geophysics

Archie’s empirical relationship By determining the formation factor and porosity (φ) on samples, Archie further established the following relationship: F = R0/Rw = φ-m

(EQ 6.17)

The exponent m was found to range between 1.8 and 2.0 in consolidated sandstones and to decline to values around 1.3 in clean unconsolidated sands. Because of its dependence on the state of consolidation, m is often referred to as cementation exponent.


Slide S6-17

Borehole Geophysics Archie’s empirical relationship Source: Archie (1942)

Relation of formation factor to porosity for (a) consolidated sandstones from the Gulf coast and (b) lowpermeability sandstone from Louisiana. Shown on the left are the original data from Archie’s paper form 1942.


Slide S6-18

Borehole Geophysics Archie’s empirical relationship Modifications to Archie’s original equation Additional studies showed that the original equation was too strict and no reasonable fit to the formation-factor/porosity plot could be achieved. An extra order of freedom was gained by introducing a scalar a to the equation: F = R0/Rw = a φ-m

(EQ 6.18a)

R0 = a Rw φ–m

(EQ 6.18b)

From the definition of F we would expect F is equal to 1 at 100% porosity (only water, no rock). It is common to determine the two parameters a and m statistically from measurements of porosity and F and not restrict a=1 (which would imply that F=1 at 100% porosity).


Slide S6-19

Borehole Geophysics Archie’s empirical relationship Effect of partial saturation If part of the pore space is occupied by non-conducting fluid (e.g. oil, gas, gas hydrate) the water saturation Sw is less than 1 and the rock resistivity increases. Experimental results show that a power-law relation is a useful approximation: Rt/R0 = Sw-n

(EQ 6.19)

or Rt = a Rw φ-m Sw-n

(EQ 6.20)

Using the previously defined Archie-relationship. Rt is the ‘true’ resistivity of the partially saturated rock sample, and R0 is the resistivity of the same rock when fully water saturated. The saturation exponent n varies with rock type, but is usually defined from discrete measurements on core samples. EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])

Slide S6-20