# 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
(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.
EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])
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 &times; B
(EQ 6.9)
The vector v&times;B is orthogonal to both v and B. The product of q with this
vector v&times;B is called Lorentz force.
EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])
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 &times; 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
∇&times; 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.
EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])
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
EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])
Slide S6-9
Borehole Geophysics
Maxwell’s equations
Name
Differential form
Integral form
Gauss's law:
Gauss' law for magnetism
(absence of magnetic
monopoles):
Amp&egrave;re'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 &times; 105
ohm-meter at 17.6&ordm;C.
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
EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])
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
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 &gt; 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.
EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])
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
EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])
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: &gt; 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.
EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])
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.
EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])
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).
EPS-550 / Winter-2008 – Professor Michael Riedel ([email protected])
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
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