Geophysics 223, 2009
Geophysics 223 - Final exam formula sheet
B : Electrical resistivity methods
Ohm’s Law
V = IR
V = voltage (V) ; I = current (A); R = resistance (Ω)
J = electric current density (Am-2); E = electric field (V/m);
σ = conductivity (S/m);
J = σE
Voltage around a single current electrode
Voltage (V) due to a single current electrode:
V =−
Iρ
2πr
ρ = resistivity of Earth (Ωm); I = current (A); r = distance from electrode (m);
Resistivity and resistance
R=
ρL
A
R = resistance (Ω); ρ = resistivity (Ωm); L = length of object (m); A = area (m2)
Apparent resistivity for a Wenner array
ρa =
2πaΔV
I
a = electrode spacing (m);
ΔV = voltage measured across potential electrodes (volts);
I = current flowing through Earth (amps)
Archie’s Law
The resistivity (ρ) of a rock that contains fluid with resistivity (ρw) can be
calculated with Archie’s Law. The rock has a porosity (Ф) and saturation (S).
The cementation factor is in the range 1 < m < 2;
S = saturation exponent; n = 1
ρ = ρ w S − nφ − m
Resistivity of brines
-0.85
ρw = 4.5 TDS
;
TDS is the amount of total dissolved solids in g/litre;
ρw = water resistivity (Ωm);
1
Geophysics 223, 2009
C : Magnetic methods
B = magnetic flux density;
μ = magnetic permeability;
H = magnetic field strength;
For free space μ = μ0 = 4π x 10-7 H/m
B= μH
At high magnetic latitude, half-width of total magnetic field anomaly
Monopole : x 12 = 0.766 z
Dipole : x 12 = 0.5 z
D : Electromagnetic methods
Skin depth
δ = skin depth
δ =
500
where f = frequency and σ = conductivity
σf
Low-induction number EM systems
Apparent conductivity of a 2-layer Earth with conductivities σ1 and σ2, separated by an
interface at depth d
_
_
Apparent conductivity σ v = σ 1 (1 − Rv ( z )) + σ 2 Rv ( z )
RV ( z ) =
z=
σ h = σ 1 (1 − Rh ( z )) + σ 2 Rh ( z )
1
(4 z + 1)
2
1
R H ( z ) = (4 z 2 + 1)
2
d
s
1
2
− 2z
s = TX-RX distance
h : denotes vertical dipoles
v : denotes horizontal dipoles
Time domain EM systems
3
5
dB z (t ) Iμ 0 A ⎛ μ 0σ ⎞ 2 − 2
=
⎜
⎟ t
dt
20 ⎝ π ⎠
Decay at late time
Bz = magnetic field (T);
I = transmitter current (amps)
A = transmitter area (m2);
σ = conductivity of Earth (S/m)
1
Depth of penetration at time t
⎡ 2t ⎤ 2
δT = ⎢
⎥
⎣ σμ 0 ⎦
2
Geophysics 223, 2009
E: Ground penetrating radar
Velocity of radar wave
c
v=
c = speed of light in vacuum = 0.3 m/ns
μrε r
εr = relative (dielectric) permittivity
μr = relative (magnetic) permeability
Variation of soil moisture content (θv) with relative permittivity (εr)
ε r = 3.03 + 9.3θ v + 146θ v 2 − 76.7θ v 3
θ v = −5.3 x10 −2 + 2.92 x10 −2 ε r − 5.5 x10 −4 ε r 2 + 4.3 x10 −6 ε r 3
Reflection and transmission at normal incidence
R=
Ar v 2 − v1
=
Ai v 2 + v1
T=
At
2v1
=
Ai v 2 + v1
GPR travel time curves
Two layer Earth with velocity v1 and v2 separated by interface at depth d
x = distance between TX and RX
Direct wave, travel time = t dir =
x
v1
Reflected wave, exact travel time = t ref =
Reflected wave, normal moveout, NMO =
Travel time at normal incidence, t 0 =
x 2 + 4d 2
v1
x2
2v12 t 0
2d
v1
3