Geophysics Final Exam Flashcards
Rock Physics Checklist
I V P G I M
Identity Volume fraction Physical Properties Geometry Interactions Methods
Combining Rock Physics
X^a = sum(f*X^a)
Inverse Problem
Use the measured geophysical response, physics, and prior knowledge to interpret what’s going on in the subsurface
Wave-based Imaging Methods
GPR (EM) waves
Seismic (mechanical) waves
Forward Problem
Know the structure, physics, etc. to predict the geophysical response
Wave
V = wavelength * frequency
Amplitude (A)
Maximum magnitude of displacement from rest
Wavelength
Distance required to complete one cycle of displacement (peak to peak or trough to trough)
Period (T)
Time required to complete one cycle of displacement [time]
Frequency (f)
The number of cycles completed in a given time period [Hz]
Attenuation
The loss of energy as the wave propagates
1) Geometric spreading
2) Intrinsic attenuation (e.g. loss of energy to heat)
3) Scattering
Wavefront
A curve (or surface) connecting points of constant phase (i.e. connecting peaks or troughs)
Wavefield
The collection of all waves at one point in time (e.g. photo of ripples on a pond)
Ray
The path traced out by following a fixed point on a wavefront. Perpendicular to wavefront
Dispersion
When the velocity of a wave depends on frequency
most waves are made up of many frequency components
Compression
An applied stress causes a change in the volume of the rock
Bulk Modulus
K = -dP / dV/V
Inverse of compressability
Compressability
B = -dV/dP / V
Poisson’s Ratio
u = -e1 / e3
Ratio of perpendicular strain to parallel strain (relative to applied stress)
Related to Bulk: K = E / 3(1-2u)
Related to Shear: G = E / 2(1-u)
Shear
Applied stress causes deformation (E) (strain) but no change in volume occurs
Shear Modulus
G = shear stress / shear strain
rigidity
Young’s Modulus
E = uniaxial applied stress / observed strain
Surface Waves
Rayleigh wave – elliptical retro-grade motion with exponentially decaying amplitude with depth
Love wave – shear motion polarized in the plane of the surface (side to side)
Properties that control Radar
1) Electrical conductivity, sigma
2) Magnetic permeability, u
* 3) Dielectric Permittivity, e
Dielectric Permittivity (e)
Relates charge separation (polarization) to the applied electric field: p = e*E
e = K*e_o
Dielectric Constant (K)
Describes the ability of a material to store energy due to charge polarization. Is the relative permittivity of a dielectric material
K = e / e_o
Dielectric Constant (K) vs. Dielectric Permittivity (e)
K = e / e_o
e = K*e_o
Dielectric Constant and Velocity
V = c / (K^1/2)
c = speed of light = 3E8 m/s
WARR
Wide Angle Reflection and Refraction
– move receiver but keep source fixed
CMP
Common Midpoint
– keep the center fixed, move source and receiver from center
COP
Constant Offset Profile
– move source and receiver together at same interval
P-wave
Primary wave
Vp = (K + 4/3G / density)^1/2
S-wave
Secondary wave
Vs = (G / density)^1/2
Vp / Vs
(K/G + 4/3)^1/2 = (1-u / 1/2-u)^1/2
Slope and Intercept
Square the traveltime eq: t^2 = 1/V2 x^2 + 4h^2 / V1^2
Plot x^2 vs t^2 to get a straight line:
slope = 1 / V1^2
intercept = 4h^2 / V1^2
Arrivals
Radar: Air first
Seismic: Air last
Both: Direct waves straight, reflection curved to join ground wave, refraction separates from reflection at ground wave
t vs. x:
Radar or Seismic?
Radar = no groundroll
Seismic = air last
- units
- calc velocities
Frequency vs. Resolution
Higher frequencies:
- better resolution
- shallower
- more attenuation
Lower frequencies:
- poor resolution
- deeper
- less attenuation
Absorption
Loss of E due to (a) intrinsic attenuation (heat loss) and (b) scattering
Ii=I1e^(-q (ri-r1) )
Reflection Strength
Depends mostly on:
1) Reflection coefficient
2) Attenuation
Radar Reflection Coefficient (R) (normal incidence)
R = amplitude of reflection / amplitude of incidence
R = V2 - V1 / V2 + V1 = sqrt(K2) - sqrt(K1) / sqrt(K2) + sqrt(K1)
Seismic Reflection Coefficient (R) (normal incidence)
R = p2 V2 - p1 V1 / p2 V2 + p1 V1 = z2 - z1 / z2 + z1
Energy Density / Intensity (I)
I = E/S = E / 4pi r^2
Direct wave
t = x / v
Reflection
t = 2 / Vrms ( x^2 /4 + h^2)^1/2
x^2 - t^2
Refraction
t = ( 2 h1 (V2^2 - V1^2)^1/2 ) / (V1V2) ) + x/V2
Diffraction
t = 2/V ((x - xs)^2 + h^2)^1/2
Traveltime (t)
t = 2a/V1 = 2(x^2 / 4 + h^2)^1/2 / V1
Dix Equation
Vn^2 = ([Vrms^2]n tn - [Vrms^2]n-1 tn-1) / (tn - tn-1)
Resistivity of Rocks
Surface charge:
– clays = conductive
Porosity:
– high = LOW resistivity, HIGH conductivity
2nd Layer Velocity
V2 = sqrt( (Vrms(dt1 + dt2) - V1^2 dt1) / dt2)
2nd Layer Thickness
h2 = dt2 V2
Thickness (h) vs. Wavelength (w)
h»_space; w:
distinct reflections apparent
h = w:
bottom layer is mirrored
h = w/2:
not as clear
h = w/4:
looks like 1 reflection overall
Reflection Coefficient (R)
Quantifies the fraction of energy returned to the surface by a contrast in properties
Mulitples
Primary reflection: tn = 2h/V1
1st multiple: tm1 = 2t_p = 4h/V1
2nd multiple: tm2 = 3t_p = 2h/V1
Snell’s Law
Sin O2 = V2/V1 sin O1
O1 = angle of incidence O2 = angle of transmission
Critical Angle (Oc)
Oc = sin^-1 (V1/V2)
Critical Distance (Xc)
Distance at which a refracted wave is generated
Xc = 2h / [(V2/V1)^2 - 1]^1/2
N Layer Refraction Traveltime
tn = X/Vn + 2/Vn sum( (hi (Vn^2 - Vi^2)^(1/2)) / Vi )
Ohm’s Law
V = I*R
V = voltage I = current R = resistance
Resistivity vs. Conductivity
resistivity = 1 / conductivity
conductivity = 1 / resistivity
Resistance (R)
R = L / A * p
p = resistivity A = cross sectional area L = length
p = A/L V/I = K V/I
Archie’s Law
nonclays:
Peff = a * O^-m * Sw^-n * Pw
clays:
Peff = [(a * O^-m * Sw^-n * Pw)^-1 + o_surface]^-1
O = porosity Sw = saturation Pw = fluid resistivity a = 0.41 - 2.13 m = 1.64 - 2.23 n = 1.1 - 2.6
Fluid Conductivity
Ow = 10^3 * F * sum( abs(Zi) * Ci * ui
F = 9.648E4 C/mol Zi = valence of ion Ci = concentration (mol/L) ui = ionic mobility ((m/s)/N)
Voltage
at point p:
Vp = pI/2pi *( 1/r1 - 1/r2)
p = dVmeas/I * 2pi[1/r1 - 1/r2 - 1/r3 + 1/r4]^-1
Depth of Resistivity Arrays
Wenner: h = a/2
Schlumberger: h = L/3
Dipole-Dipole: h = na
Electrode Array Geometric Factors (K)
Wenner: K = 2pi*a
Schlumberger: K = pi/a^2 [1 - b^2/4a^2]
Dipole-Dipole: K = pi * n * (n + t) * (n + 2) * a
Square: K = pi*a *(2 + sqrt(2))
Direct Current (DC)
Current flows continuously in direction of applied voltage
Applied Current (AC)
Current switches direction with applied voltage
Pure Capacitor (C)
Ability to hold a charge
C = Q / V
C = capacitance Q = charge V = applied voltage
Chargeability
= Vp / Vo –> difficult to get Vp accurately
Apparent = A / Vo
A = integral (area) of Vp over a specified time window
Spectral (Frequency-Domain) Induced Polarization (SIP)
Apply a sine wave of a particular frequency. Compare the amplitude and phase (i.e. time shift) of the observed voltage relative to the applied current
Complex Conductivity (sigma = o)
o(f) = o’(f) + io’‘(f)
o' = real conductivity --> energy loss (conduction) i = sqrt(-1) o'' = imaginary conductivity --> energy storage (polarization)
Electrical Double Layer
Mineral surfaces are usually charged.
Causes an imbalance of charge near the pore walls
Membrane / Electrolytic Polarization
Pore throat constriction - charge on mineral surface leads to a build-up at pore throats
Constriction by clay particles - ions accumulate on either side of a charged particle in the pore space
Self Potential (SP)
Redox reactions = electrochemical
Groundwater flow = electrokinetics
Tomography Traveltime (t)
through each cell (tj):
tj = sum( Lj / Vj ) = sum( Lj sj )
sj = 1 / Vj = slowness of cell j Lj = length of ray in cell j
Least-Squares Data Fitting
Steps:
1) Collect data –> (x,y)
2) Define a model –> y=mx+b
3) Define a measure of error –> “least squares”
4) Find parameters of the model that minimize error
Tomography Forward Problem
t = L s
d = G m
d=data
G=design matrix
m=vector with all model parameters
Tomography Inverse Model
m = ( G^T G + a I )^-1 ( G^T y - a mo )
a = trade-off/regularization parameter I = MxM identity matrix mo = best guess of m
Not as good (will blow up):
m = ( G^T G )^-1 G^T y
Newton’s Law
Fg = G m1 m2 / r^2
G = 6.672E-11 m^3/kgs^2
Fg = gravitational force = 9.8 m/s^2
m1, m2 = masses of objects
r = distance between the centers of the objects
Units of Gravity
1 Gal = 0.01 m/s^2
g = 981 Gal
Absolute Gravity (g)
The actual value of acceleration due to gravity measured at a point in space
Relative Gravity (dg)
CHANGE in gravity from a background value
=> useful for measuring variations in density
Spherical Earth vs. Non-spherical Earth
Spherical: r1=r2, g1=g2
Non-spherical: r1 x=x r2, g1 x=x g2
Polar radius is ~21km shorter than the equatorial radius
Forward Model for Gravity Anomaly
dg = G (4/3 pi a^3) (dp) z / ((x-xs)^2 + (z-zs)^2)^3/2
G = 6.672E-11 m^3/kgs^2
xs, zs = position and depth of “sphere” center
a = radius of sphere
dp = p2 - p1 = density difference between inclusion & background
Inverse Problem
- Fit data using prior knowledge
- Collect additional data
- Reduce the problem to remove non-uniqueness
Half Max Gravity Anomaly (dg_1/2max)
dg_1/2max = G (4/3 pi a^3) dp z / ( X_1/2max^2 + z^2)^3/2
Corrections to Gravity Observations
B I L T F I T
1) Bouger correction
2) Instrument drift
3) Latitude
4) Terrain correction
5) Free-air correction
6) Isostatic correction
7) Tides
Bouger Anomaly
Reported gravity anomaly
> signal actually related to ground inclusion
dg = g_obs - sum( corrections - g_base )
Gravity Anomaly (dgz)
dgz = G SSS dp(x,y,z) z/r^3 dxdydz
dgi
1 block
dgi = A/r^3 + B/2r^5 ( (5z (3z^2 - r^2)/r^2) - 4z ) + 3C(x^2 - y^2) / r^5
r^2 = x^2 + y^2 + z^2
x, y, z = distances between measurement point and block i
A = 8G dp_i abc
B = A (2c^2 - a^2 - b^2) / 6
C = A (a^2 - b^2) / 24
2a, 2b, 2c = lengths of block i in x, y, z direction
Resistivity Array Choice
Depends on:
1) Type of structure to be mapped
2) Sensitivity of the resistivity meter
3) Background noise level
Things to consider:
1) Depth of investigation
2) Sensitivity of the array to vertical & horizontal structures
3) Data coverage
4) Signal strength
Chargeability
= Vp / Vo > difficult to get Vp accurately
Apparent = A / Vo
A = integral (area) of Vp over a specified time window
Spectral (Frequency-Domain) Induced Polarization (SIP)
Apply a sine wave of a particular frequency.
Compare the amplitude and phase (i.e. time shift) of the observed voltage relative to the applied current
Complex Conductivity (sigma = o)
o(f) = o’(f) + io’‘(f)
o’ = real conductivity > energy loss (conduction)
i = sqrt(-1)
o’’ = imaginary conductivity > energy storage (polarization)
Self Potential (SP)
Redox reactions = electrochemical
Groundwater flow = electrokinetics
Least-Squares Data Fitting
Steps:
1) Collect data > (x,y)
2) Define a model > y=mx+b
3) Define a measure of error > “least squares”
4) Find parameters of the model that minimize error
Relative Gravity (dg)
CHANGE in gravity from a background value
> useful for measuring variations in density
Spherical Earth vs. Non-spherical Earth
Spherical: r1=r2, g1=g2
Non-spherical: r1 x=x r2, g1 x=x g2
Polar radius is ~21km shorter than the equatorial radius
Inverse Problem
– Fit data using prior knowledge
– Collect additional data
– Reduce the problem to remove non-uniqueness
Center of Mass (xs, zs) and Mass of Inclusion (m)
xs = get from peak of dg curve
zs = 1.305X_1/2
? m = 255dgmax(X_1/2)^2 ?
Gravimeter
– Sensitive - can detect change of 0.01 mGal
– Rely on a mass pivoted on a beam attached to a spring
– Buildings/mtns/etc. influence gravity, measurements
Gravitational Potential (u)
Represents the WORK ( / energy ) required to bring a unit mass from infinity to a position, r, away from the Earth
Gravity Forward Model
dg = M dp
[dg1; dg2;…dgN] = [K11, K12,…K1M; K21, K22,…K2M…] * [dp1; dp2; dpN]
dp = (M^T M + I a) M^T dg
K = constant that depends on model properties (grid size) and measurement location
Spectral (Frequency-Domain) Induced Polarization (SIP)
Apply a sine wave of a particular frequency.
Compare the amplitude and phase (i.e. time shift) of the observed voltage relative to the applied current
Tomography Forward Problem
t = L s
d = G m
d=data
G=design matrix
m=vector with all model parameters
Tomography Inverse Model
m = ( G^T G + a I )^-1 ( G^T y - a mo )
a = regularization parameter I = MxM identity matrix mo = best guess of m
Not as good (will blow up):
m = ( G^T G )^-1 G^T y
Forward Model for Gravity Anomaly
dg = G (4/3 pi a^3) (dp) z / ((x-xs)^2 + (z-zs)^2)^3/2
G = 6.672E-11 m^3/kgs^2
xs, zs = position and depth of “sphere” center
a = radius of sphere
dp = p2 - p1 = density difference between inclusion and background
