Forward Modeling of Applied Geophysics Methods

Using Comsol and Comparison with Analytical and

Laboratory Analog Models

S.L. Butlera,∗, G. Sinhaa

aDepartment of Geological Sciences, University of Saskatchewan, 114 Science Place,

Saskatoon, SK, S7N 5E2, Canada

Abstract

Forward modeling is useful in geophysics both as a tool to interpret datain a research setting and as a tool to develop physical understanding in aneducational setting. Gravity, magnetics, resistivity and induced polarizationare methods used in applied geophysics to probe Earth’s subsurface. In thiscontribution, we present forward models of these geophysical techniques us-ing the finite-element modeling package, Comsol. This package allows forrelatively easy implementation of these models and, as part of the AC/DCmodule, allows for exterior boundaries to be placed at infinity: a boundarycondition that is frequently encountered in geophysics. We compare the out-put of the finite-element calculations with analytical solutions and, for theresistivity method, with laboratory scale analog experiments and demon-strate that these are in excellent agreement.

Keywords: gravity magnetics resistivity finite element

1. Introduction1

The gravity, magnetic, resistivity and induced polarization (IP) tech-2

niques are geophysical methods used to infer the structure and composition3

of Earth’s subsurface from surface or airborne measurements (Telford et al,4

1990). Forward modeling of these techniques is useful, both because model5

∗Corresponding author. Fax:+1-306-966-8593Email addresses: sam.butler@usask.ca (S.L. Butler), gunjan.sinha@usask.ca (G.

Sinha)

Preprint submitted to Computer & Geosciences August 25, 2011

parameters can be varied in order to fit observations as part of an inversion6

routine, and because physical intuition can be developed when model param-7

eters are varied by students. Most forward potential fields models are based8

on integral equations where relatively simple formulas can be used when the9

anomalous density and magnetization distributions are represented in terms10

of polygons. The formulas were derived for 2D gravity and magnetics by11

Talwani et al. (1959) and Talwani and Heirtzler (1964) and for arbitrar-12

ily shaped 3D objects by Talwani and Ewing (1960) and Talwani (1965).13

2.5 D models, where the anomalous density or magnetization is of finite14

length along strike, are popular and the ‘end’ correction formulas for these15

were derived by Shuey and Pasquale (1973) and Rasmussen and Pedersen16

(1979). The popular commercial software package GMsys (GMsys User’s17

Guide, 2004) is based on these integral formulas. In this paper, the forward18

gravity and magnetics problems will be addressed by solving Poisson’s equa-19

tions with the appropriate boundary conditions. The numerical solution of20

Poisson’s equation to determine the gravitational acceleration from a density21

distribution was used previously by Zhang et al. (2004) and Farquharson and22

Mosher (2009) and a finite volume discretization of the appropriate form of23

Maxwell’s equations was used by Leliever and Oldenburg (2006) to solve for24

the anomalous magnetic field due to a given magnetic susceptibility distribu-25

tion. An advantage of the solution of Poisson’s equation is that the potential26

is computed everywhere and does not need to be computed again if a new27

point of investigation is selected.28

Resistivity modeling consists of solving Laplace’s equation with source29

terms representing the current electrodes and appropriate boundary condi-30

tions. Published models of the resistivity method include models that are31

discretized based on finite difference (e.g., Dey and Morrison, 1979; Zhoa,32

1996), finite element (e.g., Coggon, 1971; Pridmore et al., 1980; Rucker et33

al., 2006; Ren and Tang, 2010) finite volume (e.g., Pidlisecky and Knight,34

2008) and integral equation based models (e.g., Ma, 2002).35

Comsol Multiphysics (Comsol Multiphysics Users’ Guide, 2009) is a gen-36

eral finite element modeling environment. While previous forward modeling37

efforts used specialized software for each geophysical method, in this paper,38

we demonstrate that a single flexible modeling tool can be used for all of the39

methods. Comsol allows users to develop complex numerical models with40

different geometries and multiple governing equations quickly using a GUI.41

As will be shown in the subsequent sections, models of the above mentioned42

geophysical techniques can be quickly implemented in Comsol and, because43

2

of Comsol’s multiphysics capability, many different techniques can be mod-44

eled simultaneously for the same anomaly geometry. The AC/DC module45

of Comsol contains the capability to model boundaries at infinite separation46

from the model domain which is the most common boundary condition en-47

countered in geophysical problems. We will demonstrate the effectiveness48

of these infinite boundaries by comparing Comsol’s solutions with analytical49

solutions.50

Cardiff and Kitanidis (2008) recently showed how Comsol can be used to51

calculate adjoint state-based sensitivities for inverse modeling in hydrogeol-52

ogy while Kalavagunta and Weller (2005) used Comsol to calculate geomet-53

rical factors for laboratory scale resistivity experiments. Other cases where54

researchers have used Comsol to model electrical geophysical phenomena in-55

clude Park et al. (2010) who modeled controlled source electromagnetics56

(CSEM), Volkmann et al. (2008) who developed a microscopic model of the57

frequency-domain induced polarization effect, Stoll (2005) who modeled the58

bore-hole resistivity method and Braun et al. (2005) who modeled the prop-59

agation of electromagnetic waves in a conducting medium with application60

to magnetic resonance sounding.61

In what follows, we will first present the theory, implementation and62

results for a gravity model in Comsol and compare its results with an ana-63

lytical solution. Subsequently, we will present similar sections for magnetics,64

resistivity and induced polarization. Following this, we will present a sec-65

tion analyzing the accuracy of the gravity and resistivity calculations as a66

function of various model parameters.67

2. Methods, Theory and Results68

In all of the models presented here, tetrahedral elements were used and69

potentials were represented using quadratic Lagrange basis functions except70

for two simulations listed in section 3. All of the calculations were carried out71

on a desktop PC using a single 2.87 GHz Intel Xeon processor. Comsol has a72

number of linear solvers to choose from. The method of conjugate gradients73

is the default solver for the electrostatics, magnetostatics and conductive74

media modules that were used in this study and this solver was used in all75

of the calculations except for the induced polarization calculations. Induced76

polarization calculations required the use of complex numbers which is not77

supported by the conjugate gradient solver in Comsol so the UMFPACK was78

chosen for these calculations instead. The models were initially implemented79

3

using Comsol 3.5. Some of the models were tested and found to work similarly80

with some modifications in Comsol 4.2.81

2.1. Gravity82

Variations in subsurface density produce small variations in the vertical83

component of the acceleration due to gravity that can be measured at Earth’s84

surface. Comsol does not have a built-in gravity calculation module. How-85

ever, it does have an electrostatics module. Since gravity and electrostatics86

are both governed by Poisson’s equation, a gravity model can be created from87

an electrostatics model by changing the value of the electrical permittivity.88

Poisson’s equation relating the gravitational potential, U in J kg−1 , to the89

mass density distribution, ρ in kg m−3 , is90

∇2U = −4πGρ (1)

where G = 6.67 × 10−11 J m kg−2 is the universal gravitational constant91

(e.g., Telford et al., 1990). Here and throughout the rest of the paper, MKS92

units will be used. The Poisson equation in electrostatics relating electrical93

potential, V in Volts, to the charge density distribution, ρc in C m−3, is94

∇2V =−ρcε

(2)

where ε is the electrical permittivity (e.g., Jackson, 1998). By setting ε = 1

4πG95

and setting ρc = ρ and V = U a model for gravitational potential energy is96

easily produced.97

In order to test the model, a cylinder with radius, a=1000 m, and density98

contrast with the surrounding material, ∆ρ = 100 kg m−3, was used. The99

geometry is shown in figure 1a. The cylinder lies horizontally along the y100

axis at mid-height and in the middle in the x direction in a cubic solution101

domain that is 10 km on a side. The rectangular prisms along all of the outer102

faces, edges and corners are 2 km thick and are used as infinite elements. The103

infinite elements are set to stretch the coordinates to infinity in particular104

directions. The corner infinite elements are stretched in all three directions105

while the edge infinite elements are stretched in the two directions that are106

normal to the edge faces while the face infinite elements are stretched only107

in the direction normal to the face.108

The finite element mesh consisted of 153 723 tetrahedral elements. For109

comparison, another calculation was run in which the outer 2 km rectangular110

4

prisms that were infinite elements in the previous calculation were simply111

regular parts of the computational domain. In this latter case, a constant112

potential, U = 0, was imposed as a boundary condition on all of the outer113

boundaries (at x, y, z = ±7000 m). Calculating the solution used 635 Mbytes114

of memory and required 55 s when no infinite elements were used and 84 s115

when infinite elements were used.116

Profiles of the vertical component of gravitational acceleration, perpen-117

dicular to the long axis of the cylinder at 2 and 4 km height, above the centre118

of the cylinder in the y direction, are plotted in figure 1b. The dashed line119

is plotted from the analytical solution for the vertical component of gravity120

from a cylinder:121

gzcyl =2πG∆ρa2z

x2 + z2. (3)

The solid and dotted lines are plotted from numerical solutions. The solid122

line was calculated for a model for which the infinite elements were employed123

and the density anomaly extended to y = ±∞ while the dotted line did not124

use infinite elements so the cylinder stopped at ±7 km. As can be seen, the125

numerical solution with infinite elements agrees very well with the analytical126

solution at both z=2 and 4 km. Evaluating gz at different values of z mimics127

the effect of the cylinder being placed at different depths in the Earth or of the128

measurement sensor being at different altitudes. The well-known effect that129

gravity anomalies have reduced amplitude and become broader with distance130

to the source (Telford, 1990) can also be clearly seen. The numerical solution131

without infinite elements can be seen to have an amplitude that is too small,132

particularly for the z = 4 km case. The cause of the reduced amplitude133

is both due to the potential being forced to 0 for x < ∞ by the boundary134

condition and because the cylinder has only a finite length. A calculation135

with infinite elements but for which the cylindrical mass anomaly stops at136

y = ±5 km was also carried out (not shown) which had a similar amplitude137

to the case with no infinite elements which indicates that the latter effect is138

more significant. The larger discrepancy with height can be understood by139

the fact that as the observer moves farther from the anomaly, the effects of140

the mass that are far away laterally become increasingly important.141

Although the example presented here used a simple cylindrical anomaly142

for the purpose of comparison with an analytical solution, arbitrarily compli-143

cated mass density distributions can be modelled. Comsol has the capability144

to import CAD files and in the future we intend to import digital elevation145

models for the purpose of computing terrain corrections.146

5

−5000 −2500 0 2500 50000

0.5

1

1.5

2

2.5

x (m)

g z (m

Gal

)

z= 2 km

z= 4 km

b

Figure 1: a) The solution geometry. Distance units are in 104 m. The central cylinderhas anomalous density ∆ρ. The outer rectangular prisms of thickness 2 km are “infinite-elements” that place the outer boundary effectively at infinity. b) Profiles of gz at heights2 and 4 km above the cylinder along the x axis and over the centre of the cylinder. Solid,dotted and dashed lines indicate numerical solutions with and without infinite elementsand the predictions of equation 3

6

2.2. Magnetics147

Small variations in the magnetic field measured above the Earth’s surface148

are caused by variations in the degree of magnetization in Earth’s crust. The149

variations in the degree of magnetization are, in turn, caused by differences150

in the magnetic susceptibility between different Earth materials.151

In the absence of free currents and time varying magnetic and electric152

fields, the magnetic field H in A m−1 can be written as the gradient of a153

scalar potential, Vm (measured in C s−1). It can be shown that for the154

case of materials whose magnetization is linearly related to the strength of155

the magnetic field, M = kH, where M is magnetization in A m−1and k156

(dimensionless) is magnetic susceptibility, the magnetic scalar potential can157

be solved from the following equation (Jackson, 1999).158

∇ · [(1 + k)∇Vm] = 0. (4)

Equation 4 was solved in a cubic domain of size 40 by 40 m where k = 0159

except in a target object where k > 0 using the magnetostatics capability160

in the AC/DC module. The normal component of the magnetic field was161

specified on all boundaries and was constant over the boundaries to mimic162

the effects of Earth’s core field. The solution domain consisted of 105 335163

tetrahedral finite elements. The solution required 689 Mbytes of memory and164

took 36 s to run. Infinite elements were not used when modeling magnetics165

as they were not found to work well when the solution does not go to 0 at166

infinity.167

As a test of the model, a sphere of radius 1 m with constant magnetic168

susceptibility k = 1000 was used in the centre of the model. A sphere in a169

constant external field will have a constant magnetization and a sphere with170

constant magnetization has the same magnetic field as a magnetic dipole171

located at its centre. The horizontal, Hx − Hxext, and vertical, Hz − Hzext,172

components of the anomalous magnetic field calculated from the numerical173

model 2 m above the centre of the sphere are plotted along a line where y = 0174

in figure 2 (solid lines) which are compared with the analytical expressions175

for a magnetic dipole from equations 5. Hxext and Hzext represent the x176

and z components of the externally imposed magnetic field. The analytical177

expression for the horizontal and vertical components of a dipole field are178

given by (e.g., Telford et al., 1990)179

Hx =m

4π

(2x2 − z2) cos I − 3xz sin I

(x2 + z2)5/2

7

−15 −10 −5 0 5 10 15−10

−8

−6

−4

−2

0

2

4

x (m)

Mag

netic

Fie

ld (

A/m

)

Hx

Hz

Figure 2: Profiles of anomalous Hx and Hz calculated from the numerical model (solidline) and from equations 5 (dotted line) along a line where z = 2 m and y = 0.

180

Hz = −m

4π

(2z2 − x2) sin I − 3xz cos I

(x2 + z2)5/2(5)

where m = keff |Bext|V/µ0 is the dipole moment and the effective perme-181

ability for an object with demagnetization is given by keff = k/(1+ kNgeom)182

where the demagnetization factor, Ngeom = 1/3, for a sphere. In the above183

expression, I is the magnetic inclination which was set equal to 63.4o (corre-184

sponding to 45o magnetic latitude) and |Bext| is the magnitude of the external185

magnetic B field which was set to 50 µT . As can be seen, except for some186

small amplitude numerical noise, the numerical solution is in perfect agree-187

ment with the analytical expression. Because k >> 1 in this calculation,188

keff ≈ N−1geom and the amplitude of the anomalous field is significantly less189

than it would have been if the effects of demagnetization were not present.190

A further comparison was made with the results of Lelievre and Oldenburg191

(2006) where a prolate ellipsoidal object of semimajor axis 9 m and semiminor192

axes of 3 m of anomalous susceptibility was placed in the middle of the193

solution domain and oriented with its long axis horizontal. An external field194

of magnitude 50 µT and inclination I = 58.3o was applied with its horizontal195

component parallel to the long axis of the ellipsoid. For this simulation, the196

solution domain was increased in size to a cube with 80 m side lengths because197

of the large size of the magnetized object. Profiles of total field anomaly (the198

8

0 10 20 30 40 50

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

r(m)

Tot

al A

nom

alou

s F

ield

(S

cale

d)

Figure 3: Profiles of the total field anomaly over a prolate ellipsoidal object of anomalousmagnetic susceptibility. Dash-dot and dashed lines are from calculations with k = 10−3

from Lelievre and Oldenburg (2006) and from this study, respectively, while dotted andsolid lines are from calculations with k = 10 from Lelievre and Oldenburg (2006) and fromthis study.

magnitude of the total magnetic field minus the magnitude of the background199

field) are plotted at a height 8 m above the object and parallel to its long200

axis in figure 3. The total field anomaly is typically what is measured in201

exploration geophysics surveys. The dashed and solid lines are total field202

anomalies calculated from the present model with k = 10−3 and k = 10 while203

the dash-dotted and dotted lines are the results of Lelievre and Oldenburg204

(2006) with the same magnetic susceptibilities. As can be seen, the current205

model agrees very well with the results of Lelievre and Oldenburg (2006).206

The curves have been normalized so that their maximum amplitudes are 1.207

Lelievre and Oldenburg (2006) reported the ratio of the maximum amplitude208

of the curve calculated with k = 10 to that with k = 10−3 to be 2225 which is209

identical to what we calculated. The fact that the maximum amplitude does210

not scale linearly with the magnetic susceptibility arises due to the effects211

of demagnetization. Lelievre and Oldenburg (2006) also reported that for212

the k = 10 case, the magnetization had an inclination angle of 31.7o degrees213

while we calculated 31.5o.214

The effects of demagnetization were further investigated using a 1 m by215

1 m by 15 m rectangular prism of anomalous magnetic susceptibility. The216

geometry can be seen as the black lines in figures 4a and b. The rod was cen-217

9

tered in the solution domain with its long axis parallel to the x axis. The ex-218

ternal magnetic B field was specified with [x, y, z] components [22.4, 0,−44.7]219

µT to mimic Earth’s magnetic field at latitude 63.4o in the northern hemi-220

sphere. In figure 4a we show the results of a calculation with magnetic221

susceptibility k = 1, a value similar to an object with a high magnetite con-222

centration. The red lines are magnetic field lines while the cones indicate223

the direction of magnetization and the colour contour plot shows a slice of224

the total field anomaly which is plotted on a surface of constant height, in-225

dicating what would be measured in the field over such an anomaly. As can226

be seen, for this low value of magnetic susceptibility, the bar is magnetized227

roughly parallel to the external field. The total field anomaly shows a high228

to the south and a low to the north which is typical of magnetic anomalies229

measured in the northern hemisphere (Reynolds, 1997).230

Figure 4b shows the results of an identical calculation except that the231

magnetic susceptibility has been increased to 1000, a value appropriate for232

an iron bar. As can be seen, although the external magnetic field is at233

45o to the long axis of the bar, the bar is now magnetized parallel to its234

long axis because of the effects of demagnetization. Comparison of the total235

field anomalies shown in figure 4a and b also reveals that the magnetic field236

amplitude is not scaling linearly with the magnetic susceptibility and that the237

pattern of the total field anomaly is significantly different. As can be seen,238

the low to the north (positive x direction) is now much more prominent.239

Figure 3 also shows an increase in the prominence of the northern low in the240

model with large susceptibility.241

Although not used in the example shown here, the model has the capa-242

bility to include effects due to remnant magnetization and more complicated243

geometry. Comsol supplies an example model of magnetic prospecting where244

the magnetic field can be calculated on an uneven surface simulating real245

topography.246

2.3. Resistivity247

The resistivity method involves injecting electrical current into the ground248

between one pair of electrodes and measuring the electrical potential between249

another electrode pair. For a given electrode geometry, an “apparent resis-250

tivity”, ρa in Ω m, can be determined from251

ρa = kg∆V

I(6)

10

11

Figure 4: Red lines show magnetic field lines due to the anomalous object. Blue conesshow the direction of magnetization while the colour contour plots shows the total fieldanomaly at a height 5 m above the anomalous rod. Units of the magnetic field are µTwhile distances are in m. Part a) shows the results for a calculation with k = 1 while inb) the results for a calculation with k = 1000 are shown.

12

where ∆V is the difference in electrical potential between the potential elec-252

trodes in Volts, I is the injected current in Amps and kg (measured in m) is253

the geometrical factor of the array that depends only on the relative positions254

of the electrodes. The apparent resistivity is the resistivity that an infinite255

half space of constant resistivity would have in order to give the same mea-256

sured ∆V for a given injected current. If apparent resistivity changes with257

a change in the lateral array position or with a change in the array spacing,258

there must be a lateral or vertical change in the resistivity of the ground259

(Telford, 1990).260

A DC conductive media module exists in Comsol that solves261

∇ · σ∇V = 0, (7)

where σ is the electrical conductivity of the ground which is the reciprocal of262

the resistivity. Electrical current density, J in A m−2, is given by J = −σ∇V .263

Current electrodes were modelled by specifying point current sources at the264

surface of the model. The rest of the top surface boundary was made elec-265

trically insulating, ∇V · n = 0, where n is the normal to the surface. As a266

test of the method, the model results were compared with those of a labora-267

tory scale resistivity model consisting of a 14 cm deep water tank of lateral268

dimensions 1 m by 40 cm. In order to model conduction in the water tank,269

a model domain with dimensions equal to the laboratory tank was chosen270

and the side walls and bottom of the domain were made electrically insulat-271

ing. In the laboratory model investigation, the electrodes were thin copper272

wires that just penetrated the top surface of the water so that the current273

electrodes approximate point current sources. The electrodes were set up in274

the Schlumberger array geometry and the current electrode spacing was in-275

creased in steps, mimicking a “depth sounding”. As the spacing between the276

current electrodes increases, current lines are constrained to a greater degree277

by the bottom and sides of the tank increasing the apparent resistivity. In278

the numerical model, the potential was calculated everywhere in the model279

domain and the potentials at the same positions as in the laboratory model280

were determined in post-processing. The apparent resistivity was then calcu-281

lated using equation 6 once kg was calculated for the given survey geometry.282

In figure 5a the magenta lines are stream lines of electrical current density283

showing current flowing from one current electrode to the other while the284

surface plot shows the electrical potential at mid-depth in the tank.285

In figure 5b, a line plot of the voltage calculated along the top is shown286

for calculations where the current electrodes were placed at ±0.15 m and in-287

13

finite elements were used on the sides and bottom of the domain. At a point288

current source, the potential is infinite which can only be approximated in289

numerical methods. The dotted line indicates the potential from a solution290

employing 14 435 roughly equally spaced finite elements while the solid line291

employed 55 364 finite elements. The dashed line shows the analytical solu-292

tion. It can be seen that the solution employing a larger number of elements293

better approximates the potential in the vicinity of the current electrode.294

Further simulations employing a refined grid around the current electrodes295

gave potentials that were indistinguishable from the analytical expression296

when plotted as in figure 5b. The apparent resistivity was calculated using297

the potential measured at some distance from the current electrodes and was298

not generally found to be significantly affected by the mesh resolution. In299

section 3 we present a further analysis of the effects of model resolution.300

In figure 6 we show the apparent resistivity as a function of half of the301

current electrode spacing, AB/2, for a Schlumberger array. The apparent302

resistivity has been normalized by the resistivity of the water. The + symbols303

represent laboratory data while the open circles represent the results of the304

numerical simulations with insulating side and bottom boundary conditions.305

As can be seen, the numerical simulations are in excellent agreement with306

the laboratory data. The analytical expression for the apparent resistivity of307

an infinite half space with two vertical layers with contact at depth z where308

the lower layer is infinitely resistive for a Schlumberger array is309

ρa = [1+AB +MN

MN

∞∑

m=1

(1+(4mz

AB −MN)2)−0.5−

AB −MN

MN

∞∑

m=1

(1+(4mz

AB +MN)2)−0.5]

(8)(Telford et al., 1990). The analytical solution gives the results represented by310

the dashed line when the interface is at a depth of 14 cm. This situation can311

be modeled numerically by using infinite boundaries on the side walls but312

maintaining the bottom boundary as electrically insulating. The results of313

these numerical simulations are shown by the asterisks. As can be seen, the314

numerical solutions are in very good agreement with the analytical solution.315

Finally, apparent resistivity for an infinite half space of constant resistivity316

is constant, regardless of the array spacing. This situation can be modelled317

by including infinite elements on the bottom boundary as well. These sim-318

ulation results are indicated by the black × symbols. As can be seen, the319

calculated apparent resistivity remains close to a constant value of 1 indi-320

cating that the infinite boundaries are providing a good approximation of321

14

placing the outer boundaries at infinity. The greatest discrepancy between322

the numerical and analytical solutions occurs when the current electrodes323

are close to the potential electrodes so that the potential is being measured324

where it is changing rapidly.325

In figure 7 the results of a series of simulations are presented where two326

blocks of resistivity 1 Ωm and 10 Ωm are used where the contact between the327

blocks has a normal in the x direction and so the contact face is vertical. The328

model domain has dimensions 1 by 0.4 by 0.14 m as before. The position329

of the contact was varied in order to simulate translating a Schlumberger330

array across a vertical geological contact. The current electrodes were placed331

at ±0.15m while the potential electrodes were placed at ±0.04m. Infinite332

elements of thickness 0.2 m were placed on the side and bottom boundaries.333

On the plot, when x = −0.5 the entire domain has a resistivity of 10 Ω m334

while when x = 0 the left side of the domain has resistivity 1 Ω m while335

the right side has resistivity 10 Ω m. When x = 0.5 the entire domain has336

resistivity 1 Ωm.337

The results are compared with the predictions of an analytical model338

based on the method of images for an infinite half-space with a vertical339

contact between two regions with different electrical resistivities (Telford et340

al., 1990, equation 8.48). As can be seen, the simulations reproduce the341

analytical solution essentially perfectly including the counter-intuitive aspect342

that the apparent resistivity increases as more of the array falls within the343

less resistive medium when the interface is between A and M and when it is344

between N and B. Similar calculations were undertaken without the infinite345

elements and the apparent resistivity calculated from the numerical model346

was shifted upward by roughly 60% but many of the gross features of the347

variation of apparent resistivity with contact position were still correct.348

2.4. Frequency Domain Induced Polarization349

The frequency domain induced polarization method uses the same equip-350

ment as the resistivity method only an AC current is injected into the ground351

at the current electrodes while the amplitude and phase of the voltage differ-352

ence between the potential electrodes is measured. In ground that exhibits353

an induced polarization response, the voltage measured at the potential elec-354

trodes may be phase shifted relative to the input current. The in-phase and355

out-of-phase components of the measured voltage can be represented as the356

real and imaginary parts of a complex voltage. The equation governing the357

frequency-domain induced polarization method is then the same as for the358

15

16

Figure 5: a) Magenta lines are streamlines of electrical current density. The surface plotshows the electrical potential at mid-depth in the domain. The current sources are placedat ±0.15 m. Units of potential are V and dimensions are in m. b) The electrical potentialalong the top surface, through the current electrodes from a simulation with 14 435 finiteelements (dotted line), with 55 364 finite elements (solid line) and the analytical solution(dashed line).

17

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

1

2

3

4

5

6

7

8

9

10

AB/2 (m)

ρ a

experiment

numerical−insulating walls

numerical−infinite side walls

numerical−infinite side walls and bottom

Figure 6: The normalized apparent resistivity, ρa, as a function of half of the currentelectrode spacing, AB/2. Open circles are numerical model results in a box with allinsulating boundaries while ’+’ symbols are the results of laboratory experiments. Thedashed line represents the analytical solution for resistivity in an infinite half space withtwo layers where the lower layer is infinitely resistive while the asterisks show the resultsof numerical solutions with infinite side walls but insulating top and bottom boundaryconditions. The × symbols are from numerical simulations with infinite elements on thesides and bottom while the dotted line is a constant value of 1.

18

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

7

8

9

10

11

Position (m)

ρ a

A M N B

Figure 7: The apparent resistivity as a function of the position of the vertical contactbetween two regions with resistivities 1 and 10. The circles are the results of numericalsimulations while the solid line is the analytical solution based on the method of images.The positions of the current electrodes A and B and of the potential electrodes M and Nare indicated.

resistivity method (equation 7) only the electrical potential and electrical359

conductivity now have real and imaginary parts. The injected current is en-360

tirely real and so the phase of the voltage is measured relative to the injected361

current. A common model for the complex resistivity as a function of angular362

frequency, ω, for an Earth material that demonstrates an IP response is the363

Cole-Cole model:364

ρa = ρ0(1−M(1 −1

1 + (iωτ)c)). (9)

The parameters ρ0, M , c, and τ represent the DC resistivity, the “charge-365

ability”, the frequency exponent, and a time constant (Telford et al. 1990).366

A set of simulations was undertaken with the same domain as for the367

resistivity simulations with a Schlumberger array with current electrodes368

A and B at ±0.15m and potential electrodes at ±0.04m. The Cole-Cole369

model above was input for the resistivity with parameters M = 0.1, τ = 1s370

and c = 0.25. The real and imaginary parts of the potential difference371

between the two potential electrodes were calculated in post-processing and372

the real and imaginary parts of the apparent resisitivy were calculated using373

equation 6 for a number of values of the angular frequency. The magnitude374

of the apparent resistivity and the phase shift, φ, relative to the input current375

19

were then calculated. The direct linear solver “UMFPACK” was chosen for376

these simulations since this solver allows for the use of complex numbers and377

the “allow complex output from real input” tab was selected. When using378

Comsol 4.2, the direct solver “MUMPS” was used. Direct solvers requires379

significantly more memory and solutions used 2 Gbytes to solve a mesh with380

34012 tetrahedral elements and took 23.5 s. The mesh was refined in the381

vicinity of the current electrodes by specifying a maximum element size of382

10−4 at these points.383

In figure 8a the squares show the magnitude of the apparent resistivity384

as a function of frequency while the solid line shows the direct evaluation385

of equation 9. The numerical simulation results are shifted upward slightly386

compared to the direct evaluation, however, the error is significantly less than387

1%. In figure 8b, the phase shift as a function of frequency is plotted and388

the agreement between the numerical calculation and the direct evaluation389

is essentially perfect. Again, more complex geometries could be investigated.390

3. Accuracy Analysis and Sensitivity Study391

In order to investigate the accuracy of the numerical solutions, two test392

cases were devised that were compared with analytical solutions. The first393

test case chosen was the calculation of the vertical component of gravity from394

an infinite horizontal cylinder as in section 2.1. The missfit was defined as395

missfit =

∫(gz − gzcyl)

2dV∫g2zcyldV

. (10)

where the numerical integration was carried out over the volume of the do-396

main, excluding the infinite elements and the volume of the cylinder itself.397

A number of simulations were undertaken with various resolution, with and398

without infinite elements, and with different domain sizes. The missfit for399

these simulations, as well as the solution time and memory usage are sum-400

marized in table 1. As can be seen, missfit decreases substantially when401

infinite elements are used as was also demonstrated in figure 1b while the402

solution time and memory usage are only weakly affected. Increasing the403

resolution, either by increasing the number of elements or by increasing the404

order of the basis functions decreases the missfit, as would be expected, but405

at a significant cost in solution time and memory usage. However, if the406

resolution used is too low, as in the simulations with only 11 312 elements407

20

−6 −4 −2 0 2 4 60.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

log10

(ω (s−1))

|ρa|

a

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

log10

(ω (s−1))

φ (0 )

b

Figure 8: a) The magnitude of the apparent resistivity and b) of the phase angle of theresistivity as a function of the frequency of the current. Squares are the results of thenumerical simulation while the solid lines are the direct evaluation of equation 9.

21

domain # element infinite infinite solution memory missfitwidth elements order elements element time(m) width (km) (s) (Mbytes)10000 117 372 2 yes 2000 63 540 2.2 ×10−4

14000 117 372 2 no - 31 540 5.0 ×10−2

10000 421 995 2 yes 2000 457 1095 6.5 ×10−5

10000 11 312 2 yes 2000 3.1 330 6.1 ×10−2

10000 45 890 2 yes 2000 16.7 400 2.8 ×10−3

10000 21 533 2 yes 2000 7.7 372 7.4 ×10−3

10000 117 373 1 yes 2000 3.1 350 1.8 ×10−2

10000 117 373 3 yes 2000 1190 1452 1.35 ×10−4

20000 147 941 2 yes 4000 89 530 2.1 ×10−4

28000 147 941 2 no - 46 590 3.7 ×10−2

24000 129 918 2 no - 37.6 460 7.2 ×10−2

20000 129 918 2 yes 2000 79 533 9.4 ×10−3

20000 471 547 2 yes 2000 542 1041 4.0 ×10−4

20000 479 830 2 yes 4000 536 1069 1.0 ×10−4

10000 126 576 2 yes 4000 70 451 9.6 ×10−5

10000 431 306 2 yes 4000 506 1069 2.8 ×10−5

10000 420 261 2 yes 1000 407 1069 5.6 ×10−4

10000 104 138 2 yes 1000 61 580 9.7 ×10−3

Table 1: Summary of calculations for the gravity due to an infinite cylinder.

and the one with linear basis functions, the missfit increases significantly.408

When infinite elements are used, the effect of increasing the solution domain409

is fairly modest, indicating that the infinite elements are working well. How-410

ever, changing the size of the infinite elements affects the accuracy of the411

solution significantly and it can be seen that the accuracy decreases signifi-412

cantly when the thicknesses of the infinite elements are less than 20% of the413

domain dimension.414

The second test case involved comparing the electrical potential calcu-415

lated in a resistivity calculation with constant resistivity and infinite elements416

placed on the sides and bottom with the analytical expression for an infinite417

half-space of constant resistivity, ρ, and two current sources of strength, I418

22

and −I at points (x1, y1, z1) and (x2, y2, z2):419

V =Iρ

2π[

1

((x− x1)2 + (y − y1)2 + (z − z1)2)0.5−

1

((x− x2)2 + (y − y2)2 + (z − z2)2)0.5].

(11)The missfit in this case was defined in a manner similar to equation 10420

only the analytical solution above was used and compared with the potential421

calculated from the numerical model. The main purpose of this second set of422

calculations was to investigate the effects of refining the mesh in the vicinity423

of the current electrodes and the results are given in table 2. The first three424

simulations listed were carried out to look at the effect of increasing the425

resolution in a uniform mesh. As can be seen, increasing resolution again426

increases the solution accuracy only marginally while significantly increasing427

the solution time and memory usage. The mesh spacing in the vicinity of a428

point can be varied in Comsol by specifying the maximum element size at429

the point. This value is given in the table under “point refinement”. As can430

be seen, increasing the resolution near the current electrodes dramatically431

increases the accuracy of the solution without significantly increasing the432

solution time and memory usage. The last two simulations were carried out433

to investigate the effects of the width of the infinite elements and again, the434

accuracy of the solution was seen to increase significantly when the infinite435

elements were made larger.436

4. Discussion and Conclusions437

The models presented in the previous sections all took only a very short438

time to implement using Comsol’s GUI, unlike previous models of applied439

geophysical techniques which require extensive time writing and testing code.440

This makes this modelling approach particularly useful in a classroom set-441

ting as students can develop models from the equations from start to finish.442

The modeling approach is also very flexible and regions with different ma-443

terial properties and different boundary conditions (such as the insulating444

walls of the resistivity tank) can be easily implemented and changed. The445

multiphysics capability of Comsol also allows various techniques for the same446

anomalous material to be modeled simultaneously and even allows for inter-447

actions between the physical phenomena although interactions are unlikely448

for the physical systems modeled here.449

The capability to put boundaries at infinity using infinite elements is450

extremely useful when modeling applied geophysics techniques.451

23

# point infinite solution memory missfitelements refinent element time

width (m) (s) (Mbytes)20 762 no 0.4 17.5 250 1.0 ×10−2

81 520 no 0.4 97 390 8.5 ×10−3

431 706 no 0.4 1092 1230 3.5 ×10−3

23 292 10−2 0.4 22 262 1.4 ×10−3

28 320 10−3 0.4 20.4 373 1.55 ×10−4

34 025 10−4 0.4 24.6 347 4 ×10−5

16 655 10−4 0.4 6.3 280 1.02 ×10−4

12 052 10−4 0.4 4 270 1.57 ×10−4

98 580 10−4 0.4 106 483 2.3 ×10−5

455 656 10−4 0.4 989 1328 1.1 ×10−5

28 903 10−4 0.3 17.2 322 6.7 ×10−5

27 318 10−4 0.2 16.4 338 1.4 ×10−4

Table 2: Summary of calculations for the potential in a resistivity calculation in an infinitehalf-space with two current sources.

In the future, we intend to compare measured magnetic and resistivity452

anomalies over objects of known geometry in the field with anomalies calcu-453

lated using the models described above. We also intend to produce models454

of induction-based electromagnetic techniques using the AC/DC module of455

Comsol.456

5. Acknowledgements457

We would like to acknowldege Jim Merriam for his contribution to the458

resistivity experiments. We would also like to acknowledge reviewers Colin459

Farquharson and Michael Cardiff for their helpful comments that greatly460

improved this manuscript.461

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26