Physical and Physio-chemical Properties of Water . . . . . . . . . . . . . . 79 ...... Glenn and Ward (1976) for one-dimensional (ID) resistivity inversion. A critical review of ...
Assessment of Integrated Geophysical Groundwater Prospecting Methods on Aquifer Structures in The Basement Complex in NE Nuba Mountains - Sudan
Nuha Elzein Mohamed December2007
Abstract Integrated geophysical methods were applied to map the groundwater aquifers on complex geological settings, in the crystalline basement terrain in northeast Nuba Mountains, which is structurally controlled.
The water ow is controlled by the northwest-southeast extensional
faults as one of several in-situ deformational patterns that are attributed to the collision of the Pan-African oceanic assemblage of the Nubian shield against the pre-Pan African continental crust to the west. The used assessments are the electrical resistivity tomography (ERT), very low frequencyresistivity (VLF-R), and the audio magnetotelluric (AMT) soundings, vertical electrical soundings (VES) in addition to water quality analysis and petrophysical measurements. These measurements were designed to be overlapped in order to prove the reproducibility of the geophysical data and to provide better interpretation of the hydrogeological setting in the aquifer complex structure. Dierent inversion schemes accomplished by means of a cluster analysis were attempted for the synthetic and observed ERT data to study their reliability to map the dierent geometries in the complex subsurface such as fault and graben structures. The sedimentary sequence is not symmetrical due to the seasonal variations of the ow direction.
Variations of the basement
types were also recorded by their resistivity values and consequently their weathered products in uence to the water quality. The VLF-R data with Transverse Electric (TE) mode has been collected using the transmission stations GBR, GBZ and RHA of frequencies 16.0, 19.6 and 23.4 kHz respectively and it was two dimension (2D) laterally-constrained inverted in a smoothly 2D resistivity distribution model after manual incorporation of a priori data . The inverted ERT and VLF-R lines con rmed the fracture zones and the deeper basins. Eight AMT soundings were obtained, true resistivity values and their relevant thickness were calculated and viewed as three dimension (3D) surface images. The VES data was conducted, where ERT survey was not accessible, and inverted smoothly and merged with the ERT in the 3D resistivity grid. Synthetic magnetotellurics MT data enabled to test the inconsistencies between the interpreted geophysical results and the geologic expectation across AlBetira fold. Hydrochemical analysis was applied to 42 water samples collected from the dug wells in the study area, extremely high saline zones were recorded due to dierent reasons. The petrophysical measurements of arti cial saturated aquifer, using dierent soil types and various water salinities, provide a range for the porosity, formation factor and resistivity of the bulk formation and the later can be represented in term of grain size distribution. A combination between the electric conductivity (EC) data and iso-resistivity horizons provide better resolution of the target aquifer location, type and grain size distribution. The formation factor was estimated from the EC map and the corresponded bulk resistivity from the depth slices and it varies from 1 to 6.7, except of AbuGebiha which reaches 18. This result is con rmed by the previous petrophysical measurements and in addition to the water quality analysis, drainage patterns, and faults locations derived from VLF-R and ERT data.
2
New aquifer targets were suggested in the the north eastern part of AlTrtr area and along both Khor Baggara and Khor BanGadid in AlBetira area with respect to the water quality in both localities, and the southern parts of AbuGebiha town is considered as reasonable targets for groundwater supply.
3
Acknowledgment
This PhD thesis has been completed with the contribution of many people who had provided their help in valuable ways. I am thankful for Prof. Dr. Ugur Yaramanci and Prof. Dr. Uwe Troeger at the Technical University of Berlin (TUB) for supervising this research providing me with the perfect work conditions to cover the study objectives with in the planned time schedule. Their fruitful discussions and encouragements guided me for higher scienti c understanding. I am grateful to Prof.
Dr.
Badrel Din Khalil, the Ex Dean of the Faculty of Science and
Technology at Al Neelain University, who agreed to be my internal supervisor, his geological discussions greatly improved the quality of my work, also his continuous administration help during the two conducted eld trips has contributed in successful eld works and good data quality. I thank Dr. A. Eltom for his co supervision and for the enjoyable team work, fruitful discussions and successful days in the eld. Thanks and appreciations are due to the German Academic Exchange Services (DAAD) for the nancial support all over the period of the study I like to thank Dr. Braun for her nonstop help especially during Matlab work, Marco Heigel, Firas Alali, and Frau Biba at the Hydrogeology Department for her assistance during the hydrochemical analysis, and to all the colleagues at the Department of Applied Geophysics and Applied Hydrogeology at the TUB to provide the friendly working atmosphere. My appreciation reaches to Dr. Hennri Brasse at the Free University (FUB) during the interpretation of the electromagnetic data, and Dr. Oliver Ritter and Nasir Mugbel at the Applied Geophysical Center (GFZ) and the Magnetotelluric Group in our monthly meetings and the valuable discussions. I am gratefull for Dr. Thomas Gunter for providing me his software DC2DInvRes and his comments during the two dimensional modeling and inversion of the resistivity data. My indebted thanks to all my colleagues at the School of Applied Earth Scineces of Al Neelain University, Dr.
Abdella Alhaj (the revised Head of the School), and to Dr.
Gamal Abdella,
Mohammed AbdelGalil, Isam Ali, Azza Fuad, Hago Ali during the eld works.
My thanks
reach Dr. khalid Kheirallah for his great help and the fruitful discussion during loading the three dimensional models in GoCad. My appreciation goes to the eld joined team from the Geological Research Authority in Sudan (GRAS) represented by the geophysists Awad Alkareem Abdallah and Mohammed Mustafa for providing the Resistivity Imaging equipment and their great help in the second eld work.
1 am greatly indebted to the National Corporation of Rural Water
Development (NCRWD) o ces in Khartoum and El Obeid for providing the data and through discussion and motivation especially Muhai Aldin and Salah Mahgoub. Finally, I thank my husband for helping me in many ways, his valuable comments during our geological discussions, and foremost, for his endless patience, love and support.
4
Contents
1. Introduction
15
1.1.
Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.2.
Physiographic Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.2.1.
Topography
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.2.2.
Drainage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.2.3.
Climate
16
1.2.4.
Vegetations
1.2.5.
Socio-economic Activities
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
. . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.3.
Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.4.
Study Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.5.
Plan of the Thesis
18
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Geology and Geotectonic Setting 2.1.
21
Geology of The NE Nuba Mountains . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.1.1.
The high-grade gneisses
21
2.1.2.
The Kabous ophiolitic melange
2.1.3.
The low-grade volcano-sedimentary sequence
2.1.4.
Quaternary Cover
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
. . . . . . . . . . . . . . . .
23
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.2.
Structural Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.3.
Remote Sensing Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
3. Geoelectric Measurements
29
3.1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.2.
Theoretical 1D Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.2.1.
Limitations
30
3.2.2.
1D Resistivity Inversion
3.3.
3.4.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
Theoretical 2D Electrical Resistivity Tomography (ERT) . . . . . . . . . . . . . .
31
3.3.1.
Forward Modeling
32
3.3.2.
2D Resistivity Inversion
3.3.3.
Least-squared smooth inversion
3.3.4.
Ridge regression inversion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(L2)
32
. . . . . . . . . . . . . . . . . . . . .
35
. . . . . . . . . . . . . . . . . . . . . . . .
37
1D Field Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.4.1.
Data Acquisition
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.4.2.
Data Inversion and Interpretation . . . . . . . . . . . . . . . . . . . . . . .
38
3.4.3.
2D Pro ling of Field VES Data . . . . . . . . . . . . . . . . . . . . . . . .
40
3.4.4.
General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
(L1)
5
Contents
3.5.
2D ERT Field Data
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.5.1.
2D Synthetic Data
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.5.2.
Field Data Inversion Results . . . . . . . . . . . . . . . . . . . . . . . . . .
50
3.6.
Interpretation of 2D Inversed Data
. . . . . . . . . . . . . . . . . . . . . . . . . .
52
3.7.
3D resistivity modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
3.8.
General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
4. Electromagnetic Measurements 4.1.
4.2.
4.3.
4.4.
55
Elementary Electromagnetic Theory
. . . . . . . . . . . . . . . . . . . . . . . . .
55
4.1.1.
Theoretical Background
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.1.2.
Magnetotelluric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
4.1.2.1.
Homogeneous Earth . . . . . . . . . . . . . . . . . . . . . . . . .
57
4.1.2.2.
Layered Earth
59
4.1.2.3.
MT Mode decomposition in the 2D case
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
Magnetotelluric MT measurements . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.2.1.
Numerical 1D-MT modeling and Inversion
. . . . . . . . . . . . . . . . .
61
4.2.2.
Synthetic 2D-MT across Betira fold
. . . . . . . . . . . . . . . . . . . . .
62
4.2.3.
Inversion of the synthetic 2D-MT data . . . . . . . . . . . . . . . . . . . .
63
4.2.4.
MT Field Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
Very Low Frequency VLF Field Measurements
. . . . . . . . . . . . . . . . . . .
63
4.3.1.
Fraser - KarousHjelt lters
. . . . . . . . . . . . . . . . . . . . . . . . . .
68
4.3.2.
Observed VLF anomaly and DC pro les . . . . . . . . . . . . . . . . . . .
70
4.3.3.
Laterally constrained 2D inversion
73
4.3.4.
Pseudo current density cross sections
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
5. Hydrogelogy and Petrophysics 5.1.
5.2.
5.3.
Field Applications
79
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
5.1.1.
Water Sampling
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
5.1.2.
Physical and Physio-chemical Properties of Water . . . . . . . . . . . . . .
79
5.1.3.
Groundwater Classi cation
. . . . . . . . . . . . . . . . . . . . . . . . . .
83
5.1.4.
Pumping Test Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
Petrophysical Relations 5.2.1.
Theoretical Background
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
5.2.2.
Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
5.2.3.
Speci c internal surface
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
5.2.4.
Permeability
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
5.2.5.
Relationships between
5.2.6.
Electrical properties of rocks
5.2.7.
k, Φ,
and S
. . . . . . . . . . . . . . . . . . . . . .
94
. . . . . . . . . . . . . . . . . . . . . . . . .
95
Synthetic Petrophysical Measurements . . . . . . . . . . . . . . . . . . . .
98
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103
General Remark
6. 3D Integrated Interpretation
104
6
Contents
6.1.
3D Integrated Resistivity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
104
6.2.
EC map and Iso-Resistivity Horizons . . . . . . . . . . . . . . . . . . . . . . . . .
105
6.3.
Potential GW-Targets
106
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7. Conclusions and Outlook
110
Bibliography
112
A. Appendix to Chapter 2
119
B. Appendix to Chapter 3
121
C. Appendix to Chapter 4
122
D. Appendix to Chapter 5
133
7
List of Abbreviations
General notations Khor
Water stream or Wadi
DEM
Digital Elevation Model
AlTrtr
Al Terter village
AbuGeris
Abu Geris village
AlBetira
Al Betira village
AbuGebiha
Abu Gebiha town
DC
Direct Current Method
VES
Vertical Electrical Sounding
ERT
Electrical Resistivity Tomography
VLF-R
Very Low Frequency - Resistivity method
MT, AMT
Magnetotelluric, Audio Magnetotelluric method
TE, TM
Transverse Electric, Transverse Magnetic
1D, 2D, 3D
1, 2, and 3 Dimension Geoelectric inversion symbols
F
The forward operator to compute the potential distribution
ρ
The measured apparent resistivity
ρ
0
The model resistivity in the initial inversion model
Hi
Initial depth in meters
L1
Block inversion
L2
Smooth inversion
λ
Regularization parameter
di
The data
pj
th parameter The model j
Aij
Jacobian or coeecient matrix
φ
Total inversion error in
χ
Chi^2 or model miss t
e
Data mis t
ith
parameter
d
and
p
Magnetotelluric inversion symbols
E0
θ
x
Ex
The electric vector as
Hy
The magnetic vector as
f
Frequency in kiloHertz
Zs
The penetration depth of the electromagnetic wave
ρ
Apparent resistivity of a layer at a known angular frequency
at an angle
H0
at an angle
8
to the
θ
to the
axis in the TE mode
y
axis in the TE mode
Contents
φ
Impedance phase in the top of the layer at a known angular frequency
Z(ω)
Impedance in the top of the layer at a known angular frequency
Zn
Intrinsic impedance of the nth layer in the one dimensional model
Kn
Wave number in the nth layer in the one dimensional model
hn
Thickness of the nth layerin the one dimensional model
D
Skin depth of the electromagnetic wave in a homogeneous earth Hydro-Petrophysical symbols
SAR
Sodium Absorption Ratio
IB
Ionic Balance to estimate the accuracy of the chemical analysis
Φ
Total porosity of all void types between the solid components
k
Permeability in a porous rock regarding uid ow through the pore space
σ
0
Real part of the electric conductivity for a harmonic electric eld.
σ
00
Imaginary part of the electric conductivity for a harmonic electric eld.
| ρ∗ |
Resistivity magnitude
ψ
Resistivity phase
F
Formation factor
m
Exponential material constant
9
List of Figures
1.1.
Location map and Digital Elevation Model (DEM) of the eastern part Nuba Mountains, the dashed rectangular represent the study area. . . . . . . . . . . . . . . .
19
1.2.
Methodology Chart of the recent study . . . . . . . . . . . . . . . . . . . . . . . .
20
2.1.
Regional geological map of the Nuba Mountains, modi ed after AbdelGalil (2007).
22
2.2.
Digital Elevation Model (DEM) of the four studied areas showing the drainage patterns in orders: the lowest (4th order) to the highest order (1st order).
. . . .
25
. . . . . . . . . . . . . . . . . .
27
2.3.
Drainage density and frequency of AlBetira area.
2.4.
Drainage density and frequency of AbuGebiha area.
3.1.
The common used electrode con gurations for resistivity measurements in the eld
. . . . . . . . . . . . . . . .
work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.
(a') Field ERT array.
28
30
Discretization of the subsurface into rectangualr blocks
for 2D data inversion, models obtained with: (a) a default alogrithm based on the position of the data points, and the model depth set to equal the largest data depth, (b) number of model blocks exceed the number of datum blocks, (c) extending the model to the edges of the survey line, and (d) using the sensitivity values for a homogenous earth model (Loke, 1999). . . . . . . . . . . . . . . . . . 3.3.
The soundings data show the dierent curve types, and the inverted soundings tv10 and grv11 show the equivelence problem of the 1D inversion. . . . . . . . . .
3.4.
33
39
2D pro les from soundings data in (A) AlTrtr, (B) AbuGeris, (C) AlBetira, and (D) AbuGebiha areas. Each area show resistivty depth pro le from vertical electrical soundings raw data in term of apparent resistivity (Ra) versus half the cuurent spacing (L/2) in meter. Below is the related interpreted depth pro les from the inverted soundings in term of calculated resistivity versus depth.
The fracture
zone and the various sedimentary units are well mapped. . . . . . . . . . . . . . .
41
3.5.
Guide line for typical DC data inversion routine . . . . . . . . . . . . . . . . . . .
44
3.6.
Shows apparent resistivity pseudo section of 2 horizontal layer case and models produced by smooth and block inversion methods, and a 2 layer cluster model calculated for each inversed model.
3.7.
. . . . . . . . . . . . . . . . . . . . . . . . .
47
Shows apparent resistivity pseudo section of a vertical fault case and models produced by smooth and robust inversion methods, and a 2 layer cluster model calculated for each inversed model. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8.
48
Shows apparent resistivity pseudo section of a graben structure case and models produced by smooth and robust inversion methods, and a 2 layer cluster model calculated for each inversed model.
. . . . . . . . . . . . . . . . . . . . . . . . . .
10
49
List of Figures
3.9.
Examples of the 2D-DC eld data in the four study areas. The variation in resistivity responses horizontally and vertically indicate dierential sedimentation processes within the Quaternary cover, in addition to tectonic structures as fracture zones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10. 3D resistivity grids of the four studied areas.
51
A combination of the inverted
1D soundings and 2D ERT lines. The topography and the meandering drainage pattern revealed information about ner and coarser grain accumulations. 4.1.
. . . .
54
The transverse electric (TE mode) is when the electric eld is parallel to strike, while the transverse magnetic (TM mode) is when the magnetic eld is parallel to strike . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.
MT Impedence tensor for simple 1D structure, 2D and 3D strctures. reered as the MT transfer function.
60
It is alos
. . . . . . . . . . . . . . . . . . . . . . . . .
61
. . . . . . . . . .
62
4.3.
MT Forward modeling and Inversion resulty obtain by 1D MT code.
4.4.
(a): Is the geological section made by Abdelsalam and Dawoud (1991), (b): AlBetira digital elevation model contains the location of the (NW-SE) MT pro le (P1) crosses the folded quartzite, also the block-structures of the forwarded resistivity model response, then the inverted resistivity pro le in WinGLink. . . . . . . . . .
4.5.
64
Rose diagrams of the observed resistivity and phase values of the eight AMT soudings which re ects the data quality, orthogonal resistivity and phase plot indicates the accuracy of the measurements. . . . . . . . . . . . . . . . . . . . . .
4.6.
65
3D surface images of the AMT soundings, they are showing the depth to the basement 20 meters around the MT sounding point and the EW and NE-SW blue regions indicate the prevailed low regions of the basement. exageration is applied only to justify the image view.
4.7.
The vertical
. . . . . . . . . . . . . . .
66
Gb1 pro le across AlBatha stream, this data was collected with 23.4 kHz frequency, in the top the real component is plotted versus the Fraser ltered values. Bottom is the current density pseudo section shows conductivity variation in term of the In Phase response.
4.8.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A combination of VLF and DC pro les across AlTrtr stream, to detect the fracture zone. This VLF data were collected with 16.0 kHz frequency.
4.9.
69
. . . . . . . . . . .
71
A combination of VLF and DC pro les across BanGadid and Baggara streams in AlBetira area, to detect the fracture zone.
This VLF data were collected with
16.0 kHz frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
4.10. 2D layer - laterally constrained inversion of VLF T9 in AlTrtr area. . . . . . . . .
74
4.11. A combination of VLF and ERT pro les across AbuGeris stream, to detect the fracture zone. This VLF data were collected with 16.0 kHz frequency.
. . . . . .
75
4.12. A combination of VLF and ERT pro les across AlBatha and Tandik streams in AbuGebiha area, to detect the fracture zone. This VLF data were collected with 19.6 kHz frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
4.13. Current density sections: Application Hjelt lter to VLF inphase component enables one to obtain the equivalent current densities at a constant depth which would cause a magnetic eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
77
List of Figures
5.1.
Na and Ca versus Cl, SO4, HCO3 and NO3 and concentrations as an indicator for the increase of ionic concentration due to the evaporation. . . . . . . . . . . .
5.2.
Electric conductivity variations in the four studied areas in micro-mhos/cm, it increases with the dierent ow directions. . . . . . . . . . . . . . . . . . . . . . .
5.3.
84
85
Chemical composition of the collected water samples in % mg/l. Two hydrochemical groups are distinguished: group A represents AlTrtr, AbuGeris and AlBetira areas. group B is representative of AbuGebiha area.
5.4.
. . . . . . . . . . . . . . . .
86
A salinity map of the NE Nuba Mountains in term of TDS values, which are a combination of the values obtained from this study and previous readings from the NCRWD database. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.
The static water level (SWL) map of the NE Nuba Mountains, which are obtained from previous readings found in the NCRWD database.
5.6.
87
. . . . . . . . . . . . . .
87
Porosity vs. permeability and the in uence of grain size of sediments varied from 1.very coarse and coarse grained; 2.coarse and medium grained; 3. ne-grained; 4.silty; 5. clayey (see Serra (1984)). . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7.
93
A simple pore channel model to relate between porosity, permeability and speci c internal surface of a rock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
5.8.
A schematic presentation of the two conductive component in porous shale rocks
97
5.9.
The bulk (formation) resistivity is plotted versus the water resistivity in both 15 medium sand samples and 9 samples of the eld materials.
. . . . . . . . . . . .
100
5.10. The formation factor is plotted versus the bulk resistivity in which the later can be represented in term of grain size distribution, also the formation factor is plotted against the sampled water resistivity. The reddish star sand samples while the blue circles 6.1.
•
∗
is refereed to medium
are refereed to the collected eld materials.
102
Build up of the 3D resistivity grids in AlTrtr and AbuGeris areas starting from lower depths where basement surfaces is visible and in shallower depths the sedimentation increases upwards.
6.2.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107
Build up of the 3D resistivity grids in AlBetira and AbuGebiha areas starting from lower depths where basement surfaces is visible and in shallower depths the sedimentation increases upwards.
6.3.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
108
The observed EC maps are correlated to iso-resistivity horizons of 20, 100 and 300Ω.m of the 3D resistivity build up models of AlTrtr, AlBetira and AbuGebiha areas.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
A.1. Photo plates of the study area are taken during the second eld work . . . . . . .
120
C.1. VLF transmitter locations: RHA, GBZ and GBR from the study area. . . . . . .
123
C.2. Inversion Parameters of the synthetic MT pro le across AlBetira fold.
124
. . . . . . . . .
C.3. 2D surface images of the AMT soundings show the depth to the basement
. . . . . . .
125
. . . . . . . . . . . . .
126
C.5. Raw AMT sounding data: Btr2, Gbha1 and 2. . . . . . . . . . . . . . . . . . . . . . .
127
C.6. 1D MT forward modeling code in Matlab
. . . . . . . . . . . . . . . . . . . . . .
128
. . . . . . . . . . . . . . . . . . . . . . . . . . .
129
C.4. Raw AMT sounding data: Trtr1 and 2, Gris1 and 2, and Btr1.
C.7. 1D MT inversion code in Matlab
12
List of Figures
C.8. Bt1 pro le across Khor BanGadid, this data was collected with 23.4 kHz frequency, in the top the real component is plotted versus the Fraser ltered values. Bottom is the current
.
130
. . . . . . . .
131
C.10. 2D Lay - Laterally constrained inversion of VLF-Gr3,Gb3 and Gb4 pro les . . . . . . .
132
density pseudo section shows conductivity variation in term of the In Phase response.
C.9. 2D Lay - Laterally constrained inversion of VLF-T5, T6, and T7 pro les
13
List of Tables
4.1.
List of the used transmitters for the VLF-R measurements.
5.1.
Transmissivity values (T) calculated by Finite Extent (FE) and Aquifer Test (AqT). Abbreviation
no
. . . . . . . . . . . .
means that no calculated value was available.
. . . . . .
5.2.
Magnitude of permeability of unconsolidated sediments (after Hoelting (1989))
5.3.
Contains mean values for the Archie parameters
5.4.
Petrophysical data of the medium sand formation.
5.5.
Petrophysical data of the collected clayey soils.
5.6.
Summarized petrophysical results as a range of porosities, formation factors and
68
89
.
94
. . . . . . . . . . . . .
97
. . . . . . . . . . . . . . . . .
99
. . . . . . . . . . . . . . . . . . .
101
a
and
m
resistivity of the rock formations when they are fully saturated with vary-resistivity water samples.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103
D.1. Work sheet of the water samples analysis results, which shows cations and anions concentrations, TDS, EC, pH
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
134
D.3. Identity of the groundwater samples that were used to fully saturate the sandy and the clayey formations, and the related electrical conductivities.
14
. . . . . . . . . . . . . . .
135
1. Introduction
1.1. Location The Nuba Mountains (Fig.1.1) are located in south Kordofan State and it occupies about 140.000
2 of west- central Sudan. The selected study area is located in the northeastern part of the
km
Nuba Mountains which is bounded between latitudes 11° - 12°N, and longitudes 31° - 32°E. The study focused on four areas which are considered as town or big villages: AbuGeris, AlBetira, and AbuGebiha (Fig.2.2).
These are AlTrtr,
The study area is reachable from Khartoum
through an asphalt road to Kosti then through unpaved road to AlTrtr the rst stop. It is about 550 km from Khartoum to the last stop in AbuGebiha town.
1.2. Physiographic Features
1.2.1. Topography The Nuba mountains region is generally characterized by a high lying, steeply undulating surface topography. It is characterized by scattered steep and low massifs sometimes exceed 1000m a.s.l as in J. Termi and J. Temading.
These basement rocks are complex of igneous rocks, meta-
sedimentary-meta-volcanic assemblages. The unconsolidated sedimentary cover is relatively thin and consists of the Quaternary alluvial deposits of the seasonal wadies.
1.2.2. Drainage The drainage system in NE Nuba mountains is closely related to the tectonic structures of the area (Fig.2.2).
The drainage is apparently tectonically controlled by the regional extensional
fractures which are associated with the in-situ deformational patterns with linear, angular and dense dendritic patterns which are more pronounced in areas, but generally radiates pattern can also be found, but equally they ow parallel to the extensional fractures trend (120º SE) and subsequently towards the White Nile. The most important Khors in this area are Khor Tandik, AlBatha, Kabous, BanGadid, AbuGeris and Khor AlTrtr. In addition to numerous minor Khors that are irregular and running parallel to the fault and joint systems of the basement complexes, and they have a considerable amount of running water for short periods during the rainy season and may occasionally form local lakes but after the rainy season the water apparently seeps downs or highly evaporated. Due to the dominant sand dunes (qoz) west of the White Nile, it was di cult to know if the groundwater ow in the study areas is well developed; i.e.reach the White Nile.
15
1. Introduction
1.2.3. Climate The Nuba Mountains region is characterized by a tropical summer climate. The plains are hot throughout the year and the high altitude on the hills is relatively colder, with minimum day temperature reach 15°C in January during the dry season and maximum temperature 35°C in August at the height of the rains. Rainfall varies considerably from year to year from southern to northern part of the area. Rainy seasons are relatively longer compare to other parts of Kordofan State. The rainy season starts from April in the south part of the area up to October. The average annual rainfall is approximately 700 mm per year in southern part and approximately 400 mm per year in northern part. The dierence between north and south is also re ected in the wind directions, with northerly and southerly winds dominating for six months each at Kadugli but with northerly winds dominating increasingly towards the north.
1.2.4. Vegetations The vegetation is controlled by the climate and the amount and distribution of the rainfall. The Nuba mountains are mostly covered by grassland mosaics with trees and in the central area of it is transition woodland to bushland scattered mainly around Khors. In general the vegetation cover is resistant grass, thick bushes and dierent Acacia species such as Talh (Acacia seyal), Seyal (Acacia raddiana), Kitre (Acacia meliferh), Sunt (Acacia nilotica), and Hashab (Acacia arabica), and Tabaldi trees which are used for water storage in its bottom.
1.2.5. Socio-economic Activities The Nuba Mountains region is largely inhabited by the culvated Nuba tribes and the arabic tripes such as the Baggara, Hawazma, Bororo, Fellata, and others, in the plains. The availability of water supply and other services played a major role in the distribution of this population. The Nuba Mountains is densely populated in the northern parts along the railway line and in the big Khor basins, where some irrigated agriculture is possible (Appendix A.1). Away form the railway, the population density decreases. Most inhabitants breed cattle and sheep, some tribes cultivate fruits on the wadies and cereals on the plains.
1.3. Literature Review A general account of the regional geology of all Sudan was grounds by Andrew (1948), in 1951 the Geological Survey compiled a geological map of Khartoum Sheet of scale 1:1 million which included a portion of the Nuba mountains area. Vail (1973) Outlined of the geology of the Nuba Mountains and vicinity southern Kordofan, Kroner (1984) studied the late Precambrian plate tectonics and orogeny to rede ne the term Pan-African. Shaddad et al. (1979) and Sadig and Vail (1986) have established the basement divisions of the crystalline rocks into metamorphic and plutonic igneous rocks. The geology and mineral prospecting of northeastern Nuba Mountains is described by El Ageed and El Rabaa (1980) and Brinkmann (1982) and Khalil (1982). Ahmed
16
1. Introduction
et al. (1984) analyzed the Landsat image technique for groundwater exploration in the Nuba Mountains. The Pan-African crustal accretion in northeast African was rstly published by Vail (1983) and it followed by a study of the Pan-African (late Precambrian) tectonic terrains and reconstruction of the Arabian-Nubian Shield (Vail, 1985). The presence of ultrama c rocks in the northeastern Nuba mountains region was rst noted by El Ageed (1974), and Hirdes and Brinkmann (1985), and Steiner (1987) identi ed these rocks as ophiolitic fragments through a detailed geological and tectonic investigation. Geology, structures and tectonics of northeastern Nuba mountains with special emphasis on the El Betira area (Abdelsalam, 1987); and the Kabous ophiolitic melange and its bearing on the western boundary of the Nubian Shield (Abdelsalam and Dawoud, 1991). Two major important projects have been carried out in the Nuba mountains, the rst one is the Nuba Mountains Project (NMP), during the period 1977-1982, for geological mapping, prospecting and exploration, hydrogeological and geophysical investigations.
This was a joint project
between the Department of Geology, University of Khartoum, and the Faculty of Engineering , Peoples Friendship University (PFU) of Moscow, USSR; the other project is SudaneseGerman Technical Cooperation , during the period 1979-1984, for geological mapping and mineral prospecting in the Nuba mountains. The program was executed in cooperation between the Bundesanstalt für Geowissenscaften und Rohstoe (BGR) - Federal Institute for Geosciences and natural Resources - Hannover, and the Geological and Mineral Resources Department (GMRD) of the Ministry of Energy and Mining in Khartoum.
1.4. Study Objectives The objectives of this study are two folds: To assess the impact of the resistivity and the different inversion schemes obtained from dierent geophysical measurements on a complex aquifer structure and to delineate the subsurface in order to locate new aquifer targets based on the hydrogeological-petrophysical investigation. These two objectives imply dierent steps: 1.
Applications of electrical resistivity using VES and ERT survey to observe the resistivity
response of the subsurface 2.
Electromagnetic investigation consists of VLF and AMT survey to provide a comparative
data. 3. Synthetic modeling and inversion of dierent geophysical data to demonstrate the dierent inversion schemes and their in uences on the true models. 4. Hydrogeological studies and hydrochemical analysis to estimate the aquifer parameters and the quality of groundwater. 5. Petrophysical measurements to provide the exploitable electrical responses of several synthetic aquifer formations and consequently to estimate the physical properties of the suggested aquifers. 6. Results correlation and combination to nd appropriate targets for groundwater exploration.
17
1. Introduction
1.5. Plan of the Thesis Chapter 2 introduces brie y the geology and the prevailed structural features in the study area, also it discusses the major deformational phases that are attributed to the tectonic evolution in the Nuba Mountains especially the fractures and joints that controlled the surface and groundwater ow. Several geophysical methods are used for groundwater investigations, most of them take advantages of a correlation between the dierent physical properties such as electrical conductivity. In Chapter 3 the electrical resistivity eld data, consists of one dimension (1D) vertical sounding and 2D pro ling and ERT imaging, is subjected to several schemes of inversion processes (in IPI2WIN, Res2DInv, and DC2DInvRes soft wares) to demonstrate the usefulness of both smooth and robust block inversion.
In addition to the synthetic geophysical data which are
carried out to study the instabilities of the resistivity responses and to prove the reliability and the reproducibility of the inversion routines. Chapter 4 demonstrates the VLF and the AMT measurements and their correlation to the resistivity data to determine the fractured zones and the basin locations. Synthetic MT data is forwarded and inverted smoothly in WinGlink interactive software and to simulate a geological cross section across AlBetira fold made by Abdelsalam and Dawoud (1991). In Chapter 5 the hydrogeological measurements and the visibility of the estimated aquifer parameters as porosity, transmissivity and hydraulic conductivity. Petrophysical measurements on arti cial medium sand and collected soil samples is described in this chapter in addition of an estimation of the formation factor of the synthetic formation. The most essential part is the combination of the EC map and the iso-resistivity horizons and the resistivity build up models as shown in Chapter 6 and a new estimation of the formation factor is calculated to con rm the ranges found from the petrophysical data.
In summary Chapter
7 concludes the thesis work and provide the numerical results and locations of new potential targets for groundwater aquifers.
18
1. Introduction
27° 30° 33° 36° E
N 16° 14° 12° 10° 8° 0
400Km
4°
AlTrtr AbuGeris AlBetira
AbuGebiha
Figure 1.1.: Location map and Digital Elevation Model (DEM) of the eastern part Nuba Mountains, the dashed rectangular represent the study area.
19
1. Introduction
Methodologies
Geophysics
Synthetic data (Forward + inversion)
Hydrogeology / Petrophysics
Previous Geology + Remote Sensing Field data (inversion)
1D MT sounding (Matlab code) 2D MT data (Across AlBetira fold) 2D ERT data (horz., fault, graben)
Magnetotelluric data (AMT)
Drainage Map + Lineament map
Synthetic data (Petrophysical ...)
15 Sat. Sand ( , , FF)
Electromagnetic data (VLF)
27 Sat. Obs. Samp. ( , , FF)
Elect. Resis. data (VES, Profiling, ERT)
Field data (Hydrochemical ...) Well samples (water quality) Pumping test + recovery test Elect. Cond. map
3D Res. Grid 3D Integrated Interpretation and Modeling 3 Intgrated dime Figure 1.2.: Methodology Chart of the recent study
20
2. Geology and Geotectonic Setting
2.1. Geology of The NE Nuba Mountains The Nuba Mountains in central Sudan are a crystalline basement uplift that is entirely surrounded by Mesozoic to Cenozoic rocks lling several graben (Browne and Fairhead, 1983). Studies by Vail (1973), Shaddad et al. (1979) and Sadig and Vail (1986) have established the basement divisions of the crystalline rocks into metamorphic and plutonic igneous rocks. The former comprise highgrade gneisses in the west and low-grade volcano-sedimentary sequence to the east (Fig. 2.1). The latter include syn-tectonic and late-tectonic granites, and post-tectonic igneous complexes. Geology, structures and tectonics of northeastern Nuba mountains with special emphasis on the El Betira area and the Kabous ophiolitic melange and its bearing on the western boundary of the Nubian Shield are widely illustrated (Abdelsalam and Dawoud, 1991).
2.1.1. The high-grade gneisses The high-grade gneisses occupy the western part of northeastern Nuba Mountains and are exposed as low outcrops overlain in places by thick Quaternary sediments (Abdelsalam and Dawoud, 1991). The terrain is dominated by quartz-feldspathic gneisses frequently associated with small concordant lenses of Muscovite granites of EL Obied area (Fig.
2.1) is estimated to be
2000 Ma (Harris et al., 1984), this estimate is based on isotopic interpretation. The Rashad and Abbassiya syn-tectonic granites at the eastern boundary of the high-grade gneisses gave TDM model Nd ages of l000 and 950Ma respectively (Harris et al., 1984). However, the metamorphic and structural history of northeastern Nuba Mountains supports a pre-Pan-African age for the high-grade gneissic terrain. Whether these gneisses are of Archean, early Proterozoic, or middle Proterozoic age must await further geochronological investigations.
2.1.2. The Kabous ophiolitic melange The Kabous ophiolitic melange is about 10 km wide, separating the high-grade gneissic terrane from the low-grade volcano-sedimentary sequence, and can be traced for more than 70 km in a NNE direction (Fig.2.1). The zone is characterized by imbricate thrusts and dips shallower than those in the volcano-sedimentary sequence to the east or the gneisses to the west. In contrast to the volcano-sedimentary sequence, the rocks within the melange zone consist of discontinuous bodies such that the stratigraphic continuity is destroyed. Due to the intense deformation
21
2.
30°E
Geology and Geotectonic Setting
31°E
32°E
El Obeid
Kosti
13°N
Rabak
Umm Ruwaba
El Rahad El Jebeleen Khor Abu Habil
12°N White Nile
Rashad
Abu Gubeiha
11°N N
E
W
S
100
0 PROTEROZOIC
PHANEROZOIC Wadi Deposits Colluvium Sediments
200km
Unconsolidated Quaternary Sediments
Syn- Late Orogenic Intrusions Mainly Granite, Syenite & Gabbro Volcano- Sedimentary Greenschist Assemblage
Umm Ruwaba formation
Tertiary - Quaternary Sediments
Upper Proterozoic Ophiolite Mainly Mafic to Ultra Mafic Sequence Metamorphic Rocks Maily Gneiss, Quartzite Middle Proterozoic Marble, Amphibolite & Graphite Schist Amphibolite Facies
Mainly Sandstone
Cretaceous Sediments
Undifferentiated Metamorphic Rocks
Sand Sheets, Dunes and Stabilized Dunes
Extrusive Rocks mainly Ignimbrite, Rhyolite & Basalt Phanerozoic Igneous Activities Intrusive Rocks mainly Granite, Syenite & Gabbro
Lower Proterozoic Amphibolite to Granulite Facies
Geological and Topographic Features Metamorphic Banding
Normal Fault
Shear Zone
Thurst Fault
Town
Seasonal Lake
Figure 2.1.: Regional geological map of the Nuba Mountains, modi ed after AbdelGalil (2007).
22
2.
Geology and Geotectonic Setting
and metamorphism, many of the diagnostic structures and stratigraphy of the ophiolites have been obliterated (Abdelsalam and Dawoud, 1991). However, small scale primary layering can be observed in the metagabbro, which strikes NNE and dips gently (30° to 40°) to the west. The mineral assemblage of the ultrama c bodies indicate overprinting by green schist facies metamorphism on higher grade. This higher grade of metamorphism probably occurred when these rocks were at relatively deep levels, and before being thrusted to their present positions. The existence of garnet schist and amphibolite suggest that medium amphibolite facies metamorphic conditions had been reached (Hirdes and Brinkmann, 1985).
2.1.3. The low-grade volcano-sedimentary sequence This group of rocks occupies the eastern part of northeastern Nuba Mountains where the four study areas are located (Fig.2.2) and they are characterized by a thick sequence of poly deformed meta-sedimentary and meta-volcanic rocks. few hundreds of meters.
Thicknesses of units range from a few tens to a
The metamorphic mineral assemblages of the rocks indicate green
schist facies grade of metamorphism.
Lithologies include chlorite schists (ma c meta-volcanic
rocks); quartzofeldspathic schists (felsic metavolcanic rocks); quartz and quartz mica schists (psammites); metacherts and cherty marbles; quartzites; and graphitic schists (pelites). Hirdes and Brinkmann (1985) suggested that the quartzites are metamorphosed cherts. This volcanosedimentary sequence is intruded by a late-tectonic, heterogeneous granodioritic pluton.
2.1.4. Quaternary Cover The super cial deposits exist in the Nuba mountains region as alluvials which are head water sections of Wadies. The deposits are mainly composed of sands with some clayey lenses. In the study area the thickness of these alluvials may range from 8 to 25m. The gradient of the Wadies decreases downstream with increase in the clayey content, and the alluvial thickness increases and reach the maximum at 70m and increases towards the White Nile. These Wadies ow directions are related to the geology mentioned above and they generally follow the extensional fractures of NW-SE direction. The width of the active khors reach up to 40 meter (Appendix A.1).
2.2. Structural Deformations The rocks of the northeastern Nuba Mountains record three phases of deformation and
(D3 ),
which almost completely destroyed the primary structures.
(D1 , D2
Primary layering can
be recognized only in the gabbro and from the alternation of meta-sedimentary bands which have dierent lithologies. The early phase of deformation
(D1 )
is characterized by overturned
isoclinal folds at both mappable and smaller scales which is resulted in the development of a strong schistosity
(S1 )
parallel to the bedding planes
in more competent lithologies such as quartzites. deformation are mainly
extensional
(SO ).
Broadly spaced cleavages developed
The lineations (L1 )
(Abdelsalam and Dawoud, 1991).
23
related to this phase of
2.
Geology and Geotectonic Setting
The second phase of deformation
(D2 )
is the main phase and dominates other structures of the
area. It has a dierent style between the volcano-sedimentary sequence and the Kabous ophiolitic melange zone. Within the volcano-sedimentary sequence, the
D2
deformation is characterized by
tight, slightly overturned easterly verging folds. A major easterly overturned anti-form is outlined by a quartzitic bed and dominates most of the southwestern part of the volcano-sedimentary sequence. The axial planes of these folds consistently dip between 70° to 80° to the west. The planar fabric associated with this phase of deformation is mainly axial planar crenulation and spaced cleavages, which strike NNE and dip to the west.
L2
lineations within the volcano-
sedimentary sequence are mainly intersectional and plunge 45° to 5O°NNE. The third phase of deformation
(D3 )
is mainly cataclastic and forms E-W trending wrench faults and shear
zones which can be traced over all northeastern Nuba Mountains and they indicate a dextral sense of displacement. Structural elements are very scarce in the high-grade gneisses, making it impossible to work out the structural history. Despite this, the regional setting of the gneisses suggests that they have witnessed a deformational history earlier than
D1
deformation in the
volcano-sedimentary sequence (Abdelsalam and Dawoud, 1991).
2.3. Remote Sensing Analysis The procedure for extracting a drainage network from a DEM with River Tools software is highly automated and only requires user input at a few key points. The Extract Flow Grid routine only needs to be run once for a given DEM. The Extract River Network routine uses a River Tools tree le and a user-speci ed pruning method and threshold to automatically compute and archive a large number of derived quantities like drainage area and slope for every channel link and Strahler stream in a river network. This information is stored in vector-formatted les that end in `_link.rtv' and `_stream.rtv'. River Tools software provides three dierent "pruning" methods for identifying channel sources so that the tree le can be pruned to accurately depict the heads of channels. The calculated drainage network was imported in ArcGIS Map and a geostatistical analysis was made and the DEM is divided into a number of cells to calculate the drainage density as contouring grids, in addition to the spatial distribution of the length in each cell of the grid. The drainage density is the number of stream per cell, and the drainage frequency is the total length of the streams per cell. Then the density values or length values are interpolated separately using two dierent interpolating algorithm: Kriging and Inverse Distance Weighting. The Kriging is a moderately quick interpolator that can be exact or smoothed depending on the measurement error model.
It is very exible and allows to investigate graphs of spatial
autocorrelation. Kriging uses statistical models that allow a variety of map outputs including predictions, prediction standard errors, probability, etc. a lot of decision-making.
The exibility of kriging can require
Kriging assumes that the data comes from a stationary stochastic
process, and some methods assume normally-distributed data.
24
Geology and Geotectonic Setting
AlTrtr - AbuGeris areas
AlBetira area
AbuGebiha area
2.
Figure 2.2.: Digital Elevation Model (DEM) of the four studied areas showing the drainage patterns in orders: the lowest (4th order) to the highest order (1st order). 25
2.
Geology and Geotectonic Setting
Inverse Distance Weighting (IDW) is a quick exact deterministic interpolator. There are very few decisions to make regarding model parameters. It can be a good way to take a rst look at an interpolated surface. However, there is no assessment of prediction errors, and IDW can produce "bulls eyes" around data locations. There are no assumptions required of the data. In AlBetira, the drainge density in both interpolating algorithms revealed the highest values correspond to the area is enveloped by the Quartzite fold and extended south to Khor BanGadid, while the drainage frequency maps con rmed that Khor BanGadid reaches the highest drainage frequency values.
From both drainage density and drainage frequency maps, Khor BanGadid
has the maximum potentiality of new groundwater aquifers. In AbuGebiha area, the drainage density map in both interpolating algorithms revealed the highest values correspond to the area of Khor Tandik and AlBatha, while the drainage frequency maps showed that Khor Kabous and AlBatha reach the highest drainage frequency values. From both drainage density and drainage frequency maps, Khor AlBatha and the northern parts of AbuGebiha town has the maximum potentiality of new groundwater aquifers.
26
2.
Geology and Geotectonic Setting
Figure 2.3.: Drainage density and frequency of AlBetira area.
27
2.
Geology and Geotectonic Setting
AbuGebiha Town
Figure 2.4.: Drainage density and frequency of AbuGebiha area.
28
3. Geoelectric Measurements
3.1. Introduction The resistivity method has its origin in the 1920's by Schlumberger and brothers. The purpose of the surface electrical survey is to determine the subsurface resistivity distribution, the obtained ground resistivity is related to various geological parameters such as rock forming minerals, porosity, uid content and saturation degree. The ground is injected by a current (I) through two current electrode (C1 and C2) and the resulted voltage dierence (V) is measured at two potential electrodes (P1 and P2) and the apparent resistivity (ρa ) can be calculated as
ρa = k k
V I
(3.1)
(in meter) is the geometric factor which depends on the four electrode arrangement, and the
following formula expressed the
k
values in both Wenner and Schumberger arrays
kW enner = 2.π.a
kSchlumberger
π = MN
AB 2 MN 2 ( ) −( ) 2 2
The apparent (measured) resistivity is not the true resistivity when the ground is not homogeneous under the dierent electrode con guration, and it has a complex relation with the true
ρ.
The determination of the true resistivity needs an inversion of the measured
resistivity survey con guartions are shown in Fig.(3.1) with their corresponding
ρa . k
Traditional
values.
3.2. Theoretical 1D Resistivity The conventional vertical electrical sounding survey (VES) was used for quantitative interpretation where the center point of the array remain xed and the electrode spacing increase relatively for deeper penetration (Koefoed (1979), Telford (1990), and Loke (1999)). Graphical plotting of AB/2 versus the apparent resistivity
ρa
is basically matched with formalized master curves,
now variable PC-based programs are available which allow a rapid computation of an arbitrary number of layers for interpreter-de ned underground model that incorporate known geological
29
3. Geoelectric Measurements
Figure 3.1.: The common used electrode con gurations for resistivity measurements in the eld work.
or stratigraphical parameters through an iteration process to adjust the theoretical model and the measured curve, and the calculation will stop either when it reaches the best t or lowest RMS error or, preferentially, by the interpreter to select one of the solutions with respect to the geological reliability.
3.2.1. Limitations Equivalence of physical solutions is a basic principle in geophysics, but it can consider as a problem especially in VES survey.
That the sounding curve is basically related with many
physically equivalent models which are considerably dierent. Another limitation of the vertical sounding survey is the thin layers which are suppressed in deep sounding survey.
Another
limitation of electrical survey is the resistivity anisotropy in bedded sediments or schisty rock and correspondingly the resistivity parallel resistivity
ρL
ρp
to this features does not equal the longitudinal
(Kirsch, 2006). Despite the obvious limitations, two reasons make the VES survey
is common that it the lack of proper eld equipment to carry out intensive 2D or 3D data and the the lack of of practical computer interpretation tools to handle the complex 2D and 3D models, although the recent development of multi-electrode resistivity instruments (Gri ths et al., 1990) and fast computer inversion software (Loke, 1994).
30
3. Geoelectric Measurements
3.2.2. 1D Resistivity Inversion The inverse problem in resistivity interpretation was reported as early as the 1930s (e.g. Slichter (1933), Vozo (1958)and many more) over a layered earth. Backus and Gilbert (1967) introduced a linear inverse theory for geophysical problems discussing model resolution, least-squares data t, and solution uniqueness even for noisy or insu cient data, and they quanti ed the trade-o between resolution and stability for solutions to inverse problems. But as do many others suers the di culty in smoothness-degree estimation.
Inman et al. (1973) introduced the method of
generalized linear inverse theory for the resistivity problem and they described the linearization of inverse resistivity problems and minimization in a least-squares manner to nd the best possible solution but they did not discuss the singularity of the system matrix that arises with small but non-zero eigenvalues. Hoerl and Kenndrd (1970a) showed that linear estimation from non-orthogonal data (i.e.
nearly singular data) could be re ned or improved by using biased
estimators; this technique has been named ridge regression .
Marquardt (1970) established
the similarities between the generalized inverse method and ridge regression method, and proved the suitability of ridge regression methods for problems with small eigenvalues. Because most common resistivity problems involve small eigenvalues, Inman (1975) introduced ridge regression for inversion of resistivity data. Later, the method was also used by Petrick et al. (1977) and Glenn and Ward (1976) for one-dimensional (ID) resistivity inversion. A critical review of least-squares inversion techniques and their application to layered geophysical problems has been done by Lines and Treitel (1984). Marcuello and Queralt (1985) proposed an iterative technique for inverting ID resistivity data, where an error function is minimized. They tried to resolve equivalence in the resistivity data, and were able to obtain a more realistic model by applying a regularization process.
The drawback of this method that it requires that the
initial assumed model be close to the true solution. Parker (1984) addressed the non-uniqueness of ID inverse resistivity problems using models consisted of layers having uniform thickness, and the solution determined the optimum number of layers and layer thicknesses that would minimize the deviation in a least-squares manner. Assal and Mahmoud (1987) developed an algorithm for interpretation of resistivity data over a layered model, setting the thickness of each successive layer equal to the required depth resolution. They derived spectral re ection coe cients at the earth's surface for the model. These coe cients are functions of the resistivity ratios of adjacent layers and are used in the evaluation of model parameters.
Their results indicated that the
algorithm is suitable for continuously varying resistivity with depth.
3.3. Theoretical 2D Electrical Resistivity Tomography (ERT) For many years DC sounding and pro ling conventional measurements were obtained by much eort for every single data, in the early 1990s multi-electrode systems were developed.
They
allow for employing arbitrary electrode combinations of a pre-installed electrode array. Nowadays, many single data can be obtained shortly making the DC resistivity pro ling method one of the standard investigation techniques for near-surface tasks. In analogy with the impedance tomography in medical imaging, direct current (DC) resistivity measurements are typical appli-
31
3. Geoelectric Measurements
cations arise in the hydrogeological and environmental elds and it is called Electric Resistivity Tomography ERT (Fig.3.2). The earth often shows three-dimensional characteristics, which can limit a 2D interpretation. As multi-electrode systems were developing rapidly, e.g., by the use of multichannel recorders, it became more interesting to carry out measurements allowing for a three-dimensional reconstruction. Moreover, the rapid advancements of personal computers allow for the application of multi-dimensional inversion algorithms, and since boreholes measurements have been carried for many areas, they help to improve the quality of resolved structures at depth.
3.3.1. Forward Modeling An essential part of every inversion scheme is the numerical simulation of measurements for a given parameter distribution. This forward procedure is generally represented by the solution of partial dierential equations as in the. Numerical modeling techniques for surface electrode arrays (e.g. Madden (1967), Coggon (1971), Lee (1975), Jepsen (1969), Mufti (1976), and Dey and Morrison (1976)) have been extensively used on a trial-and-error basis to interpret resistivity data in terms of two-dimensional (2D) geologic structures.
Numerical modeling of resistivity
mapping using subsurface electrodes permitting better accuracy and resolution than surface electrode use only is described by Daniels (1977; 1978; 1981; 1983). Trial- and-error modeling (i.e. optimization of a model based on a forward solution) for interpreting the resistivity eld data is rather di cult and time consuming. At the same time, forward modeling does not yield much information on resolution. Thus, there is a need to develop an inverse technique to interpret resistivity data in terms of 2D or 3D subsurface structures.
3.3.2. 2D Resistivity Inversion Smith and Vozo (1984)and Tripp et al. (1984) proposed a 2D resistivity inversion using a nite dierence technique, their schemes are similar and suitable for complex 2D models but do not incorporate the eects of topography.
Tong and Yang (1990) developed an algorithm
for 2D resistivity inversion where topography is considered in the model, allowing for the direct inversion of resistivity data obtained from a rough terrain, without applying external corrections in advance.
Shima (1990) developed an algorithm to invert 2D resistivity data gathered over
complex 2D structures, formulating the forward problem by the alpha centers method which is a non-linear transformation, and the inverse problem by steepest descent and Gauss-Newton methods.
His eld investigations show that the method has good application for resistivity
surveys in steep, mountainous areas. Recently, several papers describing imaging of subsurface resistivity structures have appeared in the literature. Fry and Neuman (1985) introduced a technique to image subsurface features for resistivity problems using an impedance-computed tomography algorithm.
The paper of
Loke and Barker (1996) was the basis of the commercially available resistivity inversion program
32
3. Geoelectric Measurements
(a')
Figure 3.2.: (a') Field ERT array. Discretization of the subsurface into rectangualr blocks for 2D data inversion, models obtained with: (a) a default alogrithm based on the position of the data points, and the model depth set to equal the largest data depth, (b) number of model blocks exceed the number of datum blocks, (c) extending the model to the edges of the survey line, and (d) using the sensitivity values for a homogenous earth model (Loke, 1999).
33
3. Geoelectric Measurements
RES2DINV, which has been proved in practice.
The foundations for the forward calculation
based on nite dierences were given by Dey and Morrison (1976), Spitzer and Wurmstich (1995) and other authors. The introduction of improved boundary conditions by Zhang et al. (1995) and the singularity removal technique by Lowery et al. (1989) improved the quality of the modeling results signi cantly. Speed and accuracy of diferent discretization schemes and equation solvers have been investigated by Spitzer and Wurmstich (1995). As a result of the rapid advancement of computers, it is now possible to carry out accurate computations for large models with high resistivity contrasts.
The central objectives of the thesis are methods for modeling, inversion
and resolution analysis of DC resistivity data from a complex aquifer structure and mapping precisely locations of water-bearing formation. A cell based inversion technique is commonly used; it subdivided the subsurface into a number of rectangular cells (Fig.3.2) whose positions and sizes are xed (Loke et al., 2003). An inversion routine is then used to determine the resistivity of cells that provides a model response that agrees with the observed data. A number of inversion techniques (Treitel and Lines, 2001) have been used in the interpretation of geophysical data.
These include the least-squares (Inman,
1975), conjugate gradient (Rodi and Mackie, 2001), Singular Value Decomposition (Muiuane and Pedersen, 2001) methods. A commonly used inversion for 2D and 3D resistivity inversion is the regularized least-squares optimization method (Sasaki, 1989; deGroot Hedlin and Constable, 1990; Oldenburg and Li, 1994). This method allows the user to include available information about the surface as constraints on the inversion procedure, so that it will produce results that are closest to the actual subsurface geology. The use of the regularized least-squares optimization method with two dierent constraints will be discussed in this chapter.
deGroot Hedlin and
Constable (1990) have used a version of the regularized least-squares optimization method named as the smoothness-constrained or
L2
norm method.
This method minimizes the sum of the
squares of the spatial changes in the model resistivity and the data mis t. The optimal result of the subsurface geology exhibits a smooth variation such as the diusion boundary of the chemical plume (Barker, 1996). However, in cases of expected sharp transition in the subsurface resistivity such as vertical dikes or hard rock terrains, the
L2
method will smear out these boundaries. An
alternative method which minimize the absolute values of the spatial changes in the model resistivity and the data mis t, which is named the blocky of
L1
norm optimization method,
it produce models with regions that are piecewise constant and separated by sharp boundaries (Ellis and Oldenburg, 1994). The next section gives outline of the mathematical formulations used in the
L2
and the
L1
norm optimization methods. This will followed by the results from
the inversion of the observed eld data and the synthetic models. Model Parameter (m) Measured Data (d)
→
Forward Operator
→Inverse
Inversion problem: to nd
Operator
m = F −1 (d)
→
→
Field Data
Earth Model
⇒ F (m) = d
⇒ F −1 (d) = m
is not so easy:
Analytical solution exists only for 1D problem
Non linear problem
Model parameters > data→ underdetermined set of equations,
→model discretization
→linearization (Taylor expansion)
34
Regularization
3. Geoelectric Measurements
Cycles of inverse and forward operator→Iterative algorithm.
3.3.3. Least-squared smooth inversion (L2) Least-squares solution Consider a set of data containing N measurements over a model containing M parameters; this can be described in the form of the following equation 3.2:
di = F (pi , p2 , p3 ... pN ) where and
F
i = 1, 2, 3...N.
(3.2)
is the dimension of the data space
represents the forward operator to compute the potential distribution.
The non-linear relationship between the data and model parameters can be linearized using a Taylor series expansion which, after neglecting second and higher-order terms, yields equation 3.3
di = d0i +
M X ∂F δpj ∂pj
(3.3)
j=1
In the above relation,
di
is the
i
th measurement and
d0i
is the
ith
calculated datum. Rewriting
equation 3.3 in matrix form gives the main inverse equation 3.4:
δd = A δp where
δd = di − d0i
and
(3.4)
∂F ∂pj is the jacobian or coeecient matrix as
δp = pj − p0j ,
and
∂F1 ∂p1 ∂F2 ∂p1
∂F1 ∂p2 ∂F2 ∂p2
. . .
.
.
. . .
.
.
.
. . .
.
.
.
. . .
.
∂FN ∂p1
∂FN ∂p2
. . .
∂FN ∂pM
A=
Aij =
. . .
∂F1 ∂pM ∂F2 ∂pM
(3.5)
The solution of equation 3.4 is given by various workers as follows:
Least-squares solution (Inverse) The
L2
solution using the linear inverse theory is (Menke 1984):
δp = (AT A)−1 AT δd where the estimate of data mis t
e
is
35
(3.6)
3. Geoelectric Measurements
T
e = δd δd =
N X
(di − d0i )2
(3.7)
i=1
Weighted least-squares solution This approach is also known as Gauss-Newton's method of least-squares minimization (Menke, 1984). Large data errors or outliers in the data can severely disturb (bias) the LSQ solution. To take that into account, data weights are de ned which are the inverse of the standard deviation
W
of the data (error bars): where
is a squared diagonal matrix containing the reciprocal of the
data variances.
δp = (AT W A)−1 AT W δd
(3.8)
where W matrix is
1 σ12
0
.
.
.
0
W =
0
1 σ22
0
.
.
.
0
1 σ32
.
.
.
.
.
.
.
.
.
.
.
.
0
.
.
. 0
. . = σ −2 I . 0
The estimate of model mis t
χ
1 2 σN
(i.e. reduced chi-squared error) is
χ2R =
M X (di − d0i )2 1 N −M σi2 i=1
Generalized damped least-squares solution The solution of the LSQ method requires computation of the inverse part 3.8.
(AT W A)−1 in equation
This expression can become unstable, particularly if applied iteratively.
A more stable
solution oers the generalized damped least-squares by:
δp = (AT W A + λC T C)−1 AT W δd where
(3.9)
λ is the damping factor which controls the model roughness (some times called Marquardt
operator). The generalized damped (regularized) least-squares inversion approach has dierent names depending on the nature of the C and the criteria for selection of the damping coe cients. The important one is Marquardt-Levenburg inversion. identity matrix
If the matrix
C
is considered as the
I , then the Marquardt-Levenburg inverse solution (Marquardt, 1963) is (eq.3.10)
36
3. Geoelectric Measurements
δp = (AT W A + λI T I)−1 AT W δd
(3.10)
Then the solution given by equation 3.9 is called Occam's inversion or smooth inversion. Matrix C is called smoothing operator by
Constable et al. (1987),
C=
0
0
0
.
.
0
and it has the follwoing stucture
. −1 1 . . . . . . . . . . . . 0 . . 0 −1 1
−1
1
0 . . 0
0
.
.
3.3.4. Ridge regression inversion (L1) Tikhonov regularization is the most commonly used method of regularization of ill-posed problems of linear equations. In some elds, it is also known as ridge regression. Hoerl and Kenndrd (1970a) showed that linear estimation from non-orthogonal data
1 (i.e. nearly singular data) could
be re ned or improved by using biased estimators; this technique has been named ridge regression.
Marquardt (1970) established the similarities between the generalized inverse method
and ridge regression method, and proved the suitability of ridge regression methods for problems with small eigenvalue. Because most common resistivity problems involve small eigenvalue, Inman (1975) introduced ridge regression for inversion of resistivity data which is called also Tikhonov regularization. For a simple form of ill-conditioned sytem of linear equations is given by correspondance with the previous equations with
n
entries, and
A
is an
m×n
F (p) = d
and in
d is data vector with mentries, p is the model vector
matrix. The problem is seeking a minimization of equation
3.11
φ(p) =k Ap − d k2 +α k p k2 for some suitably chosen Tikhonov factor
α>
0, here
k.k
(3.11)
is the Euclidean norm. This improves
the conditioning of the problem, thus enabling a numerical solution. An explicit solution, denoted by
(δp)
inverted parameter, is given by
p = (AT A + α2 I)−1 AT d where,
I is
the
n×n
identity matrix. For
over determined problem where
1
α
(3.12)
= 0 this reduces to the least squares solution of an
m > n.
In matrix theory, an orthogonal matrix is a square matrix A whose transpose is its inverse:
37
AAT = AT A = I
3. Geoelectric Measurements
3.4. 1D Field Data
3.4.1. Data Acquisition Vertical Electrical Soundings were carried out at 67 sites in a Quaternary cover of about 126
km2
in a complex aquifer structure with the objective of delineating the aquifer layers, weathered
zones and ssured zones and to map the depth to the bedrock. A Schlumberger array vertical electrical sounding was performed in the study area during November of 2004. The centre point for the sounding was the expected fracture zone, with the array laid out in a northeast/southwest orientation. Terrameter SAS1000 VES transmitter was the current source, and resulting electrical potentials were measured with a multi-meter. The Schlumberger array (Fig.3.1.g) consists of an interior potential dipole (MN), and an exterior current dipole (AB). Both dipoles are arrayed symmetrically in a straight line running through the sounding centre point (Telford et al., 1976). The potential dipole was held constant at 1 m for the rst eight AB spacings and then increased to 10, 20. 50 and 100 m for the next readings. AB spacings were chosen to provide six, ten, and tweleve readings in the rst, second and third decades respectivly of the of increased separation, spaced evenly on a logarithmic scale. During the acquisition of Schlumberger sounding data, the measured potential decreases as the AB spacing increases. Eventually this value will decrease to the point where ambient electrical noise overwhelms the broadcast signal. At this point, expanding the potential dipole will improve S/N and the current dipole was again lengthened. One of the results of this tactic is that apparent resistivity (ρa ) values calculated for the same AB spacing at dierent MN spacings are rarely the same. The dierence (or clutch on the sounding curve) is attributed to local heterogeneity in electrical properties (Ward, 1990). The values determined using small MN spacings were adjusted to match the large MN spacing values at an overlapping point. This adjustment was preferred as the larger MN spacings is less aected by extremely localised conductivity variations.
3.4.2. Data Inversion and Interpretation A log-log plot of
AB/2
ρa versus AB/2 for each sounding was plotted simulateniously and the maximum
was 300 m.
As a general behaviour of the sounding curves that most of them reach a
steady 45 degree slope increases with
ρa
which indicate reaching a continues increasing-resistivity
region, where some curves, especially from AbuGeris area, ended with a steady 45 degree slope decreases with
ρa although
AB/2 exceeds 200 meter as in soundings (tv4) and (gv11) which
indicate reaching a region of continues decreasing-resistivity.
The data processing for this type of survey is minimal which sets the least possible (within the equivalence limits) value for the selected model parameter. apparent resistivity is computed by equation (3.1).
For the Schlumberger array, the
These data are interpreted using either a
curve matching procedure or by applying a computer-based inversion scheme.
38
3. Geoelectric Measurements
Figure 3.3.: The soundings data show the dierent curve types, and the inverted soundings tv10 and grv11 show the equivelence problem of the 1D inversion.
39
3. Geoelectric Measurements
In the curve-matching method, theoretical curves are compared with the reduced data until a close match is obtained. The earth resistivity and depth are then calculated from parameters for the theoretical curves. Computer analysis requires that the interpreter estimate number of layers and approximate resistivity values from the plotted data. A theoretical curve is derived from the estimated parameters and plotted against the eld data. This then forms an iterative process, where the interpreter varies the input parameters to obtain a close match of the data. Once the result is satisfactory, the input parameters were taken to represent the true earth parameters. The inversion software that was used for the present analysis (IPI2WIN) takes the rst parameter estimation and uses a least squares approach to attempt to minimise the dierence between the input data and the theoretically derived curve. Due to handy controls the interpreter is able to choose from a set of equivalent solutions the one best tting both geophysical data (i. e. providing the least tting error) and geological data (i. e. geologically sensible resistivity cross-section). The inverted results from soundings tv10 and grv11 are shown in Figure (3.3) and listed in a table below each sounding, also several cross sections based on soundings lateral combination was made to map the depth to the basement and locate the fracture zones which control the surfacewater and groundwater ow in the study area (Fig. 3.4).
3.4.3. 2D Pro ling of Field VES Data Resistivity depth pro les from vertical electrical soundings.
the geoelectrical data re ect the
geological setting beneath the pro les, and the quaternary sandy aquifer partly covered with clayey materials, also the tectonic graben as a fractured zone is positivly located in these pro les. The interpreted resistivity depth model, from individual soundings, suggests two broad electrostratigraphic zone.
The upper layer is of relatively moderate resistivity, and occurs from 0 to
average 33 m depth. The lower layer is relatively higher in electrical resistivity; and its base is not identi ed. This conductivity structure is consistent with unconsolidated sediments overlying basement rocks. Generally, ner-grained materials (especially mineralogical clays) are lower in resistivity than are coarse grained materials ((McNeill, 1980a)).
The upper electrical layer is
comprised of four zones suggestive of small scale lateral and vertical lithological variations. Zone 1 is an unsaturated zone immediately at the surface (dry top soil) of very thin thickness < 2m and its resistivity varies between 1 to 300
Ω.m,
the shallow zone 2 is (immediately below the top
soil) and zone 4 (overalain the lower layer) suggested ner-grained materials that their electrical resistivity is low, and their thicknesses vary in a wide range and they can correspond to clay metrials or weathered basement terrains. The higher resistivity values in zone 3 are attributed to a local accumulation of more coarse-grained material (coarse sand - gravel). Zone 3 is probably very localised and consider hydrogeologically as a good target, while zone 1 is broadly distributed but contributes little to the gross electrical structure as it is extremely thin. the lower layer behaved in two dierent ways, either as a very high resistive layer which contributes to the hard rocks or as a very low resistivity (< 3Ω.m) of a contineous decrease as the current dipole was extended which is probably the graphite schist or its calyey-rich weathering products that it represents one of the dominant rock type in the study area.
40
20
30 50 70 100
Tv10 400
Grv11
Grv3
100 77.43 59.95 46.42 35.94 27.83 21.54 16.68 12.92 10
20
30 50 70 100 140
Tv12 700 m
Grv5 0
Grv1 Grv3 200 400 600
Grv12 1000
800
1200
Grv11 1400 m
10
10
16
0 D E P -10 T H -20 -30 (m) -40
46 254 72 5-30
(m) -20
9-54 119
-50 -30
300-3985
-60
1012-5674
(A) Ra,L/2
1.6 2.5 4 6.5 10 16
Ra
40 65 100 160 250
Bv4 Bv9 0Bv3 300
Bv13 600 900
1200
Bv10 1500 1800
Ra
2 3
100 77.43 59.95 46.42 35.94 27.83 21.54 16.68 12.92 10
25
Pseudo cross-section
Grv10
Bv11
Bv10
Bv13
Bv4
Bv9
Pseudo cross-section
Grv8
Ra,L/2
(B)
Grv9
0 D E P T -10 H
100 77.43 59.95 46.42 35.94 27.83 21.54 16.68 12.92 10
5 7 10 14 20
30 50 70 100 140
Gv10 0
Bv11 2100m
40 D E 20 P 0 T H -20
Ra
2 3 5 7 10 14
100 77.43 59.95 46.42 35.94 27.83 21.54 16.68 12.92 10
5 7 10 14
Grv1
Grv5
Tv12
Tv10
Tv11
Ra
2 3
Tv11 0
Pseudo cross-section
Ra,L/2
Pseudo cross-section
Ra,L/2
Grv12
3. Geoelectric Measurements
300
Gv8 600 900
1200
Gv9 1500 1800m
10 D 0 E P T -10 H -20 (m) -30
6-23 16
76-131
10
(m) -40 -60 6186
-80
5-20 74-136
2-34
-40 881-1663
-100
-50
(C) Top dry soil
Clayey materials
(D) Basement Rock
Coarse grained aquifer
Figure 3.4.: 2D pro les from soundings data in (A) AlTrtr, (B) AbuGeris, (C) AlBetira, and (D) AbuGebiha areas.
Each area show resistivty depth pro le from vertical electrical
soundings raw data in term of apparent resistivity (Ra) versus half the cuurent spacing (L/2) in meter.
Below is the related interpreted depth pro les from the
inverted soundings in term of calculated resistivity versus depth. The fracture zone and the various sedimentary units are well mapped.
41
3. Geoelectric Measurements
3.4.4. General Remarks The VES provide good local estimates of the earth conductivity in depth, but take a signi cant amount of time to acquire for each sounding (on the order of 3 hours per sounding). Inverting VES data for conductivity structure also introduces the equivalence problem as shown in gure (3.3) in sounding grv11. In hard rocks where stratigraphic standard thicknesses and rock resistivities are frequently well known and constant over large areas, VES modeling and interpretation may be easier leading to a detailed knowledge of the tectonic settings. The resistivity depth pro les from the VES data provide apparent resistivity distribution models in the study area which are geologically and hydrogeologically reasonable and with out discripancies with drillng data or other geophysical results should be obtained. For comparative purposes, a combination of other geophysical techniques was chosen for this study in order to provide some redundancy and so that the limitations of one technique might be counter-balanced by a strength in another technique. The Electrical Resistivity Tomography (ERT) has been chosen to improve the data quality of the apparent resistivity response of the area subsurface in two dimensions distribution and will be discussed as follow.
3.5. 2D ERT Field Data
3.5.1. 2D Synthetic Data The program Res2DInv ((Loke and Barker, 1996)) was employed for the observed electric resistivity tomography ERT lines using the smoothness-constraint inversion, that it has the same principle of the regularized least-squares optimization method (L2 norm) in deGroot Hedlin and Constable (1990). The alogrithm of the smoothness-constrained inversion used in Res2DInv is decribed in Appendix B-1.
Synthetic Data and Initial models The apparent resistivity pseudo section data have been generated in Res2Dmod by Loke and Barker (1996) and inverted in DC2DInvRes byGuenther (2006).
The programs used a nite
dierence approaches to solve for the potential distribution, due to a point of sources of current, which is converted to apparent resistivity values. The data was created using wenner alpha array to simulate the eld survey, each line is 500m and it consists of 25 electrodes and 20 meter as an electrode spacing (A) in order to reduce the number of the considered model parameters, a single resistivity contrast of 1:10 between the top layer
ρ1 = 100 Ω.m
and
ρ2 = 1000 Ω.m).
ρ1
and the basement layer
ρ2
(where
In Res2Dmod, the subsurface is divided into a large number
of small rectangular cells, the apparent resistivity values are calculated and to simulate eld observations a Gauss distributed random noise of 5% standard deviation was added to all models responses and the calculated resistivity values can be read in Res2DInv which is useful to study the resolution and inversion instability. The initial interface (Hi ) varies through the models from 5 to 60 m (normalized to equal 0.25A to 3A). More complicated structures were added to the
42
3. Geoelectric Measurements
simple 2-layers model starting with vertical fault varies in downthrow lengths from 20, 30, 40, and 60 meters. A graben structure of dierent widths and depths was also tested. These models represent common targets in groundwater and environmental investigations in areas underlain by crystalline basement rocks. Trial tests on several synthetic data from typical 2D geological models were investigated include: 2 Horizontal layers, vertical fault, and graben structure, to see the eect of applying dierent damping factor values (λ), and smoothness versus robust constraints inversion.
The start 2D
model used in the inversion part consists of a number of rectangular blocks whose arrangement is loosely tied to the distribution of the datum points in the pseudo section. the distribution and size of the blocks are automatically generated by the program so that the number of the blocks does not exceed the number of the datum. The eect of both the depth to bedrock interface and the type of initial models (either homogeneous or layered) on the inversion results will be described.
Inversion parametrization 1. The Gauss-Newton (with explicit regularization) was the regularization algorithms is applied to the recent study. 2. In order to de ne model parameter weighting by setting the matrix C, the smoothness constraints were selected which are minimizing rst or second order derivatives of the model. The 2nd order constraints have 3 variants (Dirichlet, Numan and Mod Numan.) which dier in the handling of boundaries. 3. A reduction of free parameters often leads to ore detailed results. This can be done by deleting bad covered data (whose coverage is below minimum coverage) or by combining cells in greater depths. Both can be combined and applied to this resistivity data (as a default option). 4. A line search is carried out to determine an appropriate step length using a linear interpolation. It can signi cantly improve converge speed for high non-linearity. However, it needs one additional forward calculation. 5. If the model mis t chi^2 is much greater than 1 and the model shows less structure, lambda has to be decreased. To much structure and over- tted data require an increasment of lambda. Smoothness constraints of 2nd order are useful for delineating the boundaries of small bodies, whereas 1st order behaves better for broad resistivity structures as undulating basement. 6. A guide line in Fig.3.5 displayed the typical routine of eld DC data inversion. The initial model can be homogeneous or multi layers/bodies, based on the priori information from the studied area.
Then setting the appropriate inversion parametrization and a new calculated
resistivity model will be produced, the rms between the observed and calculated models will prove if the inversion parametrization was satis ed, otherwise a new updating of the initial model should take place. 7. Two inversion types are used for each synthetic data set: Least-squared smoothness constraint
(L2)
and Ridge regression inversion
version
(L2)
(L1)
as Robust inversion.
The Least-squared smooth in-
attempts to minimize the sum of the squares of the spatial changes in the model
43
3. Geoelectric Measurements
Figure 3.5.: Guide line for typical DC data inversion routine
resistivity that results in a smooth variation in the resistivity values, where Robust inversion
(L1)
attempts to minimize the sum of the absolute values of the spatial changes in the model
resistivity that tends to produce models with regions that are piecewise constant and separated by sharp boundaries. 8. Cluster analysis is provided in DC2DInvRes program to obtain a simpli ed model concept. Besides the resistivity (and ip phase if present) the geometrical position, i.e.
x and z of the
midpoint, is included to result in more compact models. This analysis uses euclidean distances and complete linkage for the hierarchical separation.
After this the cluster function is shown
(cluster number vs. total distance) and the user is asked to specify the cluster number. If not determined by geology or visual, it is often searched a L-shaped corner as for the selection of regularization parameter. Then the median value of each cluster is associated to all its members. In this study and as the true model consists of 2 layers, the cluster number was speci ed and a 2 layer-cluster model is produced from the inversed data.
2 Horizontal layers The rst test model was 2 horizontal layers where the top horizontal overburden layer is
100 Ω.m
with an initial depths
ρ2 = 1000 Ω.m.
Hi
ρ1 =
of 20 or 40 meters embedded in a high resistive bedrock of
The smooth inversion of the synthetic data, based on a homogeneous initial
model, shows a typical smeared-out image with a gradational boundary along the interface between the two layers and the data mis t was below the noise level was used for this comparison
44
3. Geoelectric Measurements
and the regularization parameter resistivity values for
ρ1
λ
was chosen to be very small .
For
Hi =
was overshoot the true value to a minimum of 131
Ω.m
increasing from 975 to 1150Ω.m as the value of the regularization parameter
Hi
deeper while
ρ2
λ
±2 Ω.m
while
ρ2
was
decreases. For of the true
ρ1 ,
has a recognizable undershoot values to a maximum of 563Ω.m. The robust inversion
results, at
ρ2 of
of 40m the smooth inversion shows model resistivity values
20m, the model
Hi = 20m, produced overshoot model resistivity of ρ1 reached 129Ω.m and a maximum
779Ω.m. while for larger initial depths,
108Ω.m and a maximum
ρ2
Hi
= 40m, the robust inversion showed
ρ1 =
106 to
of 481Ω.m. The horizontal sharp boundary between the overburden
and the bedrock was correctly positioned from the robust inversion models as indicated by the cluster models of the inverted sections.
However if the initial depth is too deep the inversion
results can be unstable, and in such case the depth of the interface of the lower layer begins to undulate as it shown in Fig.(3.6). Oldenburg and Li (1999) and Olayinka and Yaramanci (2000) have also reported such unusual inversion eects at the edges of the 2D structures.
Fault Structure Since the study area has a complex aquifer structures, the second test model was simply a vertical fault where the depth to the top of the fault is 20m and 40m and the fault throw is 60m (= 3A) and
ρ1 = 100 Ω.m
and
ρ2 = 1000 Ω.m.
The apparent resistivity data shows a steeping of the
iso-resistivity contours at the position of the fault. The resistivity data set was inverted based on a homogeneous initial model, and due to the smearing eect produced by the smooth inversion
Hi = 20m the model resistivity values overshoot the true value to a maximum of 109Ω.m
and for for
ρ1 ,
while for the bedrock resistivity
ρ2 was
to a maximum of 1338Ω.m. For
Hi
= 40m, the
smooth inversion model showed undershoot resistivity values reaching to a maximum 99Ω.m. for
ρ1 while
for
ρ2 was
to a maximum 679Ω.m. (Fig.3.7 b, c, j, & k).
The robust inversion results were slightly overshoot that when
Hi
= 20m,
ρ1
and a considerable undershooting in
ρ2
,
ρ1 values were of a maximum to 106Ω.m and of a maximum 794Ω.m, while
for larger initial depths as
Hi
= 40m,
ρ1 = 103Ω.m
, while
ρ2
reached a maximum of 469Ω.m.
The sharp interface between the overburden and the faulted bedrock was correctly positioned, as indicated by the cluster models of the inverted sections, but the geometrical identi cation was better in the robust inversion of both shallow and deeper
ρ2 values,
Hi
although it produced lower
that the smooth inversion produced the fault interface shallower than the true depths
(Fig.3.7 d, e, l, & m).
0
The robust inversion was tested using 2 horizontal layers initial model (ρ1 and
ρ02 )
of dierent
resistivity contrasts and initial depths (Hi ) to the basement interface, generally the inversion results were more aected by the degree of the resistivity contrast than the initial depths. It was not possible to obtain a reasonable interpretation in the case of the shallower
Hi = 1A, among all
test the best- t model in this study was shown in gure (3.7 f ) consists of initial model resistivity
ρ01 of
50Ω.m and
ρ02
of 1000
Ω.m
with an interface at 20 meter from the surface and the data
mis t value does not exceed the noise level 5%. The fault contact was correctly positioned in the inverted section and its geometry was better resolved than in the smooth inversion while the resistivity contrast was reliable than the robust inversion that based on starting homoge-
45
3. Geoelectric Measurements
Hi
neous model. Generally deeper
in the initial model reproduced lower resistivity although the
resistivity contrast in the 2-layer initial model is increased. The robust inversion of the second synthetic fault model, when
Hi
is 40m, was tested using
2-layers initial model and the error of the estimate of the basement resistivity was very low when of the initial top layer resistivity was with
Hi
ρ01 of
50
Ω.m
overlain by a
ρ02 of
Ω.mresistive
1500
layer
at 40 meter (Fig.3.7n) and the data rms mis t converged to about the same level as the
amount of noise in the pseudo section data. The geometry of the faulted basement had a lateral extension with an adulatory interface (Fig.3.7 o & p) as the resistivity contrast increases than 1:30, also when
ρ02
was smaller.
Graben structure The last test model was a graben structure where the depth to the top of the graben is 20m with width 80m (= 4A) and the graben throwdown for 40m (= 2A) and as the previous examples
ρ1 =100Ω.mand ρ2 =1000Ω.m.
The apparent resistivity data shows abrupt vertical and horizontal
changes in the iso-resistivity contours with low resistivity anomaly at the position of the graben. The resistivity data set was inverted based on a homogeneous initial model, and due to the smearing eect produced by the smooth inversion and the model resistivity values overshoot the true value to a maximum of 118Ω.m for
ρ1 ,
while for the bedrock resistivity
ρ2
was lower than
the true value to a maximum of 874Ω.m. The robust inversion results of
ρ1
were to a maximum of 128Ω.m while
ρ2
showed a considerable
undershooting to a maximum of 659Ω.m. The geometrical boundaries of the graben were obviously positioned, as indicated by the cluster models of the inverted sections, but the geometrical identi cation was better in the robust inversion that shown in gure (3.8), that the smooth inversion did not agreed with the true width of the graben although its model resistivity (118 and 796Ω.m) were the closest to the true values. The robust inversion of the synthetic graben model was also tested using several initial models consists of two horizontal layers, and the layer resistivity in the initial model for the robust inversion have been prescribed based on the range of the eld pseudo section data. The best- t model, in which the estimate error of the basement resistivity was very low, is found when of the top layer was
ρ01
of 50
Ω.m
overlain by a
ρ02
of 500
Ω.m
resistive layer and the depth to the
basement was at 20 meter (Fig.3.7n). The geometry of the graben structure was better resolved and no interface undulation is observed (Fig.3.8 g & h) and the data rms mis t converged to a closer level to the amount of noise in the pseudo section data. This study showed that the inversion results are aected by the layer resistivity in the staring model.
This study employ
3-layers initial models comprising of 50Ω.m at 20m and 300Ω.m at 60m overlain the 500Ω.m. It can be observed that the resistivity of the rst layer was reliably estimated and the graben structure was well resolved when the data rms miss t was lower than noise level.
46
3. Geoelectric Measurements
Initial depth of the top layer = 20m
Initial depth of the top layer = 40m
(a) Forward model
(f) Forward model
(g) Smooth Inv.
(b) Smooth Inv.
957 ( c) Smooth cluster
(h) Smooth cluster 131 Ωm
98 Ωm
957 Ωm
563 Ωm
(d) Robust Inv.
(i) Robust Inv.
(e) Robust cluster
(j) Robust cluster 129 Ωm
108 Ωm
623 Ωm
481 Ωm
Figure 3.6.: Shows apparent resistivity pseudo section of 2 horizontal layer case and models produced by smooth and block inversion methods, and a 2 layer cluster model calculated for each inversed model.
47
3. Geoelectric Measurements
Initial depth of the top layer = 20m
Initial depth of the top layer = 40m
(a) Forward model
(i) Forward model
(b) Smooth Inv.
(j) Smooth Inv.
( c) Smooth cluster
(k) Smooth cluster
109 Ωm
1287 Ωm (d) Robust Inv.
(l) Robust Inv.
(e) Robust cluster
(m) Robust cluster
105 Ωm
794 Ωm
103 Ωm
469 Ωm
(f) 2lay initial model
(n) 2lay initial model 50 Ωm
50 Ωm 1000 Ωm
1500 Ωm (o) Robust _2lay initial model
(g) Robust _2lay initial model
(h) Robust cluster_2lay initial model
860 Ωm
99 Ωm
679 Ωm
RMS = 5%
(p) Robust cluster_2lay initial model
103 Ωm
RMS = 6%
102 Ωm 633 Ωm
Figure 3.7.: Shows apparent resistivity pseudo section of a vertical fault case and models produced by smooth and robust inversion methods, and a 2 layer cluster model calculated for each inversed model.
48
3. Geoelectric Measurements
Start depth of the top layer: 20m, graben width: 80m, downthrown: 40m (a) Forward model
(b) Smooth Inv.
( c) Smooth cluster 111 Ωm 834 Ωm (d) Robust Inv.
(e) Robust cluster 124 Ωm 596 Ωm (f) 2lay initial model 50 Ωm 500 Ωm (g) Robust _2lay initial model
RMS = 6%
(h) Robust cluster_2lay initial model 115 Ωm 668 Ωm
Figure 3.8.: Shows apparent resistivity pseudo section of a graben structure case and models produced by smooth and robust inversion methods, and a 2 layer cluster model calculated for each inversed model.
49
3. Geoelectric Measurements
General Remarks The inversion of the dierent synthetic pseudo section data was tested, employing smooth and robust constraints, based on an initial homogeneous model and an initial 2layer model.
The
selected structure settings have dierent electric resistivity responses but in the smooth inversion results
ρ02 varies up to twice the true ρ2 value with remarkable inversion eects at the edges of the
2D structures. The smooth inversion provide a good agreement in the resulted resistivity values but it was di cult to x the actual geometry of the overburden-basement interface. The robust inversion provide better geometry of the second layer (basement terrain) but lower resistivity value than the true with deeper true
ρ2
although the true
Hi was
shallower, and the undulating interface increase
Hi .
The start initial model was modi ed based on a priori information from wells data that the basement is reachable between 20 and 40 meters where the fracture zones are the targets of the ERT mapping, and in this case the robust inversion improved the inversion quality to a better agreement and stability except of the deeper basement interfaces where the smooth inversion of the fault model has more reliable resistivity and geometry. Moreover a re-interpretation of the apparent resistivity data of the graben structure employing 3-layers initial model with a boundary layer of 200Ω.m between the top 50Ω.m layer and the 500Ω.m basement layer. The corresponding smooth inverted model has a more gradual change in resistivity values across the graben boundaries, robust-inverted model provide the lowest data rms mis t.
3.5.2. Field Data Inversion Results Based on the VES results and hydrogeological considerations, 16 imaging pro les were initially measured using wenner con guration and they crossed the NE-SW stream ow which are prominent fracture zones. The apparent resistivity data were inverted applying both smooth inversion and 2 layer-initial model robust inversion. The inverted models are shown in the form of contoured sections that help in visualizing the geological structures. These models are geoelectrically subdivided into two zones: less than 200Ω.m and higher resistive layer up to 4000Ω.m. The smooth inversion model shows a vertical low resistivity zone with a width vary between 50 to 120 meter below the top 200Ω.m layer, that is probably a fracture zone in the basement. Also the smooth inversion model shows a low resistivity zone with smooth and sloping boundaries on one side of the inverted sections (Fig.3.9) except of AbuGebiha example that its observed data did not reach the basement rocks and the high resistive lenses were well distinguished in the pseudo sections, but it was di cult to place the sharp boundary between the overburden and the basement layer. Consequently, as a nal step in the interpretation, and for comparative reasons, robust inversion method of the data was carried out. It was found more useful in determining the exact location of the boundaries of the fracture zone with more-uniform resistivity values within the fracture zone. According to drilling information from several sites close to the ERT sections, the depth to the basement rocks in the vicinity varies between 20 and 40 meters, and the data rms mis t converged between 3 to 6% in all the models tested, which is in a very good agreement with the results from the smooth inversion.
50
3. Geoelectric Measurements
Smooth Inv.
Trtr img1
2Lay-Robust Inv
Smooth Inv.
Abugeris img2
2Lay-Robust Inv
Smooth Inv.
Albetira img2
2Lay-Robust Inv
Smooth Inv.
Abugebiha img4
2Lay-Robust Inv
Figure 3.9.: Examples of the 2D-DC eld data in the four study areas. The variation in resistivity responses horizontally and vertically indicate dierential sedimentation processes within the Quaternary cover, in addition to tectonic structures as fracture zones. 51
3. Geoelectric Measurements
3.6. Interpretation of 2D Inversed Data The interpretation procedure described above in the synthetic data section has been tested with several eld ERT data obtain from the crystalline basement area of northeast Nuba Mountains. The eld data demonstrate the usefulness of both smooth and robust inversion schemes in the eld data set to determine the gross layered structure, and new targets of groundwater aquifers. The major rock type in the study area include quartzite, graphite/mica schist, and undierentiated meta-sediments of pre to Upper Cambrian age. These are often overlain by Quaternary sedimentary cover derived from the weathered basement rocks. The resistivity of the upper layer (representing the Quaternary cover) shows a slight increase as the thickness of the upper layer in the initial model was increased, and the high resistive structure is prominent between 900 to 3000Ω.m, this coinciding with the outcrops of the fractured quartzite and schist materials. Classi cation of the the dierent resistivity zones of the Quaternary cover is assessed by the petrophysical data analysis in clay and sand materials that provide an estimation to the dry and wet sedimentary formations to be comparable values to the inverted models and suggest consequently potential groundwater aquifers, this will be discussed in Chapter 6. The dierent inversion methods described can be viewed as complementary tools and the interpreter can employ to obtain the most consistent and reasonable results for a given data set (Olayinka and Yaramanci, 2000).
3.7. 3D resistivity modeling Field ERT and the soundings data were merged in a 3D grid in GOCAD 2.0 interactive program. The data were decompositioned in points and surfaces which were not connected, the topography was extracted from the digital elevation model (DEM) data, and the top of the basement was de ned from the inverted soundings and ERT data as unconnected points and curves respectively, then boundary constraints were de ned before the 3D interpolation took place.
River Tools
software provided a specialized set for the analysis of topography, watershed and river networks. The procedure for extracting a drainage network from DEM is highly automated in River Tools and is only required to identify a few key points as an input from known mapped drainage in the DEM data. Then for each area number of informations were established and it consists of points set from soundings, curves set from ERT, DEM surface, depth to the basement surface, and the drainage pattern that prevailed the region. For comparative reasons a plot of all the hand wells that are available in the study area overlain the DEM surface. For the four studied areas, a stratigraphic grid (Sgrid) was created based on the corresponding above data set, The logarithmic resistivity varies between 0.5 to 3.5 ('3 to 3200Ω.m). interpolation resolved the subsurface successfully in parts of high dense data.
The
A detailed in-
terpretation and discussion will take place in Chapter 6 after compiling these grids with other collected geophysical data: the very low frequency data and the magnetotelluric soundings (VLF
52
3. Geoelectric Measurements
and MT). Also the petrophysical analysis provide a resistivity range of the number of formations fully saturated with the collected water samples, this resistivity range can be compared to the 3D grid resistivity and will be discussed in Chapter 6.
3.8. General Remarks The synthetic data vary in initial depths, and structures, but the smooth inversion results showed how shallower interfaces increase the model basement resistivity to overshoot the true values, while in deeper interfaces the inversion was not stable with undulating surfaces although the resistivity contrast changes.
Due to the wide gradual changes in resistivity within the sharp
boundary between the overburden and the basement in the smooth inversion, it was preferred to employ the robust constrain inversion and positively the inversion results was improving while the inverted deeper-layer resistivity decreases than the true model, an unusual inversion eects at the edges of the 2D structure of the fault and graben examples were reported. It was chosen to start the robust inversion with an initial model consists of 2 horizontal layers varied in the depth to the basement
Hi ,
the inverted results based on this initial model was
better than both smooth and robust inversion based on homogeneous model.
The resolution
of the structure boundaries and good agreement with true resistivity values characterized this inversion. But when start with deeper depth to basement
Hi
in the initial model, it reproduced
lower resistivity although the resistivity contrast in the 2-layer initial model is increased. As a conclusion to the DC measurements and their inverted results, the application of the DC resistivity techniques in the study area improved the vision of the complex subsurface that the synthetic data was simulated the observed data in resistivity values or contrasts. The inverted data provide a range of resistivity based on grain size distribution from extended layer of 20Ω.m as clayey or weathered products, to 80-120Ω.m as coarser-grain limited formation at depths between 20 to 40m below surface, and nally higher resistive and deeper zone of more than 300 up to few thousands
Ω.m.
53
3. Geoelectric Measurements
AlTrtr 3D grid and cross sections
AbuGeris 3D grid and cross sections
AlBetira 3D grid and cross sections
AbuGebiha 3D grid and cross sections
Figure 3.10.: 3D resistivity grids of the four studied areas.
A combination of the inverted 1D
soundings and 2D ERT lines. The topography and the meandering drainage pattern revealed information about ner and coarser grain accumulations.
54
4. Electromagnetic Measurements
The natural source MT method is based on measuring the natural EM eld uctuations at the earth
´s
surface, which induce electric currents (or telluric currents) under the earth's surface.
These electromagnetic uctuations are caused by solar winds, which processes ow of charged particles in the earth magnetosphere; and the lightning activity that discharges in the earth ionosphere causing eld disturbance. The inductive mechanism is an electromagnetic eld propagated with slight attenuation over large distances in the space between the ionosphere and earth surface, at large distances from the source this is a plane wave of variable frequency (from
10−5
Hz up to the audio range at least) (Telford et al., 1976). In such case these frequencies travel vertically downward, incide normally to the air-earth surface, the magnetic eld is horizontal and perpendicular to the direction of the EM wave propagation.
4.1. Elementary Electromagnetic Theory
4.1.1. Theoretical Background The electromagnetic theory is developed to describe the magnetotelluric (MT) wave propagation and attenuation, but better understanding of Maxwell's equations is needed which are relating the electric and magnetic eld vectors (Telford, 1990)
∇×E=−
∂B ∂t
∇×H=J+
F araday‘ Law
∂D ∂t
(4.1)
Ampere‘s Law
(4.2)
2
where J is the current density (A/m ), E is electric eld intensity (V/m), B is the magnetic ux density (Tesla [T]), H is the magnetic eld intensity (A/m) and D is electric displacement
2
(C/m ). Using the vector identity
∇.∇ × A = 0
in equations 4.1 and 4.2 to get time-varying
elds where :
∇.B = 0
∇·J = −
∂ (∇.D) ∂t
∇.D = Q
(4.3)
and Q is accumulation rate of charge density. But in nite conductivity regions, these charges do not accumulate to any extend during current ow and thus Q = 0 so that the electric eld relations and the magnetic eld relation are:
∇.J = 0
∇·D = εε0 ∇.E= 0 55
B = µµ0 H
(4.4)
4. Electromagnetic Measurements
where and
µ and µ0 are the relative magnetic permeability of a medium and the free space respectively
µ0 = 4π × 10−7 H/m
, where
free space respectively, and
εε0 are
the relative magnetic permittivity of a medium and the
ε0 = 8.85 × 10−12 F/m.
In a homogeneous isotropic media, these relations can be expressed with Ohm's law as follows:
B = µH where
σ
D = εE
J = σE
(4.5)
is the conductivity of the media. A simpli cation for equations 4.1 and 4.2 as:
∇ × E = −µ∂H/∂t
(4.6)
∇ × H = σE + ε∂E/∂t
(4.7)
Taking the curl of the last two relations, they become
∇2 × E = −µ
∂E ∂2E ∂ ∇ × H = −µσ + µε 2 ∂t ∂t ∂t
∇2 × H = −σ(∇ × E) − ε
(4.8)
∂ ∂H ∂2H (∇ × E) = µσ − µε 2 ∂t ∂t ∂t
(4.9)
In MT work the sinusoidal time variations are used and thus the last two relations can be written as
where
ω = 2πf
E(t) = E0 ej ωt
and
∂E = jωE ∂t
(4.10)
H(t) = H0 ejωt
and
∂H = jωH ∂t
(4.11)
is the angular frequency of the eld, thus equations
∇2 × E and ∇2 × H
are
simpli ed to
∇2 × E = j ωµσE − ω 2 µεE
(4.12)
∇2 × H = j ωµσH − ω 2 µεH
(4.13)
The right hand term represent the conduction and displacement current respectively. These are the electromagnetic equations for propagation of electric and magnetic eld vectors in an isotropic homogeneous medium having conductivity permittivity
σ,
relative permeability
ε.
56
µ,
and relative dielectric
4. Electromagnetic Measurements
an identical relation would be hold for
H
also. As a result one can say that the part corresponds
to the displacement current is negligible, and in air and poorly conducting
∇2 × E ≈ 0,
∇×H≈0
(4.14)
where in a good conductor, it can written
∇2 × E = µσ
∂E = j ωµσE ∂t
(4.15)
∇2 × H = µσ
∂H = j ωµσH ∂t
(4.16)
This is the diusion equation. To solve equations 4.15 and 4.16, we assume the wave propagate along the z axis, and xy plane is the polarization plane, then
H
is (the magnitude of
H) expressed
as
H = Hy (z, t) = H0 ejωt+mz where
m2 = jωµσ
and
1/2 a = ( ωµσ 2 )
and thus
∇2 H =
∂ 2 Hy ∂ z2
= m2 H
so,
Hy = H0 e−az cos(ωt − az) The exponential term is the attenuation of the wave and can be written (taking
4π ×
(4.17)
µ = µ0 =
10−7 H/m) as Hy /H0 ≈ e−2×10
−3 z
√
f /ρ
(4.18)
A commonly used criterion for the EM penetration is the skin depth (zs ), which is the signal is reduced by
1/e
or 37%.
zs ≈ 500
p
ρ/f
meter
(4.19)
4.1.2. Magnetotelluric Fields 4.1.2.1. Homogeneous Earth The applications of Magnetotelluric (MT) theory to determine the electrical conductivity within the earth was originally described by Cagniard (1953). To adapt the wave equations to MT, it is necessary to make some simpli ed assumptions (Telford, 1990): 1. The frequencies are low that displacement currents are negligible.
57
4. Electromagnetic Measurements
2. For such plane wave, the horizontal variations in E and H are smaller compared to the vertical variations. 3. Taking the xy plane as horizontal and z positive downward. If the magnetic vector as component are
H0
at an angle
Hx0 = H0 cosθ
and
Ex
Ex =
and
p
Ey
to the
x
Hy0 = H0 sinθ
Hx = H0 cosθ e−az cos(ωt − az) From equation (4.7)
θ
axis, so that the magnitude of the magnetic and applied in equation (4.17) we can write
Hy = H0 sinθ e−az cos(ωt − az)
and
(4.20)
can be extracted and they are
2 σa (H0 sinθ) e−az cos (ωt − az + π4 )
and similarly,
r
π a 2 (H0 cosθ) e−az cos (ωt − az + ) σ 4
Ex =
(4.21)
Dividing equations (4.21)by equations (4.20), the squares of the ratios become
| replacing
∂/∂z = D,
and
a 2 Ey 2 Ex 2 | =| | =2 = ωµρ Hx Hy σ
ω = 2π/T ,
(4.22)
the last equations then give (clearly the x and y axes can
be interchanged)
D= setting
µ = µ0
T Ex | | 2πµ Hy
and
ρ=
T Ex 2 | | 2πµ Hy
(4.23)
, and substituting 4.23 in 4.9, we get nally
D≈
1 p 5ρT km 2π
ρ ≈ 0.2T |
and
E 2 | Ωm H
(4.24)
When electromagnetic monochromatic plane waves propagate downwards (along the z-axis) in a homogeneous and isotropic medium, the electric and magnetic eld vectors are orthogonal and the ratio of the electric to magnetic eld intensity is a characteristic measure of the electromagnetic properties of the medium, often called characteristic impedance Frischknecht, 1966).
This characteristic impedance
Z
Z
(Cagniard, 1953; Keller and
is described by the following equation:
Z = Ex /Hy = −Ey /Hx Thus, the plane-wave scalar apparent resistivity
ρ(ω) =
1 |Z(ω)|2 ωµ0
and
ρ,
and impedance phase,
φ = tan−1 (
58
φ
, are given by
Out P hase Z(ω) ) In P hase Z(ω)
(4.25)
4. Electromagnetic Measurements
Thus, apparent resistivity can be calculated from simultaneous measurement of Ex and Hy (or Ey and Hx) at dierent frequencies, while the phase dierence
φ
of the two orthogonal components
provides additional information about the resistivity structure of the earth. In a uniform earth, the apparent resistivity has to be the same at every frequency, and the phase diers by all frequencies (Vozo, 1972). A measured phase dierence that is not
π/4
π/4
at
is indicative of the
ground being non-uniform. Phase is proportional to the slope of the apparent resistivity curve on a log-log scale from a baseline at 45 degrees. For a non homogeneous earth, apparent resistivity
ρ
and phase
φ
are de ned as
ρ = 0.2T |Z0 |2 where
Z0
φ = arg(Z0 ) 6= 45
and
(4.26)
is the impedance at the surface.
4.1.2.2. Layered Earth For horizontally N-layered earth the plane wave impedance is given by the recursive formula (Ward and Wannamaker, 1983):
ˆ N = ZN = ωµ Z kN 0 Zn = ωµ kn p kn = (iωµσn )
where
hn =
ˆ ˆ n−1 = Zn−1 Zn + Z n−1 tanh (k1 h1 ) Z Zn−1 + Zˆn tanh (k1 h1 )
(4.27)
(intrinsic impedance of the nth layer; with
Thickness of the nth layer
ˆN= Z
Impedance at the top of the nth layer and Z1 = Z0 is that on the surface
For two-layered earth , equation 4.24 becomes
ˆ ˆ 1 = Z1 Z2 + Z1 tanh (k1 h1 ) Z Z1 + Zˆ2 tanh (k1 h1 ) where
Z = Z0 =
For large period
impedance at the surface, and
T
(i.e.
k1 h 1)
k1 =
and therefore,
√
(iωµσ1 ) =
(4.28)
q
i2πµ ρ1 T
tanh(ik1 h1 ) ≈ ik1 h1 .
Equation 4.27 can be
written as
Z0 = Z1
Zˆ2 + i Z1 k1 h1 Z2 + i Z1 k1 h1 = Z1 ˆ Z 1 + i Z2 k1 h1 Z1 + i Z2 k1 h1
4.1.2.3. MT Mode decomposition in the 2D case 1. Let the earth be two-dimensional derivations w.r.t. y vanish
59
(4.29)
4. Electromagnetic Measurements
Figure 4.1.: The transverse electric (TE mode) is when the electric eld is parallel to strike, while the transverse magnetic (TM mode) is when the magnetic eld is parallel to strike
2. An Ey thus creates a Hx and Hz (TE mode), 3. A Hy creates Ex and Ez (TM mode). 4. Both modes are independent of each other and are de ned by the sources.
Two independent modes of the impedance are analyzed for the 2D earth analysis in a right hand Cartesian coordinate system with y parallel to strike and x perpendicular to strike. Transverse electric TE mode is when the electric eld is parallel to strike. Transverse magnetic TM mode is when the magnetic eld is parallel to strike (Fig.4.1). Diagonal terms of the impedance tensor for a perfectly 2D earth are zero:
" Z=
0
Zxy
Zyx
0
# ,
where Zxy = ZT E = Ex /Hy
and
Assuming that the data are acquired in the user coordinate system
Zyx = ZT M = Ey /Hx
(x, , y , ) to (x, y) is θ,
then we
apply the rotation matrix:
" R=
cosθ
−sinθ cosθ
In practice, we do not know
θ
sinθ
θ
# , such that E = RE , and H = RH ,
exactly, and the geology is not strictly 2D. Therefore, we estimate
by minimizing the diagonal terms
Zxy and Zyx of
the MT transfer function as shown in Fig.4.2.
60
the rotated impedance tensor, also referred as
4. Electromagnetic Measurements
Figure 4.2.: MT Impedence tensor for simple 1D structure, 2D and 3D strctures. It is alos reered as the MT transfer function.
4.2. Magnetotelluric MT measurements The Magnetotelluric (MT) method is a passive surface geophysical technique which uses the time varying earth's natural electric and magnetic elds to study electrical resistivity of the subsurface. Magnetic and electrical elds are measured in the frequency band
10−4
to
101
kHz,
with high frequencies (>1 Hz) coming from thunderstorm activities in the equatorial belt while low frequencies (
