The DC Resistivity Method Modelling & Inversion

1. ERTH2020Introduction to Geophysics The DC Resistivity Method Modelling & Inversion

2. History In 1912 Conrad Schlumberger, using very basic equipment, recorded the first map of equipotential curves in his Normandy estate near Caen. http://www.slb.com/about/history.aspx

3. Applications • investigation of lithological underground structures, • estimation of depth, thickness and properties of aquifers, • mapping of preferential pathways of groundwater flow, • determination of the thickness of the weathered zone • detection of fractures and faults in crystalline rock, • localization and delineation of the horizontal extent of dumped materials, • estimation of depth and thickness of landfills, • detection of inhomogeneities within a waste dump, • mapping contamination plumes, • monitoring of temporal changes in subsurface electrical properties, • detection of underground cavities, • classification of cohesive and non-cohesive material in dikes, levees, and dams.

4. Goal • Derive general formula for the “apparent resistivity”: Potential difference (Voltage) Apparent resistivity Geometric factor Resistance Electrode spacing Source current & applications

5. DC Resistivity Method DC Resistivity survey across a circular mound, thought to contain Irish archaeological burial chambers. geotomosoft.com

6. DC Resistivity Method DC Resistivity survey mapping lithology of near-surface soil materials to examine possible contamination from fertilisers and pesticides. Loke, 2000

7. DC Resistivity Method DC Resistivity survey mapping a dolerite dyke causing a prominent high resistivity zone surrounded by shales. Loke, 2000

8. DC Resistivity Method DC Resistivity survey mapping caves within a limestone bedrock. A known air-filled cave causes a high resistivity anomaly near the centre. In the course of this survey a new cave was discovered. Causing a high resistivity anomaly near the bottom left corner. Loke, 2000

9. Conduction of electricity in rocks • Electrical conduction in rocks occurs in three ways ELECTRONICby motion of electrons ELECTROLYTICby movement of ions DIELECTRIC by displacement of electrons and positively-charged atomic nuclei • Most rocks are partial ELECTRONIC conductors • However, the bulk conductivity of rocks in the upper few km of the crust is mainly due to ELECTROLYTICconduction

10. Ohm’s Law (1827) Circuit theory: General form: is the electric field intensity vectorin V/m is the current density vector in A/m2 is the electrical resistivityin Ω m (s in S/m)

11. Ohm’s Law (1827) DC Resistivity Method “equivalent circuit” - + I completely described by Ohm’s law resistance

12. Outline • The electric field and potential • Potential and current flow in the subsurface • DC resistivity measurements • Common electrode configurations • Resistivity sounding and profiling • Modelling and inversion

13. The electric field • Electric fields may be generated by charges. • A stationary charge Q+ creates an electric field E. • The electric field intensity E is defined as the force F exerted by Q+ on a test charge q0 at position r. The magnitude is given by Coulomb’s law Q (i) A test charge does not alter the electric field – it is an idealized quantity whose physical properties are assumed to be negligible except for the property being studied. (ii) Note that electric fields also exist without charges (for example as light which is an electromagnetic wave)

14. The electric field • In accordance with Coulomb’s law, the E-field falls off as the square of the distance of the test-charge. • Adding or removing charges changes the E-field. • The E-field is a vector field as it varies from point to point (A vector has a direction and a magnitude). Electric dipole field of opposite charges. The E-field is visualized by “field lines” which are tangential to the E-field vectors.

15. KEA230 Lecture G9 The electric potential • Instead of the electric field – a vector quantity – we can use a scalar quantity to describe and calculate DC electric phenomena.: the electric potential. • The electric potential at a point is equal to theelectric potential energy of a charged particle divided by its charge. • Because it is a scalar, it simplifies many calculations.

16. The electric potential Va Vb a b E + + y x d • Static (DC, zero-frequency) electric fields are conservative: • The work done in moving a charge from point a to point b depends only on the start and end points, not on the path taken

17. The electric potential • Relationship between: Work ~ E-Field ~ Electric Potential and • The work done in moving a test charge q0 from point a to point b can be defined in terms of scalar potentials Va and Vb. • Therefore, the electric field can be given in terms of a Potential Difference:………………………………………

18. The electric potential (Jackson, 1999)

19. KEA230 Lecture G9 The electric potential • The simple example given on the previous slides can be generalised to give • That is, the electric field is the gradient of the potential function (V) • The potential V obeys Laplace’s Equation • This is the same equation which governs magnetic and gravitational (potential) fields! “Steepest ascent” “Average bending”

20. The electric field and potential • Potential and current flow in the subsurface • DC resistivity measurements • Common electrode configurations • Resistivity sounding and profiling

21. Current flow due to a single electrode Potential about a single electrode on the surface of the earth • The most simple solution can be obtained for a uniform half-space with resistivity “Rho” for a single point-source electrode on the surface, with current of “I” Ampere. • Because of this special setup, we have a radial symmetry with respect to the point source. This solution is the basis for all subsequent derivations. *A half-space is a simplified model of the local earth – it subdivides the “space” into two halves, where the upper half is air and the lower half consist of the uniform property under investigation. Lowrie, 2007, p. 212

22. Current flow due to a single electrode Potential about a single electrode on the surface of the earth Solution for a single point electrode Lowrie, 2007, p. 212; c.f. Telford, 1991, p.523

23. Current flow due to a single electrode Potential about a single electrode on the surface of the earth • The potential V about a single (point) electrode on the surface of a homogeneous half-space: ρ: half-space resistivity I: electrode source current r: radial distance from source “Equipotential” (surface of equal potential) I Surface of half-space* r r

24. Current flow due to a single electrode Potential about a single electrode on the surface of the earth • The E-field anywhere within the half-space can be calculated using the equation on the previous slide. Electric field = gradient of the potential • The current density J within the half-space can then be calculated using Ohm’s law. General form of Ohm’s Law Note: Current flows parallel to E

25. Current flow due to a single electrode Potential about a single electrode on the surface of the earth potential current T. Boyd, Colorado School of Mines Equipotential • Potential decreases away from the electrode • Current flows perpendicular to the equipotential contours • Logistically difficult to set up in practice: 2nd current electrode at ‘infinity’ (in practice, 10 times spacing of potential electrodes) •  Use two current electrodes closer together

26. Current flow due to two electrodes Potential about two electrodes on the surface of the earth • Because potential is a scalar, the total potential due to two current electrodes C1 and C2 can be calculated by adding the potentials for electrodes carrying current +I (C1) and -I (C2) respectively. • The potential at any point P within the half-space is then: • Once the potential at any point is known, the current density can be calculated using exactly the same procedure as for a single electrode r1 and r2 are the distances of point P from electrodes C1 and C2 respectively

27. Current flow due to two electrodes Potential about two electrodes on the surface of the earth C2 C1 In a half-space current always flows perpendicular to the equipotential contours Equipotentiallines T. Boyd, Colorado School of Mines Current streamlines

28. Current flow due to two electrodes Potential about two electrodes on the surface of the earth Knödelet al, 2007, p. 205

29. The electric field and potential • Potential and current flow in the subsurface • DC resistivity measurements • Common electrode configurations • Resistivity sounding and profiling

30. DC Resistivity measurementsEquipment resistivity meter and battery cables (1.26 km) with 64 electrodes Revil et al, 2012

31. DC Resistivity measurementsEquipment Contact resistance between the stainless steel electrodes and the ground is decreased by adding salty water (right) Cable layout (above) and reels for cable (right) Revil et al, 2012

32. KEA230 Lecture G10 DC Resistivity measurementsApparent resistivity • The theoretical discourse so far solved current flow and the electric potential in the subsurface for a homogeneous ground (half-space). • With these preparations we can now solve for the apparent resistivity for a known source current, measured potential (voltage) and the assumption of a homogeneous half-space. Apparent resistivity  is defined as the resistivity of homogeneous ground that would give the same voltage-current relationship as measured over a inhomogeneous ground*. • Apparent resistivity is a useful data transformation to provide a ‘normalised’ data set which accounts for system configuration. • Only for a homogeneous ground the apparent resistivity equals the true resistivity. • The exact form of the formula for the apparent resistivity depends on the relative positions of the electrodes.

33. DC Resistivity measurementsApparent resistivity • inject current into a half-space through current electrodes C1 & C2 • estimate the resistivity by measuring the potential difference across a pair of potential electrodes P1 and P2 (Telford et al., 1990)

34. DC Resistivity measurementsApparent resistivity • Potential difference DV between P1 and P2 is measured: • DV = (Potential at P1) - (Potential at P2) = V1-V2 • Ohm’s Law (Telford et al., 1990) Assumption of Geometry !

35. DC Resistivity measurementsApparent resistivity 2πa = K = geometric factor (depends on electrode array) • Rearrange to give (Telford et al., 1990)

36. DC Resistivity measurementsApparent resistivity • The general formula for the resistivityrof a uniformly resistive, homogenous Earth is: • In an inhomogeneous Earth, it is the apparentresistivityra Example: apparent resistivity curve over a two-layered earth 25m r1 = 250 Wm r2 = 50 Wm

37. DC Resistivity measurementsApparent resistivity DC Resistivity (Wenner), Fingal, Tasmania • The basic equipment consists of 2 (active) current and 2 (passive) potential electrodes and a recording instrument, the Terrameter. The Terrameter is designed to measure the resistance of the ground with high accuracy. This is done by balancing the internal resistor of the instrument so that it completely nullifies any current flow within the potential electrodes. Terrameter and Wenner electrode array

38. DC Resistivity measurementsApparent resistivity Observed field data: • during measurement record the resistance values for the associated “a”–spacing • calculate the apparent resistivities (spread sheet) • These observed (raw) data are input to the modelling program. The apparent resistivity is the resistivity of the homogeneous half-space which would produce the observed instrument response for a given electrode spacing.

39. Estimated Layer boundaries DC Resistivity measurementsApparent resistivity ra~ 1000 Wm (asymptotic) ra~ 600 Wm ra~ 50 Wm 600 Wm 50 Wm 1000 Wm DC Resistivity (Wenner), Fingal, Tasmania

40. The electric field and potential • Potential and current flow in the subsurface • DC resistivity measurements • Common electrode configurations • Resistivity sounding and profiling

41. Common electrode configurations General formula for 4-point electrode layout Lowrie, 2007, p. 213

42. Common electrode configurations 1) Wenner array Simple setup but cumbersome in practice 2) Schlumberger array A and B: current electrodes Mand N: potential electrodes A,B move; M,N fixed Convenient as only 2 electrodes move Knödelet al, 2007, p. 210

43. Common electrode configurations 3) Pole-dipole array Good for mapping of confined conductors 4) Dipole-dipole array minimum coupling between current and potential wires Knödelet al, 2007, p. 210

44. Common electrode configurations 5) Pole-Pole array a gives wide horizontal coverage and good depth coverage however suffers from poor resolution 6) In boreholes Knödelet al, 2007, p. 210

45. Common electrode configurationsSensitivity patterns Wenner Schlumberger Knödelet al, 2007, p. 211; cf. Loke, 2000, pp.10

46. Common electrode configurationsSensitivity patterns Pole-dipole Dipole-dipole Knödelet al, 2007, p. 211; cf. Loke, 2000, pp.10

47. Common electrode configurationsSensitivity patterns Pole-Pole Borehole (dip-dip) Knödelet al, 2007, p. 211; cf. Loke, 2000, pp.10

48. Common electrode configurations Median depth of investigation (). is the length of the array. • These median depths are strictly only valid for a homogeneous earth model • This depth does not depend on the measured apparent resistivity or the resistivity of the homogeneous earth model. Loke 2000, p.13

49. Common electrode configurations Comparison of most common arrays Reynolds, 2011, p.298

50. The electric field and potential • Potential and current flow in the subsurface • DC resistivity measurements • Common electrode configurations • Resistivity sounding and profiling