1. Environmental and Exploration Geophysics I
Resistivity II
tom.h.wilson
Department of Geology and Geography
West Virginia University
Morgantown, WV

2. Objectives for the day Review basic computation of V,  and G
Review in-class problem 1
Begin discussion of problems 5.1 through 5.3
Discuss current reflection, transmission and refraction across resistivity boundaries.
Define the reflection coefficient for current flow
Relate above ideas to problem 5.3
Questions?
and – time at the end for EM lab work/questions

3. A look at the calendar
Terrain Conductivity Lab due this Thursday. How many words?
Complete “in-class” problem and hand in before leaving today
Continue your reading of Chapter 5.
Look over Frohlich’s paper (linked in the resistivity section). We will be reanalyzing some of the data from his paper in the resistivity lab. It’s a tough paper, skim/review for the geological application. This is not a paper for summary!
Bring any additional questions about problems 5.1 through 5.3 to class next time. They will be due next Thursday (24th).
I’ll provide an initial review of these problems today! See today’s slides. These are relatively simple problems.

4. - Review of basic ideas presented last time A + B d1 d4 d2 d3 V
1. Assume a homogeneous medium of resistivity 120 ohm-m. Using a Wenner electrode system with a 60m spacing, Assume a current of amperes.
A. What is the measured potential difference?
B. What will be the potential difference if we place the sink (negative-current electrode) at infinity? A + - B d1 d4 d2 d3 V

5. We know in general that
For the Wenner array the geometrical factor G is 2a and the general relationship of apparent resistivity to measured potential difference is
In this problem we are interested in determining the potential difference when the subsurface resistivity distribution is given.

6. - In part A) we solve for V as follows and A + V B d1 d4 d2 d3
In part B) what happens to d2 and d4?

7. Part B)
… d2 and d4 go to . We really can’t think of this as a simple Wenner array any longer. We have to return to the starting equation from which these “array-specific” generalizations are made.
What happens when d2 and d4 go to ?

8. - +  A B V d1 d2 d3 d4
d1= 60 m and d3 = 120 m. V is now only 0.1 volts.

9. 1B

10. There are many types of arrays as shown at left
There are many types of arrays as shown at left. You should have general familiarity with the method of computing the geometrical factors at least for the Wenner and Schlumberger arrays.
The resistivity lab you will be undertaking models Schlumberger data and many of the surveys conducted by Dr. Rauch and his students usually employ the Wenner array.

11. Note that when conducting a sounding using the Wenner array all 4 electrodes must be moved as the spacing is increased and maintained constant. The location of the center point of the array remains constant (despite appearances above).

12. Conducting a sounding using the Schlumberger array is less labor intensive. Only the outer two current electrodes need to be moved as the spacing is adjusted to achieve greater penetration depth. Periodically the potential electrodes have to be moved when the current electrodes are so far apart that potential differences are hard to measure - but much less often that for the Wenner survey

13. Homework problem 5.1a (See Burger et al. p. 341)
source
sink
20m
12m 4m P1 P2
Surface
=200-m
Depth
Find the potential difference between points 1 and 2.
What kind of an array is this? What are d1, d2, d3 and d4 ?

14. Use basic equations to solve for the potential difference.
The critical point here is that you accurately represent the different distances between the current and potential electrodes in the array.

15. In problem 5.2 Actually measure the incidence angles!
5.2. Current refraction rules
In problem 5.2 Actually measure the incidence angles!
Given these resistivity contrasts - how will current be deflected as it crosses the interface between layers? Measure the incidence angle and compute the angle of refraction.

16. 5.2. Current refraction rules
Measure
compute
Given these resistivity contrasts - how will current be deflected as it crosses the interface between layers? Measure the incidence angle and compute the angle of refraction.

17. The governing relationship

18. What’s your guess?
2 > 1
tan  increases with increasing angle
2 < 1

19. 1 ?
2 > 1 2
2 varies as 1 and 1 varies as 2

20. ?
1 << 2

21. ?
2 < 1

22. ?
1 << 2

23. Let’s consider the in-class problem handed out to you last lecture.
Incorporating resistivity contrasts into the computation of potential differences.
Let’s consider the in-class problem handed out to you last lecture. m
1=200-m
1=50 -m C1 P1 
5.3. Calculate the potential at P1 due to the current at C1 of 0.6 amperes. The material in this section view extends to infinity in all directions. The bold line represents an interface between mediums with resistivities of 1 and 2.

24. Current reflection, refraction and transmission
In-Class/Takehome Problem 2
In the following diagram -
Suppose that the potential difference is measured with an electrode system for which one of the current electrodes and one of the potential electrodes are at infinity. Assume a current of 0.5 amperes, and compute the potential difference between the electrodes at
PA and . Given that d1 = 50m, d2 = 100m, 1 = 30-m, and 2 = 350-m.

25.  Current reflection and transmission Sink Source Electrode 1=30-m
One potential electrode d1 PC a PA PB b
2=350-m
d2 = a+b
Image point

26. At PA
1=30-m
Some current will be transmitted across this interface and a certain amount of current (k) will be reflected back into medium 1. d1 PA a ? PA PA b
2=350-m
d2 = a+b
Reflection point
Image point

27. Use of the image point makes it easy to estimate the length along the reflection path
Path length is distance from image point to PA.

28. Potential measured at A
k is the proportion of current reflected back into medium 1. k is also known as the reflection coefficient.

29. Potential measured at point B
1-k is the transmission coefficient or proportion of current incident on the interface that is transmitted into medium 2.

30. Potential measured at point C

33. Potentials a hair to the left or right of the interface should be approximately equal.

34. What distances do you need to measure?
To calculate the potential at P1 what do you need to know? m
1=200-m
1=50 -m C1 P1 
What else do we need to consider?
Two things …

35. Another look at the calendar
Turn the 2nd in-class problem before leaving today
Continue your reading of Chapter 5.
Get started on problems 5.1 through 5.3. They are due next Thursday with last chance for questions next Tuesday.
Resistivity papers are in the mail room.
TC lab due this Thursday 17th

36. Exam October 1st in rm 325 Brooks Hall
Just over two weeks away, but not too soon to start reviewing
We will devote part of the class on September 29th to review,