Environmental and Exploration Geophysics I Resistivity II tom

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

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

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 (24 th). • I’ll provide an initial review of these problems today! See today’s slides. These are relatively simple problems.

Review of basic ideas presented last time 1. Assume a homogeneous medium of resistivity 120 ohm-m. Using a Wenner electrode system with a 60 m spacing, Assume a current of 0. 628 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 V d 1 d 2 d 3 d 4

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.

In part A) we solve for V as follows - + A B V d 1 and d 2 d 3 d 4 In part B) what happens to d 2 and d 4?

Part B) … d 2 and d 4 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 d 2 and d 4 go to ?

- + A d 1 B V d 2 d 3 d 4 d 1= 60 m and d 3 = 120 m. V is now only 0. 1 volts.

1 B

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.

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).

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

Homework problem 5. 1 a (See Burger et al. p. 341) 20 m source Depth sink Surface 4 m 2 m =200 -m P 1 12 m P 2 Find the potential difference between points 1 and 2. What kind of an array is this? What are d 1, d 2, d 3 and d 4 ?

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.

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.

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.

The governing relationship

What’s your guess? tan increases with increasing angle 2 > 1 2 < 1

1 2 2 varies as 1 and 1 varies as 2 2 > 1

Incorporating resistivity contrasts into the computation of potential differences. 5. 3. Calculate the potential at P 1 due to the current at C 1 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. 0 5 m C 1 P 1 1=50 -m 1=200 -m Let’s consider the in-class problem handed out to you last lecture.

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 d 1 = 50 m, d 2 = 100 m, 1 = 30 -m, and 2 = 350 -m.

Current reflection and transmission Source Electrode Sink 1=30 -m One potential electrode d 1 PC a PA PB b d 2 = a+b 2=350 -m Image point

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. d 1 PA a ? PA d 2 = a+b PA b Reflection point 2=350 -m Image point

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.

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

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.

Potential measured at point C

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

What distances do you need to measure? To calculate the potential at P 1 what do you need to know? 0 5 m C 1 P 1 1=50 -m 1=200 -m What else do we need to consider? Two things …

Another look at the calendar • Turn the 2 nd 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 17 th

Exam October 1 st 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 29 th to review,