Environmental and Exploration Geophysics I. Resistivity II tom.h.wilson

Transcription

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

2 Objectives for the day Review basic computation of, and G Review in-class problem Begin discussion of problems 5. 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?

3 A look at the calendar 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. Bring any additional questions about problems 5. through 5.3 to class next time. They will be due next Tuesday. 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. Assume a homogeneous medium of resistivity 20 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 d d 2 d 3 d 4

5 We know in general that a G i For the Wenner array the geometrical factor G is 2a and the general relationship of apparent resistivity to measured potential difference is a 2a i 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 as follows ia 2a and (0.628amperes )(20 2 (60m) m) 0.2volts( ampere ) + - A B d d 2 d 3 d 4 In part B) what happens to d 2 and d 4?

7 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. i 2 d d2 d3 d 4 What happens when d 2 and d 4 go to?

8 d d i d d d d i 3 2 d d i A + - B d d 4 d 2 d 3 d = 60 m and d 3 = 20 m. is now only 0. volts. What is the geometrical factor for this electrode configuration?

9 3 2 d d i m m m amperes ) )(20 (0.628 m m amperes 20 2 ) )(20 (0.628 ) (. 0 ampere volts B

10 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 3 2 d d i M + - N d = na a Geometrical factor for the pole-dipole array with d =na & d 3 =na+a A B 2 i na na a d 3 = na+a

12 From n large to n= ( ) for n large 2 for n= 2 2 i na na a i a na n i na i a

13 Sounding and profiling 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.

14 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

15 The geometrical factor is array dependent If you don t have one in your tool kit you have to go back to the starting point and figure it out. G 2 d d d d d and d 2 usually represent distances from the left potential electrode to the source and sink, while d 3 and d 4 are distances from the right potential electrode to source and sink, respectively.

16 For the Schlumberger array 2 b l b l b l b l G S terms and two terms : l b l b Combine and reduce You have two

17 Transforming data into apparent resistivities Just for reference not an assignment Wenner array What are the geometrical factors? 2a

18 We ll illustrate the data window and show possible solutions next time Apparent resistivity? a G i

19 Homework problem 5.a (See Burger et al. p. 34) 20m source 4m Surface sink Depth 2m =200-m 2m P P 2 Find the potential difference between points and 2. What kind of an array is this? What are d, d 2, d 3 and d 4?

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

21 Current refraction

22 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.

23 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.

24 What s your guess? tan increases with increasing angle 2 > 2 tan2 tan 2 <

25 2 > 2 2 tan2 tan 2 varies as and varies as 2

26 << 2

27 Path of least resistance 2 <

28 A tough one! << 2

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

30 Current reflection, refraction and transmission Let s go through the take-home problem part of the handout from last time. The main concepts developed in this problem are those associated with partial transmission and reflection of electrical current across resistivity boundaries and the idea of the reflection coefficient. 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 P A and. Given that d = 50m, d 2 = 00m, = 30-m, and 2 = 350-m.

31 Current reflection and transmission Source Electrode Sink =30-m One potential electrode d a P C P A P B d 2 = a+b b 2 =350-m Image point

32 At P A The blue lines below represent current flow paths Some current will be transmitted across this interface and a certain amount of current (k) will be reflected back into medium. Reflected current? P A d a =30-m P A what happens at the boundary? Transmitted current P A d 2 = a+b b 2 =350-m Reflection point Image point

33 Use of the image point makes it easy to estimate the length along the reflection path Path length is distance from image point to P A.

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

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

36 Potential measured at point C d 2 P c d d Magnified view d d d 0 2 in the lim the points converge onto the boundary

37 Look at point C in the limit that d 0 d 2 Magnified view P c d in the lim the points d 0 converge onto the boundary d C-d d d 2 C+d Cd Cd

38 Use relationship for A & B derived earlier A i k 2 d d2d & B i k 2 2 d 2d in the lim the points converge onto the boundary d 0 Cd Cd Cd i k 2 d d2d Cd i k 2 2 d 2d

39 In the limit that d0 in the lim the points converge onto the boundary d 0 Cd Cd Cd i k 2 d Cd i k 2 2 d or i k i 2 k 2 d 2 d

40 Rearrange to solve for k, the reflection coefficient i k i 2 k 2 d 2 d k k 2

41 What did that analysis tell us about the problem in the book? What distances do you need to measure? To calculate the potential at P what do you need to know? Use scale bar to make an accurate estimate of distances 0 5m C P =50 -m =200-m Think this over and bring questions to class for next time

42 For now, complete the in-class work and take home problem from last time Calculate Wenner and Schlumberger. Make sure you can solve for k and complete the computation of P A using the equation developed today. Hand in before leaving

43 Another look at the calendar Turn in-class problems and problem from last time in before leaving today Continue your reading of Chapter 5. We will have another problem related to 5.3 next time Get started on problems 5. through 5.3. They are due next Tuesday with last chance for questions this Thursday. Special note on problem 5.2: measure incidence angles and calculate refraction angles.

44 Exam September 29 th in rm 325 Brooks Hall Coming soon, so start reviewing We will devote part of the class on September 27 th to review,