Environmental and Exploration Geophysics I

tom.h.wilsontom.wilson@mail.wvu.edu

Department of Geology and GeographyWest Virginia University

Morgantown, WV

Resistivity II

For next class complete in-class problems and hand in next period (Thursday – Sept. 24th).

No class next Tuesday (September 22nd).

Read over problems 5.1 through 5.3 in the text and be prepared to ask questions about them next Thursday.

I’ll summarize the approach to these problems in class today. They will be due the following Tuesday (Sept. 29th).

Redo and cross check with lec8.pdf (9/16/2010)

Review of basic ideas presented last time

1. Assume a homogeneous medium of resistivity 120 ohm-m. Using a Wenner electrode system with a 60m 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

d1

d4

d2

d3

V

i

GVa We know in general that

i

aVa

2

For the Wenner array the geometrical factor G is 2a 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.

a

iV a

2In part A) we solve for V as follows

)(2.0)60(2

)120)(628.0(

amperevolts

m

mamperesV

A

+ -

V B

d1

d4

d2

d3

and

In part B) what happens to d2 and d4?

… 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.

4321

1111

2 dddd

iV

What happens when d2 and d4 go to ?

Part B)

0

10

1

2 31 dd

iV

4321

1111

2 dddd

iV

31

11

2 dd

iV

A

+ -

V B

d1

d4

d2

d3

d1= 60 m and d3 = 120 m. V is now only 0.1 volts.

31

11

2 dd

iV

mm

mamperesV

120

1

60

1

2

)120)(628.0(

m

mamperesV

120

1

2

)120)(628.0(

)(1.0 amperevoltsV

1B

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 5.1a (See Burger et al. p. 341)

What kind of an array is this? What are d1, d2, d3 and d4 ?

Find the potential difference between points 1 and 2.

2m

source sink

20m

12m

4m

P1 P2

Surface

=200-mDepth

432121

1111

2 dddd

iVV

432112

1111

2 dddd

iV

The critical point here is that you accurately represent the different distances between the current and

potential electrodes in the array.

Use basic equations for the potential difference.

5.2. Current refraction rules

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.

Actually calculate the angles!

tan increases with increasing angle

1

2

2

1tan

tan

What’s your guess?

2 > 1

2 < 1

1

2

2

1tan

tan

1

2

2 varies as 1 and 1 varies as 2

2 > 1

2 > 1

2 < 1

2 > 1

Incorporating resistivity contrasts into the computation of potential differences.

Let’s consider the in-class problem handed out to you last lecture.

0 5m

1=200-m1=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.

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.

Current reflection and transmission

d1

d2 = a+b

1=30-m

2=350-m

Image point

a

b

PA

PB

PC

One potential electrode

Source Electrode

Sink

d1

d2 = a+b

1=30-m

2=350-m

Image point

a

b

PA

PA

PA

Reflection point

Some current will be transmitted across this interface and a certain amount of current (k) will be reflected back into medium 1.

?

At PA

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.

Incorporating resistivity contrasts into the computation of potential differences.

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.

Locate an image electrode and incorporate reflection process,

0 5m

1=200-m1=50 -m

C1

P1

Looking ahead

In the remainder of today’s class – last chance for questions about the Terrain Conductivity

lab. Put in my box Wednesday, 30th.

Uncontaminated fresh water acquifer

Not how the apparent conductivities drop with

increased depth of penetration

20

30

4050

Fresh water aquifer

Near-surface clay

Basal silty clay

Constructing a cross section view of your model results