GG 450Feb 27, 2008

Resistivity 2

Resistivity: Quantitative Interpretation - Flat interface

Recall the angles that the current will take as it hits an interface:

€

tan θ1( )

tan θ 2( )=ρ 2

ρ1+

--

Notice that where the current line goes through the interface it is bent towards a smaller angle, implying that 2 is greater than 1. Also note that deeper down the path goes through vertical.

Problem: Obtain formula that will yield the apparent resistivity for a model with a single interface separating two layers of different resistivity.

x

Current source

12

x

P1

P2

We want to obtain the potential at arbitrary points P1 and P2, above above the interface and one below. This may seem like an impossible task, since the resistivity changes at the boundary, and energy will be both reflected from and transmitted through the boundary.

We can get a solution, though, by using the principle of IMAGE SOURCES. We reflect the actual source to the other side of the boundary, generating an image virtual source:

x

Current source

12

x

P1

P2

d

d

Image source

We need the potential at P1 for all possible ways that energy from the source can be seen there - as though the interface was a partly reflecting mirror.

x

Current source

12

x

P1

P2

d

d

Image source

r1

r2

r3

We need to calculate the potential at P1 from the paths r1 and r2, and r3for the potential at P2. You can convince yourself that r2 has the same length as the path to P1 from the current source that is reflected at the interface. We do need to modify the potential where it reflects from or passes through the interface.

The energy reflected will be less than might be expected because some energy goes through the interface, with a reflection coefficient (k) and transmission coefficient (1-k). If the point where the potential is to be determined is across the boundary from the source, then the transmission coefficient will be used.

Recall the potential:

€

V =iρ

4πD

where D is the distance from the current electrode to the point of measurement. So, at P1,

€

VP1 =iρ1

4πr1+kiρ1

4πr2 and VP2 =

i 1− k( )ρ 2

4πr3

What’s the value of k? If both points P1 and P2 are on the interface, then the potentials should be equal, and:

€

VP1 = VP2 =iρ1

4πr1+kiρ1

4πr2=i 1− k( )ρ 2

4πr3, so

k =ρ 2 − ρ1

ρ 2 + ρ1

These formulas give us the basics for calculating the potential difference between two electrodes, and thus the apparent resistivity.Note that the model above doesn't include the earth's surface, which it must to be realistic. Note that above the earth’s surface, 1 is very large, so k1,0=-1.BUT…. now we have multiple reflections and image sources to worry about:

x

Current source

12

P1d

d

Image source

r1

r2

Image source

There are an infinite number of reflections that will contribute to the potential at P1, but the energy in each successive reflection is less because energy is leaking into the lower layer, assuming that k1,2 is not 1 ( 2 very large).

From the above we note that the first reflection from the upper surface contributes to the potential by:

firstupperVP1 =

ρ1ik0,1

4π r2 + 2m( )2 and first

lowerVP1 =ρ1ik1,2

4π r2 + 2 z−m( )[ ]2

All other reflections will involve reflections from both the top and bottom boundaries, and the squares and higher powers of the reflection coefficients. For the Wenner array:

We can use this equation to calculate the potentials at the surface for various cases and generate models of what me might expect as the electrode spacings are changed. See resist.m. (Wenner array ONLY.)

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a = ρ1 1+ 4k1,2n

sqrt 1+ 2nz /a( )2

[ ]− 2

k1,2n

sqrt 1+ nz /a( )2

[ ]n=1

∞

∑n=1

∞

∑ ⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

In the figures below, the electrode configuration is the Wenner Array, where the geometry of the current and potential electrodes is constant, and the spacing as shown below:

V

I

a a a

Measurements of apparent resistivity are taken at a range ofvalues for a to obtain data to compare with model situations.

The above model shows a high resistivity below a low resistivity. Note that even though the depth of the layer is only 4 m, the complete picture requires the spacing BETWEEN electrodes to be over 300 m – that’s a total spread length of almost 1 km!

This is the opposite of the case above, with the higher resistivity on top. Now the spacing isn’t so bad, with total adequate spread length of only about 300 m.

z=4z=9

z=16z=25

z=36

0=10001=100Wenner Array

1000

600

300

100

a

1 10 100 1000

a, meters

What if the resistivity of the top layer changes without changing the depth to the 2nd layer?

103

104

105

1 10 100 1000

Electrode spacing, m

High over Low Resist ivityLayer depth: 12 m

2000

This is quite easy. The resistivity of both layers and the depth are easily recognized.

What if the resistivity of the deeper layer changes with a constant depth to the interface?

103

104

105

1 10 100 1000

Electrode spacing, m

Low over High Resist ivityLayer depth: 12 m

4000

Note that the depth of the interface is well constrained by the region where the resistivity first starts to change, but if the resistivity of the deeper layer is very high, spread lengths will need to be very long.

You usually don't need to go out to very large distances if all you care about is the depth to a layer of known resistivity. If all you care about is its depth, then it isn't necessary to go beyond the slope break, or a total array length of about 150m and 80m respectively in the models above.

What if you have more than one resistivity interface below? In this case, the resistivity curve gets more complicated, but still fairly easy to interpret. Three cases for a three-layer model are shown below. The top one is for increasing resistivity with depth. Note that it looks VERY similar to the 2-layer case above.

You would have to be very careful to detect the intermediate layer!

Another model is shown below, with the medium resistivity at the top and the low at the bottom. Here, it's pretty obvious that there's something between the two layers.

The model below has the medium resistivity at the top and the high at the bottom:

Resistivity Uses

Profiling

Archeology

Groundwater

Pollution

MAGMA