Locating the bottom of a borehole

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Problem 4.23a

Barton (1929) discusses shooting into a geophone placed in a borehole (Figure 4.23a) to determine where the bottom is located. Point is vertically above the well geophone at , and are equidistant from in the cardinal directions, and the traveltimes from sources and to are equal.

Assuming straight-line travelpaths at the velocity , derive expressions for and in Figure 4.23a(ii) in terms of the traveltimes to from and and .

Solution

Since , , and , point must be in the vertical plane through and must lie on the straight line . Then, letting , we have


(4.23a)


(4.23b)

Subtracting, we find


(4.23c)

Adding equations (4.23a) and (4.23b) gives


(4.23d)

Since all quantities on the right are known, we can find .

Figure 4.23a.  Mapping a crooked borehole (from Barton, 1929). (i) Plan view; (ii) vertical section.

Problem 4.23b

What are the values of and for m/s, m, m?

Solution

From equation (4.23a),

Problem 4.23c

How sensitive is the method, that is, what are / and ? For the specific situation in part (b), how much change is there in and per millisecond error in ?

Solution

From equation (4.23c), assuming fixed, we get


(4.23e)

Differentiating equation (4.23a) and using equation (4.23e) gives

Using values from part (b), we obtain

Figure 4.23b.  Snell’s law raypaths.

For ms, m.

Also, .

For ms, m.

Problem 4.23d

Assume a velocity of 1500 m/s for the first 500 m and 3500 m/s below 500 m. What are the traveltimes now and how would these be interpreted if straight raypaths are assumed?

Solution

By trial and error we find that the angles should be as shown in Figure 4.23b. Then

Interpreting these results as in part (a), we get

Thus, varies only 5%, mainly because we subtract the squares of traveltimes, thus partially canceling errors. However, the change in is more than 20%.

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