Deducing fault geometry from well data

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Problem 10.2a

Well B is 500 m due east of well A and well C is 600 m due north of A. A fault cuts A, B, and C at depths of 800, 1200, and 600 m, respectively. Assuming that these wells are vertical and the fault is planar and extends to the surface, find the surface trace and strike of the fault.

Background

It is shown in Sheriff and Geldart, 1995, problem 15.9a that the direction cosines of a straight line satisfy the equation


(10.2a)

and Sheriff and Geldart, 1995, problem 15.9b gives the equation of a plane whose perpendicular from the origin has length h and direction cosines as


(10.2b)
Figure 10.1b.  Conventional display of seismic line shown in Figure 10.1a.

Solution

We take the origin at well A, the x- and y-axes being positive towards the east and north, respectively, and the z-axis positive vertically downward. The coordinates of the points of intersection of the fault plane with wells A, B, and C are, respectively (0,0,800), (500, 0, 1200), and (0, 600, 600) and these three points all lie on the fault plane. Hence, equation (10.2b) shows that

In addition to these equations we have equation (10.2a) so that we can solve for the four unknowns. Thus , , , Solving these equations we get , , . Using equation (10.2a) we have

We now get , , , and the equation of the fault plane is


(10.2c)

The surface trace is obtained by setting in equation (10.2c), giving

The trace intersects the -axis at , that is, west of A, and cuts the -axis at north of A. The strike is .

Problem 10.2b

At what depth would you look for this fault in well D located 500 m . of well C?

Solution

The coordinates of the wellhead at D are

Substituting these values in equation (10.2c) gives

Problem 10.2c

Another fault known to strike N20W cuts wells A and C at depths of 1300 and 1000 m, respectively. Where should it cut well B?

Solution

Let the equation of the fault plane be . With four unknows (, , and ) we need four equations. The fault intersection in well A gives the equation and the intersection in well C gives . The strike is N20W, so the slope of the strike line relative to the -axis is [see equation (4.2)], so . We use these three equations to find , and in terms of , and then the sum to get .

Solving the first three equations gives ,

Then , ,

The equation of the fault plane is thus

Substituting the coordinates of well B, we get

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