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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(l,\; m,\; n\right) of a straight line satisfy the equation


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} l^{2} +m^{2} +n^{2} =1, \end{align} (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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(l,\; m,\; n\right) as


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} lx+my+nz=h. \end{align} (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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 800n=h\; ,\ h; 500l+1200n=h\; ,\; 600m+600n=h. \end{align}

In addition to these equations we have equation (10.2a) so that we can solve for the four unknowns. Thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n/h=1/800=1.25\times 10^{-3} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): l/h+2.40n/h=1/500=2.00\times 10^{-3} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m/h+n/h=1/600=1.67\times 10^{-3} , Solving these equations we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): l/h=-1.00\times 10^{-3} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m/h=0.42\times 10^{-3} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n/h=1.25\times 10^{-3} . Using equation (10.2a) we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} l^{2} +m^{2} +n^{2} =1=h^{2} \times 10^{-6} \left(1.00^{2} +0.42^{2} +1.25^{2} \right)\; ,\; h=604\ {\rm m}. \end{align}

We now get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): l=-0.604, , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m=0.254 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n=0.755 , and the equation of the fault plane is


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} -0.604x+0.254y=0.755z=604. \end{align} (10.2c)

The surface trace is obtained by setting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): z=0 in equation (10.2c), giving

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} -0.604x+0.254y=604. \end{align}

The trace intersects the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x -axis at $ x=-1000\ {\rm {m}} $, that is, west of A, and cuts the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y -axis at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): y=2380\ {\rm m} north of A. The strike is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \tan^{-1} (1000/2380)={N}22.8^{\circ} {\rm E} .

Problem 10.2b

At what depth would you look for this fault in well D located 500 m Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \hbox{N}30^{\circ}\hbox{W} . of well C?

Solution

The coordinates of the wellhead at D are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} x=-500\sin 30^{\circ} =-250\ {\rm m},\; y=600+500\cos 30^{\circ} =1030\ {\rm m}. \end{align}

Substituting these values in equation (10.2c) gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} -0.604\times \left(-250\right)+0.254\times 1039+0.755 z=60,\\ {\rm so} \qquad\qquad\qquad\qquad z=\left(604-0.604\times 250-0.254\times 1039\right)/0.755=253\ {\rm m}. \end{align}

Problem 10.2c

Another fault known to strike N20Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): ^{\circ} W 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): l'x+m'y+n'z=h' . With four unknows (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): l' , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m' , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n' and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h' ) we need four equations. The fault intersection in well A gives the equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 1300n{'} = h' and the intersection in well C gives $ 600m{'}+1000n{'}=h' $. The strike is N20Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): ^{\circ} W, so the slope of the strike line relative to the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x -axis is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \tan (-20^{\circ})= {\rm d}y/{\rm d}x=-m'/l' [see equation (4.2)], so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m'/l'=0.364 . We use these three equations to find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): l' , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m' and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n' in terms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h' , and then the sum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): l^{'2} +m^{'2} +n^{'2} =1 to get $ h' $.

Solving the first three equations gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n'/h'=7.69\times 10^{-4} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} m'/h'=\left(1-0.769\right)/600=\left(0.231/600\right)=3.85\times10^{-4},\\ l'/h'=m'/0.364h'=10.58\times 10^{-4}. \end{align}

Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(10.58^{2} +3.85^{2} +7.69^{2} \right)\times 10^{-8} h^{'2} =1 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h'=733\ {\rm m} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} l'=0.776,\; m'=0.282,\; n'=0.564. \end{align}

The equation of the fault plane is thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 0.776 x+0.282y+0.564z=733. \end{align}

Substituting the coordinates of well B, we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 500\times 0.776+0.564z=733\; ,\; z=612\ {\rm m}. \end{align}

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