Plotting raypaths for primary and multiple reflections

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Problem 4.7

A well encounters a horizon at a depth of 2.7 km with a dip of $ 7^{\circ } $. Sources are located 2 km downdip from the well and a geophone is placed at depths of 1000 and 2600 m. Plot the raypaths and calculate the traveltimes for the primary reflection from the 3-km horizon and its first multiple. Assume $ V=3.0 $ km/s.

Figure 4.7a.  Constructing raypath for a multiple from a dipping bed.

Background

Image points were used in Figures 4.1a and 4.2a to find travel paths in constant velocity situations. Figure 4.7a illustrates their use for the more complicated situation of multiples. The image point $ I_{1} $ gives the raypaths for the wave after reflection at the dipping horizon. When the wave is then reflected at the surface $ I_{1} $ is equivalent to a new source, so the image point $ I_{2} $ is located on the perpendicular from $ I_{1} $ to the surface as far above the surface as $ I_{1} $ is below. If this multiple is reflected a second time at the horizon, the image point $ I_{3} $ is located on the perpendicular to the horizon as far below the horizon as $ I_{2} $ is above it.

Solution

With the origin at the source, the vertical depth of the reflector beneath the source 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/":): z = 2700 + 200\tan 7^{\circ}=2725 m. The perpendicular from the source to the reflector meets the reflector 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/":): x = 2725\sin 7^\circ , 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 = 2725 \cos 7^\circ , i.e., 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/":): (-330, 2700) , and the source image for the primary 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/":): I_1 is 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/":): x , 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=(-660, 5400) . The image for the surface multiple $ I_{2} $ (reflected at the reflector and then at the surface) is 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/":): -660, -5400 .

The arrival time of the primary reflection at the phone 1000 m deep 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} (460^2 + 4400^2)^{1/2}/3000 = 1.475\ \mathrm{s}, \end{align}

and the surface multiple arrives 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/":): \begin{align} (460^2 + 6400^2)^{1/2}/3000 = 2.139\ \mathrm{s}. \end{align}

The arrival time of the primary at the phone 2600 m deep 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} (460^2 + 2800^2)^{1/2}/3000 = 0.948\ \mathrm{s}, \end{align}

and the multiple arrives 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/":): \begin{align} (460^{2} + 8000^{2})^{1/2}/3000 = 2.671\ \mathrm{s}. \end{align}

The source and geophone stations in the well are drawn to scale but displaced to the right in Figure 4.7b. The raypath to the 2600 m phone for the primary (the solid line) and for the surface multiple (the dashed line) are shown. The reflection point moves updip as the distance between the phone and the reflector increases, i.e., as the phone becomes shallower. The reflection points for the multiples lie still further updip.

Figure 4.7b.  Raypaths for well survey.
Table 4.7a. Reflection traveltimes.
Traveltimes
Depth Primary Multiple
1000 m 1.475 s 2.139 s
266 0.946 2.671

The traveltimes for intermediate depths are calculated in the same way. The results are tabulated in Table 4.7a.

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