Locating the bottom of a borehole
 | |
| Series | Geophysical References Series |
|---|---|
| Title | Problems in Exploration Seismology and their Solutions |
| Author | Lloyd P. Geldart and Robert E. Sheriff |
| Chapter | 4 |
| Pages | 79 - 140 |
| DOI | http://dx.doi.org/10.1190/1.9781560801733 |
| ISBN | ISBN 9781560801153 |
| Store | SEG Online Store |
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 $ C $ is vertically above the well geophone at $ C^{\prime },A,B,D $, and $ E $ are equidistant from $ W $ in the cardinal directions, and the traveltimes from sources $ D $ and $ E $ to $ C^{\prime } $ are equal.
Assuming straight-line travelpaths at the velocity $ V $, derive expressions for $ C^{\prime }C $ and $ CW $ in Figure 4.23a(ii) in terms of the traveltimes to $ C^{\prime } $ from $ A $ and $ B,\,t_{A} $ and $ t_{B} $.
Solution
Since 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/":): EW = DW , 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/":): t_{DC^{\prime}} = t_{EC^{\prime}} , 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/":): AB \perp DE , point 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/":): C^{\prime} must be in the vertical plane through 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/":): AWB 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/":): C must lie on the straight line 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/":): AWB . Then, letting 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 = C^{\prime}C, y = CW, x = AW = BW = DW = EW , 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} (V t_{AC^{\prime}})^2 = (x - y)^{2} + h^2, \end{align} ()
$ {\begin{aligned}(Vt_{BC^{\prime }})^{2}=(x+y)^{2}+h^{2}.\end{aligned}} $ ()
Subtracting, we 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/":): \begin{align} V^{2}(t^{2}_{BC^{\prime}}-t^{2}_{AC^{\prime}})=4xy, \end{align}
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} \hbox{hence} \qquad\qquad CW=y=\left(V^{2}/4x\right)\left(t^{2}_{BC^{\prime}-t^{2}_{AC^{\prime}}}\right). \end{align} ()
Adding equations (4.23a) and (4.23b) 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} &V^{2}\left(t^{2}_{AC^{\prime}}+t^{2}_{BC^{\prime}}\right)=2(x^{2}+y^{2}+h^{2}),\\ \hbox{and} \qquad C^{\prime}C=h&=\left[(V^{2}/2)\left(t^{2}_{AC^{\prime}}+t^{2}_{BC^{\prime}}\right)- (x^{2}+y^{2})\right]^{1/2}. \end{align} ()
Since all quantities on the right are known, we can 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/":): C^{\prime}C .
Problem 4.23b
What are the values 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/":): t_{AC^{\prime}} 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/":): t_{BC^{\prime}} for 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/":): V=2500 m/s, 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/":): AW=BW=C^{\prime}C=1000 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/":): CW=200 m?
Solution
From equation (4.23a),
$ {\begin{aligned}(Vt_{AC^{\prime }})^{2}=(x-y)^{2}+h^{2},\end{aligned}} $
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} \hbox{so} \qquad\qquad t_{AC^{\prime}} = [(1.00 - 0.20)^2 + 1.00^{2}]^{1/2}/2.50 = 0.512\,\mathrm{s},\\ (V t_{BC^{\prime}})^2 = (x + y)^{2} + h^2 = 1.20^2 + 1.00^2, t_{BC^{\prime}} = 0.625\,\mathrm{s}. \end{align}
Problem 4.23c
How sensitive is the method, that is, what 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/":): \Delta(CC^{\prime} /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/":): \Delta t_{AC^{\prime}} 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/":): \Delta(CW)/\Delta t_{AC^{\prime}} ? For the specific situation in part (b), how much change is there in 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/":): CW 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/":): C^{\prime}C per millisecond error in 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/":): t_{AC^{\prime}} ?
Solution
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} \Delta(CC^{\prime})/\Delta t_{AC^{\prime}} \approx \mathrm{d}h/\mathrm{d}t_{AC^{\prime}},\quad \Delta(CW)/\Delta t_{AC^{\prime}}\approx \mathrm{d}y/\mathrm{d}t_{AC^{\prime}}. \end{align}
From equation (4.23c), assuming 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/":): t_{B} fixed, we get
$ {\begin{aligned}\mathrm {d} y/\mathrm {d} t_{AC^{\prime }}=-V^{2}t_{AC}/2x.\end{aligned}} $ ()
Differentiating equation (4.23a) and using equation (4.23e) 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} V^{2}t_{AC^{\prime}} &= (x - y)(-\mathrm{d}y/\mathrm{d}t_{AC^{\prime}}) + h\, \mathrm{d}h/\mathrm{d}t_{AC},\\ \mathrm{d}h/\mathrm{d}t_{AC^{\prime}} &= (1/h)[V^{2}t_{AC^{\prime}} - (x - y)V^{2}t_{AC^{\prime}}/2x] = (V^2t_{AC^{\prime}}/h)[1 - (x - y)/2x]\\ &= (V^{2}t_{AC^{\prime}}/2h)(1 + y/x). \end{align}
Using values from part (b), we obtain
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} \mathrm{d}y/\mathrm{d}t_{AC^{\prime}} = -2.50^2 \times 0.512/(2 \times 1.00] = -1.60\,\mathrm{km/s} = -1.6\,\mathrm{m/ms}. \end{align}
For 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/":): \Delta t_{AC^{\prime}} = 1 ms, 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/":): CW = -1.6 m.
Also, 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/":): \mathrm{d}h/\mathrm{d}t_{AC^{\prime}} = [2.50^2 \times 0.512/(2 \times 1.00)](1 + 0.20/1.00) = 1.92\,\mathrm{km/s} .
For 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/":): \Delta t_{AC^{\prime}} = 1 ms, 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/":): \Delta h = 1.9 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
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} t_{AC^{\prime}} = 500/(1500 \cos 22.5^{\circ}) + 500/(3500 \cos 51.2^{\circ}) = 0.582\,\mathrm{s},\\ t_{BC^{\prime}} = 500/(1500 \cos 22.5^{\circ}) + 500/(3500 \cos 63.2^{\circ}) = 0.678\,\mathrm{s}. \end{align}
Interpreting these results as in part (a), 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} \overline{V}&=(1/2)(1500+3500)=2500\,\mathrm{m/s},\\ y&=(\overline{V}^{2}/4x)\left(t_{BC_{\prime}}^{2}-t^{2}_{AC^{\prime}}\right)\\ &=(2500^{2}/4\times 1.00)(0.678^{2}-0.582^{2})=189\,\mathrm{m},\\ h^{2}&=(\overline{V}^{2}/2)(t^{2}_{AC^{\prime}}+t^{2}_{BC^{\prime}})-(x^{2}+y^{2})\\ &=(2500^2/2)(0.582^{2}+0.678^{2})\\ &\quad-(1.00^{2}+0.19^{2}),\\ h&=1210\,\mathrm{m} \end{align}
Thus, $ y $ varies only 5%, mainly because we subtract the squares of traveltimes, thus partially canceling errors. However, the change in 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 is more than 20%.
Continue reading
| Previous section | Next section |
|---|---|
| Time-distance curves for various situations | Two-layer refraction problem |
| Previous chapter | Next chapter |
| Partitioning at an interface | Seismic velocity |
Also in this chapter
- Accuracy of normal-moveout calculations
- Dip, cross-dip, and angle of approach
- Relationship for a dipping bed
- Reflector dip in terms of traveltimes squared
- Second approximation for dip moveout
- Calculation of reflector depths and dips
- Plotting raypaths for primary and multiple reflections
- Effect of migration on plotted reflector locations
- Resolution of cross-dip
- Cross-dip
- Variation of reflection point with offset
- Functional fits for velocity-depth data
- Relation between average and rms velocities
- Vertical depth calculations using velocity functions
- Depth and dip calculations using velocity functions
- Weathering corrections and dip/depth calculations
- Using a velocity function linear with depth
- Head waves (refractions) and effect of hidden layer
- Interpretation of sonobuoy data
- Diving waves
- Linear increase in velocity above a refractor
- Time-distance curves for various situations
- Locating the bottom of a borehole
- Two-layer refraction problem
External links
| find literature about Locating the bottom of a borehole |
|
|
|
|
|
|
|