--- title: "Limestone true thickness: A 600 m N28E, C 400 m S10W" subject: "Engineering Geology II" chapter: 6 key: "semester_4/engineering_geology_ii:bh-limestone-true-thickness-a600" source: https://www.bctengineer.com/q/semester_4/engineering_geology_ii/bh-limestone-true-thickness-a600 site: BCT Engineering --- # Limestone true thickness: A 600 m N28E, C 400 m S10W Engineering Geology II · Chapter 6: Measurement, analysis and interpretation of structural geological data - Asked in: 2073 Bhadra, 5 marks; 2070 Bhadra, 4 marks (asked 2 times) - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/engineering_geology_ii/bh-limestone-true-thickness-a600 - Whole chapter: https://www.bctengineer.com/md/semester_4/engineering_geology_ii/chapter-6 ## Question Three boreholes A, B and C were drilled for limestone reserve calculation. Borehole A lies at 600 m distance due N28°E from borehole B. Borehole C lies at 400 m distance due S10°W from borehole B. The top and bottom of the limestone bed were encountered at the following depths of the given boreholes. | Borehole | Top (m) | Bottom (m) | |---|---|---| | A | 200 | 260 | | B | 220 | 280 | | C | 240 | 300 | Calculate the true thickness of the limestone bed. ## Short answer From the three borehole depths, vertical thickness is 60 m. The top-of-bed plane dips 3.9° towards SSE (dip direction 147°), strike N57.4°E. True thickness = 60 × cos(3.9°) = 59.9 m. ## Answer **Method:** the top of the limestone is at 200 m, 220 m and 240 m depth in A, B and C. All three boreholes are vertical and the bed is 60 m thick in each (260 - 200 = 280 - 220 = 300 - 240 = 60 m), so 60 m is the **vertical thickness**. First find the attitude of the bed (three-point problem), then convert vertical thickness to true thickness. #### Step 1: coordinates (B at origin, x east, y north) $$ \begin{aligned} A &: x = 600\sin 28^\circ = 281.68\ \text{m},\ y = 600\cos 28^\circ = 529.77\ \text{m} \\ C &: x = 400\sin 190^\circ = -69.46\ \text{m},\ y = 400\cos 190^\circ = -393.92\ \text{m} \end{aligned} $$ #### Step 2: plane of the top of the bed Depth to the top: $d = 220 + a x + b y$. Substituting A and C: $$ \begin{aligned} 200 &= 220 + 281.68a + 529.77b \\ 240 &= 220 - 69.46a - 393.92b \end{aligned} $$ Solving gives $a = 0.0366$ and $b = -0.0572$ (metres of depth per metre). #### Step 3: dip and dip direction $$ \tan\delta = \sqrt{a^2+b^2} = 0.0680 \Rightarrow \delta = 3.89^\circ $$ The bed becomes deeper towards the direction $(a,b)$, so the dip direction is $\arctan(a/b)$ measured from north = $147.38^\circ$ (toward SSE). The strike is perpendicular to this: $57.38^\circ$, i.e. **N57.4°E**. **Attitude of the bed: N57.4°E / 3.9° towards SSE** (dip direction 147°). #### Step 4: true thickness $$ t = t_v\cos\delta = 60\cos 3.89^\circ = 59.86\ \text{m} $$ (The bed is almost flat, so the true thickness is only slightly less than 60 m.) **Answer: true thickness of the limestone = 59.86 m (about 59.9 m); bed dips 3.9° towards SSE.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is 60 m not the true thickness here? A. The boreholes are inclined B. It is a vertical thickness measured in vertical holes, and the bed is dipping C. The bed is overturned D. The depths are wrong Answer: B. Vertical holes give vertical thickness; true thickness = vertical thickness x cos(dip). (Key point: Vertical vs true thickness) 2. Which method finds the bed attitude? A. Three-point problem using tops at A, B and C B. Using only borehole A C. Mohr circle D. Averaging depths Answer: A. The top depths at three non-collinear points define the plane. (Key point: Method) 3. The attitude is about 3.9° dipping SSE. What is the true thickness? A. 60 m B. 60 sin 3.89° = 4.07 m C. 60 / cos 3.89° = 60.14 m D. 60 cos 3.89° = 59.86 m Answer: D. t = tv cos(delta) = 59.86 m, slightly less than 60 m. (Key point: Dip) 4. With B at the origin, how are A's coordinates found? A. x = 600 cos 28°, y = 600 sin 28° B. x = 600 sin 28°, y = 600 cos 28° (N28°E, x east, y north) C. x = 400 sin 10° D. x = y = 600 Answer: B. A is 600 m at azimuth 28°: x = 600 sin 28° = 281.68 m, y = 600 cos 28° = 529.77 m. 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