Find. (a) The dogleg severity of the correction, and whether it is within the 8°/100 ft casing-running limit. (b) The tool face angle and its orientation relative to high side.
Fig. 3 — Tool-face orientation: the required setting sits mostly toward high side (building inclination) with a 38.6° offset to the left (turning the azimuth toward N20W).
Approach. Compute the space-vector dogleg angle from the two inclination/azimuth pairs and the standard 3-D survey formula, convert to a per-100-ft severity, then use the tool-face vector formula (built from the same inclination/azimuth data) to find the setting angle measured clockwise from high side.
Dogleg angle. $$\cos\beta=\cos I_1\cos I_2+\sin I_1\sin I_2\cos(Az_2-Az_1)=\cos10^\circ\cos13^\circ+\sin10^\circ\sin13^\circ\cos(-10^\circ)$$ giving $\boxed{\beta=3.59^\circ}$ over the 80-ft course.
Part (a) — dogleg severity. $$DLS=\frac{\beta}{L}\times100=\frac{3.59}{80}\times100\Rightarrow\boxed{DLS=4.49^\circ/100\ \text{ft}}$$ Since $4.49<8$, $\boxed{\text{yes — casing can be safely run through this section}}$.
Part (b) — tool-face vector components. Using the standard vector form (clockwise from high side, $0^\circ=$ build straight ahead): $$c_h=\sin I_2\cos I_1\cos(Az_2-Az_1)-\cos I_2\sin I_1,\qquad c_r=\sin I_2\sin(Az_2-Az_1)$$ Substituting gives $c_h=+0.157$, $c_r=-0.125$ (both computed from the same $10^\circ$, $13^\circ$, $-10^\circ$ inputs as Step 1).
Solve for the tool-face angle. $$TF=\operatorname{atan2}(c_r,c_h)\Rightarrow\boxed{TF=321.4^\circ}$$ (measuring clockwise from high side, wrapped to $0$–$360^\circ$). Since $321.4^\circ=360^\circ-38.6^\circ$, this is $\boxed{38.6^\circ\ \text{to the LEFT of high side}}$ — consistent with the required motion, which is mostly a build (inclination $10\rightarrow13^\circ$) with a left turn (azimuth decreasing from $350^\circ$ to $340^\circ$, i.e. toward N20W).
Check: azimuths are converted using the standard quadrant-bearing-to-azimuth rule (N$\theta$W $=360-\theta$); the tool-face convention used here takes $0^\circ$ = high side (pure build), increasing clockwise through $90^\circ$ = right turn, $180^\circ$ = drop, $270^\circ$ = left turn, the standard directional-drilling convention (Bourgoyne/Rabia).
Quantity
Value
Dogleg angle, $\beta$
3.59° over 80 ft
(a) Dogleg severity, $DLS$
4.49°/100 ft — within the 8°/100 ft limit, casing CAN be run
