Find. (a) The dogleg severity and whether it is within the casing program's limit; (b) the toolface angle and BHA orientation relative to high side.
Fig. 4 — Toolface angle is measured clockwise from the wellbore's high side, looking in the direction of travel.
Approach. Convert both quadrant bearings to azimuths, get the total dogleg angle from the spherical law of cosines for inclination/azimuth, convert to DLS and compare to the casing limit, then resolve the toolface angle from the component of the new wellbore direction that is perpendicular to the old one.
Total dogleg angle. $$\cos\beta=\cos I_1\cos I_2+\sin I_1\sin I_2\cos(\Delta Az)$$ $$\cos\beta=\cos10\,{}^{\circ}\cos12\,{}^{\circ}+\sin10\,{}^{\circ}\sin12\,{}^{\circ}\cos(-30\,{}^{\circ})=0.9946$$ so $\boxed{\beta=5.98\,{}^{\circ}}$.
(a) Dogleg severity. $DLS=\dfrac{\beta}{\Delta MD}\times100=\dfrac{5.98}{75}\times100$, so $\boxed{DLS=7.98\,{}^{\circ}/100\ \text{ft}}$. Since $7.98<10\,{}^{\circ}/100$ ft, $\boxed{\text{yes, casing can be run through this well}}$ after the correction.
(b) Toolface angle. Resolving the new wellbore direction into components along the old high side ($h$) and the old right-side ($r$) directions: $$c_h=\sin I_2\cos I_1\cos(\Delta Az)-\cos I_2\sin I_1=0.0076,\qquad c_r=\sin I_2\sin(\Delta Az)=-0.1040$$ $$TF=\operatorname{atan2}(c_r,c_h)$$ so $\boxed{TF=274.1\,{}^{\circ}}$, measured clockwise from high side.
Orientation. $274.1\,{}^{\circ}$ sits just short of $270\,{}^{\circ}$ (due left), so the bend sub/mud-motor toolface should be set essentially to the LEFT of high side, with a small bias back toward high side — consistent with the required correction turning the azimuth counter-clockwise (S10W $\to$ S20E) while also building inclination.
Check: toolface convention used is the standard oilfield definition — angle measured clockwise from the high side of the wellbore, looking in the direction of travel; a different regional convention (e.g. measured from magnetic north on a surface gyro toolface) would require a separate conversion not requested here.
