6.0 in $\times$ 3.0 in ID, 1,400 ft (bottom of string)
Capacities
DP int. 0.0142; DC int. 0.0087; DC$\times$OH ann. 0.0272; DP$\times$OH ann. 0.0425; DP$\times$csg ann. 0.0590 bbl/ft
Pit gain (methane kick, constant-volume slug)
40 bbl
SICP / SIDPP
1,000 psi / 300 psi
Reduced-rate circulating friction
950 psi at 250 gpm
Volume pumped so far (1st circulation, original mud)
125 bbl
Surface temperature and geothermal gradient are supplementary nomenclature, not needed under the constant-volume kick assumption used below.
Find. (a) Length of the kick zone; (b) casing pressure at surface; (c) equivalent mud density at the casing shoe; (d) pit gain; (e) drillpipe pressure schedule for the 2nd (kill-mud) circulation — all after 125 bbl has been pumped into the well during the 1st circulation.
Fig. 1 — Wellbore schematic (not to scale): the 40-bbl methane slug starts at the bottom of the hole spanning the DC–OH and DP–OH annuli, then rises and shortens (constant volume, larger annular capacity) as 125 bbl of original mud is pumped in during the 1st (Driller's-method) circulation.
Approach. Establish the formation pressure (BHP) and the kick's own hydrostatic gradient from the shut-in data, track the kick's position and length (constant volume, piecewise across annulus capacities) as mud is pumped, then back out casing pressure and equivalent mud density from a BHP balance. The 2nd-circulation schedule follows the standard Driller's-method kill-mud-weight relations.
Formation pressure (BHP). The drill string carries only mud (no gas), so $$BHP=SIDPP+0.052\,MW\,D=300+0.052(13)(14{,}000)\Rightarrow\boxed{BHP=9{,}764\ \text{psi}}$$
Kick geometry at shut-in. The 40-bbl slug fills the DC$\times$OH annulus first ($1{,}400\times0.0272=38.08$ bbl), then the remaining $1.92$ bbl into the DP$\times$OH annulus ($1.92/0.0425=45.2$ ft), giving kick length $1{,}400+45.2=1{,}445.2$ ft and kick top at $14{,}000-1{,}445.2=12{,}554.8$ ft.
Kick's own hydrostatic gradient (back-derived from shut-in data). With mud above the kick to $12{,}554.8$ ft: $$g_k=\frac{BHP-SICP-0.052\,MW(12{,}554.8)}{1{,}445.2}\Rightarrow\boxed{g_k=0.192\ \text{psi/ft}}$$ (this reproduces $SICP=1{,}000$ psi exactly when checked against the shut-in annulus balance).
Part (a) — kick position and length after 125 bbl pumped. Mud enters the annulus at the bit, so the whole column (kick included) shifts up by exactly the volume pumped, measured from the bottom: $38.08$ bbl fills the DC$\times$OH annulus, leaving $125-38.08=86.92$ bbl into the DP$\times$OH annulus ($86.92/0.0425=2{,}045.2$ ft), so the new kick BOTTOM is at $12{,}600-2{,}045.2=10{,}554.8$ ft — entirely within the DP$\times$OH annulus. The 40-bbl slug now occupies a single capacity, so its length shortens: $$L_{kick}=40/0.0425\Rightarrow\boxed{L_{kick}=941.2\ \text{ft}}$$ with the kick top at $10{,}554.8-941.2=9{,}613.6$ ft — still below the 9,000 ft shoe.
Part (b) — casing pressure at surface. Sum the annulus hydrostatic (mud above the kick $+$ the kick itself $+$ mud below the kick, down to the bit) and subtract from BHP: $$P_{csg}=BHP-\big[0.052(13)(9{,}613.6)+0.192(941.2)+0.052(13)(14{,}000-10{,}554.8)\big]\Rightarrow\boxed{P_{csg}=755.9\ \text{psi}}$$
Part (c) — equivalent mud density at the casing shoe. The kick is still below the shoe, so only 13-ppg mud lies between surface and shoe: $$EMW_{shoe}=\frac{P_{csg}+0.052(13)(9{,}000)}{0.052(9{,}000)}\Rightarrow\boxed{EMW_{shoe}=14.62\ \text{ppg}}$$
Part (d) — pit gain. Under the "kick moves as a slug" (constant-volume, no gas expansion) convention, the closed circulating loop returns exactly what is pumped in at every instant, so the mud-tank level increase stays at its initial shut-in value: $$\boxed{\text{Pit gain}=40\ \text{bbl (unchanged)}}$$
Part (e) — kill mud weight and 2nd-circulation pressure schedule. Kill mud weight $MW_{kill}=13+300/[0.052(14{,}000)]=13.412$ ppg. Initial circulating pressure $ICP=SIDPP+950=1{,}250$ psi (held constant through the entire 1st circulation, which is what keeps BHP at the formation value while the ORIGINAL mud weight is still circulating). Final circulating pressure once kill mud fully fills the $191.1$-bbl drill string ($12{,}600\times0.0142+1{,}400\times0.0087$): $$FCP=950\times\frac{13.412}{13}\Rightarrow\boxed{FCP=980.1\ \text{psi}}$$ During the 2nd circulation the drillpipe pressure is stepped down LINEARLY with volume of kill mud pumped into the drill string, from $ICP=1{,}250$ psi at $V=0$ to $FCP=980.1$ psi at $V=191.1$ bbl, then held constant at $980.1$ psi for the remainder of the circulation (once kill mud fills the entire string).
Check: (1) the kick's own hydrostatic gradient is back-derived from the shut-in SICP/SIDPP data rather than assumed as the textbook 0.1 psi/ft methane simplification — the 0.192 psi/ft value found here is internally consistent (it exactly reproduces the given SICP) and is physically reasonable for high-pressure methane at these depths; (2) annulus friction pressure is not separately itemized in the source data and is neglected in the casing-pressure balance.
