6.5 in. OD $\times$ 3.0 in. ID, 89 ppf, 750 ft (bottom of string)
Pit gain
20 bbl
SICP / SIDPP
400 psi / 200 psi
Kill mud rate, $q$
5 bbl/min
Reduced (kill-rate) pump pressure
600 psi
Surface temperature (60 °F) and geothermal gradient are supplementary nomenclature and are not needed under the isothermal, constant-volume kick assumption used below.
Find. (a)–(f) above, at the instant the top of the kick zone reaches the 4,500 ft casing shoe.
Fig. 2 — The kick occupies the drill-collar annulus at shut-in, then a longer, thinner length once it crosses into the larger-capacity open-hole/drillpipe annulus.
Approach. Bottom-hole pressure equals formation pressure and is held constant throughout ($BHP=SIDPP+0.052\,MW\,D$). Treat the kick as a constant-volume slug (length set only by local annulus capacity, the standard simplification with no PVT/Boyle's-law data supplied), back its own hydrostatic gradient out of the shut-in data, and track the kick and kill-mud front positions as Engineer's Method circulates kill-weight mud from the very start (plug flow, no old-mud gap once kill mud reaches the bit).
Annulus/string capacities. DC–open-hole annulus: $(8.75^2-6.5^2)/1029.4=0.03333$ bbl/ft. Open-hole–DP annulus: $(8.75^2-5^2)/1029.4=0.05009$ bbl/ft. DP internal: $4^2/1029.4=0.01554$ bbl/ft; DC internal: $3^2/1029.4=0.00874$ bbl/ft. Top of collars sits at $10{,}000-750=9{,}250$ ft.
Formation (bottom-hole) pressure — held constant. $BHP=SIDPP+0.052\,MW\,D=200+0.052(10)(10{,}000)$, so $\boxed{BHP=5{,}400\ \text{psi}}$.
Kick's own gradient, backed out of the shut-in data. At shut-in the kick sits in the DC annulus, length $L_0=20/0.03333=600.0$ ft, top at $10{,}000-600.0=9{,}400$ ft (still within the DC zone). $BHP=SICP+0.052\,MW(9{,}400)+g_k L_0$: $g_k=[5{,}400-400-0.052(10)(9{,}400)]/600.0$, so $\boxed{g_k=0.187\ \text{psi/ft}}$.
Kill mud weight. $MW_{kill}=MW+SIDPP/(0.052D)=10+200/(0.052\times10{,}000)$, so $\boxed{MW_{kill}=10.385\ \text{ppg}}$.
(a) Track the kick top from 9,400 ft to the shoe. 150 ft through the DC-annulus ($\times0.03333=5.0$ bbl) plus 4,750 ft through the OH–DP annulus ($\times0.05009=237.9$ bbl): $242.9$ bbl pumped so far (part d). Since the kick is a constant-volume slug and both its ends now sit in the OH–DP annulus, its length is simply $20/0.05009$, so $\boxed{L_{kick}=399.3\ \text{ft}}$ (kick bottom at $4{,}899.3$ ft).
(c) Casing pressure at the surface. Sum the hydrostatic column from bottom-hole up: kill mud below the kick ($10{,}000-4{,}899.3=5{,}100.7$ ft at $10.385$ ppg $=2{,}754.4$ psi) $+$ the kick itself ($399.3$ ft at $g_k=0.187$ psi/ft $=74.5$ psi) $+$ original mud above it ($4{,}500$ ft at $10$ ppg $=2{,}340.0$ psi). $$P_c=BHP-\Sigma(\text{hydrostatic})=5{,}400-(2{,}754.4+74.5+2{,}340.0)$$ so $\boxed{P_c=231.1\ \text{psig}}$.
(b) Equivalent mud density at the shoe. The pressure at the shoe, from the surface side, is $P_{shoe}=P_c+0.052(10)(4{,}500)=231.1+2{,}340.0=2{,}571.1$ psi. $$EMW_{shoe}=\dfrac{P_{shoe}}{0.052\times4{,}500}$$ so $\boxed{EMW_{shoe}=10.99\ \text{ppg}}$ — comfortably above the 10 ppg original mud weight, confirming the shoe is not being underbalanced during this phase of the kill.
(d) Volume of mud pumped. Already found while tracking the kick top in step 5: $\boxed{V_{pumped}=242.9\ \text{bbl}}$ (about 48.6 min at the 5 bbl/min kill rate).
(e) Total pit gain. Because the well is a closed, incompressible circulating loop and the kick is carried as a constant-volume slug, every barrel pumped in at the standpipe is matched, at that same instant, by a barrel returning at the choke — the surface pit level does not rise any further once circulation is underway. $\boxed{\text{Total pit gain}=20\ \text{bbl}}$, unchanged from the value recorded at shut-in.
(f) Drillpipe pressure schedule. $ICP=SIDPP+P_{reduced}=200+600=800$ psi. $FCP=P_{reduced}(MW_{kill}/MW)=600(10.385/10)$, so $FCP=623.1$ psi. String capacity (volume to bit) $=9{,}250(0.01554)+750(0.00874)=150.3$ bbl ($\approx30.1$ min at 5 bbl/min). $$DPP(V)=ICP-(ICP-FCP)\dfrac{V}{150.3},\quad 0\le V\le150.3\ \text{bbl}$$ then $\boxed{DPP=FCP=623.1\ \text{psi, held constant}}$ for the remainder of the job, including the moment (at $V=242.9$ bbl) when the kick top reaches the shoe.
Check: (1) the kick is treated as a constant-volume slug (length changes only through local annulus capacity), the standard simplification with no PVT/Boyle's-law data supplied; (2) the kick's own hydrostatic gradient is backed out of the shut-in SIDPP/SICP/pit-gain data rather than assumed from a textbook table, and is held constant as the kick migrates; (3) surface temperature and geothermal gradient are not needed under the isothermal, constant-volume assumption used here and are treated as supplementary nomenclature.
Quantity
Value
Bottom-hole (formation) pressure, $BHP$
5,400 psi
Kill mud weight, $MW_{kill}$
10.385 ppg
(a) Length of kick zone
399.3 ft
(b) Equivalent mud density at shoe
10.99 ppg
(c) Casing pressure at surface
231.1 psig
(d) Volume of mud pumped
242.9 bbl
(e) Total pit gain
20.0 bbl (unchanged)
(f) $ICP\rightarrow FCP$
800 psi $\rightarrow$ 623.1 psi, linear over first 150.3 bbl, then constant
