Equations: The rig and the drilling process

The equations of the “The rig and the drilling process” lessons in “Drilling engineering”, with their conditions and units, and what each symbol means in them.

Annular volume and lag time

V_a = \frac{\pi}{4}\sum_i \left(D_i^2 - d_i^2\right) L_i, \qquad t_{lag} = \frac{V_a}{Q}

SI. It is handy to calculate the annular capacity in litres per metre as $\frac{\pi}{4}(D^2 - d^2)\cdot 10^{-3}$ with the diameters in millimetres

Holds when: The lengths of the intervals are measured along the hole, not vertically, because the mud travels the whole path. The mud and the cuttings move at the same average speed. In reality the cuttings lag behind the mud, especially in deviated holes. The open hole has its nominal (design) diameter. Washouts (places where the hole is wider than the bit) add volume and time, so we check the calculation with a tracer. The pump rate is constant and known. We calculate it from the number of pump strokes and the volume of one stroke, allowing for the volumetric efficiency (the share of the stroke volume that the pump really delivers).

How a well is drilled: the process and the people involved →

Mechanical, technical and commercial drilling speed

v_\text{mech} = \frac{H}{t_\text{bot}}, \qquad v_\text{tech} = \frac{720\,H}{T_\text{prod}}, \qquad v_\text{com} = \frac{720\,H}{T_\text{cal}}

metres and hours; speeds in m/h and metres per rig-month (m/rig-month)

Holds when: $T_\text{prod}$ and $T_\text{cal}$ do not include preparatory work, rig-up or testing. A speed that includes them is called the cycle speed. For a group of wells, use the total footage and the total time. The trip-averaged rate of penetration describes one bit. It is the footage of a bit run divided by the drilling and tripping time of that run (measures of bit performance).

Drilling time and cost →

Cost of a well by type of cost

C = C_\text{lump} + r\,T + \sum_j c_j\,L_j

any currency; in the example, thousand US dollars and days

Holds when: $T$ is the number of days from the time estimate. The drilling contract decides how the time of repairing the rig is paid. The boundary between the types is not sharp: bits depend on both metres and time, and mud consumption depends on the volume of the hole and on losses. Prices are those of the date of the estimate. For a programme lasting several years, we allow for rising prices (escalation).

Drilling time and cost →

Cost of an event and the mean contingency from a risk register

\Delta C_k = r\,\Delta t_k + c_k, \qquad \mathrm{E}[\Delta C] = \sum_k p_k\,\Delta C_k

thousand US dollars and days

Holds when: The events are independent: each happens at most once, and its consequences do not depend on the others. Linked events are combined, or they are given by a conditional probability. An example is lost circulation, after which stuck pipe is more likely. The register is incomplete, so we add a contingency for what it does not cover. Means add up whatever the relation between the events, but P90 values do not. We find the P90 of a sum from the whole distribution, by listing the combinations or by simulation (why P90 values are not added).

Drilling time and cost →

The learning curve (Brett–Millheim model)

t_n = C_1\,\e^{C_2(1-n)} + C_3

days

Holds when: The wells are of one type in one area, with one rig and one contractor. A break in the work, a change of crew or a change of casing programme sets the series back. We fit the constants to the actual wells of the series. It is more reliable to estimate $C_3$ separately, from the technical limit or the best composite well (fitting the curve). Non-productive time improves less with learning than productive time, so it is often analysed separately.

Drilling time and cost →

Complete or abandon

V_\text{well} > C_\text{compl} - C_\text{aband}

thousand US dollars at the date of the decision

Holds when: The cost of drilling to the planned depth has already been spent and does not depend on the decision, so it is not part of the comparison (sunk cost). The abandonment of a completed well at the end of its life is included in $V_\text{well}$ . It comes years later, so it weighs less after discounting (converting future money into today's value). If the well is also needed for other purposes, such as injection or observation, add their value to $V_\text{well}$ .

Drilling time and cost →

Required electric power and fuel consumption

\begin{gathered} N_{\text{el}} = \frac{1}{\eta_{\text{drive}}}\left(\frac{pQ}{\eta_{\text{mech}}} + M\omega\right) + N_{\text{aux}},\\ G_{\text{f}} = g_{\text{f}}\,N_{\text{el}}\,t \end{gathered}

kW, MPa × L/s = kW, kN·m × rad/s = kW

Holds when: The loads are constant during the calculation time. In reality they change with connections, circulating and trips, so the daily consumption is calculated from a load schedule. The specific fuel consumption (the fuel burned per kilowatt-hour) depends on the generator load. For modern diesel generators near the rated load it is about 0.2 kg/(kW·h), and noticeably more at a light load.

Drilling rig: power and hoisting systems →

Block efficiency of the reeving

\eta = \frac{W}{n\,F_{\text{fl}}} = \frac{K^n - 1}{K^n\,n\,(K - 1)}

dimensionless

Holds when: This is the ordinary rig reeving: the dead line passes over the stationary sheave of the crown block, and the rope goes round as many rotating sheaves as there are lines strung. If there are extra rotating sheaves in the path of the rope, their total number $S$ replaces $n$ in the exponent of the denominator, giving $K^S$ . The block moves upward at constant speed. Acceleration, braking and jerks add dynamic loads that the formula does not include. $K$ is a calculated value. It covers friction in the bearings and the loss in bending the rope. The sheave manufacturer may give its own value.

Drilling rig: power and hoisting systems →

Tension in the fast line and the dead line

F_{\text{fl}} = \frac{W}{\eta\,n}, \qquad F_{\text{dl}} = \frac{W}{n}

any units of force

Holds when: The dead line does not move, so its tension is taken as static, $W/n$ . When the block moves, friction in the sheaves makes the tension smaller when hoisting and larger when lowering (see the advanced section). For the derrick load when hoisting, $W/n$ errs on the safe side. The ratio $F_{\text{fl}}/W = 1/(\eta n)$ is called the fast line factor.

Drilling rig: power and hoisting systems →

Design factor of the drilling line

k_{\text{df}} = \frac{R}{F_{\text{fl}}}

any identical units of force

Holds when: The minimum values by API RP 9B are: 3 for the drilling line while drilling and tripping; 2 when running casing, when working stuck pipe and in other rare operations; and 2.5 for the line used to raise the mast. For a worn rope and under shock loads, the actual factor is lower than the calculated one. Work near the minimum is possible only with a sound line and equipment, and without jerks.

Drilling rig: power and hoisting systems →

Power at the drawworks drum and hoisting speed

\begin{gathered} N_{\text{drum}} = F_{\text{fl}}\,v_{\text{fl}} = \frac{W v}{\eta}, \qquad v_{\text{fl}} = n\,v,\\ N_{\text{in}} = \frac{N_{\text{drum}}}{\eta_{\text{dw}}} \end{gathered}

kN × m/s = kW

Holds when: The motion is steady. Accelerating a string of hundreds of tonnes needs extra power, so acceleration and braking are made smooth. When lowering, energy flows the other way: the load itself pulls the line, and the brakes or the motors in braking mode absorb the power $Wv$ (less friction).

Drilling rig: power and hoisting systems →

Derrick load

F_{\text{der}} = W + F_{\text{fl}} + F_{\text{dl}} = W\,\frac{1 + \eta + \eta\,n}{\eta\,n}

any identical units of force

Holds when: The fast line and the dead line run down from the crown block almost vertically. Slanting lines and wind add horizontal forces, which are allowed for in the design of the derrick itself. The dead line is taken as static ( $W/n$ ). The load is shared unevenly among the legs of the derrick (see the advanced section).

Drilling rig: power and hoisting systems →

Work of the drilling line in a full round trip

T_{\text{rt}} = \frac{D\,(L_{\text{st}} + D)\,w + 4D\,(W_{\text{trav}} + C/2)}{9807}

kN·m divided by 9807 kN·m per t·km (tonne-force × kilometre)

Holds when: The drill pipe is taken to reach the bottom, and the excess weight $C$ accounts for the heavy bottom part. For each stand, the travelling equipment moves up and down over the length of the stand when hoisting, and the same when lowering. This gives $4D$ in total per round trip. Friction is the same when lowering and when hoisting. Accelerations, vibrations and drag in a deviated hole are not included. This is a comparative measure of wear, not exact work.

Drilling rig: power and hoisting systems →

Line work in drilling, running casing and a short trip

\begin{gathered} T_{\text{drill}} = 3\left[T_{\text{rt}}(D_2) - T_{\text{rt}}(D_1)\right],\\ T_{\text{short}} = T_{\text{rt}}(D_{\text{low}}) - T_{\text{rt}}(D_{\text{up}}),\\ T_{\text{csg}} = \frac{D\,(L_{\text{jt}} + D)\,w_{\text{csg}} + 4D\,W_{\text{trav}}}{2\cdot 9807} \end{gathered}

t·km; depths and lengths in m, weights in kN and kN/m

Holds when: Drilling goes in a cycle: drill the length of a joint or stand, pick up off bottom, ream (work the bit through the new section again), pick up for the connection, make the connection, and lower. In this cycle the line does work equal to three full round trips over the increase in depth. If the string is not reamed or lifted before the connection, it is two. In a short trip, the whole string is raised and lowered over the interval between two depths. The work is exactly the difference between the work of full round trips at those depths. Casing is only run in, so the work is half that of a full round trip. There are no collars in a casing string.

Drilling rig: power and hoisting systems →

Equivalent derrick load and its utilisation factor

\begin{gathered} F_{\text{eq}} = 4\left(\frac{W}{4} + \frac{W}{n}\right) = W\,\frac{n + 4}{n},\\ E_{\text{der}} = \frac{F_{\text{der}}}{F_{\text{eq}}} = \frac{1 + \eta + \eta\,n}{\eta\,(n + 4)} \end{gathered}

any identical units of force

Holds when: The derrick has four legs, and the load from the working lines is shared equally among them. The dead line is anchored at one leg and the drawworks is midway between two others. For another arrangement the distribution is different, and masts have their own scheme.

Drilling rig: power and hoisting systems →

Output of a mud pump per stroke and per unit of time

\begin{gathered} V_{\text{stk}} = 3\cdot\frac{\pi}{4}\,d^2 L\,\eta_v \ \ \text{(triplex)}, \qquad V_{\text{stk}} = \frac{\pi}{2}\left(2d^2 - d_{\text{rod}}^2\right) L\,\eta_v \ \ \text{(duplex)},\\ Q = \frac{V_{\text{stk}}\,n}{60} \end{gathered}

SI. It is convenient to take $d$ and $L$ in decimetres, because then $V_{\text{stk}}$ comes out directly in litres

Holds when: The volume swept by the piston is calculated from the full stroke. In a duplex pump the return stroke displaces the volume of the ring-shaped space between the liner and the rod. The volumetric efficiency accounts for leakage through the valves and pistons, incomplete filling of the cylinders and compression of gas in the mud. For a pump in good order with a charge pump it is close to 1. For a worn pump it falls to 0.9 and below. With several pumps the outputs add up, and the total counter counts the strokes of all the pumps.

Rig circulating and rotary systems →

Allowable liner pressure and the power limit

p_{\max} = \frac{F_{\text{rod}}}{\pi d^2/4}, \qquad N_{\text{hyd}} = p\,Q \le \eta_{\text{mech}}\,N_{\text{drive}}

MN/m² = MPa; MPa × L/s = kW

Holds when: The strength of the fluid end, and not only the rod load, may also limit the pressure. For small liners the rated pressure can be lower than the one calculated from the rod load. Pump data sheets give the allowable pressure and the output at the highest stroke rate for each liner in a table. The formula explains where these numbers come from.

Rig circulating and rotary systems →

Strokes to the bit and up the annulus

Z_{\text{bit}} = \frac{V_{\text{string}}}{V_{\text{stk}}}, \qquad Z_{\text{ann}} = \frac{V_{\text{ann}}}{V_{\text{stk}}}, \qquad t = \frac{V}{Q}

SI; the volume of the hole and of the pipe is $\frac{\pi}{4}(D^2 - d^2)$ per metre of length

Holds when: The pipe and the hole have their nominal diameters. Washouts increase the volume of the annulus, so the actual time is checked with a tracer. The mud is incompressible and is not lost anywhere. With lost circulation or an influx the volumes stop adding up, and that is the warning sign.

Rig circulating and rotary systems →

Power of rotating the string

N_{\text{rot}} = M\,\omega, \qquad \omega = \frac{2\pi\,n_{\text{rot}}}{60}

kN·m × rad/s = kW

Holds when: The rotation is steady. The torque at the surface is the sum of the torque at the bit and the friction of the string against the walls. In a deviated well the friction is usually the larger part (drag and torque). The power at the input of the drive is higher by its efficiency.

Rig circulating and rotary systems →

Symbols

Symbol	Meaning and context
$720$	hours in a rig-month: 30 days of work of one rig, 24 h each, hDrilling time and cost
$C$	cost of the well according to the estimate, US$ thousandDrilling time and cost excess weight of the drill collars and heavy-weight pipe in mud over the weight of drill pipe of the same length, kNDrilling rig: power and hoisting systems
$C_1$	by how much the first well exceeds the limit: $t_1 = C_1 + C_3$ , dDrilling time and cost
$C_2$	rate of learning, —Drilling time and cost
$C_3$	the limit that the time approaches with full learning, dDrilling time and cost
$C_\text{aband}$	the cost of abandonment that will be avoided if the well is completed, US$ thousandDrilling time and cost
$C_\text{compl}$	the cost of the production casing and completion, US$ thousandDrilling time and cost
$c_j$	unit price of the $j$ -th material: a metre of casing, a bit, a tonne of cement, US$ thousand per unitDrilling time and cost
$c_k$	direct cost of the event: materials, fishing (recovering stuck or lost tools), tools left in the hole, US$ thousandDrilling time and cost
$C_\text{lump}$	lump-sum costs: the site, mobilisation, wellhead equipment, service operations paid per job, US$ thousandDrilling time and cost
$d$	liner diameter, mRig circulating and rotary systems
$D$	depth (length of the string along the hole), mDrilling rig: power and hoisting systems
$d_i$	outside diameter of the drill pipe or drill collars in that interval, mHow a well is drilled: the process and the people involved
$D_i$	diameter of the hole in the $i$ -th interval. In open hole (the part with no casing), use the bit diameter, or the real diameter from the caliper log (a log that measures the hole diameter). In cased hole, use the inside diameter of the casing, mHow a well is drilled: the process and the people involved
$d,\ d_{\text{rod}}$	inside diameter of the liner (piston diameter) and diameter of the rod, mRig circulating and rotary systems
$E_{\text{der}}$	derrick utilisation factor — the ratio of the actual load to the equivalent one, —Drilling rig: power and hoisting systems
$\mathrm{E}[\Delta C]$	mean (expected) overspend, US$ thousandDrilling time and cost
$F_{\text{der}}$	derrick load — the vertical force on the crown block, kNDrilling rig: power and hoisting systems
$F_{\text{dl}}$	tension in the dead line, taken equal to the static value, kNDrilling rig: power and hoisting systems
$F_{\text{eq}}$	equivalent load, which is the load the derrick would carry if all the legs were loaded as heavily as the most heavily loaded one, kNDrilling rig: power and hoisting systems
$F_{\text{fl}}$	tension in the fast line when hoisting, kNDrilling rig: power and hoisting systems tension in the fast line, the line with the highest tension when hoisting, kNDrilling rig: power and hoisting systems fast-line tension allowing for the block efficiency, kNDrilling rig: power and hoisting systems
$F_{\text{rod}}$	largest rod load according to the pump data sheet, MNRig circulating and rotary systems
$G_{\text{f}},\ g_{\text{f}},\ t$	fuel consumption, specific consumption of the generators and running time, kg, kg/(kW·h), hDrilling rig: power and hoisting systems
$H$	footage: the metres drilled by the well (or by a group of wells), mDrilling time and cost
$\Delta C_k$	extra cost if the $k$ -th event happens, US$ thousandDrilling time and cost
$\Delta t_k$	time lost, dDrilling time and cost
$K$	ratio of the tensions on the two sides of a rotating sheave: 1.04 for roller bearings, 1.09 for plain bearings, —Drilling rig: power and hoisting systems
$k_{\text{df}}$	design factor, —Drilling rig: power and hoisting systems
$L$	length of the piston stroke, mRig circulating and rotary systems
$L_i$	length of the interval along the hole, mHow a well is drilled: the process and the people involved
$L_j$	quantity of the $j$ -th material — metres of casing, pieces, tonnes, unitsDrilling time and cost
$L_{\text{jt}},\ w_{\text{csg}}$	length of a casing joint and weight of a metre of casing in mud, m, kN/mDrilling rig: power and hoisting systems
$L_{\text{st}}$	length of a stand, mDrilling rig: power and hoisting systems
$M$	torque on the string at the surface, kN·mRig circulating and rotary systems
$M,\ \omega$	torque and angular speed of the top drive or rotary table ( $\omega = 2\pi n_{\text{rot}}/60$ at a rotary speed $n_{\text{rot}}$ rpm), kN·m, rad/sDrilling rig: power and hoisting systems
$n$	number of the well in the series: 1, 2, 3 …, —Drilling time and cost number of working lines, —Drilling rig: power and hoisting systems strokes per minute, strokes/minRig circulating and rotary systems
$N_{\text{aux}}$	auxiliary consumers, kWDrilling rig: power and hoisting systems
$N_{\text{drum}}$	power at the drawworks drum, kWDrilling rig: power and hoisting systems
$N_{\text{el}}$	electric power that the generators must produce, kWDrilling rig: power and hoisting systems
$N_{\text{hyd}}$	hydraulic power that the pumps give to the mud, kW (MPa × L/s)Rig circulating and rotary systems
$N_{\text{in}},\ \eta_{\text{dw}}$	power at the input shaft of the drawworks and the efficiency of its transmissions, kW, —Drilling rig: power and hoisting systems
$N_{\text{rot}}$	power at the shaft of the rotary table or top drive, kWRig circulating and rotary systems
$p_k$	probability of the event on this well, fractionDrilling time and cost
$p_{\max}$	allowable discharge pressure for a liner of diameter $d$ , MPaRig circulating and rotary systems
$p,\ Q$	discharge pressure and pump rate, MPa, L/sDrilling rig: power and hoisting systems
$Q$	pump rate, m³/sHow a well is drilled: the process and the people involved
$r$	spread rate: the rig with its crew and all the services paid by the day, US$ thousand/dDrilling time and cost spread rate, US$ thousand/dDrilling time and cost
$R$	nominal breaking strength of the line according to the standard or certificate, kNDrilling rig: power and hoisting systems
$T$	days of work on the well according to the time estimate, dDrilling time and cost
$t_\text{bot}$	time of mechanical drilling, with the bit on the bottom, hDrilling time and cost
$T_\text{cal}$	calendar time of drilling and casing: productive plus non-productive, hDrilling time and cost
$T_{\text{csg}}$	work in running casing to depth $D$ , t·kmDrilling rig: power and hoisting systems
$T_{\text{drill}}$	work in drilling from depth $D_1$ to $D_2$ , t·kmDrilling rig: power and hoisting systems
$t_{lag}$	lag time, sHow a well is drilled: the process and the people involved
$t_n$	duration of the $n$ -th well of the series, dDrilling time and cost
$T_\text{prod}$	productive time of drilling and casing, hDrilling time and cost
$T_{\text{rt}}$	work of the line for hoisting and lowering a string to depth $D$ , t·kmDrilling rig: power and hoisting systems
$T_{\text{short}}$	work in a short trip between the lower depth $D_{\text{low}}$ and the upper $D_{\text{up}}$ (for example, to the casing shoe and back), t·kmDrilling rig: power and hoisting systems
$t,\ V,\ Q$	time to pump a volume $V$ at pump rate $Q$ , s, m³, m³/sRig circulating and rotary systems
$V_a$	volume of the annulus from the bottom of the hole to the surface, m³How a well is drilled: the process and the people involved
$V_{\text{ann}}$	volume of the annulus from the bit to the wellhead, by sections, m³Rig circulating and rotary systems
$v_\text{mech}$	mechanical speed, the rate of penetration (in the lesson Bit performance and cost per metre, the same quantity for one bit run is $\bar R$ ), m/hDrilling time and cost
$V_{\text{stk}}$	output per stroke (revolution of the shaft), m³Rig circulating and rotary systems actual output per stroke (with the volumetric efficiency); with several identical pumps, the output per stroke of any one of them, and the strokes are counted in total, m³Rig circulating and rotary systems
$V_{\text{string}}$	internal volume from the pumps to the bit: surface lines, drill pipe, drill collars and the rest of the assembly, m³Rig circulating and rotary systems
$v_\text{tech},\ v_\text{com}$	technical and commercial speed, m/rig-monthDrilling time and cost
$V_\text{well}$	the value of the well's production: the NPV (net present value, the value today) of its future cash flows, leaving out the costs already spent and the cost of completion, US$ thousandDrilling time and cost
$v,\ v_{\text{fl}}$	speed of the travelling block and speed of the fast line, m/sDrilling rig: power and hoisting systems
$w$	weight of a metre of drill pipe in mud, kN/mDrilling rig: power and hoisting systems
$W$	load on the lines: the weight of the string on the hook plus the weight of the travelling equipment, kNDrilling rig: power and hoisting systems
$W_{\text{trav}}$	weight of the travelling equipment: travelling block, hook, top drive, bails, elevator, kNDrilling rig: power and hoisting systems
$W,\ n,\ \eta$	load on the lines, number of lines and block efficiency, kN, —, —Drilling rig: power and hoisting systems
$W,\ n,\ \eta,\ F_{\text{der}}$	as in the main section, kN, —, —, kNDrilling rig: power and hoisting systems
$W,\ F_{\text{fl}},\ F_{\text{dl}}$	load on the lines, tension in the fast line and in the dead line, kNDrilling rig: power and hoisting systems
$W,\ F_{\text{fl}},\ \eta,\ n$	load on the lines, fast-line tension, block efficiency, number of lines, kN, kN, —, —Drilling rig: power and hoisting systems
$Z_{\text{bit}},\ Z_{\text{ann}}$	strokes to the bit and up the annulus from the bit to the wellhead, strokesRig circulating and rotary systems
$\eta$	block efficiency — the ratio of the load to the ideal force $nF_{\text{fl}}$ , —Drilling rig: power and hoisting systems
$\eta,\ n$	block efficiency and number of lines, —Drilling rig: power and hoisting systems
$\eta_{\text{drive}}$	efficiency of the electric drive, meaning the motors with their converters, —Drilling rig: power and hoisting systems
$\eta_{\text{mech}}$	mechanical efficiency of the pumps, —Drilling rig: power and hoisting systems
$\eta_{\text{mech}},\ N_{\text{drive}}$	mechanical efficiency of the pump and the power of its drive, —, kWRig circulating and rotary systems
$\eta_v$	volumetric efficiency (fill factor) — the fraction of the calculated volume that is actually delivered, —Rig circulating and rotary systems
$\omega,\ n_{\text{rot}}$	angular velocity and rotary speed, rad/s, rpmRig circulating and rotary systems