Equations: Formation pressures and well control

SI

Holds when: The fluid is at rest. When the mud circulates, friction adds pressure losses to the hydrostatic pressure (see Circulating pressure losses). The density is the same throughout the column. If it changes with depth (several fluids, gas, heating), we add up the pressure interval by interval: $p = p_0 + \sum_i \rho_i\, g\, \Delta h_i$. $h$ is specifically the vertical height, not the length along the hole.

SI

Holds when: The pressure is a gauge pressure, and the datum is the same one from which the mud column is measured (usually the rotary table). Equivalent mud weight is a notional (imaginary) quantity. It describes the pressure at one depth and does not mean that a fluid of that density actually lies above.

any density units, the same for all terms

Holds when: The pores are completely filled with fluid. If the pores contain several fluids, $\rho_f$ is their average density, weighted by the share of pore volume each one fills (saturation). $\rho_{ma}$ is the average grain density, allowing for the mineral composition of the rock.

SI; in metric units $\sigma_v\,[\text{MPa}] = 0.00981\sum_i \rho_{b,i}\,[\text{g/cm}^3]\;\Delta h_i\,[\text{m}]$

Holds when: The vertical stress equals the weight of the overlying rock. The layers are roughly horizontal, and we can ignore the support from neighbouring rock masses. On land the summation starts at the ground surface. Offshore, the first interval is the layer of sea water (about 1.03 g/cm³) from sea level to the sea floor. Depths are true vertical depths. To compare with the mud pressure, we measure the depth from the rotary table for the overburden pressure, as we do for the pore pressure.

any pressure units, the same for all terms

Holds when: The pore space is connected, and the pore pressure is uniform within the volume of rock considered. In pore pressure and drilling problems we normally assume $\alpha = 1$. In calculations of how a reservoir deforms, we use a measured value.

SI

Holds when: The mud is at rest (static pressure). During circulation the bottomhole pressure is higher by the pressure losses in the annulus. The mud column reaches the wellhead. If the level has dropped, the bottomhole pressure is lower. After choosing a density, we check that the mud pressure, including the ECD, does not exceed the fracture pressure anywhere in the open hole.

SI; in metric units the factor $g$ is replaced by 0.00981 (densities in g/cm³, pressure in MPa)

Holds when: A normally compacted sequence of a single rock type, with no overpressured zones. The exponential trend is an empirical approximation (it comes from observation, not from theory). We fit its parameters to the well logs of the area.

SI; pressures are absolute

Holds when: The gas is at rest. The temperature and $Z$ are constant along the column (in practice we use average values). The column is filled with gas only.

$\phi$ in fraction, $\sigma'_v$ in MPa, $c$ in MPa⁻¹

Holds when: an empirical relation for one lithology: the shales of the given basin. The parameters are fitted to logs (measurements made along the hole) and to cores (rock samples cut from the well); it describes only compaction as the effective stress grows (the “loading curve”). When the stress falls, the porosity hardly recovers.

oilfield units: $R$ in ft/h, $N$ in rpm, $W$ in lbf, $B$ in inches

Holds when: an empirical formula for one lithology, uniform shales. The other influences (the type of bit and its wear, the hydraulics of flushing, the properties of the mud) are taken as unchanged. If they change sharply, we build the trend again; we hold the weight and the rotary speed as constant as possible, and we take points at small steps of depth; the logarithms in the numerator and the denominator are taken to one base (usually base 10). Both arguments are less than one, both logarithms are negative, and $d$ is positive; the constants inside the logarithms are not dimensionless: the formula is used strictly in the units for which the constants are written.

densities in any identical units

Holds when: a simple empirical correction, not a law of physics. It is enough to remove from the plot the jumps caused by making the mud heavier.

any identical units of pressure; all the pressures refer to one depth, so the formula holds both for gradients and for equivalent mud weights

Holds when: the high pressure was created by undercompaction. With unloading and transfer mechanisms the method underestimates the pressure; the normal trend is built from uniform shales with normal pressure and is reliably continued down to the depth of the estimate; the exponent 1.2 is empirical. For other data we use other exponents: 1.2 for the ratio of the resistivities of shales $R_o/R_n$, and 3.0 for the ratio of the interval transit times $\Delta t_n/\Delta t_o$ from the sonic log (the time that a sound wave takes to cross a unit length of rock). In each area the exponent and the trend are checked against pressure measurements.

any identical units of pressure; true vertical depths

Holds when: the pressure was created by undercompaction: the shale “got stuck” on the loading curve. With unloading the method, like the Eaton method, underestimates the pressure; we compare the same lithology, and the normal trend is reliably continued.

any identical units of density

Holds when: the pores of the cuttings are filled with water, and the cuttings do not swell or dissolve during the measurement; there are no pieces of cavings or particles of mud in the sample.

any identical units of pressure at one depth, so the formula holds both for gradients and for equivalent mud weights

Holds when: the smallest principal stress is horizontal (extension or strike-slip); the fracture pressure is taken equal to the smallest stress: the fracture already exists, or the tensile strength of the rock (how much pulling it can take before it breaks) is small. This is the fracture propagation pressure (also called the fracture extension pressure) and a cautious estimate of the leak-off pressure; $K$ is fitted for an area. Tectonic stresses, salt and temperature change it.

any identical units of pressure at one depth; the same for gradients and for equivalent mud weights

Holds when: the horizontal stresses are created by the weight of the rocks, with the rock prevented from expanding sideways and with no tectonic additions. The rock is elastic and isotropic (the same in all directions); the fracture pressure equals the smallest horizontal stress (the tensile strength and the concentration of stress at the wall are not counted); $\nu$ usually grows with depth: in young shaly sections from about 0.25 at the surface to 0.4–0.5 at depth. As $\nu \to 0.5$ the ratio tends to one, and the fracture pressure tends to the overburden pressure.

SI

Holds when: the mud in the well during the test is at rest and uniform in density; the pressure under the shoe is $p_{LO} + \rho_{\text{test}} g h_{\text{shoe}}$, the sum of the wellhead pressure and the hydrostatic pressure. There are almost no friction losses at slow pumping; all the pressures are gauge pressures (pressures above the atmosphere), like the reading of the gauge at the wellhead. The absolute pressure is greater by the atmospheric pressure (about 0.1 MPa), but we calculate the equivalent mud weight from the gauge pressure; the result refers to the rock under the shoe, and not to the whole open hole.

SI; in metric units $g h$ is replaced by $0.00981\,h$ (g/cm³, MPa, m)

Holds when: to check the shoe we take only the losses above the shoe. For the bottom of the hole, we take the losses in the whole annulus and the depth of the bottom.

SI; in metric units $\text{MAASP}\,[\text{MPa}] = 0.00981\,(\rho_{LOT} - \rho_m)\,h_{\text{shoe}}$ (g/cm³, m)

Holds when: the annulus from the wellhead to the shoe is filled with mud of density $\rho_m$. If there is gas in it, or mud of another density, we calculate the pressure at the shoe from the real column; the weakest place is the rock under the shoe. If there is a weaker bed in the open hole, it sets the limit.

any identical units of pressure

Holds when: a vertical well, with the vertical stress as one of the principal stresses; the rock is elastic and uniform; the mud does not penetrate the rock before the rupture. If it does, the breakdown pressure is lower; the temperature of the wall equals the temperature of the formation. Cooling of the wall by the mud lowers the breakdown pressure.

SI; it is convenient to take $d_s$ in L/m (1 L/m = 0.001 m²) and to calculate $\Delta p\,[\text{MPa}] = 0.00981\,\rho_m\,[\text{g/cm}^3]\;\Delta h\,[\text{m}]$

Holds when: the pipe is pulled “dry”: the mud drains from it back into the well, and the level goes down equally in the pipe and in the annulus; the denominator is the section of the hole less the section of the metal of the pipes that stay in the well at the level of the mud; mud does not go into the formation and does not come from it. The deviation of the actual fill-up volume from the calculated one is exactly the sign of trouble.

any identical units of pressure; the pressures are absolute, the logarithm is natural

Holds when: the gas is ideal (it follows the simple gas law), and the temperature is constant along the hole; the gas moves with the mud and does not dissolve in it, and its mass is negligible; the gas-cutting is steady: the same amount of gas for each unit volume of mud along the whole annulus; $p_{bh}$ stands under the logarithm, so we may put in its place the pressure of the gas-free column. The error is negligible.

any identical units; the pressure and temperature are absolute

Holds when: the mass of the gas does not change: the gas does not dissolve in the mud and does not come out of it; the approximate form is for a constant temperature and $Z$. At the bottom the gas is hotter, so as it rises to the wellhead its volume grows a little less than by the approximate formula, and the change of $Z$ may shift the result in either direction.

SI; the pressures are absolute

Holds when: the same as for the formula of the fall of the bottomhole pressure: an ideal gas, a constant temperature, and gas that moves with the mud and does not dissolve in it.

SI; the pressures are gauge pressures, as on the gauges

Holds when: the string is on the bottom, the influx is only in the annulus, and the pipe holds mud of one known density; the pressures have stabilised, and the gas has not yet started to raise them by its rise. If both pressures keep rising at the same rate, this is the rise of the gas, and for the calculation we take the value from before this growth; if there is a float valve in the string (a one-way valve), it does not pass pressure from below. We find the pressure under it by pumping a little mud into the pipe with care until the valve opens; the pressures are gauge pressures. The mud column is also counted in gauge pressure, so atmospheric pressure does not enter the formula.

SI; in metric units $g\,h_k$ is replaced by $0.00981\,h_k$ (g/cm³, MPa, m)

Holds when: the influx fills the annulus at the bottom as one pocket and is not mixed with the mud; the hole is vertical. In an inclined hole the length of the pocket along the hole is converted into a true vertical height; the capacity of the annulus is known. In an open hole with washouts it is larger than the nominal one, and the pocket is in fact shorter than calculated; if the volume of the influx is larger than the volume of the annulus opposite the drill collars, we calculate the height by sections with different capacity.

SI

Holds when: the same as for the calculation of the formation pressure from the pipe pressure; a margin above balance, if it is needed, is set separately: by the mud density or by the choke pressure; $h$ is the true vertical depth. The length of an inclined hole does not enter the formula.

any identical units of pressure; the densities in identical units

Holds when: the pump rate is constant and equal to the rate at which $p_{\text{SCR}}$ was measured; when the mud is changed, the friction losses change in proportion to the density. This is an approximation for turbulent flow in the pipe and in the bit nozzles. The viscosity of the kill mud is close to the original; $p_{\text{SCR}}$ includes the losses in the whole system, including the annulus. During killing the losses in the annulus add to the bottomhole pressure as a small margin (the derivation is in the advanced section); with a subsea wellhead (a wellhead on the sea floor), friction losses in the long choke line (the pipe that carries the returns from the blowout preventer to the choke) are added to the pressures. That is a separate calculation.

pressures in any identical units, volumes in identical units

Holds when: the well is vertical, and the string has one internal diameter. A short section of heavy pipes at the bottom hardly changes the schedule (see the example below). For a string of long sections and for an inclined well, we build the schedule by sections (see the advanced section); the formula holds for $N \le N_{\text{bit}}$. After that we hold the pressure equal to $p_{\text{FCP}}$ until the kill mud comes out at the surface; the volume of the surface lines from the pump to the string is small, and for simplicity it is not counted.

SI; in metric units $\Delta p\,[\text{MPa}] = 0.00981\,\rho_m\,\Delta V/C_a$ (g/cm³, m³, m³/m)

Holds when: mud is bled off in small portions, and its volume is measured accurately, for example in the trip tank; the hole is vertical, and the capacity of the section with gas is known; the gas does not dissolve in the mud. In oil-based muds a large part of the gas is dissolved, and the method needs corrections.

SI; the pressures are gauge pressures, except for the sum $p + p_a$ in Boyle's law

Holds when: the gas is ideal, and the temperature and the gas deviation factor are constant (Ideal and real gas: the gas deviation factor (Z)); the gas moves with the mud, does not rise relative to it, does not dissolve and is not stretched along the hole; the mass of the gas is constant, so its density is inversely proportional to its volume; friction losses in the annulus and in the choke line are not counted. The bottomhole pressure is held exactly equal to the formation pressure.

any consistent units of pressure and length

Holds when: A force balance on the rod: the well pressure pushes it out with the force $p_{\text{well}} A_{\text{rod}}$, and the piston pushes with the force $p_{\text{ctrl}} A_{\text{pis}}$. The friction of the seals and the force needed to compress the rubber are not included. In reality, more pressure is needed. Shearing a pipe needs a much greater force. We estimate it from the manufacturer's data for the particular pipe.

SI; the pressures are gauge pressures

Holds when: The well is closed and filled with gas from the formation to the surface. We take the gas density with a margin on the low side. The pore pressure is taken from the upper bound of the forecast. If the rock under the shoe will clearly fracture first, the fracture pressure (the pressure at which the rock breaks) limits the wellhead pressure. We take the fracture pressure with a margin on the high side and subtract the pressure of the gas column up to the shoe (the same load case for the casing).

SI

Holds when: A liquid without gas flows through the choke. The flow is turbulent, and the losses come from accelerating the jet in the restriction. The formula does not apply to gas or to a gas–liquid mixture. Gas is compressible, and the speed in the restriction is limited by the speed of sound. It is an order-of-magnitude estimate. We take the discharge coefficient of a real choke from the manufacturer's data.

any consistent units of pressure and volume; the pressures are absolute

Holds when: Nitrogen is an ideal gas (it follows the simple gas law exactly), and the temperature is constant: $p_0 V_{\text{bottle}} = p\,(V_{\text{bottle}} - V_{\text{L}})$, where $V_{\text{L}}$ is the volume of fluid in the bottle. The discharge is slow. In a fast discharge the gas cools, and the usable volume is noticeably smaller (see the advanced section). The pressures are absolute: add the atmospheric pressure to the gauge reading.

any consistent units; the pressures are absolute

Holds when: An ideal gas, expanding with no heat exchange with the walls. The bottle is charged slowly, at constant temperature (isothermally). At high pressures nitrogen differs noticeably from an ideal gas, and a precise calculation uses the equation of state of a real gas.