Equations: Inflow and lift: nodal analysis

The equations of the “Inflow and lift: nodal analysis” lessons in “Oil and gas production”, with their conditions and units, and what each symbol means in them.

Pressure budget of the production system

\bar p_R - p_{sep} = \Delta p_{\text{res}} + \Delta p_{\text{comp}} + \Delta p_{\text{tbg}} + \Delta p_{\text{choke}} + \Delta p_{\text{line}}

any pressure unit, the same for all terms

Holds when: Steady state (nothing changes with time): the same flow passes through every element, with the volumes given at the same conditions. Each term depends on the rate and on the fluid properties. So the equality holds at only one rate: the operating rate. The weight of the fluid column counts as part of the tubing loss. In a vertical well it is usually the largest term.

Nodal analysis of the production system →

Inflow curve to the node

p_{node} = \bar p_R - \Delta p_{\text{up}}(q)

any consistent units of pressure and rate

Holds when: Steady state: the reservoir pressure does not change during the calculation. All rates are given at the same conditions.

Nodal analysis of the production system →

Outflow curve from the node

p_{node} = p_{sep} + \Delta p_{\text{down}}(q)

any consistent units of pressure and rate

Holds when: The separator pressure is held constant. The losses in each element are worked out for the real composition of the flow: gas, oil and water.

Nodal analysis of the production system →

Outflow curve for liquid without gas in vertical tubing

p_{wf} = p_{wh} + \rho g H + \lambda\,\frac{H}{d}\,\frac{\rho v^2}{2},\qquad v = \frac{4q}{\pi d^2}

Holds when: Single-phase incompressible liquid (its volume does not change with pressure): the pressure along the whole length of the tubing is above the bubble point pressure, and there is no free gas. Vertical tubing of constant diameter. In a deviated (slanted) well, the weight of the column uses the true vertical depth, and friction uses the length along the hole. The density and the friction factor are constant. Speeding up of the flow is ignored. The flow is turbulent (swirling, not smooth): the Reynolds number $Re = \rho v d/\mu$ is above a few thousand. $\lambda$ depends on $Re$ and on how rough the pipe wall is. English-language literature also uses the Fanning friction factor, which is four times smaller: $f_F = \lambda/4$ .

Nodal analysis of the production system →

Condition for a stable operating point (quasi-static)

\frac{\dd p_{\text{out}}}{\dd q} > \frac{\dd p_{\text{in}}}{\dd q}

any consistent units

Holds when: A simplified quasi-static treatment (each state is taken as if the flow had time to settle): the inertia of the flow and the compressibility of the gas in the wellbore are ignored. Real swings of the operating mode (for example, repeated surges of gas and liquid) are studied with dynamic models.

Nodal analysis of the production system →

Productivity index

J = \frac{q}{\bar p_R - p_{wf}}

metric oilfield units (m³/d, MPa)

Holds when: Stabilised flow. Single-phase inflow: $p_{wf} \ge p_b$ . The pressures are converted to the same depth. For a difference, absolute pressure (measured from zero) and gauge pressure (measured above atmospheric pressure) work equally well, as long as both are of the same kind.

Inflow performance relationship (IPR) →

Inflow to a well in pseudo-steady state

q = \frac{2\pi k h\,(\bar p_R - p_{wf})}{\mu B\left[\ln\dfrac{r_e}{r_w} - \dfrac{3}{4} + S\right]}

Holds when: Pseudo-steady state in a circular drainage area with a closed (impermeable) outer boundary and the well at the centre. For an area of another shape, the term $\ln(r_e/r_w) - 3/4$ is replaced by an expression with a shape factor. Single-phase inflow of a slightly compressible liquid, by Darcy's law. A uniform reservoir (the same properties everywhere). Imperfect penetration (the well does not open the whole pay thickness) and damage to the near-wellbore zone are included in $S$ .

Inflow performance relationship (IPR) →

Vogel's equation

\frac{q}{q_{max}} = 1 - 0.2\,\frac{p_{wf}}{\bar p_R} - 0.8\left(\frac{p_{wf}}{\bar p_R}\right)^2

dimensionless form: the rates in any one unit, and the pressures likewise

Holds when: The reservoir pressure is not above the bubble point pressure ( $\bar p_R \le p_b$ ); solution gas drive. Stabilised inflow. The equation is an empirical generalisation (a rule drawn from many calculated cases) of results for wells with an undamaged near-wellbore zone ( $S = 0$ ). Low water cut (little water in the produced liquid). With a gas cap (a layer of free gas above the oil), a high water cut or high-viscosity oil, the accuracy is lower. It is better to take absolute pressures. At reservoir pressures of tens of megapascals, the difference from gauge pressures is small.

Inflow performance relationship (IPR) →

Composite IPR

q = \begin{cases} J\,(\bar p_R - p_{wf}), & p_{wf} \ge p_b,\\[6pt] q_b + \dfrac{J p_b}{1.8}\left[1 - 0.2\,\dfrac{p_{wf}}{p_b} - 0.8\left(\dfrac{p_{wf}}{p_b}\right)^2\right], & p_{wf} < p_b, \end{cases}\qquad q_b = J\,(\bar p_R - p_b)

any consistent units (for example, m³/d and MPa)

Holds when: $\bar p_R > p_b$ : above the bubble point pressure the inflow is single-phase. Below the bubble point pressure the curve is of the Vogel type, but it is built from $p_b$ , not from the reservoir pressure. Stabilised flow. The effect of the near-wellbore zone is included in $J$ .

Inflow performance relationship (IPR) →

Productivity index from a test below the bubble point pressure

J = \frac{q_1}{(\bar p_R - p_b) + \dfrac{p_b}{1.8}\left[1 - 0.2\,\dfrac{p_{wf,1}}{p_b} - 0.8\left(\dfrac{p_{wf,1}}{p_b}\right)^2\right]}

as for the composite curve

Holds when: The same as for the composite curve. The bubble point pressure is known from a PVT study of an oil sample.

Inflow performance relationship (IPR) →

Fetkovich's inflow equation

q = C\,(\bar p_R^2 - p_{wf}^2)^n

$q$ in m³/d, pressures in MPa (absolute)

Holds when: Stabilised rates (a flow-after-flow test: one stable rate after another) or isochronal rates (every rate held for the same time). The pressures are absolute. The formula contains their squares, and a difference of squares of gauge pressures is not equal to a difference of squares of absolute ones. The reservoir pressure is not above the bubble point pressure. If it is above, the formula describes only the part of the curve below $p_b$ .

Inflow performance relationship (IPR) →

Gas inflow equation with an exponent

q_g = C\,(\bar p_R^2 - p_{wf}^2)^n

$q_g$ in thousand m³/d, pressures in MPa (absolute)

Holds when: Stabilised or isochronal rates. Absolute pressures. An empirical relationship: extrapolation far beyond the tested drawdowns is unreliable.

Inflow performance relationship (IPR) →

Two-term gas inflow equation

\bar p_R^2 - p_{wf}^2 = a\,q_g + b\,q_g^2

pressures in MPa (absolute), $q_g$ in thousand m³/d

Holds when: Stabilised rates. $a$ and $b$ are found from a test at several rates: the dependence of $(\bar p_R^2 - p_{wf}^2)/q_g$ on $q_g$ is a straight line with slope $b$ and intercept $a$ on the ordinate axis.

Inflow performance relationship (IPR) →

Flow efficiency

FE = \frac{\bar p_R - p_{wf} - \Delta p_S}{\bar p_R - p_{wf}} = \frac{J_{\text{actual}}}{J_{\text{ideal}}}

dimensionless; the pressures in the same units

Holds when: Single-phase inflow, stabilised flow. The additional pressure loss is proportional to the rate (the skin factor does not depend on the rate).

Inflow performance relationship (IPR) →

Productivity index at zero drawdown

J^* = \frac{1.8\,q_{max}}{\bar p_R} = \frac{1.8\,J}{1 + 0.8\,p_{wf}/\bar p_R}

any consistent units

Holds when: The shape of the curve is of the Vogel type ( $\bar p_R \le p_b$ ).

Inflow performance relationship (IPR) →

Conversion to a future reservoir pressure

J^*_f = J^*_p\,\frac{\left(k_{ro}/\mu_o B_o\right)_f}{\left(k_{ro}/\mu_o B_o\right)_p},\qquad q_{max,f} = \frac{J^*_f\,\bar p_{R,f}}{1.8}

the ratio is dimensionless; $J^*$ and $q_{max}$ are in the units of the current curve

Holds when: The shape of the curve will also be of the Vogel type in the future. $k_{ro}$ comes from the saturation forecast (material balance, an accounting of the fluids in the reservoir) and the relative permeability curves. $\mu_o$ and $B_o$ come from PVT data at the future pressure. The properties of the near-wellbore zone do not change.

Inflow performance relationship (IPR) →

Vogel's equation with the flow efficiency of the well

\frac{q}{q_{max}^{FE=1}} = 1.8\,FE\left(1 - \frac{p_{wf}}{\bar p_R}\right) - 0.8\,FE^2\left(1 - \frac{p_{wf}}{\bar p_R}\right)^2

dimensionless form

Holds when: A Vogel-type inflow for an undamaged well. The additional pressure loss in the near-wellbore zone is proportional to the rate. At $FE > 1$ the formula applies as long as $p'_{wf} \ge 0$ , that is, at $p_{wf} \ge \bar p_R\,(1 - 1/FE)$ . Below that it gives a physically meaningless result.

Inflow performance relationship (IPR) →

Pressure gradient in upward multiphase flow

\frac{\dd p}{\dd z} = \rho_s\,g\cos\theta + \lambda\,\frac{\rho_{ns}\,v_m^{2}}{2d} + \left(\frac{\dd p}{\dd z}\right)_{\text{acc}}

SI: the gradient in Pa/m; 1 kPa/m = 0.0442 psi/ft

Holds when: Steady upward flow in a pipe of constant cross-section. The properties of the phases and the velocities are taken at the average pressure and temperature of a section. That is why the pipe is calculated in short sections. Friction is written in the simplest, homogeneous form (the mixture is treated as uniform): from the no-slip density and the velocity of the mixture. Different methods define the density and the friction factor in different ways. The acceleration component usually does not exceed a few per cent. It matters only at high gas velocities near the wellhead.

Multiphase flow in tubing →

Mixture density with and without slip

\rho_s = \rho_L H_L + \rho_G\,(1 - H_L), \qquad \rho_{ns} = \rho_L \lambda_L + \rho_G\,(1 - \lambda_L)

kg/m³; the fractions are dimensionless

Holds when: The liquid density is the average of the oil and water densities, weighted by their volumes at tubing conditions. Within a section the phases are mixed well enough that the density can be averaged by volume.

Multiphase flow in tubing →

Drift-flux model for upward flow

v_G = C_0\,v_m + v_d \;\Rightarrow\; 1 - H_L = \frac{v_{SG}}{C_0\,v_m + v_d}, \qquad v_d \approx 0.35\sqrt{\frac{g\,d\,(\rho_L - \rho_G)}{\rho_L}}

Holds when: Upward bubble or slug flow in a vertical pipe. The model does not suit annular flow, and it overestimates the liquid holdup there. $C_0$ and the coefficient 0.35 are empirical constants (found from experiments). The formula for $v_d$ describes the rise of large bubbles that fill almost the whole cross-section. Small bubbles rise more slowly: in water at about 0.2–0.25 m/s. In inclined pipes the drift velocity depends on the angle.

Multiphase flow in tubing →

Gradient including acceleration

\frac{\dd p}{\dd z} = \frac{\rho_s g\cos\theta + \lambda\,\rho_{ns} v_m^{2}/(2d)}{1 - E_k}, \qquad E_k = \frac{\rho_{ns}\,v_m\,v_{SG}}{p}

Holds when: The gas expands isothermally, and its volume is inversely proportional to the pressure. The liquid is incompressible (its volume does not change with pressure), and the mass flow rate of the mixture is constant along the length. The momentum flux is calculated for the homogeneous mixture — without slip of the phases.

Multiphase flow in tubing →

Darcy friction factor in turbulent flow (the Haaland formula)

\frac{1}{\sqrt{\lambda}} = -1.8\,\log_{10}\!\left[\left(\frac{\varepsilon/d}{3.7}\right)^{1.11} + \frac{6.9}{Re}\right]

dimensionless quantities; $\varepsilon$ and $d$ in the same units

Holds when: Turbulent flow, $Re$ greater than about 4000. The formula approximates the implicit Colebrook–White equation (one that must be solved by trial) with an error of the order of 1–2%.

Multiphase flow in tubing →

Critical pressure ratio for a gas

r_c = \left(\frac{p_2}{p_1}\right)_c = \left(\frac{2}{k+1}\right)^{\frac{k}{k-1}}

dimensionless

Holds when: An ideal gas, expansion without heat exchange or friction (isentropic); $k$ is constant. The velocity of the gas upstream of the choke is small compared with the velocity in the narrowest part. At $p_2/p_1 \le r_c$ the flow is critical and the flow rate does not depend on $p_2$ . At $p_2/p_1 > r_c$ it is subcritical.

Chokes, surface pressure losses and well temperature →

Gas flow rate through a choke

\dot m = C_d A \sqrt{\frac{2k}{k-1}\,p_1 \rho_1 \left[r^{2/k} - r^{(k+1)/k}\right]},\qquad r = \max\!\left(\frac{p_2}{p_1},\ r_c\right)

Holds when: Isentropic expansion of an ideal gas, corrected for the gas deviation factor at the conditions upstream of the choke; $k$ is constant. The velocity of the gas upstream of the choke is small (the orifice is much smaller than the pipe); there is no liquid in the flow. Below the critical ratio, put $r_c$ into the formula. The flow rate then does not depend on the pressure downstream of the choke.

Chokes, surface pressure losses and well temperature →

Liquid flow rate through a choke

q = C_d A \sqrt{\frac{2\,(p_1 - p_2)}{\rho}}

Holds when: An incompressible single-phase liquid. The pressure in the contracted section is above the pressure at which it boils or releases gas. The velocity upstream of the orifice is small compared with the velocity in it. If the orifice is close in size to the pipe, divide the flow rate by $\sqrt{1 - (d/D)^4}$ . $C_d$ is empirical (found from experiments). In classical hydraulics it is about 0.6 for an orifice in a thin wall and about 0.8 for a short cylindrical nozzle. For a particular choke the manufacturer gives it.

Chokes, surface pressure losses and well temperature →

Gilbert-type correlations (critical flow)

p_{wh} = \frac{b\,q\,\text{GLR}^{\,c}}{S^{a}}

oilfield units: psig, STB/d, scf/STB, 1/64″

Holds when: Critical flow only. The pressure downstream of the choke is no more than about 0.5–0.6 of the pressure upstream (for Gilbert, less than 1/1.7 ≈ 0.59). The constants come from measurements in particular fields and with particular beans. Outside these conditions the accuracy falls, so check the correlation against rate measurements. In Gilbert's original the constant is 435, with the GLR in thousands of cubic feet per barrel (Mscf/bbl). This is the same formula: $10.0\cdot 1000^{0.546} \approx 435$ . Different publications give the pressure as gauge or as absolute. At pressures of a few megapascals the difference is one to three per cent.

Chokes, surface pressure losses and well temperature →

Equivalent diameter of two parallel lines

d_{\text{eq}} = \left(d_1^{2.5} + d_2^{2.5}\right)^{0.4}

any identical units of length

Holds when: Single-phase turbulent flow with the same friction factor in both lines. The lines have equal length between common points. For a gas–liquid mixture this is only an estimate. The phases need not divide between the lines in the same proportion, and the real gain is sometimes smaller than the calculated one.

Chokes, surface pressure losses and well temperature →

Relaxation distance (Ramey's method)

A = \frac{w\,c_p\,(k_e + r U f)}{2\pi\,r U\,k_e}

Holds when: Steady flow in the wellbore. Heat capacity and thermal conductivities are constant. The heat resistances of the well design do not change with depth. The rocks are uniform. Far from the well the temperature equals the geothermal one. Not included: cooling of the gas as it expands, the work of the gas against gravity, and heating by friction. For an oil well the corrections are usually small, but because of the gas the flow is in fact slightly colder than calculated. For a gas well they are significant (see the advanced section).

Chokes, surface pressure losses and well temperature →

Flow temperature in a producing well

T_f(z) = T_0 + G_T z + G_T A\left[1 - e^{-(H - z)/A}\right]

any consistent units (°C and m)

Holds when: A vertical well and a linear geothermal gradient. The fluid enters the well at the reservoir temperature. The same assumptions as for the relaxation distance.

Chokes, surface pressure losses and well temperature →

Time function at large times

f(t) \approx \tfrac{1}{2}\ln\left(4 t_D\right) - \tfrac{\gamma_E}{2},\qquad t_D = \frac{\alpha\,t}{r_{wb}^{2}}

dimensionless

Holds when: Large dimensionless times, which in practice means after the first days or weeks of operation. This is the same line-source asymptotic formula (the long-time approximation) as in reservoir flow (Line-source solution and the pressure function). At small times the long-time formula underestimates $f$ and so overestimates the heat loss. The calculated flow temperature is then lower than the true one. For small times use the full time function, for example the approximation of Hasan and Kabir, which is valid for all times.

Chokes, surface pressure losses and well temperature →

Temperature downstream of the choke

T_2 \approx T_1 - \bar\mu_{JT}\,(p_1 - p_2),\qquad \mu_{JT} = \left(\frac{\partial T}{\partial p}\right)_{h}

°C and MPa

Holds when: Throttling without heat exchange with the surroundings. The kinetic energy before and after the choke is small, so the enthalpy is constant. The coefficient varies with pressure and temperature. At large pressure drops it is more accurate to calculate with an equation of state, keeping the enthalpy constant. For a gas with liquid, also allow for the heat capacity of the liquid.

Chokes, surface pressure losses and well temperature →

Hydrate formation temperature of a sweet natural gas (Towler and Mokhatab)

T_h = 13.47\ln p + 34.27\ln\gamma_{\text{g}} - 1.675\ln p\,\ln\gamma_{\text{g}} - 20.35

oilfield: $T_h$ in °F, $p$ in psia; degrees Celsius are $(T_h - 32)/1.8$ , pressure in psia is the pressure in MPa multiplied by 145.04

Holds when: A sweet natural gas (without appreciable hydrogen sulfide or carbon dioxide) with a specific gravity of about 0.55–1.0, in the presence of fresh water. The formula reproduces the chart by gas specific gravity. But gases of the same density and different composition may form hydrates under clearly different conditions (for a gas close to pure methane the estimate is too high by several degrees). Salts in the water and inhibitors lower the hydrate formation temperature. For design decisions, calculate from the composition.

Chokes, surface pressure losses and well temperature →

Speed of sound in a uniform gas–liquid mixture (Wood's formula)

\frac{1}{\rho_m c_m^{2}} = \frac{\alpha}{\rho_G c_G^{2}} + \frac{1-\alpha}{\rho_L c_L^{2}},\qquad \rho_m = \alpha\rho_G + (1-\alpha)\rho_L

Holds when: The phases are finely mixed and move together. The length of the sound wave is much greater than the size of the bubbles. The gas in the bubbles is compressed isothermally (at constant temperature, because it has time to exchange heat with the liquid), so that $\rho_G c_G^2 \approx p$ . There is no mass exchange between the phases.

Chokes, surface pressure losses and well temperature →

Thermal resistance of the well design

\frac{1}{2\pi\,rU} = \frac{1}{2\pi r_{ti} h_f} + \sum_i \frac{\ln(r_{i+1}/r_i)}{2\pi k_i}

SI; resistance per metre of length, K·m/W

Holds when: Steady radial heat flow through coaxial layers (layers one inside another, around the same axis). The annulus is treated as a layer that conducts heat. Natural convection of the liquid in it (and radiation, if it holds gas) increases heat transfer. Without them $rU$ is too low and the calculated flow temperature is too high.

Chokes, surface pressure losses and well temperature →

Completion efficiency

E_c = \frac{J}{J_{\text{OH}}} = \frac{X}{X + S_c},\qquad X = \ln\frac{r_e}{r_w} - \frac34

dimensionless; natural logarithm

Holds when: pseudo-steady-state flow (the pressure falls at the same steady rate everywhere in the drainage area) in a circular area. For another shape, replace $X$ with $\tfrac12\ln\big(4A/(\gamma C_A r_w^2)\big)$ , $\gamma = 1.781$ (Pseudo-steady-state flow and the shape factor) unlike the flow efficiency (FE), it includes only the completion skins, not damage and not the incomplete opening of the interval

Effect of completion on inflow →

Perforation skin by the component method

s_p = s_H + s_V + s_{wb}

dimensionless

Holds when: a uniform reservoir, perforated over the whole thickness; the tunnels are clean, with no crushed zone and no damage (these are added separately) the flow follows Darcy's law (the pressure drop is proportional to the rate); at high velocities in the tunnels an inertial skin is added (Inflow at high rates: non-Darcy flow)

Effect of completion on inflow →

Plane part

s_H = \ln\frac{r_w}{r_w'(\theta)},\qquad r_w'(\theta) = \begin{cases} l_p/4, & \theta = 0^\circ\\ \alpha_\theta\,(r_w + l_p), & \theta \ne 0^\circ \end{cases}

any units of length

Holds when: a plane problem: tunnels of infinite conductivity, seen in a horizontal cross-section

Effect of completion on inflow →

Vertical part

\begin{gathered} s_V = 10^{a}\,h_D^{\,b-1}\,r_D^{\,b},\qquad a = a_1\lg r_D + a_2,\qquad b = b_1 r_D + b_2\\ h_D = \frac{h}{l_p}\sqrt{\frac{k_H}{k_V}},\qquad r_D = \frac{r_p}{2h}\left(1 + \sqrt{\frac{k_V}{k_H}}\right),\qquad h = \frac{1}{n_s}\end{gathered}

dimensionless; $\lg$ is the base-10 logarithm; lengths in the same units

Holds when: an approximation of numerical solutions for $h_D \le 10$ and $r_D \ge 0.01$ ; outside these limits a numerical calculation is needed

Effect of completion on inflow →

Wellbore part

s_{wb} = c_1\,e^{c_2 r_{wD}},\qquad r_{wD} = \frac{r_w}{l_p + r_w}

dimensionless

Holds when: the same model as for $s_H$ ; the casing and the cement are impermeable

Effect of completion on inflow →

Skin of the crushed zone

s_c = \frac{h}{l_p}\left(\frac{k}{k_c} - \frac{k}{k_s}\right)\ln\frac{r_c}{r_p}

dimensionless; natural logarithm

Holds when: the inflow into the tunnel is uniform along its length and radial through the thin ring. In reality it is concentrated nearer the end of the tunnel, so the formula gives a lower limit for the skin the thickness and permeability of the zone are taken from charge tests under reservoir conditions, or estimated; in this lesson's examples they are assumed

Effect of completion on inflow →

Tunnels end inside the damaged zone

s_{dp} = \left(\frac{k}{k_s} - 1\right)\ln\frac{r_s}{r_w} + \frac{k}{k_s}\,s_p,\qquad r_s = r_w + l_s

dimensionless; natural logarithm

Holds when: $l_p < l_s$ : the whole zone where the flow converges on the tunnels lies in damaged rock this is an upper limit: when the ends of the tunnels are close to the edge of the zone, part of the flow reaches them from undamaged rock, and the skin is smaller

Effect of completion on inflow →

Tunnels pass right through the damaged zone

\begin{gathered} s_{dp} \approx s_H + s_V(\tilde l_p) + s_{wb}\!\left(\frac{\tilde r_w}{r_w + l_p}\right)\\ \tilde l_p = l_p - \left(1 - \frac{k_s}{k}\right)l_s,\qquad \tilde r_w = r_w + \left(1 - \frac{k_s}{k}\right)l_s\end{gathered}

dimensionless

Holds when: $l_p > l_s$ . The tunnel ends are in undamaged rock, so the plane part does not change at phasings of 45–180°. At 0° its equivalent radius $l_p/4$ depends on the tunnel length beyond the wall, and there $\tilde l_p$ is used an approximation. When $k_s \to 0$ , it is the component method for a casing of radius $r_w + l_s$ , with the skin counted from the true radius $r_w$ (an impermeable ring is the same as a thicker casing). When $k_s = k$ , it gives the usual $s_p$ . A check against a numerical solution of the plane problem shows a difference of no more than 0.15 $h_D$ calculated with the length $\tilde l_p$ must stay at 10 or less

Effect of completion on inflow →

Skin of the tunnels filled with gravel

S_g = \frac{8\,k\,h\,L}{k_g\,n\,d^2}

any consistent units: $k$ and $k_g$ in the same units, and $h$ , $L$ , $d$ in the same units too

Holds when: linear flow along tunnels of equal length and diameter; the whole inflow is divided equally among the $n$ tunnels the gravel is uniform; clogging by fine particles, mixing with the reservoir sand and leftovers of the completion fluid are allowed for by a reduced $k_g$

Effect of completion on inflow →

Inertial losses in the tunnels with gravel

D_g = \frac{32\,k h\,\beta\,\rho\,B\,L}{\pi\,\mu\,n^2 d^4}

Holds when: the Forchheimer equation in each tunnel (Darcy's law plus a term that grows with the square of the velocity; see Inflow at high rates: non-Darcy flow). For gas, $\rho B$ is replaced by the gas density at standard conditions, and the viscosity is taken at bottomhole conditions the $\beta$ of the gravel grows as it gets clogged. We estimate it from tests or from correlations with the permeability of the gravel

Effect of completion on inflow →

Measured skin of a perforated well

S = S_{pp} + \frac{h}{h_w}\left(s_p + s_c + S_{d,\text{eff}}\right) + S_g + Dq

dimensionless

Holds when: simple addition is an approximation: the parts interact (damage and perforation, as in the section on the damaged zone) $Dq$ is separated out by well tests at several rates (Inflow at high rates: non-Darcy flow); the other terms do not depend on the rate

Effect of completion on inflow →

Exact equivalent radius of tunnels in a cased well

r_w'(180^\circ) = \frac{R^2 - r_w^2}{2R},\qquad r_w'(0^\circ) = \frac{l_p^2}{4R},\qquad R = r_w + l_p

any units of length

Holds when: a plane problem; the casing and the cement are impermeable, and the tunnels have infinite conductivity

Effect of completion on inflow →

Elasticity of the rate with respect to a parameter

E_x = \frac{\Delta q/q}{\Delta x/x} \approx \frac{q(x + \Delta x) - q(x - \Delta x)}{2\,q}\cdot\frac{x}{\Delta x}

dimensionless

Holds when: small changes: the elasticity describes the response near the base case and does not replace a calculation of big changes for fractions and for quantities near zero (water cut, skin factor) a relative step is inconvenient; their effect is shown in absolute steps or in ranges

Well performance sensitivity analysis →

Linear estimate of the change of the operating rate

\Delta q \approx \frac{\delta_{\text{in}} + \delta_{\text{out}}}{1/J + s_{\text{out}}}

metric oilfield units (MPa, m³/d)

Holds when: small changes, for which the curves near the operating point can be replaced by straight lines a change of $J$ also changes the slope of the inflow; for big changes (a reservoir treatment, a change of tubing) the whole new point is calculated

Well performance sensitivity analysis →

Symbols

Symbol	Meaning and context
$\dot m$	mass flow rate of the gas, kg/sChokes, surface pressure losses and well temperature
$\Delta q$	change of the operating rate, m³/dWell performance sensitivity analysis
$(\dd p/\dd z)_{\text{acc}}$	acceleration component, Pa/mMultiphase flow in tubing
$a$	viscous flow resistance coefficient; includes the effect of the skin factor, MPa²/(thousand m³/d)Inflow performance relationship (IPR)
$A$	area of the bean orifice, $\pi d^2/4$ , m²Chokes, surface pressure losses and well temperature relaxation distance — the length over which the difference between the temperatures of the flow and of the rocks falls from its starting value by a factor of $e$ (about 2.7), mChokes, surface pressure losses and well temperature relaxation distance, mChokes, surface pressure losses and well temperature
$a_1,\ a_2,\ b_1,\ b_2$	phasing constants (table below), —Effect of completion on inflow
$b$	inertial resistance coefficient, MPa²/(thousand m³/d)²Inflow performance relationship (IPR)
$B$	formation volume factor of the oil, m³/m³Inflow performance relationship (IPR)
$B_o$	formation volume factor of the oil, m³/m³Inflow performance relationship (IPR)
$b,\ c,\ a$	constants of the correlation (table below); for Gilbert 10.0, 0.546, 1.89, —Chokes, surface pressure losses and well temperature
$B,\ \mu$	formation volume factor and viscosity, m³/m³, Pa·sEffect of completion on inflow
$C$	a coefficient reflecting the properties of the reservoir and the fluid; its dimension depends on $n$ , m³/(d·MPa²ⁿ)Inflow performance relationship (IPR) a coefficient depending on the properties of the reservoir and the gas, thousand m³/(d·MPa²ⁿ)Inflow performance relationship (IPR)
$C_0$	distribution coefficient: allows for the gas being concentrated near the axis of the pipe, where the velocity is higher; about 1.2 for slug flow in a vertical pipe, —Multiphase flow in tubing
$c_1,\ c_2$	phasing constants (table below), —Effect of completion on inflow
$C_d$	discharge coefficient of the choke, —Chokes, surface pressure losses and well temperature discharge coefficient: allows for the contraction of the jet and for friction, —Chokes, surface pressure losses and well temperature
$c_m,\ c_G,\ c_L$	speed of sound in the mixture, in the gas and in the liquid, m/sChokes, surface pressure losses and well temperature
$\Delta p_{\text{choke}}$	loss across the choke: $p_{wh} - p_{ch}$ , where $p_{ch}$ is the pressure after the choke, MPaNodal analysis of the production system
$\Delta p_{\text{comp}}$	loss in the near-wellbore zone and the completion (the perforations and gravel pack that connect the well to the rock): $p_{wfs} - p_{wf}$ , where $p_{wf}$ is the bottomhole flowing pressure, MPaNodal analysis of the production system
$d$	inner diameter of the tubing, mNodal analysis of the production system inner diameter of the pipe, mMultiphase flow in tubing tunnel diameter, mEffect of completion on inflow
$d_1,\ d_2$	inner diameters of the lines laid in parallel (a second pipe laid beside the first is called looping), mChokes, surface pressure losses and well temperature
$d_{\text{eq}}$	diameter of a single pipe of the same length with the same losses, mChokes, surface pressure losses and well temperature
$D_g$	non-Darcy coefficient of the tunnels with gravel; the inertial skin is $D_g q$ , s/m³ (for a rate at standard conditions)Effect of completion on inflow
$\Delta p_{\text{down}}(q)$	sum of the pressure losses in all elements from the node to the separator at rate $q$ , MPaNodal analysis of the production system
$E_c$	completion efficiency (also called the productivity ratio, PR), —Effect of completion on inflow
$E_k$	dimensionless acceleration number — the share of the pressure drop that goes to accelerating the mixture, —Multiphase flow in tubing
$E_x$	elasticity of the operating rate with respect to the parameter $x$ , —Well performance sensitivity analysis
$f$	dimensionless time function of heat conduction in the rocks, —Chokes, surface pressure losses and well temperature
$FE$	flow efficiency, —Inflow performance relationship (IPR)
$g$	acceleration due to gravity, 9.81 m/s², m/s²Nodal analysis of the production system
$\text{GLR}$	gas–liquid ratio of the produced fluid, scf/STBChokes, surface pressure losses and well temperature
$h$	net thickness of the reservoir, mInflow performance relationship (IPR) distance between shots in the vertical direction, mEffect of completion on inflow
$H$	true vertical depth of the node; in a vertical well equal to the tubing length, mNodal analysis of the production system depth of inflow, where the fluid enters the well at the reservoir temperature, mChokes, surface pressure losses and well temperature
$h_D,\ r_D$	dimensionless distance between shots and dimensionless tunnel radius, —Effect of completion on inflow
$H_L,\ \lambda_L$	liquid holdup and no-slip liquid holdup, fractionMultiphase flow in tubing
$h,\ l_p$	distance between shots and tunnel length, mEffect of completion on inflow
$J$	productivity index, m³/(d·MPa)Inflow performance relationship (IPR) productivity index of the single-phase part of the curve, m³/(d·MPa)Inflow performance relationship (IPR) average productivity index from a test, $q/(\bar p_R - p_{wf})$ , m³/(d·MPa)Inflow performance relationship (IPR) productivity index; $1/J$ is the magnitude of the slope of the inflow curve, m³/(d·MPa)Well performance sensitivity analysis
$J^*$	productivity index at zero drawdown, m³/(d·MPa)Inflow performance relationship (IPR)
$J_{\text{actual}}$	actual productivity index, m³/(d·MPa)Inflow performance relationship (IPR)
$J_{\text{ideal}}$	productivity index of the same well without damage ( $S = 0$ ), m³/(d·MPa)Inflow performance relationship (IPR)
$J,\ J_{\text{OH}}$	productivity of the well, and of an undamaged open hole of the same radius in the same drainage area (the part of the reservoir that feeds the well), m³/(d·MPa)Effect of completion on inflow
$k$	permeability of the reservoir to oil, m²Inflow performance relationship (IPR) adiabatic exponent — the ratio of the heat capacities of the gas at constant pressure and at constant volume, $c_p/c_v$ , —Chokes, surface pressure losses and well temperature
$k_c$	permeability of the crushed zone, mDEffect of completion on inflow
$k_e$	thermal conductivity of the rocks, W/(m·K)Chokes, surface pressure losses and well temperature
$k_g$	permeability of the gravel in the tunnels, mDEffect of completion on inflow
$k_H,\ k_V$	horizontal and vertical permeability, mDEffect of completion on inflow
$k_{ro}$	relative permeability to oil at the average saturation of the reservoir, —Inflow performance relationship (IPR)
$k_s$	permeability of the rock around the crushed zone: $k$ if the tunnel is in the undamaged reservoir, and the permeability of the damaged zone if the tunnel is in it, mDEffect of completion on inflow
$k,\ h$	permeability and thickness of the reservoir, mD, mEffect of completion on inflow
$k,\ h,\ L,\ n,\ d$	as in the formula for $S_g$ , m², m, m, —, mEffect of completion on inflow
$k,\ k_s$	permeability of the reservoir and of the damaged zone, mDEffect of completion on inflow
$L$	length of a tunnel with gravel: from the screen through the casing and cement to the reservoir wall, mEffect of completion on inflow
$l_p$	tunnel length in the rock beyond the wellbore wall, mEffect of completion on inflow
$\Delta p_{\text{line}}$	loss in the flowline and the gathering system: $p_{ch} - p_{sep}$ , MPaNodal analysis of the production system
$\ln$	natural logarithm, —Chokes, surface pressure losses and well temperature
$\log_{10}$	base-10 logarithm, —Multiphase flow in tubing
$M,\ Z_1,\ T_1$	molar mass of the gas, gas deviation factor and absolute temperature upstream of the choke, kg/mol, —, KChokes, surface pressure losses and well temperature
$n$	exponent: 1 for purely viscous flow, down to 0.5 with strong inertial losses, —Inflow performance relationship (IPR) exponent: 1 for purely viscous flow, down to 0.5 when inertial losses dominate, —Inflow performance relationship (IPR) number of open tunnels in the interval, —Effect of completion on inflow
$n_s$	shot density, shots/mEffect of completion on inflow
$p$	pressure, PaMultiphase flow in tubing absolute pressure at the cross-section, PaMultiphase flow in tubing absolute pressure of the gas, psiaChokes, surface pressure losses and well temperature
$\tilde l_p$	equivalent tunnel length: the part that lies in the damaged zone counts with a weight of $k_s/k$ , mEffect of completion on inflow
$p_1 - p_2$	pressure drop across the choke (the difference can also be taken in gauge pressures), PaChokes, surface pressure losses and well temperature
$p_1,\ p_2$	absolute pressure upstream and downstream of the choke, MPaChokes, surface pressure losses and well temperature pressure upstream and downstream of the choke, MPaChokes, surface pressure losses and well temperature
$p_1,\ \rho_1$	absolute pressure and density of the gas upstream of the choke; $\rho_1 = p_1 M/(Z_1 \mathcal{R} T_1)$ , Pa, kg/m³Chokes, surface pressure losses and well temperature
$p_b$	bubble point pressure, MPaInflow performance relationship (IPR)
$p_{\text{in}}(q)$	pressure at the node from the inflow curve, MPaNodal analysis of the production system
$p_{node}$	pressure at the node, MPaNodal analysis of the production system
$p_{\text{out}}(q)$	pressure at the node from the outflow curve, MPaNodal analysis of the production system
$p_{sep}$	separator pressure, MPaNodal analysis of the production system
$p_{wf}$	bottomhole flowing pressure needed to lift the rate $q$ , PaNodal analysis of the production system bottomhole pressure while the well is flowing, MPaInflow performance relationship (IPR) bottomhole pressure, MPaInflow performance relationship (IPR)
$p_{wf,1}$	bottomhole pressure at this rate, $p_{wf,1} < p_b$ , MPaInflow performance relationship (IPR)
$p_{wh}$	wellhead pressure, PaNodal analysis of the production system pressure upstream of the choke, gauge, psigChokes, surface pressure losses and well temperature
$p,\ f$	subscripts for the present and the future state, —Inflow performance relationship (IPR)
$q$	flow rate, usually at standard conditions, m³/dNodal analysis of the production system volumetric flow rate of the liquid at tubing conditions, m³/sNodal analysis of the production system flow rate, m³/dNodal analysis of the production system oil (liquid) rate at standard conditions, m³/dInflow performance relationship (IPR) oil rate at standard conditions, m³/sInflow performance relationship (IPR) oil rate at standard conditions at the bottomhole pressure $p_{wf}$ , m³/dInflow performance relationship (IPR) volumetric flow rate of the liquid, m³/sChokes, surface pressure losses and well temperature liquid rate (oil and water) at standard conditions, STB/dChokes, surface pressure losses and well temperature operating rate of the base case, m³/dWell performance sensitivity analysis
$q_1$	rate at the measured point, m³/dInflow performance relationship (IPR)
$q_b$	rate at $p_{wf} = p_b$ , m³/dInflow performance relationship (IPR)
$q_g$	gas rate at standard conditions, thousand m³/dInflow performance relationship (IPR)
$q_{max}$	absolute open flow — the rate at $p_{wf} = 0$ , m³/dInflow performance relationship (IPR)
$q_{max},\ \bar p_R,\ p_{wf}$	absolute open flow, and the reservoir and bottomhole pressures of the test, m³/d, MPaInflow performance relationship (IPR)
$q_{max}^{FE=1}$	absolute open flow of the same well without damage, m³/dInflow performance relationship (IPR)
$q,\ \bar p_R,\ p_{wf}$	rate, reservoir pressure and bottomhole pressure, m³/d, MPaInflow performance relationship (IPR)
$r$	ratio of the pressures downstream and upstream of the choke, but not less than the critical $r_c$ , —Chokes, surface pressure losses and well temperature
$\bar p_R$	average reservoir pressure in the drainage area, converted to the depth of the middle of the perforated interval (the part of the casing with holes), MPaNodal analysis of the production system average reservoir pressure, MPaNodal analysis of the production system
$\mathcal{R}$	universal gas constant, 8.314 J/(mol·K), J/(mol·K)Chokes, surface pressure losses and well temperature
$R$	radius of the tunnel ends, mEffect of completion on inflow
$r_c,\ r_p$	radius of the crushed zone ( $r_p$ plus its thickness) and radius of the tunnel, mEffect of completion on inflow
$r_e$	radius of the circular drainage area, mInflow performance relationship (IPR)
$r_i,\ r_{i+1},\ k_i$	inner and outer radii of the $i$ -th annular layer (tubing wall, annulus, casing, cement) and its thermal conductivity, m, W/(m·K)Chokes, surface pressure losses and well temperature
$r_p$	radius of the perforation tunnel, mEffect of completion on inflow
$r_s,\ l_s$	radius and thickness of the damaged zone (the thickness is measured from the wellbore wall), mEffect of completion on inflow
$r_{ti},\ h_f$	inner radius of the tubing and the film heat transfer coefficient from the flow to its wall, m, W/(m²·K)Chokes, surface pressure losses and well temperature
$r_w$	wellbore radius, mInflow performance relationship (IPR) wellbore radius (the radius of the bit), mEffect of completion on inflow
$r_w,\ l_p$	wellbore radius and tunnel length, mEffect of completion on inflow
$r_w'$	equivalent radius in the plane problem, including the flow around the casing, mEffect of completion on inflow
$r_w'(\theta)$	equivalent radius of the “star” of tunnels in the plane problem, mEffect of completion on inflow
$r_{wb}$	radius of the wellbore (from the bit size), mChokes, surface pressure losses and well temperature
$r_{wD}$	the share of the wellbore radius in the radius of the “star”, —Effect of completion on inflow
$\bar p_R,\ p_{wf}$	average reservoir pressure and bottomhole pressure, MPaInflow performance relationship (IPR) average reservoir pressure and bottomhole pressure, absolute, MPaInflow performance relationship (IPR) reservoir and bottomhole pressures, absolute, MPaInflow performance relationship (IPR)
$\bar p_{R,f}$	future reservoir pressure, MPaInflow performance relationship (IPR)
$Re$	Reynolds number, —Multiphase flow in tubing
$\Delta p_{\text{res}}$	pressure loss in the reservoir: $\bar p_R - p_{wfs}$ , where $p_{wfs}$ is the flowing pressure at the sandface (the wall of the well), MPaNodal analysis of the production system
$r U$	product of the radius and the overall heat transfer coefficient from the flow to the wall of the well (tubing, annulus, casing, cement); it does not depend on the choice of radius, W/(m·K)Chokes, surface pressure losses and well temperature product of the radius and the heat transfer coefficient from Ramey's formula, W/(m·K)Chokes, surface pressure losses and well temperature
$S$	skin factor — the additional resistance of the near-wellbore zone (see Skin factor), —Inflow performance relationship (IPR) bean diameter in sixty-fourths of an inch, 1/64″Chokes, surface pressure losses and well temperature skin from the well test, for the whole thickness, —Effect of completion on inflow
$\Delta p_S$	additional pressure loss in the near-wellbore zone at the given rate (from well tests), MPaInflow performance relationship (IPR)
$s_c$	skin of the crushed zone, —Effect of completion on inflow
$S_c$	completion skin: the sum of the parts in this lesson (with $Dq$ if there are inertial losses), —Effect of completion on inflow
$S_{d,\text{eff}}$	the remaining contribution of damage, after allowing for how the tunnels bypass it, —Effect of completion on inflow
$s_{dp}$	skin of a perforated completion in a damaged reservoir (without the crushed zone), —Effect of completion on inflow
$S_g$	skin from the viscous losses in the tunnels with gravel, —Effect of completion on inflow
$S_g,\ Dq$	skin of the gravel pack and the inertial skin, —Effect of completion on inflow
$s_H$	plane part: the flow converging on the tunnels in the horizontal plane, —Effect of completion on inflow
$s_H,\ s_V,\ s_{wb}$	parts of the method: $s_H$ is the same as without damage (at 0° phasing, with $\tilde l_p$ instead of $l_p$ ), $s_V$ is calculated with $\tilde l_p$ , and $s_{wb}$ with $r_{wD} = \tilde r_w/(r_w + l_p)$ , —Effect of completion on inflow
$s_{\text{out}}$	slope of the outflow curve at the operating point, MPa/(m³/d)Well performance sensitivity analysis
$s_p$	perforation skin relative to an open hole of the same radius, —Effect of completion on inflow perforation skin of the same tunnels in an undamaged reservoir, —Effect of completion on inflow
$s_p,\ s_c$	skin of the perforation and of the crushed zone in the opened interval, —Effect of completion on inflow
$S_{pp}$	pseudo-skin of partial penetration (the well opens only part of the thickness), —Effect of completion on inflow
$s_V$	vertical part: the flow converging on the levels of the shots, —Effect of completion on inflow
$s_{wb}$	wellbore part: the flow going around the impermeable casing, —Effect of completion on inflow
$t$	time the well has been producing, sChokes, surface pressure losses and well temperature
$T_0 + G_T z$	rock temperature: $T_0$ is the temperature of the neutral layer near the surface (the layer where the temperature no longer follows the seasons), $G_T$ is the geothermal gradient, °C, °C/mChokes, surface pressure losses and well temperature
$T_1,\ T_2$	temperature upstream and downstream of the choke, °CChokes, surface pressure losses and well temperature
$t_D$	dimensionless time, —Chokes, surface pressure losses and well temperature
$T_f(z)$	temperature of the flow at depth $z$ , °CChokes, surface pressure losses and well temperature
$T_h$	temperature below which hydrates form at pressure $p$ , °FChokes, surface pressure losses and well temperature
$\Delta p_{\text{tbg}}$	loss in lifting through the tubing, including the weight of the fluid column and the pressure lost in the downhole equipment: $p_{wf} - p_{wh}$ , where $p_{wh}$ is the wellhead pressure, MPaNodal analysis of the production system
$\Delta p_{\text{up}}(q)$	sum of the pressure losses in all elements from the reservoir to the node at rate $q$ , MPaNodal analysis of the production system
$v$	mean velocity of the liquid in the tubing, m/sNodal analysis of the production system
$v_d$	drift velocity — the rise of gas relative to the mixture, m/sMultiphase flow in tubing
$v_G$	actual velocity of the gas, m/sMultiphase flow in tubing
$v_m$	velocity of the mixture — the sum of the superficial velocities of the phases, m/sMultiphase flow in tubing
$v_{SG}$	superficial velocity of the gas, m/sMultiphase flow in tubing
$\tilde r_w$	equivalent wall radius: the casing is as if “extended” into the damaged zone (do not confuse it with the equivalent radius $r_w'$ ), mEffect of completion on inflow
$w\,c_p$	mass flow rate of the flow multiplied by its specific heat capacity (the heat needed to warm 1 kg by 1 K), summed over oil, water and gas, W/KChokes, surface pressure losses and well temperature
$X$	the logarithmic term of an ideal well in pseudo-steady-state flow, —Effect of completion on inflow
$x,\ \Delta x$	base value of the parameter and its small change (±5% in the lesson), units of the parameterWell performance sensitivity analysis
$z$	depth along the hole (along the tubing), measured from the wellhead downwards, mMultiphase flow in tubing
$\alpha$	thermal diffusivity of the rocks, m²/sChokes, surface pressure losses and well temperature volume fraction of gas in the mixture, —Chokes, surface pressure losses and well temperature
$\alpha_\theta$	phasing constant (table below), —Effect of completion on inflow
$\beta$	inertial resistance coefficient of the gravel, m⁻¹Effect of completion on inflow
$\gamma_E$	Euler's constant, 0.5772, —Chokes, surface pressure losses and well temperature
$\gamma_{\text{g}}$	specific gravity of the gas relative to air, —Chokes, surface pressure losses and well temperature
$\delta_{\text{in}}$	rise of the inflow curve at the same rate (for example, a growth of the reservoir pressure), MPaWell performance sensitivity analysis
$\delta_{\text{out}}$	fall of the outflow curve at the same rate (for example, from a lower wellhead pressure), MPaWell performance sensitivity analysis
$\varepsilon$	absolute roughness of the wall, mMultiphase flow in tubing
$\theta$	inclination of the hole — the deviation from the vertical; $\cos\theta\,\dd z$ is the increment of true vertical depth, —Multiphase flow in tubing
$\lambda$	Darcy–Weisbach friction factor; in turbulent flow in steel pipes usually 0.015–0.03, —Nodal analysis of the production system Darcy–Weisbach friction factor, —Multiphase flow in tubing Darcy friction factor, —Multiphase flow in tubing
$\mu$	viscosity of the oil at reservoir conditions, Pa·sInflow performance relationship (IPR)
$\mu_{JT}$	Joule–Thomson coefficient: the change of temperature with pressure at constant enthalpy $h$ , °C/MPaChokes, surface pressure losses and well temperature
$\bar\mu_{JT}$	its mean value along the expansion path, °C/MPaChokes, surface pressure losses and well temperature
$\mu_o$	viscosity of the oil at reservoir pressure, mPa·sInflow performance relationship (IPR)
$\rho$	density of the liquid in the tubing, kg/m³Nodal analysis of the production system density of the liquid, kg/m³Chokes, surface pressure losses and well temperature fluid density at reservoir conditions, kg/m³Effect of completion on inflow
$\rho_L,\ \rho_G$	densities of the liquid and the gas at tubing conditions, kg/m³Multiphase flow in tubing
$\rho_m,\ \rho_G,\ \rho_L$	densities of the mixture, the gas and the liquid, kg/m³Chokes, surface pressure losses and well temperature
$\rho_{ns}$	density of the mixture without slip — from the no-slip phase fractions, kg/m³Multiphase flow in tubing density of the mixture without slip, kg/m³Multiphase flow in tubing
$\rho_s$	density of the mixture in the pipe, allowing for slip of the phases, kg/m³Multiphase flow in tubing density of the mixture in the pipe, allowing for slip (the actual density), kg/m³Multiphase flow in tubing