Equations: Oil displacement

The equations of the “Oil displacement” lessons in “Reservoir engineering”, with their conditions and units, and what each symbol means in them.

Recovery factor in waterflooding

E_R = E_D\,E_A\,E_V

dimensionless fractions

Holds when: The breakdown is approximate: the factors depend on each other and on the volume of water pumped through, and all three grow with the time of injection. $E_R$ and $E_D$ refer to the oil at standard conditions. If the formation volume factor of the oil changes noticeably during the flood, the saturations have to be converted through $S_o/B_o$ . The product $E_AE_V$ is called the volumetric sweep efficiency. In Russian-language practice the recovery factor is written as the product of the displacement efficiency and the sweep efficiency.

Why water is injected: pressure maintenance and oil displacement →

Current voidage replacement ratio

\mathrm{VRR} = \frac{q_{wi}B_w}{q_oB_o + q_o\,(R - R_s)\,B_g + q_wB_w}

any identical units of rate; the gas–oil ratio and the solution gas–oil ratio are in m³ of gas per m³ of oil at standard conditions

Holds when: $q_o(R - R_s)$ is the free gas produced in addition to the gas dissolved in the oil. Above the bubble point pressure $R = R_s$ , and this term disappears. Injected gas, if there is any, is added to the numerator as $q_{gi}B_g$ . Water influx from the aquifer is not part of VRR: it is a natural replacement, and we take it into account separately in the balance. The cumulative replacement is the ratio of the sums of injection and withdrawal over the whole period, each converted to reservoir volumes on its own date.

Why water is injected: pressure maintenance and oil displacement →

Pressure change under incomplete replacement (a closed accumulation)

\Delta\bar p = -\frac{(1 - \mathrm{VRR})\,Q\,\Delta t}{c_t V_p}

MPa, m³/d, d, 1/MPa, m³

Holds when: The accumulation is closed (there is no aquifer), and the pressure is above the bubble point pressure, so there is no free gas. The pressure equalises across the accumulation faster than it changes (the tank model: the whole accumulation is treated as one tank with a single pressure), and the compressibility is constant over the pressure interval. With free gas the compressibility of the system is many times larger, and for the same deficit the pressure responds less, but the gas escapes into the wells.

Why water is injected: pressure maintenance and oil displacement →

Oil rate at constant injection and full replacement

q_o = \frac{Q_i}{B_o + B_w\,\dfrac{\text{WC}}{1 - \text{WC}}}

m³/d; the formation volume factors are in m³/m³

Holds when: VRR = 1, and there is no aquifer. The pressure is above the bubble point pressure, so there is no free gas. Injection is limited by the capacity of the pumps and of water treatment.

Why water is injected: pressure maintenance and oil displacement →

Material balance with injection

F = N\,(E_o + mE_g + E_{fw}) + W_e + W_{inj}B_w + G_{inj}B_g

m³ (oil and gas at standard conditions, the rest at reservoir conditions)

Holds when: The assumptions of the tank material balance. The injected volumes are converted to reservoir volumes at the current average pressure.

Why water is injected: pressure maintenance and oil displacement →

Fractional flow of water with gravity and capillary pressure

f_w = \frac{1 + \dfrac{k\,k_{ro}}{\mu_o\,u_t}\left(\dfrac{\partial p_c}{\partial x} - \Delta\rho\,g\sin\alpha\right)}{1 + \dfrac{k_{ro}}{k_{rw}}\cdot\dfrac{\mu_w}{\mu_o}}

SI (any consistent system of units)

Holds when: One-dimensional linear flow, with the same saturation across the section (diffuse flow). The rock and the fluids are incompressible, so $u_t$ does not change along the flow. There is no free gas. The phases do not mix. The relative permeabilities and the capillary pressure depend only on the saturation and on the direction of its change (for the waterflooding of a water-wet formation, the imbibition curves, which describe water soaking into the rock). Darcy's law holds for each phase: the flow is laminar (smooth, not turbulent).

Fractional flow function and mobility ratio →

Fractional flow of water through the phase mobilities

f_w = \frac{\lambda_w}{\lambda_t} + \frac{\lambda_o\lambda_w}{\lambda_t}\cdot\frac{1}{u_t}\left(\frac{\partial p_c}{\partial x} - \Delta\rho\,g\sin\alpha\right)

Holds when: The same as for the general form.

Fractional flow function and mobility ratio →

Gravity number

N_g = \frac{k\,k_{ro}'\,\Delta\rho\,g\sin\alpha}{\mu_o\,u_t}

SI (a dimensionless number)

Holds when: Diffuse one-dimensional flow along the dip: the saturation is the same over the thickness. Sign: $N_g > 0$ when the water moves up the dip.

Fractional flow function and mobility ratio →

Fractional flow of water in an inclined formation without the capillary term

f_w = \frac{\lambda_w}{\lambda_t}\left(1 - N_g\,\frac{k_{ro}}{k_{ro}'}\right)

dimensionless

Holds when: We neglect the capillary term. $N_g$ is taken from the formula above, with the same sign of the dip.

Fractional flow function and mobility ratio →

Capillary length: the scale of the smearing of the front

\ell_c = \frac{k\,k_{ro}\,\Delta p_c}{\mu_o\,u_t}

Holds when: This is a rough, order-of-magnitude estimate: if the front zone is longer than $\ell_c$ , the capillary term is small compared with the viscous part. With the end-point relative permeability $k_{ro}'$ instead of the average over the front, the estimate comes out too high.

Fractional flow function and mobility ratio →

Water balance in a slice of a linear formation

\phi A\,\frac{\partial S_w}{\partial t} = -\,q_t\,\frac{\partial f_w}{\partial x}

SI (any consistent system)

Holds when: Incompressible fluids and rock, a constant total flow rate. One-dimensional flow, with the same saturation across the section.

Buckley–Leverett theory and the Welge method →

Saturation advance equation (Buckley–Leverett)

v_{S_w} = \left(\frac{\dd x}{\dd t}\right)_{S_w} = \frac{q_t}{\phi A}\,f_w'(S_w)

SI (any consistent system)

Holds when: The assumptions of the theory (the section above). The curve $f_w(S_w)$ does not change in time.

Buckley–Leverett theory and the Welge method →

Slope of the fractional flow curve for the Corey model

f_w' = \frac{f_w\,(1 - f_w)}{1 - S_{wi} - S_{or}}\left(\frac{n_w}{S_n} + \frac{n_o}{1 - S_n}\right)

dimensionless

Holds when: The Corey model, a horizontal formation without a capillary term (the same slope is used in the lesson IMPES and fully implicit schemes). At the ends of the range of mobile saturation, where only one phase flows, the slope is zero.

Buckley–Leverett theory and the Welge method →

Front speed from the water balance at the jump

v_f = \frac{q_t}{\phi A}\cdot\frac{f_{wf} - f_w(S_{wi})}{S_{wf} - S_{wi}} = \frac{q_t}{\phi A}\cdot\frac{f_{wf}}{S_{wf} - S_{wi}}

Holds when: The water that reaches the front during the time $\dd t$ all goes to raising the saturation from $S_{wi}$ to $S_{wf}$ in the slice that the front passes in that time.

Buckley–Leverett theory and the Welge method →

Saturation at the front: the tangency condition

f_w'(S_{wf}) = \frac{f_{wf}}{S_{wf} - S_{wi}}

dimensionless

Holds when: The initial water does not move. If the initial saturation $S_{w0}$ is above the irreducible one and the water is mobile, we draw the tangent from the point $(S_{w0}, f_w(S_{w0}))$ . The $f_w$ curve is S-shaped. If over the whole range of mobile saturation it lies below the straight line joining the points $(S_{wi}, 0)$ and $(1 - S_{or}, 1)$ , the jump goes up to $1 - S_{or}$ and the displacement is piston-like.

Buckley–Leverett theory and the Welge method →

Average saturation behind the front before breakthrough (Welge)

\bar S_w = S_{wi} + \frac{1}{f_w'(S_{wf})} = S_{wf} + \frac{1 - f_{wf}}{f_w'(S_{wf})}

dimensionless

Holds when: Before breakthrough; incompressible phases; the initial water does not move.

Buckley–Leverett theory and the Welge method →

Injection, oil production and time to water breakthrough

W_{iD,BT} = N_{pD,BT} = \bar S_w - S_{wi} = \frac{1}{f_w'(S_{wf})},\qquad t_{BT} = \frac{W_{iD,BT}\,V_p}{q_i}

volumes in m³ at reservoir conditions, time in days

Holds when: The assumptions of the theory. Injection starts into an element with no free gas and with immobile water. Oil is converted to standard cubic metres by dividing by $B_o$ , and the water injection is converted by dividing by $B_w$ .

Buckley–Leverett theory and the Welge method →

Average saturation and oil production after breakthrough

\bar S_w = S_{we} + (1 - f_{we})\,W_{iD},\qquad N_{pD} = \bar S_w - S_{wi},\qquad W_{iD} = \frac{1}{f_w'(S_{we})}

dimensionless

Holds when: After breakthrough; the assumptions of the theory. Graphically: the tangent to the $f_w$ curve at the point $S_{we}$ crosses the line $f_w = 1$ at $\bar S_w$ .

Buckley–Leverett theory and the Welge method →

Ratio of the time to travel along the formation to the time to separate across the thickness

\frac{t_x}{t_v} = \frac{L}{h}\cdot\frac{k_v\,k_{ro}'\,|\Delta\rho|\,g}{\mu_o\,u_t} = \frac{L}{h}\cdot\frac{k_v}{k}\cdot\frac{N_g}{\sin\alpha}

SI (any consistent system); the ratio is dimensionless

Holds when: This is an order-of-magnitude estimate, not an exact criterion. The speed of separation is taken with the end-point relative permeability of oil, which is an upper estimate. Where the phases move towards each other, their combined mobility is smaller: for the pair water–oil with the curves of the examples of this lesson, by about an order of magnitude or more. This is why a ratio of the order of ten is required for vertical equilibrium. Gas is much more mobile than oil, and for the pair gas–oil the estimate is closer to reality. The formation is uniform vertically. Shale layers cut $k_v$ sharply and make the separation slower.

Segregated flow and gravity displacement →

Slope of a sharp interface in segregated displacement (Dietz analysis)

1 - \frac{1}{M'} = N_g\left(1 + \frac{1}{\tan\alpha}\cdot\frac{\dd z}{\dd x}\right),\qquad M' = \frac{k_{rd}'/\mu_d}{k_{ro}'/\mu_o},\qquad N_g = \frac{k\,k_{ro}'\,(\rho_d - \rho_o)\,g\sin\alpha}{\mu_o\,u_t}

dimensionless

Holds when: Vertical equilibrium and a sharp interface, with end-point saturations on both sides of it. The interface moves without changing shape (a steady tilt). A uniform formation of constant thickness, incompressible phases, a constant flow rate.

Segregated flow and gravity displacement →

Condition of gravity stability of the front

M'N_g > M' - 1

dimensionless

Holds when: The same as for the slope of the interface.

Segregated flow and gravity displacement →

Critical rate of gravity-stable displacement

q_{\text{cr}} = q_{\text{gr}}\,\frac{M'}{M' - 1} = \frac{k\,k_{rd}'\,A\,|\Delta\rho|\,g\,|\sin\alpha|}{\mu_d\,(M' - 1)},\qquad q_{\text{gr}} = \frac{k\,k_{ro}'\,A\,|\Delta\rho|\,g\,|\sin\alpha|}{\mu_o}

SI (any consistent system)

Holds when: $M' > 1$ and gravity holds the displacing phase back. For $M' \le 1$ such displacement is stable at any rate. Vertical equilibrium, a sharp interface, a uniform formation. The rate is at reservoir conditions, through the whole cross-section $A$ : for an element of a line-drive pattern it is the injection of one row over its width.

Segregated flow and gravity displacement →

Steady tilt of the interface to the bedding

\tan\beta = \tan\alpha\left(1 - \frac{M' - 1}{M'N_g}\right) = \tan\alpha\left(1 - \frac{q_t}{q_{\text{cr}}}\right)

dimensionless

Holds when: The interface has had time to reach the steady tilt. The second form is for $M' > 1$ ; for $M' \le 1$ use the first. The assumptions of the stability analysis: vertical equilibrium, a sharp interface, a uniform formation.

Segregated flow and gravity displacement →

Thickness-averaged relative permeabilities under vertical equilibrium

h_D = \frac{\bar S_w - S_{wi}}{1 - S_{wi} - S_{or}},\qquad \bar k_{rw} = k_{rw}'\,h_D,\qquad \bar k_{ro} = k_{ro}'\,(1 - h_D)

fractions

Holds when: Vertical equilibrium, a sharp interface, a formation uniform over the thickness. Behind the interface the oil is already at its residual saturation.

Segregated flow and gravity displacement →

Fractional flow of water in segregated displacement

f_w = \frac{M'h_D\,\bigl(1 - N_g\,(1 - h_D)\bigr)}{1 + (M' - 1)\,h_D}

dimensionless

Holds when: The averaged relative permeabilities above. The tilt of the interface itself does not enter the formula (it is taken into account in the advanced section).

Segregated flow and gravity displacement →

Breakthrough and full sweep in unstable segregated displacement

W_{iD,BT} = \frac{\Delta S}{M'(1 - N_g)},\qquad W_{iD}\big|_{h_D = 1} = \frac{M'\,\Delta S}{1 + M'N_g},\qquad \Delta S = 1 - S_{wi} - S_{or}

pore volumes (reservoir conditions), as in the lesson Buckley–Leverett theory and the Welge method

Holds when: $M'N_g < M' - 1$ (an unstable interface, a tongue). With a stable interface, breakthrough and full sweep coincide: $W_{iD} = \Delta S$ . Injection at a constant rate, a linear element, the assumptions of the averaged relative permeabilities.

Segregated flow and gravity displacement →

Limiting rate of gravity drainage

q_{\text{gd}} = \frac{k\,k_{ro}\,A\,(\rho_o - \rho_g)\,g\sin\alpha}{\mu_o}

SI (any consistent system)

Holds when: The pressure in the gas is hydrostatic: the gas only takes the place of the draining oil. Vertical equilibrium, a uniform formation.

Segregated flow and gravity displacement →

Vertical sweep for piston-like displacement in the layers

E_V = \frac{\sum_i \phi_i h_i\,\xi_i}{\sum_i \phi_i h_i},\qquad \xi_i = \frac{x_{f,i}}{L}

dimensionless

Holds when: The displacement in each layer is piston-like; the layers are uniform along their length. The end-point saturations $S_{wi}$ and $S_{or}$ are the same in all layers. If they are not, the weight of a layer is $\phi_ih_i(1 - S_{wi,i} - S_{or,i})$ . A linear element between the rows of wells: the areal sweep is not taken into account ( $E_A = 1$ ).

Displacement in heterogeneous reservoirs →

Front speed in a layer without crossflow

\phi_i\,\Delta S\,\frac{\dd x_{f,i}}{\dd t} = \frac{k_i\,\Delta p}{x_{f,i}/\lambda_w + (L - x_{f,i})/\lambda_o},\qquad \Delta S = 1 - S_{wi} - S_{or}

SI (any consistent system); in metric oilfield units the right side is multiplied by 0.0853, the constant of Darcy's law

Holds when: Piston-like displacement in the layer; the rock and the fluids are incompressible. There is no crossflow between the layers; the pressure difference $\Delta p(t)$ is common to all the layers. Capillary pressure and gravity are not taken into account.

Displacement in heterogeneous reservoirs →

Position of the front in a layer when a faster layer breaks through (Dykstra–Parsons)

\xi_i = \frac{M' - \sqrt{M'^2 + a_i\,(1 - M'^2)}}{M' - 1},\qquad a_i = \frac{v_i}{v_j} = \frac{k_i/\phi_i}{k_j/\phi_j} \le 1

dimensionless

Holds when: The assumptions of the front speed equation; the end points are the same in all layers. At $M' = 1$ the formula turns into $\xi_i = a_i$ .

Displacement in heterogeneous reservoirs →

Stiles method: the sweep and the fractional flow of water when layer j breaks through

E_{V,j} = C_j + (1 - F_j)\,\frac{\bar v}{v_j},\qquad f_{we,j} = \frac{M'F_j}{M'F_j + 1 - F_j},\qquad \bar v = \frac{\sum_i k_ih_i}{\sum_i \phi_ih_i}

dimensionless

Holds when: Piston-like displacement in the layers, no crossflow, the same end points. The fronts move at constant speeds. This is exact at $M' = 1$ ; for other $M'$ it is an approximation, and the method is not internally consistent: $M'$ is taken into account in the share of water but not in the movement of the fronts. Gravity and capillary pressure are not taken into account.

Displacement in heterogeneous reservoirs →

Fractional flow of water in the production of a layered section without crossflow

f_{we} = \frac{\sum_{\text{flushed}} M'k_ih_i}{\sum_{\text{flushed}} M'k_ih_i + \sum_{\text{unflushed}} k_ih_i\big/\left[1 - \xi_i\,(1 - 1/M')\right]}

dimensionless; the fractional flow of water at reservoir conditions

Holds when: The assumptions of the front speed equation: piston-like displacement, no crossflow, a common pressure difference. The common factor $\lambda_o\Delta p/L$ has been cancelled.

Displacement in heterogeneous reservoirs →

Pseudo functions of the section under vertical equilibrium

\hat S_w = S_{wi} + (1 - S_{wi} - S_{or})\,C_n,\qquad \hat k_{rw} = k_{rw}'F_n,\qquad \hat k_{ro} = k_{ro}'\,(1 - F_n)

dimensionless

Holds when: Vertical equilibrium ( $R_L \gg 1$ ); gravity and capillary pressure are negligible. Piston-like displacement in the layers; the end points are the same (in the general case the layers enter the sums with their own $S_{wi}$ , $S_{or}$ , $k_{rw}'$ , $k_{ro}'$ ). The order of flushing is by decreasing $k/\phi$ ; under strong segregation it is from the bottom up (the next section).

Displacement in heterogeneous reservoirs →

Ratio of gravity to viscous forces in the section

N_{gv} = \frac{k_v\,k_{ro}'\,\Delta\rho\,g\,L}{\mu_o\,u_t\,H}

Holds when: This is an order-of-magnitude estimate, not an exact criterion. Vertical movement is possible only if the layers communicate ( $R_L$ is not small). $k_v k_{ro}'\Delta\rho g/\mu_o$ is the velocity at which oil would rise under gravity alone.

Displacement in heterogeneous reservoirs →

A point of the fractional flow curve of the formation from the production history

S_{we} = S_{wi} + N_{pD} - (1 - f_{we})\,W_{iD},\qquad N_{pD} = \frac{N_pB_o}{V_p},\qquad W_{iD} = \frac{W_{inj}B_w}{V_p}

dimensionless; the volumes are in the same units

Holds when: The pressure is maintained and there is no free gas: injection at reservoir conditions replaces production. Injection and production refer to the same area, and no water flows out of it. $V_p$ is known with an error, which shifts all the points in the same way.

Displacement in heterogeneous reservoirs →

Fractional flow of gas

f_g = \frac{q_g}{q_g + q_o} = \frac{\lambda_g}{\lambda_t}\left(1 - N_g\,\frac{k_{ro}}{k_{ro}'}\right),\qquad N_g = \frac{k\,k_{ro}'\,(\rho_g - \rho_o)\,g\sin\alpha}{\mu_o\,u_t}

SI (the metric oilfield forms are the same as for the gravity number in the lesson Fractional flow function and mobility ratio)

Holds when: The flow is one-dimensional, and the saturation is the same over the cross-section (this is called diffuse, or spread-out, flow). The connate water does not move. The phases are immiscible, the pressure is constant, and the capillary term is neglected.

Displacement of oil by gas →

Producing gas–oil ratio from the fractional flow of gas

R = R_s + \frac{B_o}{B_g}\cdot\frac{f_g}{1 - f_g}

m³ of gas per m³ of oil at standard conditions (0.101325 MPa and 20 °C)

Holds when: The oil is saturated with gas at reservoir pressure. The free gas and the dissolved gas are converted to the same standard conditions.

Displacement of oil by gas →

Gas breakthrough and oil production after it (Welge method)

f_g'(S_{gf}) = \frac{f_{gf}}{S_{gf}},\qquad G_{iD,BT} = N_{pD,BT} = \frac{1}{f_g'(S_{gf})},\qquad N_{pD} = S_{ge} + (1 - f_{ge})\,G_{iD},\quad G_{iD} = \frac{1}{f_g'(S_{ge})}

fractions and pore volumes

Holds when: The assumptions of the Buckley–Leverett theory. There is no initial free gas (the tangent starts at the origin). The pressure is constant, and the injection rate at reservoir conditions is constant.

Displacement of oil by gas →

Gas injection that fully replaces the withdrawal

q_{g,\text{inj}} = q_o\,\frac{B_o}{B_g} + q_o\,(R - R_s) + q_w\,\frac{B_w}{B_g}

m³/d at standard conditions

Holds when: The pressure is constant and not above the bubble point pressure, so the injected gas stays free (does not dissolve in the oil). The injected gas has the same composition as the reservoir gas (the same $B_g$ ).

Displacement of oil by gas →

Symbols

Symbol	Meaning and context
$\Delta t$	time, dWhy water is injected: pressure maintenance and oil displacement
$\Delta S$	range of mobile saturation, $1 - S_{wi} - S_{or}$ , —Segregated flow and gravity displacement range of mobile saturation, —Segregated flow and gravity displacement range of mobile saturation: the amount by which the water saturation grows when the front passes, —Displacement in heterogeneous reservoirs
$\Delta p$	pressure difference between the rows of injection and production wells, common to all the layers, PaDisplacement in heterogeneous reservoirs
$\bar v$	average speed of the section: flow capacity divided by storage capacity, mDDisplacement in heterogeneous reservoirs
$A$	area of the cross-section of the formation across the flow: thickness × width of the front, m²Segregated flow and gravity displacement
$B_o,\ B_g$	formation volume factors of the oil and of the gas, m³/m³Displacement of oil by gas
$B_o,\ B_g,\ B_w$	formation volume factors of oil, gas and water at the current average reservoir pressure (surface volumes are at 0.101325 MPa and 20 °C), m³/m³Why water is injected: pressure maintenance and oil displacement formation volume factors of oil, gas and water, m³/m³Displacement of oil by gas
$B_o,\ B_w$	formation volume factors of oil and water, m³/m³Why water is injected: pressure maintenance and oil displacement
$\Delta p_c$	capillary pressure drop across the front: between the water saturation ahead of the front and behind it, PaFractional flow function and mobility ratio
$C_j$	share of the pore volume (storage capacity $\phi h$ ) of the section in the $j$ layers flushed by this moment; here the layers are numbered in the order of breakthrough, —Displacement in heterogeneous reservoirs
$C_n,\ F_n$	shares of storage capacity $\phi h$ and of flow capacity $kh$ in the $n$ flushed layers, —Displacement in heterogeneous reservoirs
$c_t$	total compressibility of rock and fluids, $S_oc_o + S_wc_w + c_f$ , at the current saturations (Flow regimes), 1/MPaWhy water is injected: pressure maintenance and oil displacement
$E_A$	areal sweep efficiency: the share of the area of the formation that the water has passed through, fractionWhy water is injected: pressure maintenance and oil displacement
$E_D$	displacement efficiency: the share of the initial oil displaced from the flushed pores; not more than $(1 - S_{wi} - S_{or})/(1 - S_{wi})$ , fractionWhy water is injected: pressure maintenance and oil displacement
$E_o,\ E_g,\ E_{fw}$	expansion of the oil with its dissolved gas, of the gas cap (for $m = 1$ ) and of the irreducible water with the pores, per cubic metre of oil in place, m³/m³Why water is injected: pressure maintenance and oil displacement
$E_R$	recovery factor: the share of the initial oil in place that has been produced by the given time, fractionWhy water is injected: pressure maintenance and oil displacement
$E_V$	vertical sweep efficiency (over the thickness): the share of the thickness of the formation, on the swept area, that the water has passed through, fractionWhy water is injected: pressure maintenance and oil displacement vertical sweep: the share of the mobile oil of the section between the rows of wells that the water has passed through, —Displacement in heterogeneous reservoirs
$F$	underground withdrawal accumulated to date, $N_p[B_o + (R_p - R_s)B_g] + W_pB_w$ ; $R_p$ is the cumulative gas–oil ratio, m³Why water is injected: pressure maintenance and oil displacement
$f_g$	the share of gas in the total reservoir flow rate of gas and oil, —Displacement of oil by gas fractional flow of gas in the reservoir flow at the bottom of the well, —Displacement of oil by gas
$F_j$	share of the flow capacity $kh$ of the section in the same layers, —Displacement in heterogeneous reservoirs
$f_w$	share of water in the total flow at reservoir conditions, —Fractional flow function and mobility ratio fractional flow of water, —Buckley–Leverett theory and the Welge method share of water in the reservoir flow rate through the cross-section, —Segregated flow and gravity displacement
$f_w'(S_w)$	slope of the fractional flow curve at this saturation, —Buckley–Leverett theory and the Welge method
$f_w'(S_{wf})$	slope of the fractional flow curve at the front point, —Buckley–Leverett theory and the Welge method
$f_w(S_{wi})$	fractional flow of water ahead of the front; it is zero if the initial water does not move, —Buckley–Leverett theory and the Welge method
$f_{we}$	fractional flow of water in the reservoir flow at the outlet, found from the water cut of the production through $B_o$ and $B_w$ , —Displacement in heterogeneous reservoirs
$f_{we,j}$	fractional flow of water at the outlet, at reservoir conditions, just after layer $j$ breaks through, —Displacement in heterogeneous reservoirs
$g$	acceleration of gravity, 9.81, m/s²Fractional flow function and mobility ratio acceleration of gravity, m/s²Displacement in heterogeneous reservoirs
$G_{iD}$	the gas injected, in pore volumes, at reservoir conditions, —Displacement of oil by gas
$h_D$	fraction of the thickness occupied by water (by gas, for gas), —Segregated flow and gravity displacement
$h_D,\ M',\ N_g$	fraction of the thickness under water, mobility ratio at the end points, gravity number, —Segregated flow and gravity displacement
$k$	absolute permeability of the rock, m²Fractional flow function and mobility ratio
$k_i,\ h_i$	permeability and thickness of the layer, mD, mDisplacement in heterogeneous reservoirs
$k_i,\ \phi_i$	permeability and porosity of the layer, m², —Displacement in heterogeneous reservoirs
$k_{rd}',\ \mu_d$	end-point relative permeability and viscosity of the displacing phase, —, Pa·sSegregated flow and gravity displacement
$k_{rg},\ k_{ro}$	relative permeabilities of the gas and of the oil at the current gas saturation (the water does not move), —Displacement of oil by gas
$k_{ro}$	relative permeability of oil where it drains: in the oil part below the gas–oil contact it is the end-point value $k_{ro}'$ , and in a zone already occupied by gas it is many times smaller, —Segregated flow and gravity displacement
$k_{ro},\ k_{rw}$	relative permeabilities of oil and water at the current water saturation, —Fractional flow function and mobility ratio
$k_{ro}'$	end-point relative permeability of oil at irreducible water saturation, —Fractional flow function and mobility ratio
$k_{ro}',\ \mu_o$	end-point relative permeability and viscosity of the oil, —, Pa·sDisplacement in heterogeneous reservoirs
$k_{ro}/k_{ro}'$	relative permeability of oil divided by its end-point value: 1 at $S_{wi}$ , 0 at $1 - S_{or}$ , —Fractional flow function and mobility ratio
$k_{rw}',\ k_{ro}'$	end points of the core relative permeability curves, —Segregated flow and gravity displacement
$k_v$	vertical permeability of the unit (the group of layers), m²Displacement in heterogeneous reservoirs
$k_v,\ k$	vertical permeability and permeability along the formation, m²Segregated flow and gravity displacement
$k,\ \Delta\rho,\ g,\ \alpha,\ \mu_o,\ u_t$	as in the fractional flow equation, SIFractional flow function and mobility ratio
$k,\ k_{ro},\ \mu_o,\ u_t$	as in the fractional flow equation; $k_{ro}$ is a typical relative permeability of oil in the front zone, SIFractional flow function and mobility ratio
$L$	distance between the rows of injection and production wells, mDisplacement in heterogeneous reservoirs
$\ell_c$	length over which the capillary term in the fractional flow equation is of the order of one, mFractional flow function and mobility ratio
$L,\ h$	distance along the formation between the rows of injection wells and production wells, and thickness of the formation, mSegregated flow and gravity displacement
$L,\ H$	distance between the rows of wells and thickness of the unit, mDisplacement in heterogeneous reservoirs
$m$	ratio of the reservoir volume of the gas cap to the reservoir volume of the oil, —Why water is injected: pressure maintenance and oil displacement
$M'$	mobility ratio of the displacing phase and the oil at the end points, —Segregated flow and gravity displacement mobility ratio at the end points, $(k_{rw}'/\mu_w)/(k_{ro}'/\mu_o)$ , —Displacement in heterogeneous reservoirs
$M',\ N_g$	mobility ratio and gravity number with its sign, as in the relation above, —Segregated flow and gravity displacement
$N$	oil in place, m³Why water is injected: pressure maintenance and oil displacement
$N_g$	gravity number: the ratio of two pressure gradients. One is due to the difference between the weights of the phases along the formation, $\Delta\rho\,g\sin\alpha$ . The other is the gradient needed to pump oil alone through the rock at the velocity $u_t$ , —Fractional flow function and mobility ratio gravity number of the lesson Fractional flow function and mobility ratio, with the density difference between the displacing phase and the oil. It is positive when gravity holds the displacing phase back: water moves up the dip or gas moves down the dip, —Segregated flow and gravity displacement the gravity number of the lesson Fractional flow function and mobility ratio, with the density difference of gas and oil, —Displacement of oil by gas
$N_g,\ \alpha$	gravity number and dip angle of the formation, as in the lesson Fractional flow function and mobility ratio, —, radSegregated flow and gravity displacement
$N_{gv}$	ratio of the time to cover the distance $L$ to the time the phases need to separate vertically over the thickness $H$ , —Displacement in heterogeneous reservoirs
$N_p,\ W_{inj}$	cumulative oil production and water injection at standard conditions, m³Displacement in heterogeneous reservoirs
$N_{pD}$	cumulative oil production in pore volumes (reservoir conditions), —Buckley–Leverett theory and the Welge method the oil produced, in pore volumes, at reservoir conditions, —Displacement of oil by gas
$N_{pD,BT}$	oil production up to breakthrough, in pore volumes, at reservoir conditions, —Buckley–Leverett theory and the Welge method
$n_w,\ n_o$	Corey exponents for water and oil (Corey model), —Buckley–Leverett theory and the Welge method
$p_c$	capillary pressure $p_o - p_w$ , PaFractional flow function and mobility ratio
$Q$	underground withdrawal per unit of time, the denominator of VRR, m³/dWhy water is injected: pressure maintenance and oil displacement
$q_{\text{cr}}$	critical rate: the largest total reservoir flow rate through the cross-section of the formation at which the interface is still stable, m³/sSegregated flow and gravity displacement
$q_{g,\text{inj}}$	rate of the injected gas at standard conditions, m³/dDisplacement of oil by gas
$q_{\text{gd}}$	largest reservoir flow rate of oil that drainage under gravity along the formation can give, m³/sSegregated flow and gravity displacement
$q_{\text{gr}}$	gravity rate: the rate at which the gravity number equals one, m³/sSegregated flow and gravity displacement
$q_i$	injection rate at reservoir conditions, m³/dBuckley–Leverett theory and the Welge method
$Q_i$	injection at reservoir conditions, $q_{wi}B_w$ , m³/dWhy water is injected: pressure maintenance and oil displacement
$q_o$	oil production rate at standard conditions, m³/dWhy water is injected: pressure maintenance and oil displacement
$q_o,\ q_w$	production rates of oil and water at standard conditions, m³/dWhy water is injected: pressure maintenance and oil displacement rates of oil and water at standard conditions, m³/dDisplacement of oil by gas
$q_t$	total flow rate at reservoir conditions, equal to the injection rate, m³/sBuckley–Leverett theory and the Welge method
$q_t,\ q_{\text{cr}}$	reservoir flow rate through the cross-section and the critical rate, m³/sSegregated flow and gravity displacement
$q_t,\ \phi,\ A$	as in the water balance, SIBuckley–Leverett theory and the Welge method
$q_{wi}$	injection rate of water at standard conditions, m³/dWhy water is injected: pressure maintenance and oil displacement
$R$	instantaneous (current) producing gas–oil ratio: all the produced gas per cubic metre of oil, m³/m³Why water is injected: pressure maintenance and oil displacement producing gas–oil ratio: the gas per cubic metre of oil at standard conditions (producing gas–oil ratio), m³/m³Displacement of oil by gas
$R_s$	solution gas–oil ratio at the current average reservoir pressure, m³/m³Why water is injected: pressure maintenance and oil displacement solution gas–oil ratio of the oil at reservoir pressure, m³/m³Displacement of oil by gas
$R,\ R_s$	producing gas–oil ratio and solution gas–oil ratio of the oil at reservoir pressure, m³/m³Displacement of oil by gas
$\bar k_{rw},\ \bar k_{ro}$	thickness-averaged relative permeabilities of water and oil, —Segregated flow and gravity displacement
$\hat k_{rw},\ \hat k_{ro}$	pseudo relative permeabilities of the section for water and oil, —Displacement in heterogeneous reservoirs
$S_{ge},\ f_{ge}$	the gas saturation at the outlet after breakthrough and the fractional flow of gas at that saturation, —Displacement of oil by gas
$S_{gf},\ f_{gf}$	the gas saturation at the front and the fractional flow of gas at that saturation, —Displacement of oil by gas
$S_n$	normalised water saturation $(S_w - S_{wi})/(1 - S_{wi} - S_{or})$ , —Buckley–Leverett theory and the Welge method
$S_w$	water saturation in the section $x$ at the moment $t$ , —Buckley–Leverett theory and the Welge method
$S_{we}$	equivalent saturation at the outlet (a calculated value, not a measured one): the abscissa (horizontal coordinate) of a point of the fractional flow curve of the formation, —Displacement in heterogeneous reservoirs
$S_{we},\ f_{we}$	water saturation at the outlet and the fractional flow of water at it, which is the share of water in the production at reservoir conditions, —Buckley–Leverett theory and the Welge method
$S_{wf},\ f_{wf}$	water saturation just behind the front and the fractional flow of water at it, —Buckley–Leverett theory and the Welge method
$S_{wf},\ f_{wf},\ f_w'(S_{wf})$	saturation, fractional flow and slope of the curve at the front, —Buckley–Leverett theory and the Welge method
$S_{wf},\ f_{wf},\ S_{wi}$	saturation at the front, the fractional flow at it, the initial water saturation, —Buckley–Leverett theory and the Welge method
$t_{BT}$	time from the start of injection to water breakthrough, dBuckley–Leverett theory and the Welge method
$t_v$	time the phases need to separate across the formation, $\phi\,\Delta S\,h/u_v$ ; $u_v$ is the speed of separation, an upper estimate, sSegregated flow and gravity displacement
$t_x$	time in which the front covers the distance $L$ along the formation, $\phi\,\Delta S\,L/u_t$ , sSegregated flow and gravity displacement
$u_t$	total velocity $q_t/A$ : the flow rate of both phases per unit area of the cross-section, m/sFractional flow function and mobility ratio total velocity along the formation, $q_t/A$ , m/sSegregated flow and gravity displacement total velocity along the formation, m/sDisplacement in heterogeneous reservoirs total Darcy velocity of gas and oil (flow rate per unit area), $q_t/A$ , m/sDisplacement of oil by gas
$u_t,\ p_c,\ \Delta\rho,\ g,\ \alpha$	as in the general form above, SIFractional flow function and mobility ratio
$v_f$	speed of the displacement front, m/sBuckley–Leverett theory and the Welge method
$v_i,\ v_j$	layer speeds $k/\phi$ , mDDisplacement in heterogeneous reservoirs
$v_j$	speed of layer $j$ , $k_j/\phi_j$ , mDDisplacement in heterogeneous reservoirs
$V_p$	pore volume of the accumulation, m³Why water is injected: pressure maintenance and oil displacement pore volume of the element, $\phi A L$ , m³Buckley–Leverett theory and the Welge method pore volume of the flooded formation or area, m³Displacement in heterogeneous reservoirs
$v_{S_w}$	speed of the section with water saturation $S_w$ , m/sBuckley–Leverett theory and the Welge method
$\mathrm{VRR}$	current voidage replacement ratio, fraction (or %)Why water is injected: pressure maintenance and oil displacement
$\bar S_w$	average water saturation of the flushed zone, from the inlet to the front, —Buckley–Leverett theory and the Welge method average water saturation of the whole element, —Buckley–Leverett theory and the Welge method average water saturation over the thickness in the cross-section, —Segregated flow and gravity displacement
$\hat S_w$	water saturation averaged over the pore volume of the section in the given cross-section, —Displacement in heterogeneous reservoirs
$W_e$	cumulative water influx from the aquifer, m³Why water is injected: pressure maintenance and oil displacement
$W_{iD}$	cumulative water injection in pore volumes, —Buckley–Leverett theory and the Welge method
$W_{iD}\big\|_{h_D = 1}$	injection after which water occupies the whole thickness at the outlet and all the mobile oil has been produced, —Segregated flow and gravity displacement
$W_{iD,BT}$	water injection up to breakthrough, in pore volumes of the element, —Buckley–Leverett theory and the Welge method water injection up to breakthrough in pore volumes; the same amount of oil has been produced by breakthrough, —Segregated flow and gravity displacement
$W_{inj},\ G_{inj}$	cumulative water and gas injection at standard conditions, m³Why water is injected: pressure maintenance and oil displacement
$\text{WC}$	water cut of the production: the share of water in the produced liquid at standard conditions, fractionWhy water is injected: pressure maintenance and oil displacement
$x$	coordinate along the flow, from the injection side to the production side, mFractional flow function and mobility ratio
$x_{f,i}$	distance covered by the front in layer $i$ ; for a flushed layer $x_{f,i} = L$ , mDisplacement in heterogeneous reservoirs position of the front in layer $i$ , mDisplacement in heterogeneous reservoirs
$z,\ x,\ \alpha$	height of the interface above the base, coordinate along the formation and dip angle, as above, m, m, radSegregated flow and gravity displacement
$\alpha$	angle of the axis $x$ to the horizontal: positive if the water moves up the dip, negative if it moves down the dip, radFractional flow function and mobility ratio dip angle of the formation (in absolute value: one formula serves for water going up the dip and for gas going down the dip), radSegregated flow and gravity displacement dip angle of the formation. For drainage in the vertical direction $\sin\alpha = 1$ , and $k$ and $A$ are the vertical permeability and the area in plan, radSegregated flow and gravity displacement angle of the flow axis to the horizontal: positive if the gas moves up the dip, radDisplacement of oil by gas
$\beta$	angle between the interface and the bedding, radSegregated flow and gravity displacement
$\Delta\bar p$	change in the average reservoir pressure over the time $\Delta t$ , MPaWhy water is injected: pressure maintenance and oil displacement
$\Delta\rho$	difference between the densities of water and oil, $\rho_w - \rho_o$ , at reservoir conditions, kg/m³Fractional flow function and mobility ratio density difference between the displacing phase and the oil, kg/m³Segregated flow and gravity displacement density difference between water and oil at reservoir conditions, kg/m³Displacement in heterogeneous reservoirs
$\lambda_g,\ \lambda_o,\ \lambda_t$	the mobilities of the gas, $k\,k_{rg}/\mu_g$ , of the oil, $k\,k_{ro}/\mu_o$ , and the total mobility, m²/(Pa·s)Displacement of oil by gas
$\lambda_o,\ \lambda_w$	mobilities of oil and water, $k\,k_{ro}/\mu_o$ and $k\,k_{rw}/\mu_w$ , m²/(Pa·s)Fractional flow function and mobility ratio
$\lambda_t$	total mobility $\lambda_o + \lambda_w$ , m²/(Pa·s)Fractional flow function and mobility ratio
$\lambda_w,\ \lambda_o$	end-point mobilities of water and oil, $k_{rw}'/\mu_w$ and $k_{ro}'/\mu_o$ , 1/(Pa·s)Displacement in heterogeneous reservoirs
$\lambda_w/\lambda_t$	viscous part: the fractional flow of water in a horizontal formation, —Fractional flow function and mobility ratio
$\mu_o,\ \mu_w$	viscosities of oil and water at reservoir conditions, Pa·sFractional flow function and mobility ratio
$\xi_i$	fraction of the length $L$ covered by the front in layer $i$ at the moment when layer $j$ breaks through, —Displacement in heterogeneous reservoirs fraction of the length covered by the front in an unflushed layer, —Displacement in heterogeneous reservoirs
$\rho_o - \rho_g$	difference between the densities of oil and gas at reservoir conditions, kg/m³Segregated flow and gravity displacement
$\sum_{\text{flushed}},\ \sum_{\text{unflushed}}$	sums over the flushed layers and over the layers where the front has not yet reached the production wells, —Displacement in heterogeneous reservoirs
$\phi,\ A$	porosity and cross-section area, —, m²Buckley–Leverett theory and the Welge method
$\phi_i,\ h_i$	porosity and thickness of layer $i$ , —, mDisplacement in heterogeneous reservoirs