Equations: Inflow to a well

any consistent units, for example m³/d and MPa

Holds when: Single-phase inflow ($p_{wf}$ above the bubble point pressure) and stabilised flow. The productivity index does not depend on the flow rate: there are no inertial losses.

pressures in MPa; the product $\rho g H$ comes out in pascals and is divided by $10^6$

Holds when: Vertical tubing down to the perforated interval; single-phase liquid without free gas, constant density. The wellhead pressure is held constant — by the choke or by the pressure in the gathering system (the network of pipes that collects the production). $K$ depends on the tubing diameter and on the density and viscosity of the liquid. The lesson Nodal analysis of the production system shows how to calculate it.

any identical flow-rate units, for example m³/d

Holds when: We consider the period before water breaks through to the production wells. The pressure is above the bubble point pressure, so only oil with dissolved gas is produced. The reservoir pressure is held constant: the withdrawal is fully replaced.

m³/d; the producing and solution gas–oil ratios are in m³ of gas per m³ of oil at standard conditions

Holds when: The flow rates are at standard conditions; the formation volume factors and the solution gas–oil ratio are at the current reservoir pressure. Free gas is the gas in excess of the dissolved gas, $(R_p - R_s)\,q_o$ at standard conditions.

SI (any consistent system of units); $\ln$ is the natural logarithm

Holds when: Steady state: a constant pressure $p_e$ is held at the outer boundary. The reservoir is homogeneous and isotropic (its permeability is the same in all directions) and has constant thickness. The well stands at the centre of a circular area and penetrates the whole thickness. The flow is single-phase. The fluid is incompressible or slightly compressible and obeys Darcy's law. The flow is horizontal. The near-wellbore zone is unchanged, so the skin factor is zero (the lesson shows below how to account for it).

SI; $\ln$ is the natural logarithm

Holds when: The same as for the Dupuit equation.

SI; $\ln$ is the natural logarithm

Holds when: The conditions of the Dupuit equation hold, and the skin is zero.

SI; in metric and oilfield units the constants are the same as in the Dupuit equation

Holds when: The altered zone is small compared with the drainage area, and the flow in it can be treated as steady-state. The skin does not depend on the flow rate (there are no inertial losses).

SI; $\ln$ is the natural logarithm

Holds when: Steady-state inflow in a circular area; porosity and thickness are constant. $r_w \ll r_e$: terms of the order of $(r_w/r_e)^2$ are left out.

SI; absolute pressures only; $\ln$ is the natural logarithm

Holds when: Steady-state inflow at constant temperature (isothermal). The reservoir is homogeneous, and the well penetrates its whole thickness. $\mu Z$ changes little between $p_w$ and $p_e$, which holds roughly for pressures up to 14 MPa. At higher pressures we use the pseudo-pressure (advanced section). Darcy's law holds. In gas wells at high flow rates, inertial losses are added (Inflow at high rates: non-Darcy flow). The skin is added to the logarithm in the same way as for oil.

SI; $\ln$ is the natural logarithm

Holds when: Constant flow rate after start-up; a homogeneous reservoir. $r$ is much smaller than the radius of investigation $\sqrt{4kt/(\phi\mu c_t)}$ (how far into the reservoir the disturbance has travelled): the disturbance reached this point long ago.

SI; absolute pressures

Holds when: Steady-state inflow at constant temperature, obeying Darcy's law. The gas properties are known from PVT data (laboratory measurements of how pressure, volume and temperature are linked).

any consistent units: for example, m³/d, 1/MPa and m³ give MPa/d

Holds when: A closed area: no fluid enters across the boundaries. Constant flow rate; single-phase fluid, constant compressibility. For the average pressure the equality holds at any time. For the pressure at each point it holds only after the pseudo-steady-state regime has set in.

any consistent units

Holds when: Pseudo-steady-state regime, well at the centre of a circular area, homogeneous reservoir. $r_w \ll r_e$.

SI; $\ln$ is the natural logarithm

Holds when: Pseudo-steady-state regime, circular area with the well at the centre. $r_w \ll r_e$: terms of the order of $(r_w/r_e)^2$ are left out. The skin is an additional pressure drop in a thin zone at the wellbore (Skin factor).

SI (any consistent system of units); $\ln$ is the natural logarithm

Holds when: The same as for the pressure distribution. The fluid is single-phase and slightly compressible, Darcy's law holds, and the skin does not depend on the flow rate.

SI; $A$ and $r_w^2$ in the same units of area; $\ln$ is the natural logarithm

Holds when: Pseudo-steady-state regime: the dimensionless time $t_{DA}$ is greater than the tabulated value for the given shape. A homogeneous isotropic reservoir, the well penetrates the whole thickness, and the boundary is impermeable along its whole perimeter.

dimensionless

Holds when: The same as for the equation with the shape factor.

SI

Holds when: A homogeneous reservoir; time is counted from the start-up of the well or from the last change of flow rate.

SI; $\ln$ is the natural logarithm

Holds when: Constant flow rate from the initial state; the pseudo-steady-state regime has already set in. A closed area, single-phase slightly compressible fluid.

any consistent units

Holds when: All wells of the block operate in the pseudo-steady-state regime at constant flow rates. Single-phase fluid; the compressibility and formation volume factor are the same throughout the block. Wells of different blocks separated by sealing faults are considered separately.

dimensionless; $L$, $W$ and $r_w$ in the same units; $\ln$ is the natural logarithm

Holds when: Pseudo-steady-state regime, homogeneous reservoir, the well penetrates the whole thickness. The well is at least the width $W$ away from both short sides. Then the error of the logarithmic term is below 0.01. For a well on the long axis ($y_D = 1/2$), half the width is enough. Closer to a short side the equation underestimates the resistance. For an off-axis well at a distance of $W/2$ the error reaches 0.1–0.2. For a well in a corner of the area the equation is unsuitable, and tabulated values are needed.

SI (any consistent system of units)

Holds when: The zone of altered permeability is small compared with the drainage area, and the flow in it can be taken as steady-state at the current flow rate. No fluid builds up in the thin zone, so $\Delta p_s$ follows the flow rate immediately and disappears when the well is shut in. The flow is single-phase and obeys Darcy's law. If there are inertial losses, a term that grows with the flow rate is added to the skin. The skin is always referred to the same $k$ and $h$ that describe the inflow in the reservoir.

any consistent units: only the ratios $k/k_s$ and $r_s/r_w$ enter

Holds when: The altered zone is a cylinder around the wellbore with uniform permeability $k_s$, and $r_s \ll r_e$. The well penetrates the whole thickness of the reservoir, and the flow in the zone is radial. The flow is single-phase and obeys Darcy's law.

any units of length (the same for $r_w$ and $r_w'$)

Holds when: The skin does not change with time and does not depend on the flow rate. The zone that the effective radius replaces is small compared with the drainage area ($r_w' \ll r_e$).

SI (any consistent system of units)

Holds when: A homogeneous reservoir, a circular drainage area with the well at the centre. The flow is single-phase and obeys Darcy's law. The well penetrates the whole thickness of the reservoir. Otherwise a component for imperfect penetration (the well opens only part of the layer) enters the skin. The second formula is for the pseudo-steady-state regime, when the pressure disturbance has already reached the boundaries of the closed area.

SI

Holds when: Pseudo-steady-state flow; in other regimes the logarithmic term changes, and the skin enters in the same way.

dimensionless; pressures in any identical units

Holds when: Pseudo-steady-state flow; both productivities refer to the same drainage area. $\bar p$ is the average pressure in the drainage area. For a new well we use instead the extrapolated pressure from the build-up curve (the pressure found by extending the build-up curve to an infinitely long shut-in).

dimensionless quantities

Holds when: Simple addition is only an approximation, because the components affect one another. If only part of the thickness $h_w$ is open, the damage skin, counted over the whole thickness $h$, grows by roughly a factor of $h/h_w$. Shallow damage that the perforation tunnels pass right through hardly hinders the inflow.

any pressure units (the same as for $m$)

Holds when: We get $m$ and $S$ from one test, on a section of radial flow. $\Delta p_s$ refers to the flow rate of that test.

SI (m, m³/m)

Holds when: Piston-like displacement; the reservoir is uniform over its thickness. Usually $r_{inv}$ is much larger than $r_w$, and the term $r_w^2$ can be left out.

dimensionless; lengths and permeabilities in identical units; $\ln$ is the natural logarithm

Holds when: A homogeneous reservoir with constant $k_h$ and $k_v$ between an impermeable top and bottom; a vertical well, one continuous interval. The formula approximates numerical solutions. For $h_D$ from 100 to tens of thousands and $h_w/h \ge 0.1$, it differs from the numerical solution with uniform pressure along the interval by no more than 0.6 (a few per cent), usually on the low side. For smaller $h_D$ (a thin reservoir) it underestimates the pseudo-skin: at $h_D = 20$ and a small penetration ratio, by a third. Then a numerical calculation is needed. For an interval in an in-between position, the value lies between the values for an interval at the top and an interval in the middle. For greater accuracy, use a numerical model.

dimensionless

Holds when: The damaged zone is thin: its radius is small compared with the interval and with the zone where the flow converges. The flow through the damaged zone is radial towards the interval. In a gas well an inertial skin is added to $S$, and partial penetration also magnifies it (Inflow at high rates: non-Darcy flow).

SI (any consistent system of units)

Holds when: A homogeneous anisotropic reservoir; the interval is short compared with the thickness. The time is later than the end of early radial flow and earlier than the start of late radial flow. Wellbore storage has ended.

metric oilfield units: $t$ in h, $h$ and $h_w$ in m

Holds when: These are estimates from a model of uniform inflow along the interval, for a penetration ratio of up to 0.3. We take the boundaries of the regimes where the derivative departs from the plateau by 5%. If we allow 10% instead, the factor $a_1$ is four times larger and $a_2$ is about 20% smaller. Wellbore storage is not taken into account: it often hides early radial flow completely.

dimensionless; $\lg$ is the decimal logarithm, angles in degrees

Holds when: The wellbore opens and perforates the whole thickness of a homogeneous reservoir. The limits are $\theta' \le 75^\circ$ and $h_D > 40$. In an isotropic reservoir $\theta' = \theta$. Allowing for anisotropy through the effective angle is a common approximation. The value refers to late, pseudo-radial flow. Other models describe horizontal wells (Horizontal wells).

dimensionless

Holds when: The damaged zone is thin and runs along the whole wellbore in the reservoir. The wellbore is perforated along its whole length in the reservoir. $\cos\theta'$ equals $\cos\theta$ (the ratio of the thickness to the length of the wellbore in the reservoir) multiplied by the ratio of $k_h$ to the effective permeability across the wellbore. In an anisotropic reservoir that permeability is smaller than $k_h$, so $\cos\theta' > \cos\theta$.

dimensionless

Holds when: A homogeneous anisotropic reservoir. A line source with uniform inflow along the interval. The pressure is averaged over the interval.

dimensionless

Holds when: A line source with uniform inflow along the interval; a reservoir unbounded in plan; no wellbore storage.

SI (any consistent system of units)

Holds when: The cone is stationary, and the water beneath it is in hydrostatic equilibrium. The interface is sharp: we ignore the capillary transition zone (the zone where oil and water change over gradually). We take the pressure at the tip of the cone to be equal to the bottomhole pressure. In fact the tip is below the interval, and the pressure there is higher than the bottomhole pressure. So the estimate understates the permissible drawdown, which errs on the safe side. For a gas cone $\Delta\rho = \rho_o - \rho_g$, and $h_c$ is the distance from the GOC to the top perforations.

SI; $\ln$ is the natural logarithm

Holds when: The inflow of single-phase oil to a partially penetrating well is steady-state, in a homogeneous reservoir. The pressure at the outer boundary is maintained. The water is stationary, and the cone does not distort the pressure field in the oil. The tip “feels” the whole drawdown at the bottomhole, including the resistance of the flow converging on the interval. This is why the result is too low. Damage skin does not enter the formula, because the pressure drop in the thin damaged zone at the wellbore does not reach the formation below the interval.

SI; $\ln$ is the natural logarithm

Holds when: Vertical equilibrium: the pressure in the oil is constant along the vertical, and the oil flows horizontally. Close to the well this assumption is rough. The reservoir is isotropic, so $k_v$ does not enter the formula. The resistance of the flow converging on the interval is not taken into account. The interval is open from the top. At the critical rate the thickness of oil above the cone at the wellbore equals the interval $h_w$.

SI (any consistent system of units)

Holds when: A homogeneous anisotropic reservoir. Steady-state inflow, with a constant pressure at the outer boundary. Stationary water. An interval open from the top. The formula approximates theoretical calculations of a stable cone for a well that is open at the top of the reservoir. $F$ grows as $r_{eD}$ decreases, so a lower vertical permeability raises the critical rate.

SI (any consistent system of units)

Holds when: The groups and the exponent $\alpha$ come from a correlation built on model experiments and calculations, for a homogeneous reservoir with bottom water and an interval open from the top. In oilfield units ($\Delta\rho$ in g/cm³, $h$ in ft, $k$ in mD, $q$ in STB/d, $t$ in days) the factors are 0.00307 (for $Z$) and 0.00137 (for $t_D$). Some reference books write the dimensionless time with the same factor as $Z$ (0.00307 instead of 0.00137). Then the breakthrough time comes out 2.2 times too small. The original definition gives 0.00137, which matches the form in consistent units above. If you use other units, work out the factor again from that form. A check of the units will not catch this error.

dimensionless

Holds when: This is an empirical correlation of model experiments. An earlier correlation with the same groups (Sobocinski–Cornelius) is $t_{D,BT} = (4Z + 1.75Z^2 - 0.75Z^3)/(7 - 2Z)$ for $Z < 3.5$. For $Z$ above 4.29 (3.5 in the second correlation) the water does not reach the interval. This is one more, “dynamic”, estimate of the critical rate. The reservoir is homogeneous. Layers, fractures and vertical heterogeneity change the result several times over, so the accuracy is at best about a factor of two.

dimensionless; thicknesses in the same units

Holds when: Both zones deliver liquid to the well side by side, in proportion to $k\,k_r h/\mu$. The pressure and the permeability are the same in both zones. As fluid is produced, the OWC rises. So $h_{aq}$ grows, $h$ decreases, and the limit rises with time. The water fraction is at reservoir conditions. At standard conditions the water cut is somewhat higher, because when oil is brought to the surface its volume shrinks more than the volume of water (the oil formation volume factor is greater).

dimensionless

Holds when: At $t/t_{BT} < 0.5$ the ratio is zero, and at $t/t_{BT} > 5.7$ it is 1. The reservoir is homogeneous with bottom water, and the liquid rate is constant. Formally the water appears already at half the calculated breakthrough time. This comes from how the correlation is scaled to the estimate of $t_{BT}$, and it is one more reminder of how accurate that estimate is. We allow separately for the rise of the limit $f_{w,\lim}$ as the OWC rises, by recalculating $h$ and $h_{aq}$ from the cumulative production (the total produced so far).

SI; $\ln$ is the natural logarithm

Holds when: Vertical equilibrium and an isotropic reservoir, as in the formula with one cone. Both phases, below and above the oil, are stationary. We find the position of the interval from the stability condition. We do not take into account the movement of the contacts during development (the cap expands, the water rises).

SI (any consistent system of units)

Holds when: The flow is single-phase (one fluid) and isothermal (constant temperature). At every point the velocity points along the pressure gradient (the direction in which pressure changes fastest). $\beta$ is a property of the rock, like $k$. We find it by experiment (see below). As $v \to 0$, the second term vanishes and the equation becomes Darcy's law.

no units; the quantities are in consistent units

Holds when: This is an estimate at one point of the flow. Near a well, $v$ and Fo change with distance.

SI; pressures absolute

Holds when: The flow is isothermal and single-phase. $Z$ and $\mu$ do not change over the range of pressures in the test. Gas slippage (the Klinkenberg correction: gas molecules slip along the pore walls, so the rock looks more permeable than it is) is small. To make sure of this, the test is run with back pressure (pressure held at the outlet), so that the mean pressure in the sample is about a megapascal. Otherwise slippage is allowed for separately (Rock permeability).

empirical: $\beta$ in m⁻¹, $k$ in mD

Holds when: The rocks are consolidated and the gas flow is single-phase. Measurements scatter around the correlation by up to an order of magnitude (a factor of ten). Irreducible water (water held in the rock that cannot be produced) or condensate that has dropped out narrows the channels. Then $\beta$ for gas increases, sometimes several times over.

SI; $\ln$ is the natural logarithm

Holds when: The inflow is steady-state and single-phase, and the liquid is slightly compressible (we treat gas below, through the mass flow rate). The reservoir is homogeneous (the same everywhere). The well is an open hole (not cased or perforated) over the whole thickness. $\beta$ and $\rho$ are constant.

no units; radii in the same units

Holds when: Steady-state radial flow, as in the previous formula. The approximate equality holds when $r_e \gg r_w$.

SI; $\ln$ is the natural logarithm

Holds when: $r_e \gg r_w$, so the term $1/r_e$ is dropped. The well is an open hole over the whole thickness, and $\beta$ at the wellbore is the same as in the reservoir. With damage and with partial penetration, $D$ is larger (see the last section).

SI

Holds when: The mass flow rate is the same at all radii, and the inflow is isothermal. The inertial losses are concentrated at the wellbore, so the viscosity is taken at the bottomhole pressure. The well is an open hole over the whole thickness, and the near-wellbore zone is undamaged.

SI; pressures absolute; $\ln$ is the natural logarithm

Holds when: The inflow is pseudo-steady-state, in a circular area with the well at the centre. For another shape of area we change the logarithmic term, as in the lesson on pseudo-steady-state flow. Using $m(\bar p)$ instead of the volume-averaged pseudo-pressure is the usual approximation. $D$ is taken with the gas viscosity at the bottom of the well. The gas in the reservoir is single-phase.

metric oilfield (or any units consistent with $a$ and $b$)

Holds when: The inflow is stabilised (the pressures no longer change with time) and the coefficients $a$ and $b$ are known. For $b = 0$ the formula becomes $q = \Delta m/a$.

any; $D$ is in units that are the inverse of the rate units

Holds when: All flow periods are interpreted with one reservoir model, and the flow capacity $kh$ is the same for all of them. The skin and $D$ do not change during the test (the well has already been cleaned up after completion).

any consistent units; $D$ and $D_0$ in the same units

Holds when: The damaged zone is a cylinder around the wellbore, and the flow in it is steady-state. The first term is the Hawkins formula (Skin factor). $r_e \gg r_s$, and the hole is open over the whole thickness.

no units; $D_0 q$ in consistent units

Holds when: The damaged zone is thin compared with the interval, and the flow near the wellbore is radial towards the interval. The inertial losses in the zone of flow convergence, farther from the wellbore, are small compared with the losses at the wall.

SI

Holds when: The pack is unconsolidated (loose), and the grains are of similar size. The constants 150 and 1.75 are empirical (found from experiments).