Equations: Rock properties

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

Porosity

\phi = \frac{V_p}{V_b} = \frac{V_b - V_{gr}}{V_b} = \frac{V_p}{V_{gr} + V_p}

any units of volume, the same for all three volumes; porosity has no units

Holds when: All three volumes refer to the same sample at the same conditions (pressure, temperature, load). What counts as the pore volume depends on the kind of porosity: all the voids (total) or only the connected ones (open); see the next section.

Porosity →

Grain volume by Boyle's law (two-chamber gas porosimeter)

V_{gr} = V_c - V_r\,\frac{p_1 - p_2}{p_2 - p_0}

all volumes in the same units; absolute pressures, all in the same units

Holds when: The process is isothermal (the temperature does not change): $p_2$ is read after the gas temperature has settled. The gas is ideal (it obeys the ideal gas law) and does not adsorb (stick to the minerals). For helium at pressures of about 1 MPa this holds with high accuracy. The volumes of the chambers and connecting channels are calibrated with steel cylinders of known volume. If the chamber with the sample starts at atmospheric pressure, you can use gauge pressures (read from a gauge, relative to the atmosphere) instead. Then $p_0 = 0$ , and the denominator equals the gauge reading $p_2$ .

Porosity →

Porosity from the density log

\phi_D = \frac{\rho_{ma} - \rho_b}{\rho_{ma} - \rho_f}

g/cm³ or kg/m³ — any units of density, the same for all three densities

Holds when: The solid framework is one known mineral, or a mixture with a known average density, and has no clay. The pores in the zone the tool reads (about 10 cm from the borehole wall) are filled with a fluid of known density — usually mud filtrate. Corrections for mudcake (a layer of drilling-mud solids on the borehole wall) and for the roughness of the borehole wall have been applied.

Porosity →

Average porosity over thickness

\bar\phi = \frac{\sum_i \phi_i\,h_i}{\sum_i h_i}

all thicknesses in the same units; porosity as a fraction

Holds when: Only reservoir layers enter the sum. Each $\phi_i$ represents its whole interval. For logs this holds automatically; for core it holds only if the samples are spread evenly.

Porosity →

Volumetric calculation of oil in place

N = \frac{A\,h\,\phi\,(1 - S_w)}{B_o}

SI: area in m², thickness in m; the result is m³ of oil at standard conditions

Holds when: The parameters are averages over the whole accumulation, and they match each other: porosity and saturation come from the same definition. The accumulation contains oil and irreducible water only, with no free gas.

Porosity →

Exponential trend of normal compaction

\phi(z) = \phi_0\,\e^{-c\,z}

$c$ and $z$ in matching inverse units (for example, km⁻¹ and km)

Holds when: Normal compaction: the pore pressure is close to hydrostatic (the pressure of a column of water). One lithology only. Cementation and dissolution do not disturb the trend.

Porosity →

Saturation

S_w = \frac{V_w}{V_p},\qquad S_o = \frac{V_o}{V_p},\qquad S_g = \frac{V_g}{V_p},\qquad S_w + S_o + S_g = 1

any units of volume, the same for all; saturations have no units

Holds when: The phase volumes and the pore volume refer to the same conditions. For reservoir saturations, these are reservoir pressure and temperature. Saturation is measured against the same pore volume as the porosity you use (open or total). Otherwise the bound water will be counted wrongly.

Fluid saturation →

Capillary pressure at a height above the free water level

p_c(h) = (\rho_w - \rho_o)\,g\,h

Holds when: The accumulation is in hydrostatic and capillary equilibrium. This is the case before production starts, or after a long shut-in (wells closed for a long time). Water and oil each form a continuous (unbroken) phase over the whole height of the accumulation.

Fluid saturation →

Saturation from extraction data

S_w = \frac{V_w}{V_p},\qquad S_o = \frac{m_1 - m_2 - \rho_w V_w}{\rho_o\,V_p}

g, cm³, g/cm³

Holds when: The water and oil are fully extracted and the solvent has been removed by drying. The salts of the formation water stay in the sample when the water evaporates, and they are not accounted for. For salty waters a correction is applied to the water volume. The oil density is known (from a sample of stock-tank oil).

Fluid saturation →

Water saturation from Archie's equation (clean reservoir)

S_w = \left(\frac{a\,R_w}{\phi^m\,R_t}\right)^{1/n}

resistivities in the same units (Ω·m); the other quantities have no units

Holds when: Only the formation water conducts current: there are no clays or conductive minerals. (Other models are used for shaly rocks: Water saturation in shaly sands.) $R_t$ has been corrected for the effect of the borehole and the invaded zone, and $R_w$ has been converted to reservoir temperature.

Fluid saturation →

Average water saturation over a section

\bar S_w = \frac{\sum_i S_{w,i}\,\phi_i\,h_i}{\sum_i \phi_i\,h_i}

all thicknesses in the same units; porosity and saturation as fractions

Holds when: Only reservoir layers enter the sum. Porosity and saturation have been obtained consistently, from the same definition of porosity.

Fluid saturation →

Displacement efficiency

E_D = \frac{S_{oi} - S_{or}}{S_{oi}} = \frac{1 - S_{wi} - S_{or}}{1 - S_{wi}}

fractions

Holds when: The volume considered is completely swept: the injected water has reached every part of it (the sweep efficiency equals one). The pressure is maintained, so the oil formation volume factor is the same at the start and at the end. If it is not, do the calculation in standard volumes (volumes at surface conditions). $S_{or}$ was measured in the laboratory for the same displacing agent, at conditions close to reservoir conditions.

Fluid saturation →

Water saturation profile from the Brooks–Corey model

S_w(h) = S_{wirr} + (1 - S_{wirr})\left(\frac{p_d}{(\rho_w - \rho_o)\,g\,h}\right)^{\lambda},\qquad h > h_d = \frac{p_d}{(\rho_w - \rho_o)\,g}

Holds when: A drainage curve: the accumulation formed as oil displaced water and has not been flushed since. The rock is the same over the whole height. For a layered reservoir, build the profile for each rock type separately. The parameters $p_d$ , $\lambda$ and $S_{wirr}$ are fitted to the capillary pressure curves of core, after conversion to reservoir conditions.

Fluid saturation →

Attenuation of an X-ray beam

I = I_0\,\e^{-\mu h}

$\mu$ and $h$ in inverse units (for example 1/cm and cm)

Holds when: A narrow beam of one energy. In a real beam with a range of energies the soft (low-energy) part of the spectrum is absorbed first, and processing allows for this. The material along the beam is uniform. For a non-uniform material the exponent holds the sum $\mu_i h_i$ over the sections of the path.

Core analysis →

Accuracy of the mean and the number of samples needed

P_s = \frac{200\,C_v}{\sqrt{N_s}}\ \%,\qquad N_0 = (10\,C_v)^2

dimensionless quantities

Holds when: The samples are independent and were taken without choosing “convenient” places. The interval ±2 standard errors of the mean corresponds to a confidence level of about 95%. With a small number of samples it is somewhat wider. The 20% threshold is a convention, not a law.

Core analysis →

Darcy's law for linear flow

q = \frac{k A}{\mu}\cdot\frac{p_1 - p_2}{L}

SI (any consistent system of units)

Holds when: Steady-state flow: the pressures and the flow rate do not change with time. Laminar (slow) flow, with inertial forces negligibly small. The pores are filled with one fluid (single-phase flow). The fluid is incompressible and does not react with the rock. The flow is horizontal. For inclined flow, instead of the pressure difference we take the potential difference, which allows for the weight of the column of fluid.

Rock permeability →

Steady-state radial flow (the Dupuit equation)

q = \frac{2\pi k h\,(p_e - p_w)}{\mu\,\ln(r_e/r_w)}

Holds when: Steady state: the pressure at the outer boundary $r_e$ is kept constant. A uniform isotropic reservoir (the same in all directions) of constant thickness; the well penetrates the whole thickness. Single-phase flow of a slightly compressible fluid (one whose volume hardly changes with pressure), laminar flow. No damage to or improvement of the near-wellbore zone, the rock close to the well (the skin factor is zero).

Rock permeability →

Mean permeability of layers working in parallel

\bar{k} = \frac{\sum_i k_i h_i}{\sum_i h_i}

any, the same for all layers

Holds when: The flow goes along the layers, and the pressure difference is the same for all the layers. There is no crossflow between the layers (fluid moving from one layer to another), or it does not change the picture.

Rock permeability →

Mean permeability of layers working in series

\bar{k} = \frac{\sum_i L_i}{\sum_i L_i / k_i}

any, the same for all layers

Holds when: The flow crosses the layers one after another, and the flow rate through all the layers is the same.

Rock permeability →

Darcy's law for linear flow of gas

q_b = \frac{k A\,(p_1^2 - p_2^2)}{2\,\mu\,L\,p_b}

SI (any consistent system)

Holds when: Steady isothermal flow of an ideal gas. Laminar flow without slippage (the Klinkenberg correction is added separately). All pressures are absolute.

Rock permeability →

Klinkenberg correction

k_g = k_\infty\left(1 + \frac{b}{\bar p}\right)

any consistent units

Holds when: Single-phase gas flow at low velocities. The pressure is absolute, not gauge.

Rock permeability →

The Forchheimer equation

-\frac{\dd p}{\dd x} = \frac{\mu}{k}\,v + \beta\,\rho\,v^2

Holds when: Single-phase flow; $\beta$ is found from a test or from correlations with permeability and porosity.

Rock permeability →

The Kozeny–Carman equation (spherical grains)

k = \frac{d^2\,\phi^3}{180\,(1 - \phi)^2}

SI ( $k$ comes out in the same square units as $d^2$ )

Holds when: An unconsolidated (not cemented) packing of spherical grains of one size. The coefficient 180 allows empirically for the winding and shape of the channels; in the Blake–Kozeny variant (and in the Ergun equation) 150 is used.

Rock permeability →

The Timur correlation (permeability from porosity and irreducible water saturation)

k \approx 10^{4}\,\frac{\phi^{4.5}}{S_{wi}^{2}}

empirical: $k$ in mD, $\phi$ and $S_{wi}$ as fractions

Holds when: An empirical relation obtained for sandstones; the constants are fitted to the core of a particular reservoir. $S_{wi}$ must really be irreducible (the reservoir is above the transition zone).

Rock permeability →

Isothermal compressibility

c = -\frac{1}{V}\,\frac{\dd V}{\dd p}

1/MPa or 1/psi: $1\ \text{MPa}^{-1} = 6.895\cdot10^{-3}\ \text{psi}^{-1}$ ; $10^{-6}\ \text{psi}^{-1} = 0.145\cdot10^{-3}\ \text{MPa}^{-1}$

Holds when: The temperature is constant. The minus sign makes the compressibility positive: as the applied pressure grows, the volume decreases. For a rock you must say which volume you mean and which pressure you change, the external or the pore pressure.

Rock compressibility and the effect of stress on core properties →

Pore volume compressibility

c_f = \frac{1}{V_p}\left(\frac{\partial V_p}{\partial p}\right)_{\sigma_v} = -\frac{1}{V_p}\,\frac{\partial V_p}{\partial \sigma'}

consistent units of pressure and compressibility

Holds when: The overburden pressure is constant, and only the pore pressure changes. The effective stress equals the difference between the overburden and pore pressures, so a growth of the stress by 1 MPa acts on the pores in the same way as a fall of the pore pressure by 1 MPa. The value refers to a certain level of stress: as the load grows, $c_f$ usually decreases.

Rock compressibility and the effect of stress on core properties →

The link between pore compressibility and bulk compressibility

c_f \approx \frac{c_b}{\phi}

any identical units of compressibility

Holds when: The grains are incompressible: the whole volume by which the rock has shrunk is lost by the pores. Both compressibilities refer to the same kind of loading.

Rock compressibility and the effect of stress on core properties →

Change of the pore volume when the reservoir pressure falls

\Delta V_p = c_f\,V_{pi}\,\Delta p,\qquad V_p = V_{pi}\,(1 - c_f\,\Delta p)

consistent units of pressure and compressibility: MPa and 1/MPa, or psi and 1/psi

Holds when: $c_f$ is constant over the range of pressure considered. For large changes, take the dependence of the pore volume on stress directly from a test. $c_f\Delta p \ll 1$ . Otherwise use the exact form $V_p = V_{pi}\,\e^{-c_f\Delta p}$ . The pressure has fallen in the whole volume considered.

Rock compressibility and the effect of stress on core properties →

Pore compressibility from a test with stepwise confining pressure

c_f^{\text{lab}} = \frac{1}{V_p}\cdot\frac{\Delta V_w}{\Delta\sigma_c}

volumes in identical units; pressure in MPa or psi

Holds when: The pore pressure is constant (atmospheric), and only the confining pressure changes. A growth of the confining pressure acts on the pores in the same way as an equal fall of the pore pressure. This is true while the grains can be considered incompressible. For tight low-porosity rocks the test is run at the real overburden and pore pressures. The sample is squeezed equally from all sides.

Rock compressibility and the effect of stress on core properties →

Converting the pore compressibility from all-round compression to uniaxial strain

c_f = c_f^{\text{lab}}\;\frac{1 + \nu}{3\,(1 - \nu)}

any identical units of compressibility

Holds when: The rock is elastic and the same in all directions, and the grains are incompressible. The overburden pressure is constant and there are no lateral strains: the reservoir is wide compared with its thickness. $\nu$ was measured on the same rock under slow loading, when the fluid has time to leave the pores.

Rock compressibility and the effect of stress on core properties →

Exponential dependence of permeability on the reservoir pressure

k = k_i\,\e^{-\gamma\,(p_i - p)}

$\gamma$ and pressure in mutually inverse units

Holds when: An empirical approximation for a limited range of pressures. $\gamma$ is found from tests on core under load, or from the history of well productivity. The overburden pressure is constant. We take the change as reversible; for weak rocks this is not so.

Rock compressibility and the effect of stress on core properties →

Compaction of the reservoir when the pressure falls

\Delta h \approx \phi\,c_f\,h\,\Delta p

consistent units of pressure and compressibility; $\Delta h$ in the same units as $h$

Holds when: The reservoir is wide compared with its thickness and is compressed only vertically. The grains are incompressible, so the volume of the rock decreases exactly by as much as the volume of the pores decreased. The pressure has fallen over the whole thickness $h$ .

Rock compressibility and the effect of stress on core properties →

Four compressibilities of a porous rock

\begin{aligned} c_{bc} &= -\frac{1}{V_b}\frac{\partial V_b}{\partial \sigma_c}, & c_{bp} &= \frac{1}{V_b}\frac{\partial V_b}{\partial p},\\[4pt] c_{pc} &= -\frac{1}{V_p}\frac{\partial V_p}{\partial \sigma_c}, & c_{pp} &= \frac{1}{V_p}\frac{\partial V_p}{\partial p}\end{aligned}

consistent units of pressure and compressibility

Holds when: The rock is elastic and uniform, and the grains consist of one mineral with compressibility $c_r$ . When one pressure changes, the other is constant. The loading is all-round.

Rock compressibility and the effect of stress on core properties →

Relations between the compressibilities

c_{bp} = c_{bc} - c_r,\qquad c_{pc} = \frac{c_{bc} - c_r}{\phi},\qquad c_{pp} = c_{pc} - c_r

any identical units of compressibility

Holds when: The same as for the definitions of the four compressibilities.

Rock compressibility and the effect of stress on core properties →

Growth of the mean effective stress in uniaxial strain

\Delta\sigma'_m = \frac{1}{3}\left(1 + \frac{2\nu}{1 - \nu}\right)\Delta p = \frac{1 + \nu}{3\,(1 - \nu)}\,\Delta p

any identical units of pressure

Holds when: An elastic isotropic rock and incompressible grains (the Biot coefficient is 1). A constant overburden pressure and zero lateral strains.

Rock compressibility and the effect of stress on core properties →

The Hall correlation for the pore compressibility of cemented rocks

c_f \approx \frac{0.2585}{\phi^{0.438}}\cdot10^{-3}\ \text{MPa}^{-1}

empirical: $\phi$ as a fraction, $c_f$ in 1/MPa

Holds when: Cemented sandstones and limestones with a porosity of roughly 2 to 25%. Not applicable to weakly cemented and loose rocks, to chalk or to fractured reservoirs. Later tests showed a scatter of several times around this relation, so it is suitable only for a preliminary estimate.

Rock compressibility and the effect of stress on core properties →

Young's equation (equilibrium at the contact line)

\sigma_{so} - \sigma_{sw} = \sigma_{ow}\cos\theta

any units of tension, the same for all three (N/m, mN/m)

Holds when: The surface is smooth, chemically uniform and non-deformable. The system has come to equilibrium: for oil with polar components this takes tens or hundreds of hours of contact with the mineral. The angle is measured through the water; measured through the oil it would be $180^\circ - \theta$ .

Wettability and interfacial phenomena →

Amott and Amott–Harvey wettability indices

I_w = \frac{V_{o1}}{V_{o1} + V_{o2}},\qquad I_o = \frac{V_{w1}}{V_{w1} + V_{w2}},\qquad I_{AH} = I_w - I_o

volumes in the same units; the indices have no units

Holds when: Spontaneous imbibition lasts the same agreed time in every test. Slow imbibition may not have time to finish, and then the index comes out too low. The volumes are measured accurately. When the moveable saturation is small, small errors change the ratios a lot. The sample keeps the reservoir wettability.

Wettability and interfacial phenomena →

USBM wettability index

W = \lg\frac{A_1}{A_2}

both areas in the same units; $\lg$ is the base-10 logarithm

Holds when: Both curves are obtained on the same sample over the same range of capillary pressure, for example from −70 to +70 kPa (about ±10 psi). Only forced displacement counts: spontaneous imbibition happens at $p_c \approx 0$ , so it adds almost nothing to the area.

Wettability and interfacial phenomena →

Laplace equation for a curved interface

p_c = p_{\text{nw}} - p_{\text{w}} = \sigma\left(\frac{1}{R_1} + \frac{1}{R_2}\right)

Holds when: The interface is stationary or moves very slowly (capillary equilibrium). The interfacial tension is the same over the whole interface.

Capillary pressure and the transition zone →

Capillary pressure in a cylindrical channel

p_c = \frac{2\sigma\cos\theta}{r}

Holds when: The channel has a circular cross-section, and the meniscus is part of a sphere. The contact angle is measured through the wetting phase, so $\cos\theta > 0$ . The system is in equilibrium: the meniscus is stationary.

Capillary pressure and the transition zone →

Height of capillary rise

(\rho_w - \rho_o)\,g\,h = \frac{2\sigma\cos\theta}{r}\quad\Rightarrow\quad h = \frac{2\sigma\cos\theta}{(\rho_w - \rho_o)\,g\,r}

Holds when: Capillary and hydrostatic equilibrium. The tube is water-wet, and water is the wetting phase.

Capillary pressure and the transition zone →

Conversion of capillary pressure from laboratory to reservoir fluids

p_{c,\text{res}} = p_{c,\text{lab}}\,\frac{(\sigma\cos\theta)_{\text{res}}}{(\sigma\cos\theta)_{\text{lab}}}

both pressures in the same units; all tensions in the same units

Holds when: The pore structure is the same in the test and in the reservoir (shaly and poorly cemented rocks are tested under reservoir confining stress, the squeeze from the surrounding rock). The contact angles are measured through the wetting fluid: for mercury this is the angle through the vacuum, about 40°, not 140°. The wettability of the reservoir is known. This is the weakest link of the conversion.

Capillary pressure and the transition zone →

Leverett J-function

J(S_w) = \frac{p_c}{\sigma\cos\theta}\sqrt{\frac{k}{\phi}}

SI (any consistent system)

Holds when: The samples belong to one rock type, with similar pore structure and tortuosity (how winding the flow paths are). $k$ , $\phi$ and the $p_c$ curve are measured on the same sample. If the contact angle is unknown, $\cos\theta$ is sometimes left out. Then it must be left out in the reverse conversion too.

Capillary pressure and the transition zone →

Normalised water saturation

S_w^{*} = \frac{S_w - S_{wirr}}{1 - S_{wirr}}

fraction

Holds when: $S_{wirr}$ is measured on the same sample, or for the same rock type, as the curve.

Capillary pressure and the transition zone →

Capillary pressure above the free water level

p_o(h) - p_w(h) = p_c(h) = (\rho_w - \rho_o)\,g\,h

Holds when: Hydrostatic and capillary equilibrium: the accumulation is not in production, or has been shut in for a long time. Both fluids are continuous over the whole height, and the formation water is not flowing (flow does not tilt the contact). For a gas accumulation, use the gas density in place of the oil density. For a gas–oil contact, use the gas–oil pair of fluids.

Capillary pressure and the transition zone →

Darcy's law for each phase in multiphase flow

q_o = \frac{k\,k_{ro}\,A}{\mu_o}\cdot\frac{\Delta p_o}{L},\qquad q_w = \frac{k\,k_{rw}\,A}{\mu_w}\cdot\frac{\Delta p_w}{L}

SI (any consistent system; in Darcy units, as for single-phase flow)

Holds when: Steady-state flow at constant saturation. Horizontal flow. For inclined flow the weight of the column of the phase is added. The pressure differences in the phases differ by the change of capillary pressure along the length. In tests we try to make this difference negligibly small.

Relative permeability →

Relative permeability

k_{rx}(S) = \frac{k_x(S)}{k_{\text{base}}}

identical units of permeability

Holds when: The base of normalisation is stated explicitly: the absolute permeability (to gas with the Klinkenberg correction, or to water), or the effective permeability to oil at irreducible water saturation $k_o(S_{wi})$ . To calculate a rate, we multiply the relative permeabilities by the same permeability that they are referred to.

Relative permeability →

The Corey model for the oil–water pair

S_n = \frac{S_w - S_{wi}}{1 - S_{wi} - S_{or}},\qquad k_{rw} = k_{rw}'\,S_n^{\,n_w},\qquad k_{ro} = k_{ro}'\,(1 - S_n)^{n_o}

saturations and relative permeabilities as fractions

Holds when: An empirical approximation: the parameters are fitted to the laboratory points. Outside the interval from $S_{wi}$ to $1 - S_{or}$ a phase is immobile: $k_{rw} = 0$ at $S_w \le S_{wi}$ and $k_{ro} = 0$ at $S_w \ge 1 - S_{or}$ . One set of parameters describes one process, drainage or imbibition.

Relative permeability →

Fractional flow of water

f_w = \frac{q_w}{q_w + q_o} = \frac{1}{1 + \dfrac{k_{ro}}{k_{rw}}\cdot\dfrac{\mu_w}{\mu_o}}

dimensionless; viscosities in identical units

Holds when: Horizontal flow: gravity is not taken into account. The gradient of capillary pressure is neglected. The flow rates are at reservoir conditions, and there is no free gas.

Relative permeability →

Water cut of the production at the surface

\text{WC} = \frac{f_w/B_w}{f_w/B_w + (1 - f_w)/B_o}

dimensionless

Holds when: Only oil and water flow in the reservoir. We convert the reservoir flow rates to standard conditions with the formation volume factors.

Relative permeability →

The mobility ratio at the end points

M' = \frac{k_{rw}'/\mu_w}{k_{ro}'/\mu_o}

dimensionless

Holds when: We take the end points from the curves of the same process, usually imbibition. The viscosities are at reservoir conditions.

Relative permeability →

Two-phase Darcy's law in differential form

u_o = -\frac{k\,k_{ro}}{\mu_o}\left(\frac{\partial p_o}{\partial x} + \rho_o g\sin\alpha\right),\qquad u_w = -\frac{k\,k_{rw}}{\mu_w}\left(\frac{\partial p_w}{\partial x} + \rho_w g\sin\alpha\right),\qquad p_o - p_w = p_c(S_w)

Holds when: Immiscible phases, laminar flow. The relative permeabilities and the capillary pressure depend only on saturation and on the direction of its change. At $\alpha > 0$ the weight of a phase holds back its movement up the dip.

Relative permeability →

Average permeability of ring zones in radial inflow

\bar k = \frac{\ln(r_e/r_w)}{\sum_{j=1}^{n} \ln(r_j/r_{j-1})/k_j}

permeabilities in any identical units; radii in identical units (only the ratios enter)

Holds when: The flow is steady-state and single-phase. The zones are coaxial cylinders (they share one axis, the well) across the whole thickness of the reservoir, and each zone is uniform. The pressure at the outer boundary is constant. With one inner zone, the formula is equivalent to the Hawkins skin factor: $\bar k = k\ln(r_e/r_w)/[\ln(r_e/r_w) + S]$ (Skin factor).

Reservoir heterogeneity and permeability averaging →

Means of log-normally distributed permeability

\bar k_{\text{A}} = k_G\,e^{\sigma^2/2},\qquad \bar k_{\text{H}} = k_G\,e^{-\sigma^2/2},\qquad \frac{\bar k_{\text{A}}}{\bar k_{\text{H}}} = e^{\sigma^2}

permeabilities in any identical units; $\sigma$ is dimensionless

Holds when: Permeability is log-normally distributed. The means refer to the whole population (a large sample, with volumes as weights). From about ten samples the formulas give only a rough guide, because small samples represent the tails of the distribution poorly.

Reservoir heterogeneity and permeability averaging →

Effective aspect ratio (length to thickness)

R_L = \frac{L}{H}\sqrt{\frac{k_v}{k_h}} = \frac{L}{I\,H}

any identical units of length and of permeability

Holds when: $k_v$ and $k_h$ are the equivalent permeabilities of the unit in the two directions. This is an estimate of the order of magnitude: the number compares scales and is not an exact criterion.

Reservoir heterogeneity and permeability averaging →

Lorenz coefficient of log-normal permeability

L_c = 2\,\Phi\!\left(\frac{\sigma_{\ln k}}{\sqrt{2}}\right) - 1

dimensionless

Holds when: Permeability is log-normal and the porosity of all the layers is the same. Otherwise the layers are sorted by $k/\phi$ and weighted by $\phi h$ , and the formula is only a rough guide. There are many layers, so the Lorenz curve is smooth.

Reservoir heterogeneity and permeability averaging →

Power mean of permeability

k_\omega = \left(\frac{\sum_i w_ik_i^{\,\omega}}{\sum_i w_i}\right)^{1/\omega}

permeabilities in any identical units

Holds when: The exponent $\omega$ is an empirical parameter of how the patches are arranged. We fit it by calculating the flow on a detailed model. For log-normal permeability, $k_\omega = k_Ge^{\omega\sigma^2/2}$ .

Reservoir heterogeneity and permeability averaging →

Symbols

Symbol	Meaning and context
$\bar p$	mean pressure in the sample, $(p_1 + p_2)/2$ , absolute, atm or MPaRock permeability
$\Delta p$	fall of the reservoir pressure, $p_i - p$ , MPaRock compressibility and the effect of stress on core properties fall of the reservoir pressure, MPaRock compressibility and the effect of stress on core properties fall of the pore pressure, MPaRock compressibility and the effect of stress on core properties
$\Delta h$	decrease of the thickness of the reservoir, mRock compressibility and the effect of stress on core properties
$\bar k$	permeability of a uniform reservoir that gives the same inflow, mDReservoir heterogeneity and permeability averaging
$(\sigma\cos\theta)_{\text{res}},\ (\sigma\cos\theta)_{\text{lab}}$	adhesion tensions of the reservoir and the laboratory pairs, N/mCapillary pressure and the transition zone
$A$	oil-bearing area, m²Porosity area of the cross-section perpendicular to the flow, m²Rock permeability
$\bar k_{\text{A}},\ \bar k_{\text{H}}$	arithmetic and harmonic means, mDReservoir heterogeneity and permeability averaging
$A_1$	area under the curve of displacement of water by oil (the capillary pressure is positive), kPa × fractions of pore volumeWettability and interfacial phenomena
$A_2$	area above the curve of displacement of oil by water (the capillary pressure is negative), in absolute value, the sameWettability and interfacial phenomena
$A,\ L$	cross-section area and length, m², mRelative permeability
$a,\ m,\ n$	empirical coefficients (found from experiments). For sandstones they are often $a \approx 1$ and $m \approx n \approx 2$ , but they should be measured on core, —Fluid saturation
$b$	slip coefficient, which depends on the gas and on the size of the channels, the same units of pressureRock permeability
$B_o$	oil formation volume factor — the number of reservoir cubic metres needed for one cubic metre of oil at standard conditions (see Black oil model: solution gas–oil ratio and formation volume factors), m³/m³Porosity
$B_w,\ B_o$	formation volume factors of water and of oil, m³/m³Relative permeability
$c$	compaction coefficient. It is fitted to the data of a particular basin, and it is larger for clays than for sands, km⁻¹Porosity compressibility, 1/MPaRock compressibility and the effect of stress on core properties
$c_b$	bulk compressibility of the rock (per unit of rock volume), 1/MPaRock compressibility and the effect of stress on core properties
$c_{bc},\ c_{bp}$	compressibility of the rock as a whole when the external and the pore pressure change, 1/MPaRock compressibility and the effect of stress on core properties
$c_{bc},\ c_{bp},\ c_{pc},\ c_{pp}$	as in the previous formula, 1/MPaRock compressibility and the effect of stress on core properties
$c_f$	pore volume compressibility, 1/MPaRock compressibility and the effect of stress on core properties pore compressibility (per unit of pore volume), 1/MPaRock compressibility and the effect of stress on core properties pore volume compressibility at reservoir conditions, 1/MPaRock compressibility and the effect of stress on core properties pore compressibility at reservoir conditions, 1/MPaRock compressibility and the effect of stress on core properties pore compressibility at reservoir conditions, mean for the range of pressures, 1/MPaRock compressibility and the effect of stress on core properties
$c_f^{\text{lab}}$	pore compressibility under all-round compression, 1/MPaRock compressibility and the effect of stress on core properties pore compressibility measured under all-round compression, 1/MPaRock compressibility and the effect of stress on core properties
$c_{pc},\ c_{pp}$	compressibility of the pore volume when the external and the pore pressure change, 1/MPaRock compressibility and the effect of stress on core properties
$c_r$	grain compressibility, 1/MPaRock compressibility and the effect of stress on core properties
$C_v$	coefficient of variation: the standard deviation divided by the mean, —Core analysis
$d$	diameter of the grains, mRock permeability
$E_D$	displacement efficiency: the share of the initial oil displaced from a completely swept volume, fractionFluid saturation
$f_w$	share of water in the total flow at reservoir conditions, —Relative permeability
$g$	acceleration due to gravity, 9.81, m/s²Fluid saturation
$h$	average net oil-saturated thickness, mPorosity height above the free water level, mFluid saturation thickness of the sample along the beam, cmCore analysis net thickness of the reservoir, mRock permeability thickness of the reservoir over which the pressure has fallen, mRock compressibility and the effect of stress on core properties height of the meniscus above the level where the pressures of the two fluids are equal, mCapillary pressure and the transition zone
$H$	thickness of the unit, mReservoir heterogeneity and permeability averaging
$h_d$	height of the OWC above the free water level; below it $S_w = 1$ , mFluid saturation
$h_i$	net thickness of the $i$ -th layer, mPorosity thickness of layer $i$ , mRock permeability
$I$	anisotropy index $\sqrt{k_h/k_v}$ , as in the lesson Horizontal wells, —Reservoir heterogeneity and permeability averaging
$I_0,\ I$	intensity of the beam before and after the sample, the same unitsCore analysis
$I_{AH}$	Amott–Harvey index, from −1 to +1Wettability and interfacial phenomena
$I_o$	oil displacement index: the fraction of the water displaced by oil that was displaced spontaneously, —Wettability and interfacial phenomena
$I_w$	water displacement index: the fraction of the oil displaced by water that was displaced spontaneously, —Wettability and interfacial phenomena
$J$	dimensionless capillary pressure, —Capillary pressure and the transition zone
$k$	permeability of the rock, m²Rock permeability permeability of the reservoir (horizontal), m²Rock permeability permeability, m²Rock permeability permeability of the sample, m²Capillary pressure and the transition zone absolute (base) permeability, m²Relative permeability
$k_\infty$	equivalent liquid permeability (the limit at infinite pressure), mDRock permeability
$k_{\text{base}}$	base permeability, mDRelative permeability
$k_g$	permeability measured with gas, mDRock permeability
$k_G$	geometric mean, which is also the median (the middle value) of the log-normal distribution, mDReservoir heterogeneity and permeability averaging
$k_i$	permeability of layer $i$ , mDRock permeability
$k_i,\ k$	permeability at the initial and at the current reservoir pressure, mDRock compressibility and the effect of stress on core properties
$k_i,\ w_i$	permeability of a patch and its weight (volume or thickness), mD, —Reservoir heterogeneity and permeability averaging
$k_j$	permeability of zone $j$ , the ring between the radii $r_{j-1}$ and $r_j$ , mDReservoir heterogeneity and permeability averaging
$k_{ro},\ k_{rw}$	relative permeabilities to oil and to water at the current saturation, —Relative permeability relative permeabilities of oil and water at the current water saturation, —Relative permeability
$k_{rw}',\ k_{ro}'$	the end points: the relative permeability of water at residual oil, and of oil at irreducible water, —Relative permeability end points of the relative permeability of water and of oil, —Relative permeability
$k_{rx}$	relative permeability of the phase, fractionRelative permeability
$k_v,\ k_h$	vertical and horizontal permeability of the unit, mDReservoir heterogeneity and permeability averaging
$k_x(S)$	effective permeability for phase $x$ (oil, water, gas) at saturation $S$ , mDRelative permeability
$k_\omega$	power mean with exponent $\omega$ , mDReservoir heterogeneity and permeability averaging
$k,\ A,\ \mu,\ L$	permeability, area, gas viscosity and length of the sample, as in Darcy's law, m², m², Pa·s, mRock permeability
$L$	length of the flow path, mRock permeability distance along the reservoir, for example between an injection well and a producing well, mReservoir heterogeneity and permeability averaging
$L_c$	Lorenz coefficient, —Reservoir heterogeneity and permeability averaging
$L_i$	length of the path in layer $i$ (for vertical flow, the thickness of the layer), mRock permeability
$m_1$	mass of the sample before extraction, gFluid saturation
$m_2$	mass of the sample after extraction and drying, gFluid saturation
$M'$	ratio of the mobility of the water behind the displacement front (the moving boundary between the injected water and the oil) to the mobility of the oil ahead of it, —Relative permeability
$N$	stock-tank oil initially in place (STOIIP): the oil in place, converted to standard conditions (surface pressure and temperature), m³Porosity
$N_0$	number of samples at which $P_s = 20$ %, —Core analysis
$N_s$	number of samples, —Core analysis
$n_w,\ n_o$	Corey exponents for water and oil, —Relative permeability
$\Delta p_o,\ \Delta p_w$	pressure differences in the oil and in the water over the length $L$ , PaRelative permeability
$p$	pressure applied to this volume, MPaRock compressibility and the effect of stress on core properties pore (reservoir) pressure, MPaRock compressibility and the effect of stress on core properties
$\Delta V_p$	decrease of the pore volume, m³Rock compressibility and the effect of stress on core properties
$p_0$	initial pressure in the chamber with the sample, usually atmospheric, kPa, absolutePorosity
$p_1$	initial gas pressure in the reference chamber, kPa, absolutePorosity
$p_1,\ p_2$	pressure at the inlet and at the outlet, PaRock permeability absolute pressure at the inlet and the outlet of the sample, PaRock permeability
$p_2$	final (equalised) pressure after the valve is opened, kPa, absolutePorosity
$p_b$	absolute pressure at which the flow rate is measured (usually close to atmospheric), PaRock permeability
$p_c$	capillary pressure: the difference between the oil and water pressures at height $h$ , PaFluid saturation capillary pressure, PaCapillary pressure and the transition zone capillary pressure at water saturation $S_w$ , PaCapillary pressure and the transition zone
$p_{c,\text{lab}}$	capillary pressure measured in the laboratory, PaCapillary pressure and the transition zone
$p_{c,\text{res}}$	capillary pressure of the reservoir pair of fluids at the same saturation, PaCapillary pressure and the transition zone
$p_c(S_w)$	capillary pressure at the current water saturation, PaRelative permeability
$p_d$	threshold (entry) pressure: the capillary pressure at which oil begins to enter the largest pores, PaFluid saturation
$p_e$	pressure at the outer boundary of the drainage area, PaRock permeability
$p_i - p$	fall of the pore pressure; at constant overburden pressure it equals the growth of the effective stress, MPaRock compressibility and the effect of stress on core properties
$p_{\text{nw}},\ p_{\text{w}}$	pressures of the non-wetting and the wetting phase at the interface, PaCapillary pressure and the transition zone
$p_o,\ p_w$	pressures in the oil and in the water, PaRelative permeability
$p_o(h),\ p_w(h)$	oil and water pressures at height $h$ , PaCapillary pressure and the transition zone
$P_s$	half-width of the interval in which the true mean lies with a probability of about 95%, as a percentage of the mean over the samples, %Core analysis
$p_w$	pressure at the well wall (bottomhole), PaRock permeability
$q$	volumetric flow rate of the fluid at the conditions of the test (in the reservoir, at reservoir conditions), m³/sRock permeability rate at reservoir conditions; the surface rate is $q/B$ , where $B$ is the formation volume factor (the ratio of the volume of the oil in the reservoir to its volume at the surface), m³/sRock permeability
$q_b$	volumetric flow rate of the gas at the measurement pressure $p_b$ , m³/sRock permeability
$q_o,\ q_w$	flow rates of oil and water at reservoir conditions, m³/sRelative permeability
$r$	radius of the channel; in a rock, the effective radius of the pore throat through which the meniscus passes, mCapillary pressure and the transition zone
$R_1,\ R_2$	principal radii of curvature of the interface (how sharply it bends in two perpendicular directions), mCapillary pressure and the transition zone
$r_e$	radius of the drainage area (the area that feeds the well), mRock permeability
$r_j$	outer radius of zone $j$ ; $r_0 = r_w$ , $r_n = r_e$ , mReservoir heterogeneity and permeability averaging
$R_L$	effective aspect ratio — the ratio of length to thickness, corrected for anisotropy, —Reservoir heterogeneity and permeability averaging
$R_t$	resistivity of the uninvaded part of the formation, beyond the invaded zone (the zone around the well that mud filtrate has entered), Ω·mFluid saturation
$r_w$	radius of the well, mRock permeability
$R_w$	resistivity of the formation water at reservoir temperature, Ω·mFluid saturation
$r_w,\ r_e$	radius of the well and radius of the outer boundary of the drainage area (the part of the reservoir that feeds the well), mReservoir heterogeneity and permeability averaging
$S_n$	normalised water saturation: 0 at $S_{wi}$ , 1 at $1 - S_{or}$ , —Relative permeability
$S_{oi} = 1 - S_{wi}$	initial oil saturation (no free gas), fractionFluid saturation
$S_{or}$	residual oil saturation for the given displacing agent (water or gas), fractionFluid saturation
$S_w$	average water saturation (see Fluid saturation), fractionPorosity
$S_w^{*}$	normalised water saturation: 0 at the irreducible value, 1 at full water saturation, fractionCapillary pressure and the transition zone
$S_w,\ S_o,\ S_g$	water, oil and gas saturation, fraction (or % of pore volume)Fluid saturation
$S_{w,i}$	water saturation of the $i$ -th layer, fractionFluid saturation
$S_{wi}$	initial water saturation; in the main part of the accumulation it equals the irreducible one, fractionFluid saturation irreducible (residual) water saturation, fractionRock permeability
$S_{wi},\ S_{or}$	irreducible water saturation and residual oil saturation, fractionRelative permeability
$S_{wirr}$	irreducible water saturation of the given sample or layer, fractionCapillary pressure and the transition zone
$S_{wirr},\ \rho_w,\ \rho_o,\ g,\ h$	as above in the lesson, —, kg/m³, m/s², mFluid saturation
$u_o,\ u_w$	flow velocities of oil and water: the flow rate per unit area, m/sRelative permeability
$v$	flow velocity (the flow rate divided by the area), m/sRock permeability
$V$	the volume considered, m³Rock compressibility and the effect of stress on core properties
$V_b$	volume of the sample or rock element including the pores, cm³, m³Porosity
$V_b,\ V_p$	volume of the rock and volume of the pores, m³Rock compressibility and the effect of stress on core properties
$V_c$	volume of the chamber with the sample (including the channels up to the valve), cm³Porosity
$V_{gr}$	volume of the solid phase: grains, cement and clay minerals, cm³, m³Porosity volume of the sample that the gas cannot reach: grains together with isolated pores, cm³Porosity
$V_{o1},\ V_{o2}$	volumes of oil displaced by water in spontaneous imbibition and then in forced displacement, cm³Wettability and interfacial phenomena
$V_p$	volume of the voids that the given kind of porosity counts, cm³, m³Porosity pore volume (as a rule the open, that is connected, pores), m³, cm³Fluid saturation pore volume of the sample measured after extraction, cm³Fluid saturation pore volume, m³Rock compressibility and the effect of stress on core properties pore volume at the beginning of the interval, cm³Rock compressibility and the effect of stress on core properties
$V_{pi},\ V_p$	pore volume at the initial and at the current pressure, m³Rock compressibility and the effect of stress on core properties
$V_r$	volume of the reference chamber (including the channels up to the valve), cm³Porosity
$V_w$	volume of water collected in the trap, cm³Fluid saturation
$V_w,\ V_o,\ V_g$	volumes of water, oil and gas in the pores at reservoir conditions, m³, cm³Fluid saturation
$V_{w1},\ V_{w2}$	volumes of water displaced by oil in spontaneous imbibition and then in forced displacement, cm³Wettability and interfacial phenomena
$\Delta V_w$	volume of water pushed out of the pores between two steps, cm³Rock compressibility and the effect of stress on core properties
$W$	USBM index, —Wettability and interfacial phenomena
$\text{WC}$	water cut: the share of water in the produced liquid at standard conditions, —Relative permeability
$z$	burial depth, kmPorosity
$\alpha$	angle of the axis $x$ to the horizontal, positive if the axis points up the dip, radRelative permeability
$\beta$	inertial resistance coefficient of the rock, m⁻¹Rock permeability
$\gamma$	coefficient of the sensitivity of permeability to pressure (permeability modulus), 1/MPaRock compressibility and the effect of stress on core properties
$\Delta\sigma'_m$	growth of the mean effective stress, MPaRock compressibility and the effect of stress on core properties
$\Delta\sigma_c$	increase of the confining pressure between the steps, MPaRock compressibility and the effect of stress on core properties
$\theta$	contact angle, measured through the water, degreesWettability and interfacial phenomena contact angle, measured through the wetting phase, degreesCapillary pressure and the transition zone
$\lambda$	pore-size distribution index: the larger it is, the more uniform the pores and the sharper the transition, —Fluid saturation
$\mu$	attenuation coefficient of the material; it grows with density and with the atomic number of the elements, 1/cmCore analysis dynamic viscosity of the fluid, Pa·sRock permeability viscosity of the fluid at reservoir conditions, Pa·sRock permeability
$\mu,\ k$	viscosity and permeability, as in Darcy's law, Pa·s, m²Rock permeability
$\mu_o,\ \mu_w$	viscosities of oil and water, Pa·sRelative permeability
$\mu_w,\ \mu_o$	viscosities of water and oil at reservoir conditions, mPa·sRelative permeability viscosities of water and of oil at reservoir conditions, mPa·sRelative permeability
$\nu$	Poisson's ratio of the rock (how much a rock bulges sideways when it is squeezed lengthwise), —Rock compressibility and the effect of stress on core properties Poisson's ratio, —Rock compressibility and the effect of stress on core properties
$\rho$	density of the fluid, kg/m³Rock permeability
$\rho_b$	bulk density of the rock measured by the tool, g/cm³Porosity
$\rho_f$	density of the fluid in the zone the tool reads, usually mud filtrate (the liquid from the drilling mud that has seeped into the rock): about 1.0 for fresh mud, up to 1.1–1.2 for salty mud, g/cm³Porosity
$\rho_{ma}$	grain (matrix) density of the framework: quartz 2.65, calcite 2.71, dolomite 2.87, g/cm³Porosity
$\rho_o,\ \rho_w$	densities of oil and water, kg/m³Relative permeability
$\rho_w,\ \rho_o$	densities of formation water and oil at reservoir conditions (for a gas accumulation, the gas density instead of $\rho_o$ ), kg/m³Fluid saturation densities of the collected water and of the stock-tank oil (oil with its gas removed), g/cm³Fluid saturation densities of water and oil, kg/m³Capillary pressure and the transition zone densities of formation water and oil at reservoir conditions, kg/m³Capillary pressure and the transition zone
$\sigma$	interfacial tension, N/mCapillary pressure and the transition zone standard deviation of the natural logarithm of permeability, —Reservoir heterogeneity and permeability averaging
$\sigma_{\ln k}$	standard deviation of the natural logarithm of permeability, —Reservoir heterogeneity and permeability averaging
$\sigma,\ \theta,\ r$	as in the formula for a channel, N/m, degrees, mCapillary pressure and the transition zone
$\sigma'$	effective stress, MPaRock compressibility and the effect of stress on core properties
$\sigma_c,\ p$	external (confining) pressure and pore pressure, MPaRock compressibility and the effect of stress on core properties
$\sigma\cos\theta$	adhesion tension of the pair of fluids, N/mCapillary pressure and the transition zone
$\sigma_{ow}$	oil–water interfacial tension, N/mWettability and interfacial phenomena
$\sigma_{so}$	tension of the mineral–oil interface, N/mWettability and interfacial phenomena
$\sigma_{sw}$	tension of the mineral–water interface, N/mWettability and interfacial phenomena
$\sigma_v$	overburden pressure, which stays constant, MPaRock compressibility and the effect of stress on core properties
$\phi$	porosity, fraction (in reports, %)Porosity average open porosity, fractionPorosity porosity (fraction), —Rock permeability porosity of the sample, fractionCapillary pressure and the transition zone
$\Phi$	the standard normal distribution function (the share of a standard bell-shaped distribution that lies below the given value), —Reservoir heterogeneity and permeability averaging
$\phi(z)$	porosity at burial depth $z$ , fractionPorosity
$\phi_0$	porosity of the sediment at the surface: about 0.6 for clay-rich muds and 0.4–0.5 for sands, fractionPorosity
$\phi_D$	porosity from the density log. It is total porosity, because isolated pores also lower the density, fractionPorosity
$\phi_i$	porosity of the $i$ -th layer (interval), fractionPorosity porosity of the $i$ -th layer, fractionFluid saturation
$\omega$	exponent: 1 gives the arithmetic mean and −1 the harmonic mean; as $\omega \to 0$ we get the geometric mean, —Reservoir heterogeneity and permeability averaging