Equations: Formation evaluation from well logs

$v$ in m/h, $b$ in bits, $R$ in bit/s, $\Delta z$ in m; 3600 is the number of seconds in an hour

Holds when: The whole channel carries the logging curves. In practice the directional survey and service data take part of it, and the interval is then larger. The rate of penetration and the telemetry rate are constant. We ignore data compression.

any units of the measured quantity, the same for all; $h$ and $L$ in metres

Holds when: The tool averages the property evenly over a window of length $L$. The real response is smoother: the middle of the window counts for more than the edges. The bed lies between thick enclosing rocks that are identical. The quantity mixes linearly. This is reasonable for GR and density, but not for resistivity, where the mixing depends on the type of sonde. For $h \ge L$ the reading in the middle of the bed is $x_c = x_b$.

noise in the units of the curve (g/cm³, API, or porosity units: one porosity unit is 0.01, or 1%, of porosity); speeds in the same units

Holds when: Counting statistics set the noise: the number of pulses per sample is proportional to $\Delta z/v$. We ignore other interference. Adjacent samples are independent and the sampling interval $\Delta z$ is the same. The bed is uniform over the averaged samples.

m

Holds when: The tool and the filter are uniform windows, as in the thin-bed model (vertical resolution). We replace the two windows by one window with the same second moment (a measure of spread): when two averagings are applied one after the other, their variances (squared spreads) add. Compared with the exact calculation of two windows, the approximation slightly exaggerates the loss of contrast (in the example below, 0.149 against 0.152).

units of the curve being checked

Holds when: Both recordings are depth-matched. We choose the depths in uniform parts of beds, not at boundaries. The errors of the two recordings are independent and equal: the difference then has a standard deviation of $\sigma\sqrt2$. The standard error of the mean difference is $s_\Delta/\sqrt N$. A mean that is several times larger than this error is a systematic offset.

m

Holds when: Between markers the shift changes smoothly (cable stretch, thermal elongation). Interpolation does not correct jerks of the tool at ledges in the hole. Above the uppermost marker and below the lowest one we take the shift as constant, equal to the shift at the nearest marker.

resistivities in the same units; $\lg$ is the decimal logarithm (logarithm to base 10)

Holds when: The grid is logarithmic across the whole width; on a split grid the formula applies only to the logarithmic track.

m, h

Holds when: We know the bit depth from the surface record. Its own errors (pipe stretch, heave at sea: the up-and-down motion of the rig) are not included here. The second formula holds as long as the rate of penetration is constant within the clock disagreement.

curves in the same units; shift in metres

Holds when: The window contains distinct beds. On a uniform stretch $r$ hardly depends on the shift, and the maximum cannot be trusted. The shift is constant within the window. The curves were recorded by tools of similar resolution.

any consistent system (here SI)

Holds when: Piston-like displacement (the filtrate pushes the fluids ahead of it like a piston): beyond the filtrate front the formation is unchanged, and we neglect the transition zone. Invasion is uniform around the circumference and over the thickness of the bed, without gravity segregation. The filtrate displaces the formation water and the moveable hydrocarbons rather than mixing with them. We take the volume of filtrate as given. In reality the mud, the cake, the overbalance and time determine it.

any, the same for all the resistivities (usually Ω·m)

Holds when: A clean formation with intergranular porosity (pores between the grains): only the water in the pores conducts current. Archie's law with a saturation exponent $n = 2$ and the same formation resistivity factor in both zones. A step profile. We obtained $R_{xo}$ and $R_t$ after corrections for the borehole, the cake and invasion. $R_{mf}$ and $R_w$ have been brought to formation temperature.

SI (any consistent system)

Holds when: Filtration at rest, without circulation. The cake is incompressible, and its permeability and composition are constant. The resistance of the formation is small compared with the resistance of the cake. The cake is thin compared with the borehole radius (a planar problem). We neglect the instant initial filtration (spurt loss). In practice an initial volume $V_{sp}$ is added to the formula.

any resistivity units, the same for all

Holds when: A step invasion profile. Corrections for the borehole and the enclosing rocks have already been applied. A conductive mud (the laterolog does not work in oil-based muds).

any resistivity units, the same for all

Holds when: A step invasion profile. Corrections for the borehole, the enclosing rocks and the skin effect have already been applied.

as indicated; $C_{\text{g/L}}/\rho$ is grams of salt per kilogram of solution

Holds when: The density and the salinity refer to the same temperature. The density formula is an approximation for NaCl solutions at room temperature. For waters of complex composition we measure the density.

°C and metres; $1\ ^\circ\text{C}/100\ \text{m} = 0.549\ ^\circ\text{F}/100\ \text{ft}$

Holds when: The gradient is constant over the section. This is a good approximation for a uniform sedimentary sequence without thick salt layers and without active groundwater movement. $T_{TD}$ is the stabilised formation temperature, not the temperature measured immediately after circulation.

°C; $t_c$ and $\Delta t$ in the same units; $\lg$ is the decimal logarithm

Holds when: The cooling of the bottom of the hole is like the removal of heat by a line source of constant power (a thin line that takes away heat) during $t_c$. The readings are taken at one depth, and the time $\Delta t$ is not too short (usually more than a few hours). We need at least three readings (two points cannot show whether the data lie on a line), recorded together with the time circulation stopped.

empirical: $R$ in Ω·m, $C$ in ppm

Holds when: An empirical approximation (a formula fitted to tables of measurements) for NaCl solutions, from hundreds of ppm to nearly saturated. The error is usually within a few per cent, and larger for nearly saturated brines. For a water of another composition, $C$ is the equivalent NaCl concentration.

°C; the resistivities in any identical units

Holds when: An empirical approximation for NaCl solutions in the usual range of formation temperatures. In logging we apply it to all aqueous solutions: formation water, filtrate and drilling mud. For mudcake and for waters of complex composition the accuracy is lower.

dimensionless (relative changes)

Holds when: The thickness and porosity of the bed do not change; only $S_w$ does. The error is small (linear approximation). The bed stays on the same side of the cut-offs. Otherwise its pore thickness changes in a jump, all at once.

units of the method: g/cm³ for density, barn/cm³ for the volumetric photoelectric absorption $U$, µs/m for interval transit time, fraction for the hydrogen index

Holds when: The sonde sees a uniform volume, the bed is thicker than the vertical resolution, and the borehole corrections have been applied. For bulk density the rule is exact, because it is a mass balance. For photoelectric absorption, it is not $P_e$ that mixes but the product $U = P_e\,\rho_e$. For the neutron method the rule is approximate: the matrix mineral, neutron absorbers and gas all have an effect. For interval transit time it is the empirical time-average equation, valid for compacted, cemented rocks. For resistivity the rule does not hold: current flows only through the water in the pores, and the relationship is nonlinear (Archie's equation, Archie's equation and saturation of clean reservoirs).

API; densities in any units, the same for all terms

Holds when: Corrections for the hole diameter, the mud and the casing have already been applied. The contributions of the minerals are independent and add up. The bed is uniform within the volume the tool sees and thicker than the tool's vertical resolution.

any consistent gamma ray units; the index is dimensionless

Holds when: The radioactivity of the bed is created only by shaly material. The shales in the bed are the same as in the reference shale packages, and the clean part is the same as in the reference clean bed. The baselines are chosen in the same hole, on the same record and close in depth.

GR in API; potassium in mass per cent, uranium and thorium in g/t

Holds when: The coefficients are approximate and depend on the tool and its calibration. Radioactive equilibrium holds in the uranium and thorium series. Borehole corrections have been applied.

SI; for a logging speed in m/h, $N = 3600\,n\,\Delta z/v$

Holds when: The count rate is constant within a sample (a uniform bed). The background radiation and the dead time of the detector (the short pause after each count, when it cannot count) are ignored. The samples are independent, with no filtering over depth.

mV; resistivities in any consistent units; $\lg$ is the base-10 logarithm

Holds when: A clean (shale-free) water-bearing bed between thick shales that behave as ideal cation membranes. The solutions are close in composition to an NaCl solution; with a large share of other salts the coefficient at 25 °C differs from 71 mV. The streaming potentials are small or cancel each other. Both resistivities and the coefficient $K$ refer to formation temperature.

mV; resistances of the sections in ohms

Holds when: This is a qualitative equivalent circuit. The currents in the bed and in the shales spread in a complicated way, and we do not measure $r_{sd}$ and $r_{sh}$ directly. The electrochemical EMF is concentrated at the shale–mud and filtrate–formation water boundaries.

Ω·m; SSP and $K$ in mV

Holds when: The SSP was taken in a thick clean water-bearing bed or corrected for the thickness and resistivity of the bed. $R_{mfe}$ and $K$ refer to formation temperature. The water and the filtrate are close to NaCl solutions, and the streaming potentials are small.

SI: $K$ comes out in volts, multiply by 1000 for millivolts; $2.303 = \ln 10$ converts the natural logarithm to the base-10 one

Holds when: The shales are an ideal membrane: they pass only cations. The bed is clean and is not itself a membrane. Both solutions are solutions of the same salt, NaCl; the transport numbers are constant over the range of concentrations considered.

$\Delta t$ in µs/m, $V$ in m/s

Holds when: The factor $10^6$ converts seconds to microseconds. The velocity and the interval transit time refer to the same wave, the compressional one unless stated otherwise.

SI: moduli in Pa, density in kg/m³, velocities in m/s

Holds when: A uniform, isotropic medium (the same in all directions) that behaves elastically (it springs back after it is deformed). The wavelength is much greater than the grains and pores, and the rock behaves as a continuous medium with averaged properties. For a liquid and a gas $G = 0$: there is no shear wave in them, and $V_p = \sqrt{K/\rho}$, where $K$ and $\rho$ are the bulk modulus and the density of the fluid itself.

times in µs, distances in m, interval transit times in µs/m

Holds when: The ray approximation: the wave travels along straight segments, from the transmitter to the wall, along the wall and from the wall to the receiver. The rock is faster than the mud ($\Delta t < \Delta t_m$); otherwise there is no head wave. The wall in the path of the wave is smooth and parallel to the tool, and the rock along it is uniform.

times in µs, span in m, $\Delta t$ in µs/m

Holds when: Both receivers trigger on the same oscillation of the same wave, the first arrival of the compressional head wave. The gaps between the tool and the wall at both receivers are equal. The rock is uniform within the span: the measured $\Delta t$ is the average over an interval of length $s$.

distances in m; interval transit times in the same units

Holds when: The ray approximation; the gap is the same at the transmitter and at the receiver. The rock is faster than the mud: $\Delta t < \Delta t_m$.

$\Delta t$ in µs/m, $h$ in m, $t$ in µs, $\bar V$ in m/s (the factor $10^6$ converts microseconds to seconds)

Holds when: The wave travels vertically; in a deviated well, vertical thicknesses are used. The velocity at logging frequencies (kilohertz) is usually somewhat higher than at seismic frequencies (tens of hertz), and the tool sees only the rock at the wall. So we calibrate the integrated time against a check-shot survey or vertical seismic profiling (VSP). Both are direct measurements of the travel time from the surface to a receiver in the well.

SI: density in kg/m³, interval transit time in s/m, modulus in Pa

Holds when: A uniform isotropic elastic rock. The moduli are dynamic: they describe the small, fast deformations in a sound wave. Static moduli, which are measured by loading core slowly, are usually lower. For engineering calculations we obtain them from relationships checked on core. $\Delta t_s$ is measured, not estimated from $\Delta t_p$.

$N$ in counts/s (natural logarithm), $\rho_e$ in g/cm³

Holds when: An approximate relationship found by experiment, for a fixed tool design, energy window and pad contact. The distance to the detector is greater than a few photon mean free paths (the average distance a photon travels between collisions). Right next to the source the relationship is reversed: a denser rock scatters more photons to the detector. The rock is uniform in the volume of investigation, with no mudcake or standoff.

densities in g/cm³; the ratio $2\sum Z/M$ is dimensionless when $M$ is in g/mol

Holds when: For a mixture of minerals and fluids the indices add in proportion to the volume fractions, as the densities themselves do.

g/cm³

Holds when: Calibration on limestone saturated with fresh water: for calcite and for water $\rho_a$ coincides with the true density. The constants 1.0704 and 0.1883 come from requiring $\rho_a = \rho_b$ for these two substances. So for other substances the equality is approximate.

barn/electron

Holds when: An approximate relationship for the photon energies at which the tool measures photoelectric absorption.

barn/electron

Holds when: The factor of a mineral is the average of the factors of its elements, weighted by the number of electrons.

barn/cm³; for several minerals $U = \sum_i V_i U_i$ by volume fractions

Holds when: Pores and minerals are distributed uniformly in the volume of investigation; there is no barite in the mud or the mudcake.

g/cm³

Holds when: The gas is methane ($1.334 \approx 1.0704\cdot 1.2467$). For a heavier gas the coefficients are slightly different. The calibration formula is linear, and the fractions of the components add up to one. So the apparent densities of the components mix by volume in the same way as the true ones. The apparent density of gas is a calculated quantity for such mixing, not a physical density. It can even turn out negative.

energies in the same units; the fraction is dimensionless

Holds when: An elastic collision (the colliding bodies keep their total energy of motion) with a nucleus at rest. In a glancing collision the loss is smaller, on average half the largest.

energies in the same units; natural logarithms

Holds when: Elastic scattering that is isotropic (equal in all directions) in the centre-of-mass frame. This is a good approximation for light nuclei at the energies of logging sources. For $A = 1$ the second term vanishes and $\xi = 1$. A medium made of nuclei of one kind. In a rock, the contributions of the nuclei add up according to their number and the probability of scattering on each.

$\rho_h$ in g/cm³; the index is dimensionless

Holds when: The atomic masses of carbon and hydrogen are rounded to 12 and 1. The hydrocarbon is described by the average formula $\mathrm{CH}_x$.

$\rho_w$ in g/cm³; the index is dimensionless

Holds when: The salt contains no hydrogen; hydrogen belongs only to the water in the solution.

fraction

Holds when: A clean formation without shale. The curve is converted to the matrix of the formation and corrected for the borehole. The excavation effect is ignored: in a gas-bearing formation a real tool shows an even lower porosity (advanced section). Neutron absorbers have no influence (or the tool is epithermal).

fraction

Holds when: An empirical rule (it comes from experience) for clean gas-bearing formations. Both curves are corrected for the borehole. For formations with liquid on limestone scales, we use the arithmetic mean $(\phi_N + \phi_D)/2$. It depends little on whether the formation is sandstone, limestone or dolomite. Charts and calculations give an accurate estimate, with the excavation effect and the gas saturation of the flushed zone (Gas, shale, carbonate and pressure effects on porosity logs).

$f$ in MHz when $B_0$ is in tesla

Holds when: Hydrogen nuclei: other nuclei have a different gyromagnetic ratio and do not respond at the frequency of the tool. The design of the tool sets the field in the sensitive zone. In a non-uniform field, different distances from the tool correspond to different frequencies.

$\mathrm{TW}$ and $T_1$ in the same units

Holds when: One fluid with a single value of $T_1$. In a rock, $T_1$ has a spread of values, and we apply the formula to each part. There is no magnetisation before polarisation starts: the rock has just entered the field of the tool, or the previous cycle has completely “erased” it.

amplitudes in porosity units (p.u.), times in ms

Holds when: The fluids are fully polarised and have a hydrogen index of 1. Otherwise the amplitudes are too low (see the section on hydrocarbons). Components with $T_2$ comparable to $\mathrm{TE}$ or shorter die away before the first echoes, so they are poorly determined. The amplitudes of the late echoes are small compared with noise. So for a reliable decomposition, we stack the signal over several cycles (add the cycles together).

SI; it is convenient to take $\rho_2$ in µm/s and $S/V$ in µm⁻¹, and their product comes out in s⁻¹

Holds when: Fast diffusion: a molecule travels around the pore many times during the relaxation time, $\rho_2 r/D \ll 1$. The pore then relaxes as a whole. The surface term refers to the wetting phase. A non-wetting phase (oil in a water-wet rock, or gas) does not touch the walls and relaxes only by bulk and diffusion relaxation. The diffusion term is written for unrestricted diffusion in a uniform gradient. In small pores the walls limit the movement, and the term is smaller. Neighbouring pores do not exchange molecules during the measurement. Otherwise their times average out.

ms

Holds when: The sum is taken over an agreed part of the distribution. In this lesson it is the effective porosity, $T_2 > 3$ ms. Use the same convention in the calculation and in the calibration on core.

fraction of bulk volume (or per cent, that is, porosity units)

Holds when: The hydrogen index of all fluids is about 1, and polarisation is complete. Otherwise parts of the distribution are too low. The echo spacing is short enough to see clay-bound water. At a $\mathrm{TE}$ of about 1 ms the tool gives only effective porosity. The cutoffs are chosen for this rock (typical values are in the table below).

empirical: $k$ in mD, $\phi_e$ in per cent; with another way of writing porosity, $C$ changes

Holds when: This is an empirical relationship proposed for sandstones. We choose $C$ (one often starts from a value of 10) and the exponents from the core of the formation. The cutoff separating BVI and FFI is calibrated for this rock. BVI is water only, and FFI is all the mobile fluids, with corrections for the hydrogen index and incomplete polarisation.

empirical: $k$ in mD, $\phi_e$ as a fraction, $T_{2LM}$ in ms

Holds when: This is an empirical relationship. We choose $a$ and the exponents from the core of the formation. Without core, one often starts from $a$ of about 4 mD/ms² for sandstones. In carbonates the coefficients are different ($a$ is much smaller). The pores are filled with water, and the distribution reflects pore sizes, not fluid properties. $T_{2LM}$ is calculated over the same part of the distribution as in the calibration.

porosity and saturation as fractions; $\mathrm{TW}$ and $T_{1h}$ in the same units

Holds when: Water-based filtrate with a hydrogen index of 1, fully polarised. The hydrocarbon signal is not lost because $T_2$ is too short (for heavy oil this does not hold). The hydrogen index of water is taken as 1; for salt water it is slightly lower.

fraction; $\mathrm{TW}$ and $T_{1g}$ in the same units

Holds when: A clean formation with a known matrix density; both tools see the same flushed zone. Water is fully polarised, and the hydrogen index of the filtrate is 1. The properties of the gas ($\rho_g$, $\mathrm{HI}_g$, $T_{1g}$) are estimated for reservoir pressure and temperature.

SI

Holds when: A tool with one working frequency. A multi-frequency tool, which measures several shells in turn, gathers data faster by about as many times as it has frequencies. The cycle time is polarisation plus the echo train. Service pauses are ignored. The distance the tool moves during stacking is simply added to the antenna length. This is a rough estimate of the vertical resolution.

amplitudes in porosity units

Holds when: The noise is random and the same for all echoes. The time grid is set in advance and covers the whole possible range, from fractions of a millisecond to several seconds. A penalty in the form of a sum of squares is one common option. Penalties on the curvature of the distribution are also used.

SI. In logging, conductivity is usually given in mS/m: $C = 1000/R$ with $R$ in Ω·m

Holds when: The sample is uniform, and the current flows along it evenly. Resistivity and conductivity refer to one temperature: in solutions and rocks they change a lot when the temperature rises.

SI

Holds when: The electrodes are points, and the return electrode $B$ is infinitely far away. The constant $K$ is derived for a uniform medium. A borehole is not uniform, so here the formula only defines what we call apparent resistivity. By the reciprocity principle, the current and measuring electrodes can swap places, and the reading does not change.

SI ($R$ in Ω·m, $f$ in Hz, $\delta$ in metres)

Holds when: A uniform non-magnetic medium. It is an estimate for a plane wave. The field of a real sonde is more complicated, but the order of magnitude is the same. Displacement currents (the part of the current that comes from a changing electric field) are neglected; at frequencies of tens of kilohertz this is justified.

any consistent units of resistivity, the same for every term

Holds when: The profile is a step: a flushed zone with $R_{xo}$ out to the diameter $d_i$, and beyond it the formation with $R_t$. There is no transition zone and no annulus. This form is for the laterolog, where resistances add up. For induction logging the conductivities $1/R$ are added in the same way, with geometric factors. The bed is thicker than the vertical resolution of the sondes. Corrections for the borehole and the enclosing rocks have already been made.

SI

Holds when: The medium is uniform, isotropic (the same in all directions) and infinite. The electrode is a point, and the return electrode is infinitely far away.

any identical units of resistivity

Holds when: A focused sonde: the measuring current crosses the zones one after the other, so the resistivities add up. The model does not suit the unfocused sondes of the microlog. A step profile: mudcake of thickness $h_{mc}$, the flushed zone up to the invasion front, and beyond it the uninvaded formation. $J_{mc}$ grows with the thickness of the mudcake, and the share of the formation grows as the front comes closer to the wall. The factors depend on the design of the tool and, to some extent, on the resistivity contrast, so real corrections come from the charts of the tool maker.

dimensionless; resistivities in the same units

Holds when: A clean formation (one without shale). In a shaly one we use models of shaly reservoirs with the same substitutions (Water saturation in shaly sands). The water in the flushed zone is practically only filtrate. If the filtrate is mixed with formation water, the resistivity of that water is lower than $R_{mf}$, and a calculation with $R_{mf}$ gives $S_{xo}$ too high. A water-based mud: the filtrate conducts current. $R_{mf}$ is brought to formation temperature.

fraction of rock volume

Holds when: $S_w$ and $S_{xo}$ are calculated with the same porosity and the same coefficients. The filtrate displaced the formation fluids as in waterflooding (water pushing oil out of the rock). The hydrocarbons it moved are counted as moveable. Before the invasion, the flushed zone was the same as the uninvaded formation: the same bed and the same saturation.

dimensionless

Holds when: A clean water-bearing bed completely flushed by the filtrate: $S_{xo} = 1$, and the water in the flushed zone is filtrate. The invasion front is deeper than the depth of investigation of the microresistivity sonde, and the mudcake correction has been applied. In shaly beds, the conductivity of the shales makes $F$ too low.

the interval transit times in the same units, µs/m or µs/ft; porosity as a fraction

Holds when: A clean rock without shale, with a known mineral composition. The rock is compacted and cemented. In loose sands the equation gives porosity that is too high. The pores are small and evenly spread, either intergranular or intercrystalline (between grains or between crystals). Vugs and fractures hardly enter the result. The pores in the zone of investigation are filled with liquid: the mud filtrate. The equation is empirical. It is not derived from the mechanics of elastic waves. It describes experimental data for such rocks.

$\Delta t_{sh}$ in µs/m; if $\Delta t_{sh}$ is in µs/ft, the divisor is 100

Holds when: Estimating $C_p$ from the neighbouring shales is a rough rule of thumb. Shales compact together with sands. A $\Delta t_{sh}$ above 328 µs/m (100 µs/ft) is taken as a sign of undercompaction. At smaller values we apply no correction ($C_p = 1$). It is more reliable to find $C_p$ in a reference bed where the porosity is known independently (see the calibration section below). The bed is clean, and its pores are filled with liquid.

the interval transit times in the same units; porosity as a fraction

Holds when: A clean rock, with the pores filled with liquid. The formula approximates an experimental relationship in the range of ordinary porosity, up to about 35–37%. In looser rocks the grains no longer make a rigid framework, and the relationship is different. For gas-bearing beds we reduce the constant (see the section on oil and gas).

the interval transit times in the same units; porosity and shale volume as fractions

Holds when: Laminated shale: the wave crosses the sand and shale layers one after the other, and the travel times add up. For thin layers this is an approximation. The shale layers are the same as the neighbouring shales from which $\Delta t_{sh}$ was taken. The shale volume $V_{sh}$ was found independently (Determining shale volume). The rock is compacted, with liquid in the pores. Shale material dispersed in the pores of a compacted sandstone changes $\Delta t$ little. For it the correction is too large.

fraction

Holds when: A rough rule of thumb for high-porosity sands with shallow invasion. The multiplier depends on the saturation of the zone of investigation and on the stiffness of the rock, so it cannot be exact. We apply the correction only when the overestimate is obvious, for example by comparison with the water-bearing part of the same bed or with density porosity. In the empirical transform, for gas-bearing beds we take a smaller constant, $C = 0.6$, instead of a multiplier.

fraction

Holds when: A clean rock of known mineral composition, with liquid in the pores. Gas and shales distort the three methods in different ways. The parameters for calculating $\phi_S$ have been checked in an interval without vugs. There $\phi_S$ must agree with $\phi_t$. Otherwise the error of the equation itself enters the difference. Vugs are not quite invisible to the wave, and they increase $\Delta t$ a little. So the difference is usually somewhat smaller than their true volume. We exclude from the calculation the points with cycle skipping, which fractures often cause. This is an index, not a measurement: the errors of the two methods add up.

the interval transit times in the same units

Holds when: An empirical relationship for clean water-bearing rocks. The porosity is less than about 37%. In looser rocks the grains no longer make a rigid framework, and the relationship is different.

g/cm³ or kg/m³: any density units, the same for all terms

Holds when: All components are included: the volume fractions add up to one. The rock is uniform within the volume the tool sees: 10–15 cm from the borehole wall and 15–45 cm along the borehole. The reading is corrected for the mudcake (the layer of mud solids on the borehole wall) and the borehole. For ordinary minerals and liquids the tool reads practically the true density; for gas, salt and gypsum it does not (why).

g/cm³

Holds when: The fractions are volume fractions, as thin sections give them (thin slices of rock studied under a microscope). Mass fractions from X-ray analysis must first be divided by the mineral densities and then normalised (scaled so that they add up to one). The composition is constant throughout the formation for which the value is adopted.

g/cm³

Holds when: The factor $1 + 0.73P$ is an approximation for NaCl solutions at 24 °C and atmospheric pressure. We assume that brine expands when heated and compresses under pressure roughly like fresh water. The uncertainty of the estimate is up to 0.01 g/cm³. For filtrates of complex composition and for oil-based muds, the density is measured or taken from the mud data.

g/cm³

Holds when: The zone of investigation contains only filtrate and residual hydrocarbons; no formation water is left, or it cannot be told from the filtrate by density. For gas, use the apparent density (the value the density tool shows, which differs from the true density of gas). For methane, $\rho_{a,g} \approx 1.334\,\rho_g - 0.188$ (derivation).

densities in g/cm³; porosity as a fraction

Holds when: The errors are small compared with the difference $\rho_{ma} - \rho_f$ (linear approximation). Independent random errors add in quadrature (as the square root of the sum of their squares). Systematic errors add with their signs.

densities in g/cm³; porosity and shale volume as fractions

Holds when: The shale in the formation is the same as in the reference shale unit; for dispersed shale this holds less well. The shale volume $V_{sh}$ has been found independently (Determining shale volume). There is no gas in the zone of investigation.

g/cm³

Holds when: The shale is laminated: the laminae are thinner than the vertical resolution of the tool (the thinnest bed it can separate), so the tool averages them over the volume. The shale laminae have the same density as the adjacent thick shales.

densities in g/cm³, resistivities in Ω·m

Holds when: The formation is clean; the flushed zone contains only filtrate and residual gas. Archie's equation is applied to the flushed zone with $a = 1$ and $m = n = 2$. With other coefficients the filtrate fraction itself depends on porosity, and the problem has to be solved by successive approximations. $R_{xo}$ is measured with a microresistivity tool and corrected for mudcake. For light oil, use its density in place of $\rho_{a,g}$.

densities in g/cm³

Holds when: The density of the bound water equals the fluid density used in the calculation. The composition of the shale is constant.

fraction

Holds when: The contributions add up in proportion to the volumes. This is an approximation that suits fluid and shale. The matrix itself contributes nothing when the reading is in units of its own matrix. If the matrix is different, first convert the reading using the tool's conversion curve. Gas lowers the reading further because of the excavation effect (see the section on gas). There are no hydrogen-bearing minerals, such as gypsum or coal, in the formation.

fraction

Holds when: The pores are filled with water or oil (hydrogen index about 1); there is no gas. The shale material in the formation is the same as in the reference shale unit. The shale volume has been found independently, for example from the gamma ray log (GR) or SP (Determining shale volume).

fraction

Holds when: The formation is clean; the reading has been corrected for environmental conditions and converted to the formation matrix. The zone of investigation contains only filtrate and residual gas. The hydrogen index of methane is $\mathrm{HI}_g = 2.25\,\rho_g$, with the gas density in g/cm³ (derivation).

fraction

Holds when: It approximates the results of neutron transport calculations. The numbers 2 and 0.04 are fitted to those results. Enter porosity and $S_{wH}$ as fractions. The formula describes the additional lowering. The lack of hydrogen itself is accounted for by the factor $S_{wH}$.

$\rho_h$ in g/cm³; the index is dimensionless

Holds when: The first formula is derived in the topic on the neutron log, with atomic masses rounded to 12 and 1. If the composition is known, take $x$ from it. For dry gas (almost pure methane), $x = 4$ and $\mathrm{HI}_h = 2.25\,\rho_h$. For oil, $x \approx 2$ and $\mathrm{HI}_h \approx 1.29\,\rho_h$. The estimate $x \approx 4 - 2.5\,\rho_h$ is empirical (it comes from experience, not from theory). It reduces light hydrocarbons to methane, and oil with a density of about 0.8 g/cm³ to the formula $\mathrm{CH}_2$.

g/cm³

Holds when: The tool is calibrated on limestone saturated with fresh water: $\rho_a = 1.0704\,\rho_e - 0.1883$ (electron and apparent density). The fraction is the ratio $2\sum Z/M$ for the group $\mathrm{CH}_x$. The group has $6 + x$ electrons (six from carbon and $x$ from hydrogen), and its molar mass is $12 + 1.01\,x$. For methane the formula becomes $\rho_{a,g} \approx 1.334\,\rho_g - 0.188$ (apparent gas density), and for oil with $x = 2$ it becomes $\rho_{a,h} \approx 1.222\,\rho_h - 0.188$.

fractions; densities in g/cm³

Holds when: The formation is clean and its matrix is known. Both curves have been corrected for environmental conditions and brought to the formation matrix. Both tools see the same zone, the first 10–30 cm behind the wall, with filtrate and residual hydrocarbons. The hydrogen index of the flushed-zone fluid $S_{wH} = 1 - k_N S_{hr}$ is the same as in the excavation formula (more detail), with $\mathrm{HI}_{mf} = 1$.

fractions

Holds when: The formation is clean and its lithology is known. Both porosities are calculated for one, correct matrix and corrected for environmental conditions. In a shaly formation, they are used after the corrections for shale volume (next section). $k_D$ and $k_N$ are calculated from the gas density at reservoir conditions. The excavation correction depends on the unknowns $\phi$ and $S_{hr}$, so repeat the calculation with the new correction until the result stops changing. Two or three repetitions are usually enough. Both tools see the same zone with the same saturation.

fraction

Holds when: The shale material in the formation is the same as in the reference shale unit. The contributions add in proportion to the volumes, as in the topics on each method. The gas shifts and the excavation effect apply to the pores outside the shale, so the solution for gas is applied to the corrected values.

densities in g/cm³, depths in km

Holds when: The same shales are compared; the borehole in the shales is smooth and density is reliable. The exponential trend follows from the compaction law $\phi = \phi_0\,\e^{-c\,\sigma'}$ at normal pressure, when the effective stress grows in proportion to depth. We build the trend from the normally compacted part of the section and extend it downwards. The error of the extrapolation grows with distance.

units of the method

Holds when: The formation is clean, the pores are filled with liquid of known density, and environmental corrections have been applied. For density the formula is an exact mass balance. For neutron porosity and interval transit time it is an approximation. The contributions of minerals do not add up in a strictly straight-line way, and on real charts the mixture lines are slightly curved.

densities in g/cm³; porosity is dimensionless

Holds when: This is not a statement that the formation is limestone. It is only a scale common to all the lines. Choose the chart by the filtrate: for fresh and salty muds the lines are different.

fractions

Holds when: The coefficients are the same as in the topic on gas, shale and carbonates, but density porosity is calculated for limestone. We estimate the gas volume $\phi S_{hr}$ from the saturation of the flushed zone (found with microresistivity tools) and a first estimate of porosity, and then refine it. For oil, $k_{D,ls}$ is about 0.15 and $k_N$ is close to zero: the point shifts one or two units upwards.

$P_e$ in barn/electron, $U$ in barn/cm³, $\rho_e$ in g/cm³ (as a measure of the number of electrons per unit volume)

Holds when: The components are mixed uniformly in the small volume behind the borehole wall that the tool sees. We take the electron density of each component from the calibration of the tool (limestone with fresh water), so the volumes mix the way the tool sees them. There is no barite or heavy minerals in the mudcake and the mud.

densities in g/cm³, $U$ in barn/cm³, $P_e$ in barn/electron

Holds when: The formation is clean, both minerals are known, and the pores are filled with liquid of known density and composition. The readings have been corrected for environmental conditions, and there is no barite in the mudcake or the mud. With gas in the flushed zone, porosity will come out too high (see the example with a gas-bearing limestone below).

g/cm³; porosity as a fraction

Holds when: The formation is clean, and the pores are filled with liquid of known density. The apparent porosity must be correct: an error of 0.01 shifts $\rho_{maa}$ by about 0.02 g/cm³ (see below).

barn/cm³

Holds when: The same as for $\rho_{maa}$; in addition, there is no barite in the mud and the mudcake.

potassium in mass %, thorium in g/t (ppm)

Holds when: The readings have been corrected for the borehole; there is no potassium chloride in the mud. The components have a constant composition. Natural minerals differ in contents several times over.

TOC in mass %, uranium in g/t

Holds when: Uranium is associated with organic matter, not with phosphates, fractures or secondary deposits. The calibration is valid for one unit and one depositional environment (the setting where the sediment settled). Do not transfer it to other units. Average the uranium curve over the interval, because individual readings are noisy.

$\Delta t$ in µs/m, density in g/cm³; $M$ is a conventional quantity (µs/ft per 1 g/cm³, divided by 100)

Holds when: A clean bed with liquid in the pores, and the readings corrected for environmental conditions. The interval transit time is linear in volumes (the time-average equation). With the empirical transform, $M$ changes slightly with porosity. The multiplier is a conventional scale: 0.01 with $\Delta t$ in µs/ft, that is, $0.01\cdot 0.3048 = 0.003048$ with µs/m. The calculation and the chart must use the same multiplier.

neutron porosity as a fraction, density in g/cm³; $N$ in cm³/g

Holds when: A clean bed with liquid. The neutron curve is in limestone units of the same tool for which the mineral points were built. $N$ does not depend on porosity only if the neutron reading is linear in volumes. For sandstone and dolomite it is not linear, and their $N$ changes by 0.02–0.04 between 0 and 30% pores.

fraction

Holds when: A clean bed with liquid, made of one known mineral. The sonic log sees only the intergranular porosity $\phi_1$ and follows the time-average equation. The density and neutron logs see the total porosity $\phi_t = \phi_1 + \phi_2$.

g/cm³

Holds when: A clean bed with liquid of known density; the apparent porosity is correct.

µs/m (or µs/ft: the same for all quantities)

Holds when: This form is for the time-average equation. The empirical transform has its own form (below). The mineral points on the chart were built with the same transform as the calculation.

g/cm³ and µs/m per unit of porosity (as a fraction)

Holds when: Small porosity errors; the derivatives were obtained by differentiating the formulas of the apparent parameters.

$w_i$ and $f_i$ are fractions; $M$ and $N$ are in their own units (see above)

Holds when: The responses of all three methods are linear in volumes. For sandstone and dolomite, which have a non-linear neutron response, we take the mineral points at the porosity of the bed.

dimensionless; $R_0$ and $R_w$ in the same units

Holds when: A clean rock: only the water in the pores conducts current, and the surface conductivity of the grains and of the shale is negligible. Intergranular porosity, more or less uniform through the volume. $a$ and $m$ are fitted for the given rock, and they are used in the range of porosity where they were found. $R_0$ and $R_w$ refer to one temperature. For a bed, this is the formation temperature.

dimensionless

Holds when: A clean rock, and only water conducts. The water forms a connected system: films and filled small pores (a water-wet rock). $R_t$ and $R_0$ refer to one rock, one water and one temperature.

dimensionless; all resistivities in the same units

Holds when: A clean (shale-free) bed with intergranular porosity. Only the formation water conducts current. $R_t$ is the resistivity of the uninvaded part of the formation, with corrections for the borehole, the enclosing rocks and invasion (Borehole environment: invaded zone and mudcake). $R_w$ is at formation temperature. $a$, $m$ and $n$ suit the rock (from core or from a checked analogy). The rock is water-wet.

dimensionless

Holds when: The same as for $S_w$. The water in the flushed zone is practically only filtrate. A water-based mud (with an oil-based mud the filtrate does not conduct current).