Equations: Sedimentary rocks and environments

percentages of the sum of the three framework constituents

Holds when: Matrix, cement, pores, micas and other grains are not included in the sum. Some schemes count chert as quartz and others count it as rock fragments, so a report should state how it was counted.

per cent

Holds when: Cement and pores are not included in the denominator. Authigenic clay minerals that grew in the pores are cement, not matrix; they can be told apart only in thin section or under an electron microscope.

SI

Holds when: The water evaporates completely and the whole dissolved component precipitates as a single mineral. The salt is not redissolved and is not carried away with the brine; nothing is lost into pores or brine.

fraction

Holds when: The points are independent: the grid spacing is greater than the grain size, otherwise one grain falls under several points and the error is greater than calculated. The thin section is representative of the sample, and the sample of the bed; the number of points does not reduce this error.

$d$ and $d_0$ in millimetres; $\varphi$ is dimensionless

Holds when: The logarithm is to base 2: $\log_2 x = \ln x/\ln 2 \approx 3.322\,\log_{10} x$. The minus sign is chosen so that sand and finer particles have positive φ values. For gravel and coarser fragments, φ is negative.

SI

Holds when: A single smooth spherical particle in still water, far from walls and other particles. The water flows slowly around the particle: the particle Reynolds number $\rho_f w d/\mu$ (a number that compares inertia with viscosity) is below about 0.5. For quartz in water at 20 °C this means particles finer than about 0.08 mm. The particles do not stick together, because a dispersant is added to the sample.

φ units

Holds when: The percentiles are read from the cumulative mass curve (the share coarser than a given size). For a distribution that is normal in φ units, $M_z$ equals the mean and also the median $\varphi_{50}$ (the middle size).

φ units

Holds when: The formula is applied only to sizes in φ units. You must measure the tails of the curve. If $\varphi_{95}$ falls on the pan (the fraction that passed through the last sieve), it cannot be found by interpolation, and you measure the fine fraction further by settling or by laser.

dimensionless; percentiles in φ units

Holds when: The sign depends on the units: in millimetres it is the opposite, so a report must say which units were used. For a curve that is symmetric in φ units, both terms are zero.

dimensionless

Holds when: For a normal distribution $K_G = 1$. The constant 2.44 is the ratio of the spread between the 5% and 95% percentiles to the spread between the 25% and 75% percentiles of the normal curve.

any identical units of length

Holds when: The grain is treated as an ellipsoid with axes $L \ge I \ge S$. For a sphere $\psi_K = 1$.

any identical units of length

Holds when: The grain is treated as an ellipsoid with axes $L \ge I \ge S$. For a sphere $\psi_P = 1$.

fraction

Holds when: The fine grains are so much smaller than the coarse ones that they pass into the pores of the framework through the narrow gaps between the grains. For close packing, this means finer than about 0.15 of the diameter of the coarse grains (size of the constrictions between grains). The fine grains pack in the pores as tightly as they would on their own, and the coarse grains as tightly as they would without the fine ones. Within each fraction, the grains are of one size and shape.

fraction

Holds when: The same assumptions as in the formula for the porosity of the mixture.

empirical: $k$ in darcies, $d_g$ in millimetres, $\sigma$ in φ units

Holds when: Derived for loose uncemented sands with an approximately log-normal size distribution, packed to a porosity of about 40%. The further the porosity is from this value, the cruder the estimate. For compacted and cemented rocks the formula greatly overestimates the permeability. At $\sigma = 0$ it is close to the Kozeny–Carman model for equal spheres at a porosity of 40% (that model gives about 30% more).

dimensionless

Holds when: The size distribution is log-normal (normal in φ units). For real samples the formulas are only approximate.

percent of the present rock volume, from a point count in thin section

Holds when: Pores inside grains and pores left by dissolved grains are not included in $P_{ig}$ — they are not intergranular. Cement that has replaced grains is counted with the grains, not in $C$.

percent

Holds when: The volume of the framework grains does not change on burial: grains neither dissolve nor are added. $P_i$ is taken from the texture of the sediment; an error in it passes directly into COPL (the example below shows by how much). Cement only fills the intergranular volume without changing it.

percent

Holds when: The same as for COPL. Cement fills the pores without pushing the grains apart.

empirical: $k$ in mD, $\phi$ as a fraction, $\lg$ is the decimal logarithm

Holds when: The samples are of one rock type and one scale. Permeability is brought to the same conditions: reservoir stress, and liquid flow (measured with liquid, or corrected for gas slip with the Klinkenberg correction). The relationship has not been checked outside the porosity range of the samples. $a$ and $b$ are found by least squares, regressing $\lg k$ on $\phi$. This is the right way when permeability is predicted from porosity. A regression of $\phi$ on $\lg k$ gives a different, steeper line.

any units of permeability

Holds when: The scatter about the line is lognormal and the same at any porosity. If the samples are a mixture of several rock types, the scatter is not lognormal, and the formula gives only the order of magnitude of the correction.

$k$ in mD, $\phi$ as a fraction; RQI and FZI in µm

Holds when: $0.0314 = \sqrt{0.9869\cdot 10^{-3}}$ converts millidarcy to square micrometres under the root. Permeability is single-phase (to liquid or with the Klinkenberg correction), porosity is effective, and the test conditions are the same for all samples. The basis is the model of a bundle of winding (tortuous) capillaries. FZI gathers the specific surface of the pores (their surface area per unit of volume), their tortuosity and shape into one number.

$k$ in mD, FZI in µm

Holds when: $1014 = 1/0.0314^2$. The formula is just the definitions of RQI and FZI solved for $k$.

empirical: $r_{35}$ in µm, $k$ in mD, $\phi$ in percent

Holds when: It is a regression on samples for which mercury curves, porosity and permeability were all available. For other rocks, especially tight ones, check it against your own data. Porosity is substituted in percent.

consistent: $k$ in µm² when $S_{gv}$ is in µm⁻¹

Holds when: The pores are a network of tortuous channels, and the flow is laminar (smooth, not turbulent). For grain packings the product $F_s\tau^2$ is close to 5. For cemented rocks and rocks with clays it is unknown and is included in FZI.

SI

Holds when: $\tau = \rho g h S$ holds for steady uniform flow in a wide channel. $\tau$ includes the resistance of the bedforms themselves, so only a part of the stress acts on the grains. The Shields number for the grains therefore comes out too high (an upper estimate).

SI

Holds when: Non-cohesive grains (grains that do not stick together) of one size on a flat bed. The experimental data are widely scattered — at the same $D_*$ the threshold varies severalfold, so the result is only a guide.

SI

Holds when: $U$ and $h$ are the averages over the cross-section of a wide flow. Flow in an open channel is laminar (smooth) at Re below about 500 and turbulent at Re above about 2000 (based on the depth or the hydraulic radius).

any identical velocity units

Holds when: The bedforms move at a constant speed and keep their shape. The angle is measured in a section parallel to the current: in an oblique section it appears smaller.

SI

Holds when: Waves of small height in deep water (depth greater than about $L/2$). In shallow water the orbits flatten into ellipses, and at the bed itself the water sways back and forth.

degrees; $\bar R$ is dimensionless

Holds when: The measurements are of one kind of structure in one formation; indicators of different kinds are averaged separately. The distribution is unimodal (it has one peak). If there are two directions, as with ebb and flood tides, the overall mean is meaningless. The azimuths are taken after the structural tilt of the beds has been removed (advanced section).

any identical length units

Holds when: The borehole is vertical and round. In an inclined borehole this gives the dip relative to the borehole axis, and the true dip is found by recalculating with the inclination and azimuth of the borehole (attitude (strike and dip)). The plane cuts the whole borehole; its dip azimuth is the azimuth of the lowest point of the sinusoid.

dimensionless vectors, angles in degrees

Holds when: The formation was tilted by rotation about a horizontal axis, without folds with an inclined hinge. The beds were deposited horizontally. On the slopes of deltas and in dunes, an original dip of a few degrees remains as an error of the method.

any identical length units

Holds when: The sets were left by dunes under weak aggradation, and their thicknesses are markedly scattered (the coefficient of variation is about 0.9 ± 0.3). The ratio of depth to dune height differs between rivers. In large rivers dunes may be lower than a tenth of the depth. So the estimate is rough, within about a factor of two.

percent of the rock volume; stresses in MPa; depth in km

Holds when: The form is empirical. It is used in published models of sandstone compaction. We fit $P_i$, $\mathrm{IGV}_f$ and $\beta$ to thin sections from wells for each type of sand, by sorting and the share of ductile grains. The sand is not cemented: cement that grew early props up the framework, and compaction goes no further. Compaction is irreversible: on uplift and erosion the intergranular volume is not restored, which is why the formula contains the greatest stress, not the present one. There is no matrix; clay matrix compacts by its own laws.

mol/(cm²·s), °C

Holds when: The rate is set by precipitation on the surface, not by the delivery of silica: the source is pressure solution nearby. Quartz grains with a clean surface; a surface covered by coats takes no part in precipitation. The constants were found for particular sandstones. For another basin, check them against the volume of cement in thin sections and the temperatures of fluid inclusions.

cm, cm², g, mol, s, °C; $V_q$ and $\phi_0$ as fractions of the rock volume

Holds when: The heating rate is constant. For a complex history, take the integral in time steps. The free surface decreases in proportion to the remaining pores: $A = A_0(\phi_0 - V_q)/\phi_0$. Mechanical compaction after the start of cementation is not taken into account — the cement props up the framework. The sandstone is quartzose; feldspars and rock fragments are not included in $f$.

numbers of moles of reactants and products

Holds when: Silica precipitates within the same volume of rock as quartz, and aluminium as kaolinite: the system is closed for silicon and aluminium. Potassium leaves with the water. On burial deeper than about 120 °C, it may itself go into the growth of illite.

numbers of moles

Holds when: All the carbonate of the dolomite is taken from the replaced calcite: only magnesium is brought in, only calcium is carried out. This is the limiting case, with the greatest gain of pores. If the solution brings carbonate as well, dolomite may replace limestone “volume for volume”, and no pores are added.

any consistent units

Holds when: The layers are horizontal and can be traced over the distance between wells. Flow is along the layers, and the contribution of a layer is proportional to its flow capacity $k_i h_i$ (averaging over layers).

dimensionless

Holds when: Beds are defined by one rule throughout the section. The simple estimate of $r_{ij}$ is approximate. Refined methods treat the ban on transitions of a facies into itself more rigorously. We test the significance of deviations with the $\chi^2$ test (the chi-squared test).

any identical units of length

Holds when: Two-dimensional dunes of the same size and spacing that migrate and climb at constant rates. The angle of climb is less than the dip of the windward slope (the windward slope is eroded). If it is greater, the windward laminae are preserved too, and the sets pass into one another. The thickness measured at right angles to the bounding surfaces is $\lambda\sin\zeta$. At small angles it is the same. In ergs with a high water table, the sand is preserved even without climbing: the wind cuts the dunes down to the damp layer, and the rise of the water table sets the thickness of the sets.

any identical units of permeability

Holds when: The laminae are flat and parallel, and each one is homogeneous. The set is much longer than the distance over which a lamina crosses it from base to top. $k_x$ is for a flow that is horizontal on average (no net vertical crossflow). If vertical crossflow is free, $k_\parallel\cos^2\delta + k_\perp\sin^2\delta$ applies instead of $k_x$, and this is an upper bound. $k_z$ is for a flow that is vertical on average. The derivation is in the advanced section.

any identical units of length

Holds when: The reach is several bends long. For a single bend, the value is random. For a multichannel river, sinuosity is calculated for the main channel or separately for each channel.

metres only: the formula is empirical, with a dimensional coefficient

Holds when: It comes from modern meandering rivers with sinuosity above 1.7. It does not apply to braided rivers or deltaic distributaries. The width-to-depth ratio grows with the size of the river, so empirical formulas that relate it to other properties of the channel are more reliable for small rivers. The thickness of a cycle is less than the channel depth if its top has been cut off by the next channel or if the sediment has compacted. In that case the width is underestimated. The thickness of a whole unit of stacked channels overestimates the depth.

metres only

Holds when: Both formulas come from published data on modern rivers. The second one comes from almost two hundred river reaches in different countries, for channels from 1.5 m to 4 km wide. The belt width of a modern river is the span of its bends now. An ancient belt over which the river wandered for a long time may be wider, and belts narrow downstream. The two formulas are two variants of an estimate. Their disagreement reminds us how rough the estimate is. It is not a call to choose the “right” one.

consistent: m, m³/(m·yr), years

Holds when: Sea level and the shape of the shore profile are constant, all the sand that arrives stays in the wedge, and compaction is ignored. When sea level rises, part of the sand goes into building the whole profile upwards, so the shoreline advances more slowly. When sea level falls, it advances faster. The sand supply is constant in time and the same along the stretch of coast.

dimensionless

Holds when: Beds are defined as deposits of separate flows, and contacts are described from core or image logs. Some studies divide by the number of beds $N$ rather than by the number of contacts $N - 1$. With few beds, the results differ noticeably. So when you compare with published values, check the definition.

SI: $k$ in 1/m, $z$ in metres

Holds when: The water is equally clear at all depths. In reality $k$ changes with depth, season and weather. In the clearest ocean water $k$ is only a few hundredths (about 0.03–0.05 1/m). In muddy coastal water it is 0.2 1/m or more.

$G$ and $G_m$ in mm/yr (1 mm/yr is 1 m in a thousand years), $I_0$ and $I_k$ in the same units of light, $k$ in 1/m, $z$ in metres

Holds when: Only light limits production. Temperature, nutrients and salinity are already built into $G_m$, which holds for one community of organisms in one set of conditions. The equation describes the build-up of the sea floor in place. Waves and currents carry part of the production away, and organisms destroy another part. So the average rate at which a carbonate sequence builds up over thousands and millions of years is much smaller than $G_m$. The parameters of the model are fitted to observations of modern reefs. In the example they are assumed for illustration.

$\dot A$, $G$ and $\dd z/\dd t$ in mm/yr (1 mm/yr = 1000 m per million years), $z$ in metres

Holds when: All the production stays on the top. Removal of sediment and destruction reduce $G$. $\dot A$ is constant. In reality the sea level oscillates with various periods.