Equations: Structural geology

The equations of the “Structural geology” lessons in “Petroleum geoscience”, with their conditions and units, and what each symbol means in them.

Stresses on an inclined plane (two-dimensional case)

\sigma_n = \frac{\sigma_1 + \sigma_3}{2} + \frac{\sigma_1 - \sigma_3}{2}\cos 2\theta, \qquad \tau = \frac{\sigma_1 - \sigma_3}{2}\sin 2\theta

any identical units of stress; the angle in degrees or radians

Holds when: The plane contains the direction of the intermediate principal stress $\sigma_2$ , so it does not enter the formulas. The stresses are uniform near the plane. The formulas follow from the balance of forces on a triangular prism (advanced section). For effective stresses, subtract the pore pressure from $\sigma_1$ , $\sigma_3$ and $\sigma_n$ . The shear stress $\tau$ does not depend on it.

Stress, strain and structural analysis →

Extension, stretch and shear strain

e = \frac{l - l_0}{l_0}, \qquad S = \frac{l}{l_0} = 1 + e, \qquad \gamma = \mathrm{tan}\,\psi

dimensionless quantities; lengths in the same units

Holds when: The quantities describe the finite (total, accumulated) strain, not the path by which it developed. In homogeneous strain (the same everywhere) they are the same throughout the volume. In inhomogeneous strain they refer to a small region.

Stress, strain and structural analysis →

Apparent dip on a vertical section

\mathrm{tan}\,\delta' = \mathrm{tan}\,\delta\cdot\sin\beta = \mathrm{tan}\,\delta\cdot\cos\omega

angles in degrees; $\tan$ is the tangent (written tg in Russian-language notation)

Holds when: The bed is flat (planar) within the area. For a curved surface, we apply the formula to the tangent plane (the flat plane that just touches the surface). The section is vertical. For a tilted section, for example along the hole of a deviated well, we need a vector calculation (advanced section).

Stress, strain and structural analysis →

True thickness of a bed from the vertical thickness

h = h_\text{v}\cos\delta

any identical units of length

Holds when: The top and base are parallel, and the dip is constant over the thickness of the bed. The well is vertical. In a deviated well, the length of the hole in the bed is related to the true thickness through the angle between the hole and the normal to the bed (advanced section).

Stress, strain and structural analysis →

Dip angle from structure contours

\mathrm{tan}\,\delta = \frac{\Delta z}{w}

$\Delta z$ and $w$ in the same units; $w$ is the distance on the ground, that is, the distance on the map multiplied by the second number of the scale (25,000 in 1:25,000)

Holds when: The surface between neighbouring contours is taken to be planar. We take the distance at right angles to the contours. In any other direction, we would get an apparent dip.

Stress, strain and structural analysis →

Angle on a section with vertical exaggeration

\mathrm{tan}\,\delta_V = V\cdot\mathrm{tan}\,\delta'

angles in degrees

Holds when: The exaggeration is the same throughout the section. On a seismic section in time (where the vertical axis is time, not depth), the vertical scale also changes with the wave velocity, so angles are distorted unevenly (Seismic interpretation).

Stress, strain and structural analysis →

Angle between the hole and the normal to the bed, and the length of hole in the bed

\cos\gamma = \left|\sin\delta\,\sin\theta\,\cos(a - \varphi) - \cos\delta\,\cos\theta\right|, \qquad h = \Delta l\,\cos\gamma

angles in degrees; lengths in the same units

Holds when: The bed is flat, and its top and base are parallel. The hole is straight over the interval of the bed. The azimuths of the bed and of the hole are measured from the same north.

Stress, strain and structural analysis →

Equal-area projection of a line onto the lower hemisphere

r = R\sqrt{2}\,\sin\frac{90° - \rho}{2}, \qquad x = r\sin\varphi,\quad y = r\cos\varphi

$r$ , $R$ , $x$ , $y$ in the same units of length; angles in degrees

Holds when: We use the lower-hemisphere projection, which is the standard one in structural geology. We show a plane either by an arc (the projections of all the lines lying in it) or by a single point, its pole. The pole is the projection of the normal, which plunges at an angle $90° - \delta$ in the direction opposite to the dip.

Stress, strain and structural analysis →

Components of the slip on a fault

D_d = D\sin r,\quad D_s = D\cos r,\quad t = D_d\sin\delta,\quad w = D_d\cos\delta

any equal units of length; angles in degrees

Holds when: The fault plane is flat within the area. The values refer to the given point of the fault, since the slip changes along it. $t$ and $w$ are parts of the slip along the dip. The component $D_s$ is horizontal and runs along the fault.

Faults →

Vertical and stratigraphic separation of an inclined bed

s = t \pm w\,\mathrm{tan}\,\delta_b,\qquad s_n = s\cos\delta_b = D_d\sin(\delta \pm \delta_b)

equal units of length; angles in degrees

Holds when: The sign “+” is for beds that dip towards the fault plane, and “−” for beds that dip the same way. The slip is along the dip. The beds are planar (not curved) and run parallel to the strike of the fault. Otherwise we take the apparent dip of the beds (the dip seen in a section that is not at right angles to the strike) in the section across the fault. The strike-slip part of the slip also changes $s$ . The well is vertical. In an inclined well, we convert the length of the interval along the hole to thickness at right angles to the beds, using the angle between the hole and the normal to the beds (inclined wells). The difference of vertical depths equals $s$ only for horizontal beds or for a hole along the strike. The stratigraphic separation at the crossing point does not depend on the well path.

Faults →

Power-law distribution of faults by throw

N(t) = N_1\left(\frac{t_1}{t}\right)^{c},\qquad c = \frac{\ln(N_1/N_2)}{\ln(t_2/t_1)}

throws in metres; the number of faults per area or per unit area; natural logarithms (the ratio of logarithms does not depend on the base)

Holds when: One population of faults: one setting and one stage of deformation. The power law holds over the whole range of extrapolation. Points below the resolution limit are not used for the fit, because the number of faults there is too low.

Faults →

Shale gouge ratio: the share of clay material in the fault zone

SGR = \frac{\sum_i V_{sh,i}\,\Delta z_i}{t}\cdot 100\,\%

thicknesses and throw in the same units; $V_{sh}$ as a fraction

Holds when: The material of the interval that has slid past is fully mixed in the fault zone. $V_{sh}$ is calculated in the same way as in the calibration data. SGR thresholds found with a different way of calculating shale volume must not be used. We use the throw at the given point. Along the fault, SGR changes together with it.

Faults →

Pressure difference that a fault holds, and the height of column above the point

\Delta p_f = 10^{\,SGR/27 - C}\ \text{bar},\qquad H = \frac{\Delta p_f}{(\rho_w - \rho_h)\,g}

$\Delta p_f$ in bar (to calculate $H$ in SI, multiply by $10^5$ Pa/bar); $H$ in m; densities in kg/m³

Holds when: The envelope is an empirical upper bound from fields of one region. A real fault may hold less. The growth of the pressure difference held with SGR is confirmed by data up to an SGR of about 40%. Above this value, buoyancy measurements show no further growth, and we do not extrapolate the envelope there. Across the fault there is a water-bearing layer of the same hydrodynamic system, and the pressure difference equals the buoyancy of the column of hydrocarbons (buoyancy). The rocks are water-wet. Calibrations for oil and for gas differ (because of the larger interfacial tension, gas needs a larger pressure difference to break through the same fault rock). Outside the region of the calibration, we check the envelope against local data.

Faults →

Extension and compression of a layer in bending

e = \frac{y}{R},\qquad e_{max} = \frac{h}{2R},\qquad \frac{1}{R} \approx \frac{\dd^2 z}{\dd x^2} \approx \frac{z_1 - 2z_0 + z_2}{a^2}

lengths in the same units; $e$ is dimensionless (multiply by 100 to get per cent)

Holds when: The neutral surface lies in the middle of the layer, strains are small and the layer does not slip within itself. The curvature from the second difference (the formula with $z_1 - 2z_0 + z_2$ ) is valid when the slope at the middle point is small (at the crest $\dd z/\dd x \approx 0$ ). Otherwise use $1/R = z''/(1 + z'^2)^{3/2}$ . We take the profile at right angles to the hinge. In any other direction the curvature comes out smaller.

Folds, salt diapirs and their relationship to faults →

Pressure difference in the salt beneath a minibasin and beneath a dome

\Delta p = \sigma_v(z_1) - \sigma_v(z_2) - \rho_s\,g\,(z_1 - z_2) = g\int_{z_2}^{z_1}\bigl(\rho_b(z) - \rho_s\bigr)\,\dd z

SI; in metric oilfield units, replace $g$ by 0.00981 (densities in g/cm³, pressure in MPa)

Holds when: Salt behaves as a fluid: the pressure in it is the same in all directions and increases with depth according to its density. The minibasin and the dome are wide enough for the weight of the column above a point to set the pressure. We ignore the strength of the roof and the friction against the walls, although they slow the flow. We compare the same depth $z_1$ . Below it both columns consist of the same salt, so the difference does not change.

Folds, salt diapirs and their relationship to faults →

Dominant wavelength of buckling

L_d = 2\pi h\,\sqrt[3]{\frac{\eta_l}{6\,\eta_m}}

$L_d$ and $h$ in the same units; only the ratio of the viscosities matters

Holds when: A single viscous layer in a uniform viscous medium; small amplitudes at the start of fold growth. The viscosity contrast is large (tens or more); with a small contrast the layer thickens instead of bending. Multilayer sequences, slip along layers and brittle rocks give other wavelengths. The formula shows a tendency, not an exact value.

Folds, salt diapirs and their relationship to faults →

Griffith envelope and the condition for a joint to form

\tau^2 = 4T\,(\sigma_n' + T);\qquad \text{joint:}\ \ \sigma_3' = -T,\quad \sigma_1 - \sigma_3 \le 4T

MPa; effective stresses use the full pore pressure, as in failure criteria

Holds when: The rock is intact, with no weak planes. On existing fractures and faults, slip starts earlier, by the law of friction (slip on a fracture). The parabola describes the tensile and weakly compressive part of the envelope. At high compressive stress, the real strength is closer to the Coulomb line, and the boundary between hybrid and shear fractures moves a little. The differential stress $\sigma_1 - \sigma_3$ does not depend on pore pressure.

Fractures and naturally fractured reservoirs →

True linear intensity of a set from observations in a well (Terzaghi correction)

P_{10} = \frac{N}{l\,\cos\gamma},\qquad s = \frac{1}{P_{10}},\qquad \cos\gamma = |\mathbf n\cdot\mathbf w|

metres and inverse metres; angles in degrees

Holds when: The fractures of the set are parallel and much longer than the diameter of the well. If the orientations scatter, we calculate the angle $\gamma$ for each fracture separately and add up the weights $1/\cos\gamma$ . When the well is almost parallel to the fractures ( $\gamma$ close to 90°), the correction is unreliable: the fractures are crossed rarely, and one chance fracture gives a huge contribution. So the weight is limited, usually when the angle between the well and the plane of the fractures is less than 10–15°. The correction cannot restore fractures that the well did not cross: a set parallel to the well may not be noticed at all.

Fractures and naturally fractured reservoirs →

The cubic law: flow rate through a fracture and permeability of a set

q = \frac{w^3 h}{12\,\mu}\cdot\frac{\Delta p}{L_f},\qquad k = \frac{\sum_i w_i^3}{12\,L},\qquad \phi_f = \frac{\sum_i w_i}{L}

SI; to convert permeability to millidarcies, divide by $0.9869\cdot10^{-15}$ m²

Holds when: Laminar flow (smooth and orderly) between smooth parallel walls. For rough fractures with contact points, we use the hydraulic aperture, which is smaller than the average mechanical one. The fractures of the set are parallel, and the flow is along them. Across the set the fractures hardly help, and the matrix decides the permeability in that direction. The matrix adds its own permeability to that along the fractures: $k_{\text{along}} \approx k_m + k$ .

Fractures and naturally fractured reservoirs →

Closure of a fracture under load (hyperbolic law)

\Delta V = \frac{\sigma_n' V_m}{K_{ni} V_m + \sigma_n'}

stresses in MPa, closure and $V_m$ in µm, $K_{ni}$ in MPa/µm

Holds when: An empirical relation. $V_m$ and $K_{ni}$ are fitted to tests on samples with a fracture, and they depend on the roughness and strength of the walls. Closure is reversible only in part: under repeated load the fracture is stiffer, while slip along it can, on the contrary, open it through the bumps of the walls. We take the change of the hydraulic aperture to equal the change of the mechanical one. For rough fractures, the hydraulic aperture can fall faster as the contacts grow.

Fractures and naturally fractured reservoirs →

Stretching from the rotation of rigid blocks (domino model)

\beta = \frac{L}{L_0} = \frac{\sin\delta_0}{\sin\delta},\qquad \delta_0 = \delta + \delta_b

angles in degrees; $L$ and $L_0$ in the same units of length

Holds when: The faults are planar (straight, not curved) and parallel, and the blocks rotate as rigid bodies about axes parallel to the strike. Before extension the beds were horizontal. We measure the tilt of the beds and the dip of the faults in a section across the strike of the faults, otherwise the angles are apparent (apparent dip). The formula describes the extension of only the upper brittle crust cut by these faults, not of the whole lithosphere.

Structural styles and tectonic history →

Directions of structures in a strike-slip zone

\alpha_R = \frac{\varphi}{2},\quad \alpha_{R'} = 90° - \frac{\varphi}{2},\quad \alpha_P = -\frac{\varphi}{2},\quad \alpha_T = 45°,\quad \alpha_F = -45°

angles in degrees

Holds when: Uniform simple shear in a wide zone. The greatest compression is horizontal and at 45° to the zone, and shears form by the Coulomb criterion (a rule for when rock fails in shear) at an angle of $45° - \varphi/2$ to it. The angles refer to the beginning of the deformation. As the displacement grows, the structures rotate towards the strike of the strike-slip fault, and the shears merge into a through-going fault. If the blocks also move towards each other (convergence), folds and thrusts lie closer to the strike of the strike-slip fault. If they move apart (divergence), normal faults lie closer to it.

Structural styles and tectonic history →

Net throw of an inverted normal fault and the null point

t_{net}(y) = t_{ext}(y) - t_{rev},\qquad t_{ext}(y) = h_{syn} - y,\qquad y_0 = h_{syn} - t_{rev}

metres

Holds when: There are no syn-rift deposits in the footwall, and the depression in the hanging wall was filled with sediment right to the top during extension. Then the throw of a horizon equals the thickness of the syn-rift layers above it. In compression the fault slides as a whole, and all horizons receive the same reverse throw. If part of the compression went into new cross-cutting reverse faults or folds, the reverse throw on the old fault is less than the total shortening.

Structural styles and tectonic history →

Expansion index and the increase in throw of a growth fault

EI = \frac{h_{hw}}{h_{fw}},\qquad \Delta t = h_{hw} - h_{fw},\qquad t_k = \sum_{i > k}\Delta t_i

thicknesses and throws in metres

Holds when: Each time, sediment filled the relief created by the fault right to the top, and the footwall was not eroded. The thicknesses in both walls are reduced to one degree of compaction. In the hanging wall the unit is buried deeper and compacted more, so we restore its thickness to the original where possible (restoring thicknesses). The youngest unit is not displaced by the fault. Otherwise its own throw is added to the sum.

Structural styles and tectonic history →

What fraction of the curvature of a fold the second difference shows

\frac{z_1 - 2z_0 + z_2}{a^2} = \sum_k A_k\left(\frac{2\pi}{\lambda_k}\right)^{2} s_k,\qquad s_k = \left(\frac{\sin(\pi a/\lambda_k)}{\pi a/\lambda_k}\right)^{2}

lengths in metres, curvature in m⁻¹

Holds when: We write the profile of the top as a sum of cosines. The formula is for a point where all the waves have a crest. At other points each wave enters the sum with its own factor, but the fraction $s_k$ is the same. The slopes are small, as in the formula for the crest in the lesson on folds. The depths have no errors. A random error in depth is divided by $a^2$ , so at a very small spacing the noise distorts the curvature more than the loss of short waves does.

Practical structural geology →

Pressure in an isolated dipping bed and the excess at its top

p_r(z) = p_{sh}(z_c) + G_w\,(z - z_c),\qquad \Delta p_{top} = p_r(z_{top}) - p_{sh}(z_{top}) = (G_{sh} - G_w)\,(z_c - z_{top})

pressures in MPa, gradients in MPa/m, depths in metres

Holds when: The body is connected and filled with water. Hydrocarbons at the top add their buoyancy (the extra pressure that comes from being lighter than water) to the pressure. The flows through the shales are steady, and the shales let water through equally well over the whole surface of the body. Then the centroid is the mean depth of the surface of the body, weighted by area (the derivation is in the advanced section). The pressure gradient of the shales is constant over the depth interval of the body. If the shales deeper down are more compacted and let less water through, the centroid moves up, and the excess at the crest is less than the calculated value.

Practical structural geology →

Normal stress on a vertical fracture

\sigma_n = \sigma_H\sin^2\theta + \sigma_h\cos^2\theta,\qquad \sigma_n' = \sigma_n - p_p

stresses and pressure in MPa, angle in degrees

Holds when: The fracture is vertical and one of the principal stresses (the three stresses that act at right angles to one another) is vertical. For an inclined fracture, the contribution of $\sigma_v$ is added (advanced section). We measure the angle $\theta$ from the strike of the fracture. In stresses on a plane the angle is measured from the normal to the fracture (the line at right angles to it), and from that line $\sigma_H$ lies at $90° - \theta$ . The effective stress on the fracture is the total stress minus the pore pressure, as in the lesson on fractures. The stresses are present-day, at a given point; near faults and salt they may differ from the regional ones.

Practical structural geology →

Shortening from the bed length

S = l_0 - W = \sum_i l_i\,(1 - \cos\delta_i),\qquad \varepsilon = \frac{S}{l_0}

lengths in metres, angles in degrees

Holds when: Flexural slip: the length and thickness of the bed do not change. Before the deformation the bed lay horizontally; the section is drawn in depth, at true scale, along the direction of transport. We account for displacements on faults within the section; they are added to the bed length.

Practical structural geology →

Excess area and the depth of the detachment

A = S\,h_d,\qquad h_d = \frac{A}{S},\qquad A(z) = S\,(z_d - z)

lengths in metres, areas in m²

Holds when: The area of the unit between the horizon and the detachment does not change: plane strain, no pressure solution or compaction. The detachment is horizontal and there is no deformation beneath it. All horizons are shortened equally. Then the points of the area–depth graph lie on a straight line.

Practical structural geology →

Symbols

Symbol	Meaning and context
$\Delta z$	contour interval — the difference in depth between neighbouring contours, mStress, strain and structural analysis
$\Delta l$	length of hole between the top and the base of the bed, mStress, strain and structural analysis
$\Delta p$	excess of the pressure in the salt beneath the minibasin over the pressure in the salt beneath the dome, at the same depth $z_1$ , PaFolds, salt diapirs and their relationship to faults
$\Delta V$	closure of the fracture compared with the unloaded state, µmFractures and naturally fractured reservoirs
$\Delta t$	increase in vertical throw during the accumulation of the unit, mStructural styles and tectonic history
$\Delta p,\ L_f$	pressure drop over the length of the path along the fracture, Pa, mFractures and naturally fractured reservoirs
$\mathbf n,\ \mathbf w$	unit vectors of the normal to the fractures and of the direction of the well, as in the vector calculation, —Fractures and naturally fractured reservoirs
$a$	spacing of the points from which the curvature is calculated, mPractical structural geology
$A$	excess area of the horizon above its regional level, m²Practical structural geology
$A_k$	amplitude of the $k$ -th wave — half the height difference between its crest and its trough, mPractical structural geology
$A_k(2\pi/\lambda_k)^2$	true curvature of the $k$ -th wave at its crest, m⁻¹Practical structural geology
$c$	exponent, —Faults
$C$	constant that depends on the greatest burial depth: 0.5 for less than 3 km; 0.25 for 3 to 3.5 km; 0 for more than 3.5 km, —Faults
$D$	slip on the fault plane (net slip), mFaults
$D_d$	slip along the dip of the fault plane, mFaults
$D_d,\ D_s$	components of the slip along the dip and along the strike of the fault plane, mFaults
$e$	relative elongation of a line (extension); a negative value is shortening, —Stress, strain and structural analysis relative elongation along the layer (strain); positive for extension, negative for shortening, —Folds, salt diapirs and their relationship to faults
$EI$	expansion index of the unit, —Structural styles and tectonic history
$\Delta p_f$	greatest pressure difference across the fault that it holds (1 bar = 0.1 MPa), barFaults
$F$	fold axes and the strike of thrusts, —Structural styles and tectonic history
$g$	acceleration of gravity, 9.81, m/s²Faults acceleration due to gravity, 9.81, m/s²Folds, salt diapirs and their relationship to faults
$G_{sh}$	pore pressure gradient of the shales over the depth interval of the body; with undercompaction at a constant effective stress, it equals the overburden pressure gradient, MPa/mPractical structural geology
$G_w$	pressure gradient of the water in the bed, MPa/mPractical structural geology
$h$	true (stratigraphic) thickness — measured along the normal to the bedding, mStress, strain and structural analysis true thickness of the bed, mStress, strain and structural analysis thickness of a layer that bends as a whole (no slip within it), mFolds, salt diapirs and their relationship to faults thickness of the competent layer, mFolds, salt diapirs and their relationship to faults size of the fracture in its plane, across the flow, mFractures and naturally fractured reservoirs
$H$	how far below the point the free water level of the accumulation can fall before the fault at this point starts to let hydrocarbons through, mFaults
$h_d$	height of the regional level of the horizon above the detachment, mPractical structural geology
$h_{hw},\ h_{fw}$	thickness of the unit in the hanging wall and in the footwall next to the fault, mStructural styles and tectonic history
$h_{syn}$	thickness of the syn-rift sequence in the hanging wall next to the fault, mStructural styles and tectonic history
$h_\text{v}$	vertical thickness — between the top and the base in a vertical well, mStress, strain and structural analysis
$\Delta z_i$	thickness of layer $i$ within the interval of height $t$ that has slid past the point: in the section of the footwall, from the point upwards by $t$ , mFaults
$k$	permeability of the rock along the fracture set, created by the fractures, m²Fractures and naturally fractured reservoirs
$K_{ni}$	initial normal stiffness of the fracture, MPa/µmFractures and naturally fractured reservoirs
$l$	length of the interval along the well, mFractures and naturally fractured reservoirs
$L$	length of the stretch across the set on which the fractures are counted, mFractures and naturally fractured reservoirs
$l_0$	length of the bed between the pin and loose lines, measured along the bed — its length before deformation, mPractical structural geology
$l_0,\ l$	length of the line before and after deformation, mStress, strain and structural analysis
$L_d$	dominant wavelength of the folds, mFolds, salt diapirs and their relationship to faults
$l_i,\ \delta_i$	length and dip angle of the $i$ -th straight segment of the bed, m, °Practical structural geology
$L,\ L_0$	present and original horizontal distance between neighbouring faults, kmStructural styles and tectonic history
$N$	number of fractures of the set crossed by the well in the interval, —Fractures and naturally fractured reservoirs
$N_1,\ N_2$	numbers of faults with throw not less than $t_1$ and $t_2$ ; both throws are at or above the resolution limit, —Faults
$N(t)$	number of faults whose greatest throw is not less than $t$ in the area, —Faults
$p$	pore pressure, MPaFractures and naturally fractured reservoirs
$P$	P shears, which are also synthetic but rotated the other way; they appear later than R shears and link them, —Structural styles and tectonic history
$P_{10}$	linear intensity along the normal to the set, m⁻¹Fractures and naturally fractured reservoirs
$p_p$	pore (reservoir) pressure, MPaPractical structural geology
$p_r(z)$	water pressure in the sand body at depth $z$ , MPaPractical structural geology
$p_{sh}(z)$	pore pressure in the surrounding shales, MPaPractical structural geology
$q$	flow rate through one fracture, m³/sFractures and naturally fractured reservoirs
$r$	distance of the point from the centre of the projection circle, mm on the drawingStress, strain and structural analysis angle in the fault plane between the strike line and the direction of slip (rake), °Faults
$R$	radius of the projection circle, mmStress, strain and structural analysis radius of curvature of the neutral surface, mFolds, salt diapirs and their relationship to faults
$R,\ R'$	synthetic and antithetic Riedel shears, —Structural styles and tectonic history
$s$	vertical separation of the bed: the vertical distance between the parts of one bed in the two walls. In a vertical well it is the thickness of the missing interval (for a normal fault) or the repeated interval (for a reverse fault), mFaults average spacing of the set, mFractures and naturally fractured reservoirs
$S$	stretch, —Stress, strain and structural analysis shortening of the section — a length; the stretch ratio $l/l_0$ from the lesson on strain here equals $1 - \varepsilon$ , mPractical structural geology shortening, mPractical structural geology
$s_k$	fraction of the curvature of the $k$ -th wave that the second difference shows, —Practical structural geology
$s_n$	stratigraphic separation: the same thickness, measured at right angles to the layers, mFaults
$SGR$	share of clay material in the fault zone at a given point of the fault plane, %Faults SGR at this point, %Faults
$t$	throw (vertical offset), mFaults greatest throw of a fault, mFaults throw of the fault at this point, mFaults
$T$	tensile strength of the rock (a positive number), MPaFractures and naturally fractured reservoirs joints and normal faults, —Structural styles and tectonic history
$t_{ext}(y)$	throw of this horizon accumulated during extension, mStructural styles and tectonic history
$t_k$	vertical throw of the top of unit $k$ — the sum of the increases of all younger units, mStructural styles and tectonic history
$t_{net}(y)$	net vertical throw on the horizon at height $y$ : positive is normal, negative is reverse, mStructural styles and tectonic history
$t_{rev}$	reverse throw added by compression; it is measured on a horizon that had not been displaced before the compression, mStructural styles and tectonic history
$t,\ w$	throw and heave, mFaults
$\Delta p_{top}$	pressure excess at the crest of the body over the pressure of the shales at the same depth, MPaPractical structural geology
$V$	exaggeration factor: how many times the vertical scale is larger than the horizontal one, —Stress, strain and structural analysis
$V_m$	greatest possible closure, µmFractures and naturally fractured reservoirs
$V_{sh,i}$	shale volume of layer $i$ from well logs (Determining shale volume), fractionFaults
$w$	horizontal distance between neighbouring contours, measured perpendicular to them, mStress, strain and structural analysis heave (horizontal offset), at right angles to the strike, mFaults
$W$	present horizontal distance between the pin and loose lines, mPractical structural geology
$w,\ w_i$	hydraulic aperture of the fracture, mFractures and naturally fractured reservoirs
$x,\ y$	coordinates of the point: to the east and to the north of the centre, mmStress, strain and structural analysis
$y$	distance from the neutral surface of the layer, which keeps its length: positive towards the convex side (the outer arc), negative towards the centre of curvature, mFolds, salt diapirs and their relationship to faults height of the horizon above the base of the syn-rift sequence in the hanging wall next to the fault, mStructural styles and tectonic history
$y_0$	height of the null point, mStructural styles and tectonic history
$z$	regional depth of the horizon, mPractical structural geology
$z_1$	depth of the top of the salt beneath the minibasin (the thickness of sediment above the salt there), mFolds, salt diapirs and their relationship to faults
$z_1,\ z_0,\ z_2$	depths of the surface at three points of a profile perpendicular to the hinge, equally spaced by a distance $a$ ; $z_0$ is at the crest, mFolds, salt diapirs and their relationship to faults depths of the top at three points a distance $a$ apart; $z_0$ is at the crest, mPractical structural geology
$z_2$	depth of the top of the salt at the dome, mFolds, salt diapirs and their relationship to faults
$z_c$	depth of the centroid, where the pressures of the body and the shales are equal; for a body whose area is spread evenly over its depth range, the middle of its height, mPractical structural geology
$z_d$	depth of the detachment, mPractical structural geology
$z_{top}$	depth of the top (crest) of the body, mPractical structural geology
$\alpha$	angle between the strike of the structure and the strike of the strike-slip fault; for a right-lateral fault it is measured clockwise, for a left-lateral one anticlockwise, °Structural styles and tectonic history
$\beta$	angle between the line of the section and the strike, °Stress, strain and structural analysis stretching factor: how many times a section across the strike of the faults has lengthened, —Structural styles and tectonic history
$\gamma$	shear strain, —Stress, strain and structural analysis angle between the axis of the hole and the normal to the bedding, °Stress, strain and structural analysis angle between the well and the normal to the fractures of the set, °Fractures and naturally fractured reservoirs
$\delta$	true dip, °Stress, strain and structural analysis dip of the bed, °Stress, strain and structural analysis dip angle, °Stress, strain and structural analysis dip of the fault plane, °Faults present dip of the faults, °Structural styles and tectonic history
$\delta,\ a$	dip angle and dip azimuth of the bed, °Stress, strain and structural analysis
$\delta'$	apparent dip on the section, °Stress, strain and structural analysis apparent dip on this section (the true dip on a section across the strike), °Stress, strain and structural analysis
$\delta_0$	dip of the faults at the time they formed, °Structural styles and tectonic history
$\delta_b$	dip of the beds in a section across the strike of the fault, °Faults tilt of the beds in the blocks — the angle through which the blocks have rotated (the notation of the lesson on faults), °Structural styles and tectonic history
$\delta_V$	the angle at which the bed is drawn on the section, °Stress, strain and structural analysis
$\varepsilon$	relative shortening, —Practical structural geology
$\eta_l,\ \eta_m$	viscosities of the layer and of the surrounding rocks, Pa·sFolds, salt diapirs and their relationship to faults
$\theta$	angle between the direction of $\sigma_1$ and the normal (perpendicular) to the plane, °Stress, strain and structural analysis angle between the strike of the fracture and the direction of $\sigma_H$ , °Practical structural geology
$\theta,\ \varphi$	inclination and azimuth of the hole, °Stress, strain and structural analysis
$\lambda_k$	wavelength of the $k$ -th wave — the distance between neighbouring crests, mPractical structural geology
$\mu$	viscosity of the fluid, Pa·sFractures and naturally fractured reservoirs
$\rho$	plunge of the line, °Stress, strain and structural analysis
$\rho_b(z)$	bulk density of the sediments at depth $z$ , kg/m³Folds, salt diapirs and their relationship to faults
$\rho_s$	density of the salt, about 2160, kg/m³Folds, salt diapirs and their relationship to faults
$\rho_w,\ \rho_h$	densities of formation water and of the hydrocarbon in the reservoir, kg/m³Faults
$\sigma_1,\ \sigma_3$	greatest and least principal stresses, MPaStress, strain and structural analysis
$\sigma_1',\ \sigma_3'$	greatest and least effective principal stress, $\sigma_1 - p$ and $\sigma_3 - p$ , MPaFractures and naturally fractured reservoirs
$\sigma_H,\ \sigma_h$	greatest and least horizontal stresses, MPaPractical structural geology
$\sigma_n$	normal stress on the plane, MPaStress, strain and structural analysis total normal stress on the plane of the fracture, MPaPractical structural geology
$\sigma_n'$	effective normal stress on the plane, $\sigma_n - p$ ; compression is positive and tension is negative, MPaFractures and naturally fractured reservoirs effective normal stress on the fracture, MPaFractures and naturally fractured reservoirs effective normal stress that squeezes the fracture shut, MPaPractical structural geology
$\sigma_v(z)$	overburden pressure of the sediments at depth $z$ (overburden pressure), PaFolds, salt diapirs and their relationship to faults
$\tau$	shear stress on the plane, MPaStress, strain and structural analysis shear stress on a plane, MPaFractures and naturally fractured reservoirs
$\varphi$	trend of the line, °Stress, strain and structural analysis angle of friction: $\mathrm{tan}\,\varphi = \mu$ (friction on a plane), °Structural styles and tectonic history
$\phi_f$	fracture porosity, fractionFractures and naturally fractured reservoirs
$\psi$	angular shear: the angle by which a line that was perpendicular to a given direction has turned away from the right angle, °Stress, strain and structural analysis
$\omega$	angle between the line of the section and the dip direction, $\omega = 90° - \beta$ , °Stress, strain and structural analysis