Equations: Statistics, volumes and reserves

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

Arithmetic mean of a sample

\bar x = \frac{1}{N}\sum_{i=1}^{N} x_i

the same as those of the measured quantity

Holds when: Each measurement represents the same share of the studied volume (the plugs are taken evenly). Otherwise we need weights, such as thicknesses or volumes. All the measurements belong to one population: one layer, one rock type.

Descriptive statistics in geology →

Geometric mean

\bar k_{\text{geom}} = \left(\prod_{i=1}^{N} k_i\right)^{1/N} = \exp\left(\frac{1}{N}\sum_{i=1}^{N}\ln k_i\right)

the same as for $k_i$ ; $\ln$ is the natural logarithm

Holds when: All the values are positive. A single zero turns the mean into zero, so instead of “zeros” we put in the measurement limit of the instrument. The plugs represent equal volumes of rock. Otherwise the logarithms are weighted by thickness.

Descriptive statistics in geology →

Harmonic mean

\bar k_{\text{harm}} = N\left(\sum_{i=1}^{N}\frac{1}{k_i}\right)^{-1}

the same as for $k_i$

Holds when: All the values are positive. The plugs represent equal thicknesses. Otherwise $\bar k_{\text{harm}} = \sum h_i / \sum (h_i/k_i)$ .

Descriptive statistics in geology →

Relation between the three means

\bar k_{\text{harm}} \le \bar k_{\text{geom}} \le \bar k_{\text{arith}}

any units, the same for all three

Holds when: All the values are positive and are taken with equal weights. Equality holds only if all the values are equal.

Descriptive statistics in geology →

Variance and standard deviation of a sample

s^2 = \frac{1}{N-1}\sum_{i=1}^{N}\left(x_i - \bar x\right)^2, \qquad s = \sqrt{s^2}

$s$ is in the units of the measured quantity, $s^2$ is in their squares

Holds when: The measurements belong to one population. The denominator is $N - 1$ , not $N$ . The deviations are measured from the mean of the same sample, so on average they are slightly smaller than from the true mean. For $N$ above a few dozen, the difference does not matter.

Descriptive statistics in geology →

Coefficient of variation

C_v = \frac{s}{\bar x}

dimensionless; sometimes it is expressed in per cent

Holds when: The quantity is positive and is measured from a natural zero (permeability, porosity, thickness). For temperature in degrees Celsius or for elevations above sea level, the coefficient of variation has no meaning. The mean and the deviation are calculated from the same measurements on the ordinary scale, not the logarithmic one.

Descriptive statistics in geology →

Cumulative shares of flow capacity and storage capacity (Lorenz curve)

F_J = \frac{\sum_{j=1}^{J} k_j h_j}{\sum_{i=1}^{I} k_i h_i}, \qquad C_J = \frac{\sum_{j=1}^{J} \phi_j h_j}{\sum_{i=1}^{I} \phi_i h_i}

fractions; $k$ , $\phi$ and $h$ are in any units that are the same for all the layers

Holds when: The layers are ordered by decreasing $k/\phi$ . Each plug represents its own layer. For equal thicknesses, $h$ cancels.

Descriptive statistics in geology →

Lorenz coefficient

L_c = 2A

dimensionless, from 0 to 1

Holds when: We find the area from the plotted curve, for example by the trapezoid rule (adding up the areas of the trapezoids between neighbouring points).

Descriptive statistics in geology →

Probability density of the normal distribution

f(x) = \frac{1}{\sigma\sqrt{2\pi}}\,\exp\left[-\frac{(x - \mu)^2}{2\sigma^2}\right]

$x$ , $\mu$ and $\sigma$ are in the units of the measured quantity

Holds when: The quantity can take any values, including negative ones. For porosity or thickness this is only an approximation, and it is good as long as $\mu$ is greater than about $3\sigma$ . The distribution is unimodal (it has one peak) and symmetrical.

Distributions and sample size →

Standardised value

z = \frac{x - \mu}{\sigma}

dimensionless

Holds when: The distribution is normal. Then $z$ follows the standard normal distribution, which has a mean of 0 and a standard deviation of 1.

Distributions and sample size →

Median, mean and mode of the log-normal distribution

x_{50} = \e^{\mu}, \qquad \bar x = \e^{\mu + \sigma^2/2}, \qquad x_{\text{mode}} = \e^{\mu - \sigma^2}

$x$ is in the units of the quantity (for example, mD); $\mu$ and $\sigma$ refer to $\ln x$ and are dimensionless

Holds when: The logarithm of the quantity is normally distributed. We calculate $\mu$ and $\sigma$ from natural logarithms (base e). For decimal logarithms, $\mu_{\ln} = 2.303\,\mu_{\lg}$ and $\sigma_{\ln} = 2.303\,\sigma_{\lg}$ .

Distributions and sample size →

Percentiles and coefficient of variation of the log-normal distribution

x_p = \e^{\mu + z_p\sigma}, \qquad C_v = \sqrt{\e^{\sigma^2} - 1}

$x_p$ is in the units of the quantity; the rest is dimensionless

Holds when: The logarithm of the quantity is normally distributed.

Distributions and sample size →

Standard error of the mean

SE = \frac{s}{\sqrt{N}}

the same as those of the measured quantity

Holds when: The measurements are independent: the value of one sample does not predict the value of another. The sample is representative: every part of the population had an equal chance of being included. For strongly skewed data, the normal approximation for the mean works only with tens of samples.

Distributions and sample size →

Confidence interval of the mean

\bar x - t\,\frac{s}{\sqrt{N}} \;\le\; \mu \;\le\; \bar x + t\,\frac{s}{\sqrt{N}}

the same as those of the measured quantity

Holds when: The same as for the standard error. The data are distributed approximately normally, or there are enough samples. A confidence probability of 95% means this: if we build a hundred such intervals from different samples, about 95 of them will cover the true mean.

Distributions and sample size →

Number of samples needed for a given precision of the mean

N_0 = \left(\frac{100\,t\,C_v}{P}\right)^2

$P$ is in per cent; the other quantities are dimensionless

Holds when: We are estimating the arithmetic mean. The measurements are independent and the sample is representative. There are enough samples for the mean to be distributed approximately normally. $C_v$ is known at least approximately, from a trial series of measurements or by analogy with similar rocks.

Distributions and sample size →

Rule for the number of samples: precision of the mean ±20% at 95% probability

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

$P_s$ is in per cent of the mean

Holds when: $t = 2$ : probability 95%, more than thirty samples. The first formula refers to a tolerance of 20%, the second to any number of samples at $t = 2$ . For very heterogeneous rocks ( $C_v$ noticeably greater than 1, for example complex carbonates), a tolerance of ±20% is often unattainable. Then we accept ±50%, and $N_0 = (4\,C_v)^2$ .

Distributions and sample size →

Dykstra–Parsons coefficient

V_{DP} = \frac{k_{50} - k_{84.1}}{k_{50}} = 1 - \e^{-\sigma_{\ln k}}

dimensionless, from 0 to 1

Holds when: Permeability is log-normally distributed. For other distributions the coefficient can be misleading. We take $k_{50}$ and $k_{84.1}$ from the line drawn through the points of the probability plot, not from individual samples. The percentage 84.1 is the proportion of samples with a greater permeability. It is the exceedance probability, not the cumulative proportion. On a plot with an axis of cumulative proportion, as in this lesson, we read $k_{84.1}$ at the 16% mark ( $z = -1$ ).

Distributions and sample size →

Confidence interval of the geometric mean

\exp\left(m - t\,\frac{s_{\ln}}{\sqrt N}\right) \;\le\; x_{50} \;\le\; \exp\left(m + t\,\frac{s_{\ln}}{\sqrt N}\right)

the limits are in the units of the quantity

Holds when: The quantity is log-normally distributed. The measurements are independent and representative.

Distributions and sample size →

Standard error of the mean for dependent measurements

SE = \frac{s}{\sqrt{N}}\,\sqrt{1 + (N - 1)\,\bar\rho}, \qquad N_{\text{eff}} = \frac{N}{1 + (N - 1)\,\bar\rho}

$SE$ is in the units of the measured quantity

Holds when: We are estimating the mean of a population that is wider than the sampled stretch: of a reservoir from one well, or of an accumulation from several wells. All the measurements have the same variance (the same spread), and the correlation between them is not negative.

Distributions and sample size →

Experimental variogram

\hat\gamma(h) = \frac{1}{2N(h)}\sum_{i=1}^{N(h)}\left[z(x_i + h) - z(x_i)\right]^2

values in their own units, distances in metres

Holds when: The property is stationary in the studied area: the mean does not change in a regular way, and the differences depend only on the lag. Irregularly placed data are grouped into classes of lag ( $h$ plus or minus a tolerance) and of direction (plus or minus an angular tolerance). An estimate from lags with few pairs is unreliable. A practical rule is to use lags no longer than half the size of the area and to keep several tens of pairs per lag.

Spatial correlation and the variogram →

Spherical, exponential and Gaussian models

\begin{aligned} \gamma_{\text{sph}}(h) &= C_0 + C\left[\frac{3h}{2a} - \frac{1}{2}\left(\frac{h}{a}\right)^{3}\right]\ \text{for}\ h \le a;\qquad C_0 + C\ \text{for}\ h > a\\ \gamma_{\text{exp}}(h) &= C_0 + C\left[1 - e^{-3h/a}\right]\\ \gamma_{\text{gauss}}(h) &= C_0 + C\left[1 - e^{-3h^2/a^2}\right] \end{aligned}

the variogram in squared units of the property, distances in metres

Holds when: A model is a sum of permitted structures with non-negative coefficients (nested structures). The nugget effect is a jump at any arbitrarily small $h > 0$ , and the variogram itself is zero at zero. The exponential and Gaussian models are often written with the parameter $a/3$ or $a/\sqrt3$ . When we move parameters between programs, we check which range is meant.

Spatial correlation and the variogram →

Relation between covariance, correlogram and variogram

C(h) = C(0) - \gamma(h),\qquad \rho(h) = \frac{C(h)}{C(0)} = 1 - \frac{\gamma(h)}{C(0)},\qquad C(0) = C_0 + C = \sigma^2

in squared units of the property; $\rho$ is dimensionless

Holds when: Second-order stationarity: a constant mean and a finite variance, which the variogram reaches at the sill. With a trend, or with a variogram that has no sill, the covariance is not defined but the variogram is, and we use the variogram.

Spatial correlation and the variogram →

Reduced distance and range in a direction for geometric anisotropy

h' = \sqrt{\left(\frac{h_{\parallel}}{a_{max}}\right)^2 + \left(\frac{h_{\perp}}{a_{min}}\right)^2},\qquad a(\alpha) = \left[\frac{\cos^2(\alpha - \alpha_0)}{a_{max}^2} + \frac{\sin^2(\alpha - \alpha_0)}{a_{min}^2}\right]^{-1/2}

distances in metres, azimuths in degrees

Holds when: Geometric anisotropy: the sill and the nugget effect are the same in all directions. The axes of the ellipse coincide with the principal directions, usually along the elongation of the bodies and across it. In three dimensions we add a vertical axis with a much smaller range. For zonal anisotropy, we add to the model a structure that acts in one direction only.

Spatial correlation and the variogram →

The ordinary kriging system

\sum_{j=1}^{n}\lambda_j\,\gamma(x_i - x_j) + \mu = \gamma(x_i - x_0),\quad i = 1, \dots, n;\qquad \sum_{j=1}^{n}\lambda_j = 1

variogram in squared units of the property

Holds when: The mean is unknown but constant in the neighbourhood of the estimate (ordinary kriging). When the mean is known, we use simple kriging: without the condition on the sum of the weights, and with the covariance instead of the variogram. The variogram is known and refers to the same quantity and the same support as the data. The estimate is a linear combination of the data. Measurement errors are included in the nugget effect.

Spatial correlation and the variogram →

The kriging estimate and its variance

z^*(x_0) = \sum_{i=1}^{n}\lambda_i\,z(x_i),\qquad \sigma_K^2 = \sum_{i=1}^{n}\lambda_i\,\gamma(x_i - x_0) + \mu

as for the property and the variogram

Holds when: The weights and $\mu$ are the solution of the ordinary kriging system. The kriging variance is correct if the variogram is correct. The distribution of the errors need not be normal.

Spatial correlation and the variogram →

Oil initially in place (volumetric method)

N = \frac{\mathrm{GRV}\cdot \mathrm{NTG}\cdot\phi\cdot(1 - S_w)}{B_{oi}}

metric: volumes in m³, the other quantities are dimensionless

Holds when: The pores of the reservoir contain only oil and connate water (the water that has stayed in the pores since the rock was formed). If there is a gas cap (a layer of free gas on top of the oil), the volume above the gas–oil contact is calculated separately, with the formula for gas. The mean $\phi$ and $S_w$ refer to the same volume of reservoir as the NTG. $B_{oi}$ corresponds to the initial reservoir conditions and to the adopted scheme of separating the oil at the surface. We use mean values for the whole accumulation. If the parameters change noticeably across the area, we calculate from maps or from the cells of the geological model and add up the results.

Volumetric estimation of hydrocarbons in place →

Free gas initially in place

G = \frac{\mathrm{GRV}\cdot \mathrm{NTG}\cdot\phi\cdot(1 - S_w)}{B_{gi}}

metric: volumes in m³

Holds when: The pores of the reservoir are filled with gas and connate water (a gas accumulation or a gas cap). $B_{gi}$ is calculated for the same standard conditions to which the volumes in place are referred. Gas dissolved in the oil is not included here. We count it separately as $N\cdot R_{si}$ , where $R_{si}$ is the initial solution gas–oil ratio of the oil (how much gas is dissolved in each cubic metre of oil).

Volumetric estimation of hydrocarbons in place →

Gas formation volume factor

B_g = \frac{p_{sc}}{T_{sc}}\cdot\frac{Z\,T}{p}

any consistent units: pressures in the same units, temperatures absolute (K or °R)

Holds when: The real-gas equation of state $pV = ZnRT$ (it links the pressure, volume and temperature of the gas); at standard conditions the gas is taken as ideal ( $Z = 1$ ). The pressures are absolute (measured from zero), not gauge (measured above atmospheric pressure). The temperatures are in kelvins or in degrees Rankine (the absolute scale for degrees Fahrenheit).

Volumetric estimation of hydrocarbons in place →

Mean porosity over the net thickness

\bar\phi = \frac{\sum_k \phi_k\,h_k}{\sum_k h_k}

any units, the same for all intervals

Holds when: We add up only the intervals of net thickness. All the intervals belong to one well (one point on the map).

Volumetric estimation of hydrocarbons in place →

Mean water saturation weighted by pore volume

\bar S_w = \frac{\sum_k S_{w,k}\,\phi_k\,h_k}{\sum_k \phi_k\,h_k}

any units, the same for all intervals

Holds when: We add up the same intervals as for the mean porosity. Saturation is a share of the pore volume, so the weight of an interval is its pore volume, not its thickness.

Volumetric estimation of hydrocarbons in place →

Hydrocarbon pore thickness

H = \sum_k h_k\,\phi_k\,(1 - S_{w,k}) = h_{\text{eff}}\,\bar\phi\,(1 - \bar S_w)

m (or ft)

Holds when: $\bar\phi$ and $\bar S_w$ are averaged by the formulas above.

Volumetric estimation of hydrocarbons in place →

Recoverable volume

Q_{\text{rec}} = N\cdot\text{RF}

the same as for $N$

Holds when: The RF refers to a particular development system and particular economic conditions. When these change, the recoverable volume changes too. For gas we use the gas recovery factor in the same way.

Volumetric estimation of hydrocarbons in place →

Value of a parameter from a random number

x = F^{-1}(u)

$u$ and $F$ are dimensionless; $x$ is in the units of the parameter

Holds when: $u$ is uniformly distributed: any value between 0 and 1 is equally likely. $F(x)$ is continuous and increasing, so that each $u$ corresponds to one value of $x$ . For independent parameters, the random numbers are independent of one another.

Probabilistic volumetrics by Monte Carlo simulation →

Distribution function of the triangular distribution

F(x) = \begin{cases} \dfrac{(x - a)^2}{(b - a)(m - a)}, & a \le x \le m \\[10pt] 1 - \dfrac{(b - x)^2}{(b - a)(b - m)}, & m < x \le b \end{cases}

$x$ , $a$ , $m$ and $b$ are in the units of the parameter; $F$ is dimensionless

Holds when: The probability density rises linearly from zero at $a$ to its largest value at $m$ , and falls linearly to zero at $b$ . Values smaller than $a$ and larger than $b$ are impossible: there $F = 0$ and $F = 1$ . $a < m < b$ . If $m$ is in the middle, the distribution is symmetrical; otherwise it is skewed.

Probabilistic volumetrics by Monte Carlo simulation →

Drawing from the triangular distribution

x = \begin{cases} a + \sqrt{u\,(b - a)(m - a)}, & u \le F(m) \\[4pt] b - \sqrt{(1 - u)(b - a)(b - m)}, & u > F(m) \end{cases} \qquad F(m) = \frac{m - a}{b - a}

$x$ , $a$ , $m$ and $b$ are in the units of the parameter

Holds when: The same as for the distribution function above. $u$ is uniformly distributed between 0 and 1.

Probabilistic volumetrics by Monte Carlo simulation →

Error of the mean and of a probability from a simulation

SE = \frac{s}{\sqrt{n}}, \qquad SE_p = \sqrt{\frac{p\,(1 - p)}{n}}

$SE$ is in the units of the result (million m³); $SE_p$ and $p$ are fractions

Holds when: The trials are independent of one another: a simple random draw. $n$ is large enough for the mean and the share to follow a distribution close to normal (hundreds of trials or more). The formulas describe the error of the calculation for the given distributions, not the uncertainty of the volume in place.

Probabilistic volumetrics by Monte Carlo simulation →

Mean and variance of the triangular distribution

\mu = \frac{a + m + b}{3}, \qquad \sigma^2 = \frac{a^2 + m^2 + b^2 - am - ab - mb}{18}

in the units of the parameter; the coefficient of variation $C_v = \sigma/\mu$ is dimensionless

Holds when: A triangular distribution with $a < m < b$ .

Probabilistic volumetrics by Monte Carlo simulation →

Mean and coefficient of variation of a product of independent quantities

\mu_N = \prod_j \mu_j, \qquad 1 + C_v^2(N) = \prod_j \left[1 + C_v^2(x_j)\right]

the means are in the units of the quantities; the coefficients of variation are dimensionless

Holds when: The factors are independent. With dependence, the mean of the product is no longer equal to the product of the means. The factor is exactly what enters the product: $1 - S_w$ , not $S_w$ , and $1/B_{oi}$ , not $B_{oi}$ . The type of the distributions does not matter: the formulas are exact for any independent factors.

Probabilistic volumetrics by Monte Carlo simulation →

A pair of linked random numbers

z_2 = \rho\,z_1 + \sqrt{1 - \rho^2}\;z', \qquad u_2 = \Phi(z_2)

all quantities are dimensionless

Holds when: $u_1$ and $u'$ are independent and uniformly distributed between 0 and 1. $\rho$ is the correlation of the normal quantities $z_1$ and $z_2$ . The correlation of the parameters themselves is close to it, but does not coincide with it exactly. The distributions of both parameters stay the same: only the way their values combine changes.

Probabilistic volumetrics by Monte Carlo simulation →

Probabilistic meaning of the categories of reserves

\Pr(Q \ge \text{1P}) \ge 0.9,\quad \Pr(Q \ge \text{2P}) \ge 0.5,\quad \Pr(Q \ge \text{3P}) \ge 0.1

any units of quantity, the same for all the estimates

Holds when: This is how estimates made by a probabilistic method are checked: 1P, 2P and 3P coincide with the P90, P50 and P10 of the distribution of recoverable quantities (Probabilistic volumetrics by Monte Carlo simulation). In a deterministic estimate, proved reserves are quantities that will be recovered with “reasonable certainty”, that is, with a high degree of confidence. For 2P, the actual recovery is equally likely to be above or below the estimate. The same thresholds apply to contingent resources (1C, 2C, 3C) and prospective resources (1U, 2U, 3U). The probabilities of prospective resources refer to the volume on the condition that the accumulation is discovered.

Classification of reserves and resources →

Chance of commerciality

P_c = P_g\cdot P_d

probabilities, fractions

Holds when: For prospective resources both factors apply. For contingent resources the discovery has already taken place, and $P_c = P_d$ . For reserves no separate chance is stated: the class needs high confidence that the project will be carried out. $P_d$ includes the size of the discovery (whether it is enough to pay back; this is how the lesson Geological risk and chance of success estimates the commercial chance of success), and also the market and the approvals.

Classification of reserves and resources →

Starting signature bonus for an area with approved reserves

B_\text{start} = \max\left(0.0004\,C + 0.0001\,C_\text{c2c3};\ 10{,}000\ \text{MCI}\right)

tenge (KZT)

Holds when: The value of the reserves is found from the arithmetic mean of the price quotations on the day before the auction conditions are published (or before the protocol of direct negotiations is signed), and at the official exchange rate on the day before the bonus is paid (Article 761 of the 2026 Tax Code). If the area is divided into blocks, then for each block after the three-hundredth the starting amount is increased by 10 MCI. The final bonus is set by the result of the auction or negotiations, and it is not lower than the starting amount.

Classification of reserves and resources →

Reserves replacement ratio

\mathrm{RRR} = \frac{\Delta R_\text{rev} + \Delta R_\text{disc} + \Delta R_\text{deals}}{Q_\text{yr}}

the same units for all quantities; reserves of one category (usually 1P or 2P)

Holds when: There is no single mandatory formula: companies state whether deals and revisions caused by prices are included. The ratio of reserves to annual production (R/P) shows for how many years the reserves will last at the present production, if they are not replaced.

Classification of reserves and resources →

Symbols

Symbol	Meaning and context
$\bar x$	arithmetic mean, an estimate of the true mean of the population, units of the quantityDescriptive statistics in geology arithmetic mean of the same measurements, units of the quantityDescriptive statistics in geology arithmetic mean, units of the quantityDistributions and sample size mean of the sample, units of the quantityDistributions and sample size
$\Pr(\cdot)$	probability of an event, —Classification of reserves and resources
$\text{1P},\ \text{2P},\ \text{3P}$	the minimum, best and maximum estimates of recoverable reserves (the cumulative categories), the sameClassification of reserves and resources
$a$	range: for the spherical model, the distance at which it reaches the sill; for the exponential and Gaussian models, the distance at which it reaches 95% of the partial sill, mSpatial correlation and the variogram smallest possible value, units of the parameterProbabilistic volumetrics by Monte Carlo simulation
$A$	area between the Lorenz curve and the diagonal, —Descriptive statistics in geology
$a_{max},\ a_{min}$	correlation ranges along the major and minor axes, mSpatial correlation and the variogram
$a,\ m,\ b$	the smallest, the most likely and the largest values, units of the parameterProbabilistic volumetrics by Monte Carlo simulation
$a(\alpha)$	correlation range in the direction $\alpha$ , mSpatial correlation and the variogram
$\bar k_{\text{arith}}$	arithmetic mean, mDDescriptive statistics in geology
$b$	largest possible value, units of the parameterProbabilistic volumetrics by Monte Carlo simulation
$B_{gi}$	gas formation volume factor at the initial reservoir pressure and temperature, m³/m³Volumetric estimation of hydrocarbons in place
$B_{oi}$	oil formation volume factor at the initial reservoir pressure and temperature, m³/m³ (reservoir per standard)Volumetric estimation of hydrocarbons in place
$B_\text{start}$	starting amount of the signature bonus under a production contract, or an exploration and production contract, for hydrocarbons, tengeClassification of reserves and resources
$C$	partial sill: the variability that is related to distance, units²Spatial correlation and the variogram value of the reserves of the commercial categories A, B and C1 approved by the state reserves commission, tengeClassification of reserves and resources
$C_0$	nugget effect, units²Spatial correlation and the variogram
$C_0 + C$	sill, units²Spatial correlation and the variogram
$C_\text{c2c3}$	value of the reserves of category C2 (approved, or taken into account from the operational estimate) and of the resources C3, tengeClassification of reserves and resources
$C_J$	cumulative share of storage capacity, fractionDescriptive statistics in geology
$C_v$	coefficient of variation, —Descriptive statistics in geology coefficient of variation of the quantity itself (not of its logarithm), —Distributions and sample size coefficient of variation $s/\bar x$ , —Distributions and sample size
$C_v(N)$	coefficient of variation of the result, —Probabilistic volumetrics by Monte Carlo simulation
$C_v(x_j)$	coefficient of variation of the factor: the standard deviation divided by the mean, —Probabilistic volumetrics by Monte Carlo simulation
$C(0) = \sigma^2$	variance of the property, the sill of the variogram, units²Spatial correlation and the variogram
$C(h)$	covariance of values a distance $h$ apart, units²Spatial correlation and the variogram
$\Delta R_\text{deals}$	shares bought minus shares sold, million tClassification of reserves and resources
$\Delta R_\text{disc}$	additions from discoveries and extensions of accumulations, million tClassification of reserves and resources
$F^{-1}(u)$	inverse function: the value below which the share $u$ of the distribution lies, units of the parameterProbabilistic volumetrics by Monte Carlo simulation
$F_J$	cumulative share of flow capacity, fractionDescriptive statistics in geology
$F(m)$	cumulative probability of the most likely value: the share of the distribution to the left of the peak, —Probabilistic volumetrics by Monte Carlo simulation
$f(x)$	probability density, 1/units of the quantityDistributions and sample size
$F(x)$	distribution function of the parameter: the probability that it does not exceed $x$ , —Probabilistic volumetrics by Monte Carlo simulation probability that the parameter does not exceed $x$ , —Probabilistic volumetrics by Monte Carlo simulation
$G$	free gas initially in place at standard conditions (GIIP), m³Volumetric estimation of hydrocarbons in place
$\bar k_{\text{geom}}$	geometric mean, mDDescriptive statistics in geology
$\mathrm{GRV}$	gross rock volume inside the oil outline: between the top and the base of the layer, above the OWC, m³Volumetric estimation of hydrocarbons in place
$\mathrm{GRV},\ \mathrm{NTG},\ \phi,\ S_w$	as in the formula for oil; GRV is the volume of rock above the gas–water (or gas–oil) contact, m³ and fractionVolumetric estimation of hydrocarbons in place
$h$	lag: the distance (and, over an area, the direction) between the points of a pair, mSpatial correlation and the variogram lag, $h > 0$ ; at $h = 0$ the variogram is zero, mSpatial correlation and the variogram
$H$	the thickness of the layer that the hydrocarbons would form if we gathered them out of the pores into a solid layer; the volume of hydrocarbons at reservoir conditions per unit area, mVolumetric estimation of hydrocarbons in place
$h_{\parallel},\ h_{\perp}$	components of the distance along the major axis of the ellipse and across it, mSpatial correlation and the variogram
$h_{\text{eff}}$	net thickness, $\sum_k h_k$ , mVolumetric estimation of hydrocarbons in place
$h_k$	thickness of this interval, mVolumetric estimation of hydrocarbons in place
$h'$	the distance between the points as a fraction of the range: the model is evaluated at $h/a = h'$ , —Spatial correlation and the variogram
$\bar k_{\text{harm}}$	harmonic mean, mDDescriptive statistics in geology
$I$	total number of layers; the sums in the denominators are taken over all the layers, —Descriptive statistics in geology
$J$	number of layers counted up to this step, —Descriptive statistics in geology
$k_{50}$	median permeability, mDDistributions and sample size
$k_{84.1}$	permeability exceeded by 84.1% of the samples — one standard deviation of the logarithm below the median, mDDistributions and sample size
$k_i$	permeability of the $i$ th plug, mDDescriptive statistics in geology
$k_j,\ \phi_j,\ h_j$	permeability, porosity and thickness of the $j$ th layer, in the order of decreasing $k/\phi$ , mD, fraction, mDescriptive statistics in geology
$L_c$	Lorenz coefficient, —Descriptive statistics in geology
$m$	mean of the natural logarithms of the sample, —Distributions and sample size most likely value (the mode), units of the parameterProbabilistic volumetrics by Monte Carlo simulation
$\text{MCI}$	monthly calculation index on the date of publishing the auction conditions or signing the protocol of direct negotiations; in 2026 it is 4,325 tenge, tengeClassification of reserves and resources
$n$	number of data points in the neighbourhood, —Spatial correlation and the variogram number of trials, —Probabilistic volumetrics by Monte Carlo simulation
$N$	number of measurements, —Descriptive statistics in geology number of plugs, —Descriptive statistics in geology number of independent measurements, —Distributions and sample size oil initially in place at standard conditions (STOIIP), m³Volumetric estimation of hydrocarbons in place oil initially in place, m³ or tVolumetric estimation of hydrocarbons in place the result, a product of factors $x_j$ ; for the volume in place these are GRV, NTG, $\phi$ , $1 - S_w$ and $1/B_{oi}$ , million m³Probabilistic volumetrics by Monte Carlo simulation
$N_0$	number of samples needed; the result is rounded up, —Distributions and sample size number of samples for a tolerance of ±20%, —Distributions and sample size
$N_{\text{eff}}$	effective number of independent measurements, —Distributions and sample size
$N_s$	actual number of samples, —Distributions and sample size
$N(h)$	number of pairs of points a distance $h$ apart, —Spatial correlation and the variogram
$\mathrm{NTG}$	net-to-gross ratio: the share of reservoir in the gross rock volume, fractionVolumetric estimation of hydrocarbons in place
$p$	reservoir pressure, absolute, MPaVolumetric estimation of hydrocarbons in place a probability estimated as a share of the trials, for example the share of results below a given value, fractionProbabilistic volumetrics by Monte Carlo simulation
$P$	tolerance — the permissible deviation of the mean of the samples from the true mean, % of the meanDistributions and sample size
$P_c$	chance of commerciality: the probability that the project will be approved and will reach commercial production, fractionClassification of reserves and resources
$P_d$	chance of bringing the discovered accumulation into development, fractionClassification of reserves and resources
$P_g$	geological chance of success (the chance of discovery): the probability of finding a significant quantity of moveable hydrocarbons (Geological risk and chance of success), fractionClassification of reserves and resources
$P_s$	precision of the mean reached with $N_s$ samples, %Distributions and sample size
$p_{sc},\ T_{sc}$	absolute pressure and temperature at standard conditions, MPa, KVolumetric estimation of hydrocarbons in place
$Q$	the quantity that will actually be recovered by the project from the date of the estimate to the end of production, t, m³ or bblClassification of reserves and resources
$Q_{\text{rec}}$	recoverable oil, m³ or t (as for N)Volumetric estimation of hydrocarbons in place
$Q_\text{yr}$	production for the year, million tClassification of reserves and resources
$\Delta R_\text{rev}$	revisions of earlier estimates, with their own sign, million tClassification of reserves and resources
$\text{RF}$	oil recovery factor, fractionVolumetric estimation of hydrocarbons in place
$\mathrm{RRR}$	reserves replacement ratio, fraction or %Classification of reserves and resources
$s$	standard deviation (SD), also called the root-mean-square deviation, units of the quantityDescriptive statistics in geology standard deviation, units of the quantityDescriptive statistics in geology standard deviation of the data, units of the quantityDistributions and sample size standard deviation of the results of the trials, units of the resultProbabilistic volumetrics by Monte Carlo simulation
$s^2$	variance of the sample, square of the units of the quantityDescriptive statistics in geology
$s_{\ln}$	standard deviation of the natural logarithms, —Distributions and sample size
$S_w$	mean initial water saturation of the reservoir; $1 - S_w$ is the oil saturation $S_o$ , fractionVolumetric estimation of hydrocarbons in place
$S_{w,k}$	water saturation of the $k$ th interval, fractionVolumetric estimation of hydrocarbons in place
$s,\ N$	standard deviation of the data and number of measurements, units of the quantity, —Distributions and sample size
$SE$	standard error of the mean, units of the quantityDistributions and sample size standard error of the mean over the trials, units of the resultProbabilistic volumetrics by Monte Carlo simulation
$SE_p$	standard error of this share, fractionProbabilistic volumetrics by Monte Carlo simulation
$t$	Student's coefficient (a table value) for the chosen confidence probability and $N - 1$ degrees of freedom (the number of independent values left after the mean has been found), —Distributions and sample size Student's coefficient; about 2 at a probability of 95% and more than thirty samples, —Distributions and sample size
$T$	reservoir temperature, absolute ( $T[\text{K}] = t[{}^{\circ}\text{C}] + 273.15$ ), KVolumetric estimation of hydrocarbons in place
$t,\ N$	Student's coefficient and number of measurements, —Distributions and sample size
$u$	random number, uniformly distributed between 0 and 1, —Probabilistic volumetrics by Monte Carlo simulation random number between 0 and 1, —Probabilistic volumetrics by Monte Carlo simulation
$u_2$	the random number for the second parameter; for the first we take $u_1$ itself, —Probabilistic volumetrics by Monte Carlo simulation
$V_{DP}$	Dykstra–Parsons coefficient, —Distributions and sample size
$x$	value of the quantity, units of the quantityDistributions and sample size value of the parameter in this trial, units of the parameterProbabilistic volumetrics by Monte Carlo simulation value of the parameter, units of the parameterProbabilistic volumetrics by Monte Carlo simulation value of the parameter that corresponds to the random number $u$ , units of the parameterProbabilistic volumetrics by Monte Carlo simulation
$x_{50}$	median; also the geometric mean, units of the quantityDistributions and sample size median (geometric mean) of the population, units of the quantityDistributions and sample size
$x_i$	the $i$ th measurement, units of the quantityDescriptive statistics in geology
$x_{\text{mode}}$	mode — the most probable value, units of the quantityDistributions and sample size
$x_p$	value below which the proportion $p$ of the values lies, units of the quantityDistributions and sample size
$z$	distance from the mean in standard deviations, —Distributions and sample size
$Z$	gas deviation factor at the reservoir $p$ and $T$ , —Volumetric estimation of hydrocarbons in place
$z^*(x_0)$	estimate of the property at the point $x_0$ , units of the propertySpatial correlation and the variogram
$z_1,\ z'$	quantiles of the standard normal distribution for two independent random numbers: $z_1 = \Phi^{-1}(u_1)$ , $z' = \Phi^{-1}(u')$ , —Probabilistic volumetrics by Monte Carlo simulation
$z_2$	a normal quantity linked with $z_1$ by the correlation $\rho$ , —Probabilistic volumetrics by Monte Carlo simulation
$z_p$	quantile of the standard normal distribution: −1.282 for 10%, 0 for 50%, +1.282 for 90%, —Distributions and sample size
$z(x_i),\ z(x_i + h)$	values of the property at the points of a pair, units of the propertySpatial correlation and the variogram
$\alpha,\ \alpha_0$	azimuth of the direction and azimuth of the major axis, °Spatial correlation and the variogram
$\hat\gamma(h)$	experimental (semi)variogram at lag $h$ , square of the units of the property, for example %²Spatial correlation and the variogram
$\gamma(h)$	variogram, units²Spatial correlation and the variogram
$\gamma(x_i - x_0)$	variogram between a data point and the point of estimation $x_0$ , units²Spatial correlation and the variogram
$\gamma(x_i - x_j)$	variogram between data points, units²Spatial correlation and the variogram
$\lambda_j$	weight of the $j$ th data point, —Spatial correlation and the variogram
$\mu$	mean; for the normal distribution it is also the median and the mode, units of the quantityDistributions and sample size mean of the natural logarithm of the quantity, —Distributions and sample size true mean of the population, units of the quantityDistributions and sample size Lagrange multiplier, a number that accounts for the condition that the weights add up to one, units²Spatial correlation and the variogram mean, units of the parameterProbabilistic volumetrics by Monte Carlo simulation
$\mu,\ \sigma$	mean and standard deviation of the distribution, units of the quantityDistributions and sample size mean and standard deviation of the natural logarithm, —Distributions and sample size
$\mu_j$	mean of the factor $x_j$ , units of the factorProbabilistic volumetrics by Monte Carlo simulation
$\mu_N$	mean of the result, million m³Probabilistic volumetrics by Monte Carlo simulation
$\bar\rho$	mean correlation coefficient between pairs of measurements (0 means independent, 1 means that they repeat one another), —Distributions and sample size
$\rho$	the specified correlation coefficient, from −1 to 1, —Probabilistic volumetrics by Monte Carlo simulation
$\rho(h)$	correlogram: the correlation coefficient of values a distance $h$ apart, —Spatial correlation and the variogram
$\sigma$	standard deviation, units of the quantityDistributions and sample size standard deviation of the natural logarithm of the quantity, —Distributions and sample size
$\sigma_{\ln k}$	standard deviation of the natural logarithm of permeability, —Distributions and sample size
$\sigma^2$	variance; the standard deviation $\sigma$ is its square root, square of the units of the parameterProbabilistic volumetrics by Monte Carlo simulation
$\sigma_K^2$	kriging variance, the expected squared error of the estimate, units²Spatial correlation and the variogram
$\phi$	mean porosity of the reservoir, fractionVolumetric estimation of hydrocarbons in place
$\Phi$	the standard normal distribution function, —Probabilistic volumetrics by Monte Carlo simulation
$\phi_k$	porosity of the $k$ th interval that passed the cut-offs, fractionVolumetric estimation of hydrocarbons in place
$\phi_k\,h_k$	pore volume of the interval per unit area, mVolumetric estimation of hydrocarbons in place