Equations: Distributed pressure measurements

The equations of the “Distributed pressure measurements” lessons in “Well testing and pressure transient analysis”, with their conditions and units, and what each symbol means in them.

Drawdown mobility at steady flow to the probe

\left(\frac{k}{\mu}\right)_d = \frac{C\,q}{2\pi\,r_{pe}\,\Delta p_d}

SI (any consistent system)

Holds when: The flow is steady: the piston works for longer than the pressure needs to level off at a constant value (advanced section). One phase (one fluid) of known viscosity flows, usually the filtrate in the invaded zone. The rock at the probe is uniform, and the flow follows Darcy's law. In an anisotropic rock (one with different permeabilities in different directions), the result reflects a mix of horizontal and vertical permeability. It usually lies between the spherical and the horizontal permeability. There is no extra resistance at the opening of the probe from leftover mudcake, damaged rock or fine particles. If there is, the mobility comes out too low.

Pressure measurement with the wireline formation tester →

Random error of a point on the pressure–depth plot

\sigma_p^2 = \sigma_{\text{gauge}}^2 + (G\,\sigma_z)^2

SI (in practice kPa, kPa/m and m)

Holds when: The errors of pressure and depth are random and independent. Systematic errors (errors that always push the same way, unlike random ones) are not included: a common shift of the depths, or a constant error of the gauge. They shift the line but do not change its slope. An error that grows with depth (cable stretch, survey error) changes the slope too.

Pressure measurement with the wireline formation tester →

Pressure gradient from the points by the least-squares method

G = \frac{\sum_i (z_i - \bar z)\,p_i}{\sum_i (z_i - \bar z)^2},\qquad s_G = \frac{s}{\sqrt{\sum_i (z_i - \bar z)^2}}

any consistent system (in practice MPa and m; the gradient in kPa/m)

Holds when: All the points belong to one continuous phase (for example, one column of oil) in hydrostatic equilibrium: the fluid is at rest, so the pressure changes only with depth. They also belong to one hydraulically connected system, in which fluid can flow between all the parts. The errors of the points are random and of equal size. Points that we know are shifted in one direction, such as a supercharged point or an unfinished build-up, have been excluded beforehand. The density of the fluid in the interval is constant. For tall gas columns this is not so (pressure in a gas column).

Pressure measurement with the wireline formation tester →

Pressure at the probe with tool storage

S\,\frac{\dd p}{\dd t} = -q_{\text{pist}} + \frac{k/\mu}{K}\,(p_{\text{res}} - p),\qquad \tau = \frac{S\,K}{k/\mu}

cm³, MPa, s; mobility in mD/(mPa·s)

Holds when: The flow to the probe is quasi-steady: in the rock at the probe the pressure adjusts much faster than in the tool. Tool storage is constant. The transient spherical flow in the formation (flow that has not yet become steady) is not included. It gives a slow “tail” to the build-up: the pressure gains the last few kilopascals not exponentially but noticeably more slowly. So the stabilisation time from this estimate is a lower limit. With a strict criterion (tenths of a kilopascal), the real time is several times longer.

Pressure measurement with the wireline formation tester →

Supercharging as a pressure divider

\Delta p_s = p_s - p_{\text{res}} = \Delta p_{\text{over}}\,\frac{R_{\text{res}}}{R_{\text{mc}} + R_{\text{res}}},\qquad R_{\text{mc}} = \frac{\mu_f\,h_{mc}}{k_{mc}}

Holds when: The flow through the cake is steady: its thickness and permeability are constant (dynamic equilibrium, or slow growth in the static period). The resistance of the rock changes slowly with time (logarithmically), so the formula is applied to the moment of the measurement.

Supercharging and permeability from formation-tester pretests →

Supercharging by the model of filtrate injection

\Delta p_s = \frac{u_2\,r_w\,\mu_f}{2k_f}\left[\frac{u_1}{u_2}\ln\frac{T_d + t_s}{t_s} + \ln\frac{k_f\,t_s}{\phi\,\mu_f\,c_t\,r_w^2} + 0.809\right]

SI; natural logarithm

Holds when: The cake sets the filtration rate: the supercharging is small compared with the overbalance (otherwise use the pressure divider above). In each stage the rate is constant. The flow of filtrate is single-phase (one fluid), in a uniform rock. The logarithmic approximation at the wall holds a few minutes after injection starts, even in a layer of 0.03 mD, long before logging. The filtrate spreads away from the wall horizontally (radially), as in a uniform formation. A thin tight layer between permeable parts partly drains into its neighbours in the vertical direction, and the supercharging in it is smaller than calculated.

Supercharging and permeability from formation-tester pretests →

Distortion of the gradient and of the contact by supercharging

G_{\text{app}} = G + \frac{\Delta(\Delta p_s)}{\Delta z},\qquad \delta z^* = \frac{\Delta p_{s,o} - \Delta p_{s,w}}{G_w - G_o}

SI (in practice kPa, kPa/m and m)

Holds when: The lines are fitted to the supercharged points as to valid ones. The supercharging is small compared with the pressure differences on which the classification of the fluids is based.

Supercharging and permeability from formation-tester pretests →

Pressure build-up after a pretest under spherical flow

p(\Delta t) = p^* - m_s\,f_s(\Delta t),\qquad f_s = \frac{1}{\sqrt{\Delta t}} - \frac{1}{\sqrt{t_p + \Delta t}},\qquad m_s = \frac{q\,\mu\sqrt{\phi\,\mu\,c_t}}{4\pi^{3/2}\,k_s^{3/2}}

Holds when: The rock is uniform (possibly anisotropic). The disturbance has moved away from the probe by more than a few hole radii. The flow to the probe is then a full sphere, because the hole is a thin impermeable tube. (For flow to a hemisphere at a flat wall, the slope would be twice as large.) The influence of tool storage has ended (see the section on tool storage). The disturbance has not yet reached the boundaries of the formation. One phase of known viscosity flows. With two rates of draw, $f_s$ is built by superposition, by the same rules.

Supercharging and permeability from formation-tester pretests →

Response of an observation probe at a constant draw rate

\Delta p_{\text{obs}}(t) = \frac{q\mu}{4\pi k_h\,\Delta z}\,\operatorname{erfc}\frac{\Delta z}{\sqrt{4\eta_v t}} \;\approx\; \frac{q\mu}{4\pi k_h\,\Delta z} - \frac{m_s}{\sqrt t},\qquad \eta_v = \frac{k_v}{\phi\mu c_t}

SI; the approximate form holds when $\Delta z/\sqrt{4\eta_v t} < 0.15$

Holds when: A point sink in a uniform anisotropic medium; the hole is not allowed for. The hole slightly increases the response (by no more than a factor of two, the limit of a flat wall), and then the first formula underestimates $k_h$ . In processing real data, the constant is taken from a model of the tool with a hole. The rate at the sink is constant from the start; the flow is single-phase.

Supercharging and permeability from formation-tester pretests →

Relation between the permeability of core to gas and to formation water at reservoir conditions

k_b = c\,k_a^{\,n}\qquad\Longleftrightarrow\qquad \lg k_b = \lg c + n\lg k_a

mD; the constant $c$ refers specifically to millidarcies

Holds when: The relation is found on samples of the same rock type, and it is applied within the permeabilities of these pairs. It already includes gas slippage (gas slipping along the pore walls, which makes gas permeability read too high; Klinkenberg correction), the stress and the swelling of clays. There is no need to separate these corrections if the pairs were measured in this way.

Comparing formation-tester, core and well-test data →

Horizontal absolute permeability from the drawdown mobility

k_h = \frac{(k/\mu)_d\,\mu_f}{k_{rf}'\,f_A},\qquad f_A = \frac{k_d}{k_h} = \frac{\pi}{2K(1 - 1/A)}

$k_h$ in mD, for $(k/\mu)_d$ in mD/(mPa·s) and $\mu_f$ in mPa·s

Holds when: The flushed zone is at the residual saturation of the displaced phase, and there is no extra resistance at the opening of the probe (cake, fine particles). Otherwise $k_h$ comes out too low. $f_A$ comes from the model of a round probe on a flat impermeable wall (derivation). Solutions with an ellipsoidal sink give values about 10% smaller. $A$ is taken from pairs of horizontal and vertical samples of the same layer. At the scale of the probe, lamination (thin layering) can give a larger anisotropy than that of the samples.

Comparing formation-tester, core and well-test data →

Flow capacity from the Horner line, converted to absolute permeability

(kh)_{\text{abs}} = \frac{(kh)_o}{k_{ro}'} = \frac{2.149\,qB\mu}{m\,k_{ro}'}

metric oilfield units; $2.149 = \ln 10/(4\pi\cdot 0.0853)$

Holds when: Only oil flows in the formation around the well, with irreducible water (the pressure is above the bubble point pressure, and the water is immobile). The slope is taken from the period of radial flow in an infinite reservoir (Diagnosis with the pressure derivative). The whole interval is open, and the layers are continuous within the radius of investigation.

Comparing formation-tester, core and well-test data →

Systematic bias and scatter of the ratios

d_i = \lg\frac{k_{\text{pt},i}}{k_{\text{core},i}},\qquad \bar d = \frac{1}{n}\sum_i d_i,\qquad s = \sqrt{\frac{\sum_i (d_i - \bar d)^2}{n - 1}},\qquad \bar d \pm t\,\frac{s}{\sqrt n}

dimensionless (decimal logarithms)

Holds when: The deviations of the points are independent and close to lognormal (their logarithms follow the usual bell-shaped normal distribution). Points with a known defect of the measurement are excluded beforehand, not because of the size of their deviation. $10^{\bar d}$ is the geometric mean of the ratios, a systematic multiplier. $10^{s}$ is the typical scatter of one point, as “by how many times”.

Comparing formation-tester, core and well-test data →

Number of points needed to tell bias from scatter

n \approx \left(\frac{z\,s}{\lg F}\right)^2

dimensionless

Holds when: The condition is $|\bar d| > z\,s/\sqrt n$ at $\bar d = \lg F$ . For small $n$ , instead of $z$ we take Student's coefficient for $n - 1$ degrees of freedom. It is noticeably larger than two (3.18 for four points), and more points are needed. This condition is for a mean equal to $\lg F$ . Because $\bar d$ itself scatters, a bias of exactly a factor of $F$ will be significant in only about half of the sets. To notice it in four sets out of five, we take $z$ as about 2.8, and about twice as many points are needed. The points are independent: two points in one core window are in effect one.

Comparing formation-tester, core and well-test data →

The pressure that the formation tester records at the oil–water contact

p_{\text{meas}}(z) = \begin{cases} p_o(z) = p_{\text{FWL}} + G_o\,(z - z^*), & z < z_{\text{OWC}} \\ p_w(z) = p_{\text{FWL}} + G_w\,(z - z^*), & z > z_{\text{OWC}} \end{cases}

SI (in practice MPa, kPa/m and m)

Holds when: The accumulation is in hydrostatic and capillary equilibrium, and the formation water does not flow. The rock is water-wet, and the oil in the transition zone is continuous. In an oil-wet rock the picture is the mirror image (advanced section). The measurements are valid ones, without supercharging and without an unfinished build-up (classes of pretests). The well is drilled with water-based mud.

Pressure gradients, fluid contacts and capillary equilibrium →

Depth of the crossing of two pressure lines

z^* = z_0 + \frac{p_a(z_0) - p_b(z_0)}{G_b - G_a}

any consistent system (MPa, MPa/m and m, or kPa, kPa/m and m)

Holds when: Both lines belong to one hydraulically connected system in equilibrium. The densities of the phases are constant with height. If the composition changes noticeably with depth, the line of the oil or gas is curved (section on fluid systems).

Pressure gradients, fluid contacts and capillary equilibrium →

Standard error of a contact from two lines

\sigma_{z^*} = \frac{\sqrt{\sigma_a^2(z^*) + \sigma_b^2(z^*)}}{|G_b - G_a|},\qquad \sigma^2(z) = s^2\left[\frac1n + \frac{(z - \bar z)^2}{\sum_i (z_i - \bar z)^2}\right]

any consistent system (in practice kPa, kPa/m and m)

Holds when: The errors of the points are random, independent and of equal size. The points within each line belong to one phase. Systematic errors are not included. A constant error of the gauge in one survey shifts both lines in the same way and does not change the crossing. A common shift of depths moves the crossing together with the logs of the same well. For lines from different wells, these errors are added separately (section on the regional water line). With few points, $s$ is estimated only roughly. The confidence interval (the range in which the true value is expected to lie) is taken with Student's coefficient for $n - 2$ degrees of freedom, and not just ±2σ (about the standard error).

Pressure gradients, fluid contacts and capillary equilibrium →

Density and gradient of formation water from analysis

\rho_{w,\text{res}} = \frac{\rho_{w,\text{sc}}}{B_w},\qquad G_w = \rho_{w,\text{res}}\,g

SI; in metric form $G_w$ [kPa/m] $= 9.81\,\rho_{w,\text{res}}$ [g/cm³]

Holds when: Dissolved gas is ignored (otherwise the density is calculated from the mass of water and gas). Salinity and temperature are the same over the interval through which the line is extended.

Pressure gradients, fluid contacts and capillary equilibrium →

GOC from the oil line and the bubble point pressure

z_{\text{GOC}} = \bar z_o + \frac{p_{\text{b}} - \bar p_o}{G_o},\qquad \sigma_{z_{\text{GOC}}} \approx \frac{\sigma_{p_{\text{b}}}}{G_o}

any consistent system

Holds when: The accumulation is in equilibrium, and the gas cap and the oil are connected. The composition of the oil is constant with height. With compositional grading, the bubble point pressure of a sample taken below the GOC is smaller than that of the oil at the contact, and the GOC from this formula comes out above the true one.

Pressure gradients, fluid contacts and capillary equilibrium →

A pressure line with a set gradient and its uncertainty

p(z) = \bar p + \rho g\,(z - \bar z),\qquad \sigma^2(z) = \frac{s^2}{n} + (g\,\sigma_\rho)^2\,(z - \bar z)^2

SI (in practice MPa, kPa/m and m); $g$ = 9.81 m/s²

Holds when: All the chosen points belong to one continuous phase, are valid, and have no noticeable supercharging. The density of the sample refers to the fluid of this line, and it is constant over the interval of extrapolation. The errors of the points and the error of the density are independent. With one point, the line passes through it, and $s$ is taken from other measurements.

Forced-gradient technique and multi-well analysis →

What the shifts of the lines of a well are made of

\delta_o = \delta + G_o\,\Delta z + \Delta_o,\qquad \delta_w = \delta + G_w\,\Delta z + \Delta_w

SI (in practice kPa, kPa/m and m)

Holds when: The lines of the field and of the well are single-phase, in equilibrium, with the same gradients. The errors are constant over the well. A depth error that grows with depth changes the slope of the lines too.

Forced-gradient technique and multi-well analysis →

How shifts of the lines move the FWL of a well

\delta z^* = \frac{\delta_o - \delta_w}{G_w - G_o},\qquad \delta p_{\text{FWL}} = \delta_o + G_o\,\delta z^*

SI (in practice kPa, kPa/m and m)

Holds when: The gradients of the phases at the well are the same as in the field. A depth error $\Delta z$ shifts the level vertically by $-\Delta z$ at the same pressure. It is allowed for separately.

Forced-gradient technique and multi-well analysis →

Weighted mean level and a check of consistency

\bar z^* = \frac{\sum_i z_i^*/\sigma_i^2}{\sum_i 1/\sigma_i^2},\qquad \sigma_{\bar z^*} = \frac{1}{\sqrt{\sum_i 1/\sigma_i^2}},\qquad \chi^2 = \sum_i\left(\frac{z_i^* - \bar z^*}{\sigma_i}\right)^2

any consistent system

Holds when: The errors of the wells are independent and correctly estimated. Errors common to all wells (the fluid density, systematic errors of one type of tool) are not reduced by this method. All the wells belong to one system. A well that does not fit the others is excluded only after the cause is explained.

Forced-gradient technique and multi-well analysis →

Free water level with a common oil line

z^* = z_{\text{ref}} + \frac{\Phi_o - \Phi_w}{G_w - G_o},\qquad z^*_{\text{B}} - z^*_{\text{A}} = \frac{\Phi_{w,\text{A}} - \Phi_{w,\text{B}}}{G_w - G_o}

any consistent system (MPa, MPa/m and m)

Holds when: The oil is at rest and connected between the wells, so its datum pressure is one. At the contact in each well the water is continuous. The densities of the phases are constant. For gas, the gas gradient is taken instead of $G_o$ . The depths are vertical, from one level. The pressures are absolute, from calibrated gauges.

Perched and tilted contacts, trapped water →

Excess of the pressure of perched water over the main aquifer

\Delta p_{\text{perch}} = (G_w - G_o)\,(z^*_{\text{main}} - z^*_{\text{perch}})

any consistent system

Holds when: The water of the block is closed on all sides, and since it was separated it has not leaked through seals and faults. The densities of the phases in the blocks are the same.

Perched and tilted contacts, trapped water →

Tilt of the contact from the gradient of the datum water pressure

\tan\beta = \frac{|\nabla\Phi_w|}{G_w - G_o} = \frac{\rho_w}{\rho_w - \rho_o}\,|\nabla h|

a gradient in kPa/km and pressure gradients in kPa/m give $\tan\beta$ in m/km

Holds when: The oil is at rest, the flow of water is steady, and the densities are constant. The datum water pressures are compared on one datum and at one water density (otherwise, see the advanced section). Capillary forces are ignored: this is about the FWL. The OWC from logs is parallel to it, at the height $h_d$ (transition zone).

Perched and tilted contacts, trapped water →

Height of the accumulation in a dome with a tilted contact

\begin{aligned} h(x, y) &= H_{\text{eff}} - a_x\,(x - x_0)^2 - a_y\,(y - y_0)^2, \\ x_0 &= \frac{t_x}{2a_x},\quad y_0 = \frac{t_y}{2a_y},\quad H_{\text{eff}} = H + \frac{t_x^2}{4a_x} + \frac{t_y^2}{4a_y}, \\ V &= \frac{\pi\,H_{\text{eff}}^2}{2\sqrt{a_x a_y}} \end{aligned}

coordinates in km, depths and heights in m

Holds when: The top is an elliptic paraboloid with axes along $x$ and $y$ . The formation is thicker than the accumulation (there is bottom water), so the rock volume is the volume between the top and the contact. The contact is a plane. The trap does not spill: the outline of the accumulation does not reach the spill point (the lowest point through which oil can leave).

Perched and tilted contacts, trapped water →

Pressure gradient with vertical flow

\frac{\dd p}{\dd z} = \rho g + \frac{\mu\,u_z}{k_z}

Holds when: single-phase flow; the velocity $u_z$ and the permeability $k_z$ are constant over the measured interval, so the profile is a straight line if $k_z$ changes with depth, the viscous component changes too: a thin tight layer gives a step (see the section on barriers) depths are true vertical depths, and the fluid density is known independently (from the analysis of a sample)

Pressure measurements in producing reservoirs: barriers and accuracy →

Significance of the deviation from the hydrostatic gradient

G_{\text{visc}} = G - \rho g,\qquad \sigma_{G_{\text{visc}}} = \sqrt{s_G^2 + (g\,\sigma_\rho)^2}

SI (in practice kPa/m)

Holds when: the errors of the slope and of the density are random and independent; the points of the line are valid and belong to one unit a systematic scale error of the depths (a wrong directional survey) is not in the formula and is checked separately we call the flow detected if $|G_{\text{visc}}|$ is larger than two or three of its errors. With few points the error of the slope is only a rough estimate, so the threshold is taken from Student's t coefficient (a factor from statistical tables that widens the threshold when there are few points)

Pressure measurements in producing reservoirs: barriers and accuracy →

Potential step across a barrier

\delta\Phi_b = \frac{\mu\,u_z\,b}{k_b}

Holds when: the barrier is uniform and continuous; its storage capacity (how much fluid it gives up as the pressure falls) is small, so the flow through it is almost steady-state the units above and below the barrier are in vertical equilibrium and are filled with one fluid

Pressure measurements in producing reservoirs: barriers and accuracy →

Material balance of two units with crossflow

\begin{aligned} c_tV_1\,\frac{\dd\Phi_1}{\dd t} &= -q_1 - q_{12}, \\ c_tV_2\,\frac{\dd\Phi_2}{\dd t} &= -q_2 + q_{12}, \qquad q_{12} = T\,(\Phi_1 - \Phi_2) \end{aligned}

metric oilfield: MPa, m³, m³/d, 1/MPa, days

Holds when: the potential is the same everywhere inside each unit: the unit is much more permeable than the barrier, and the drawdown around the wells (the extra pressure drop that drives fluid into a well) is ignored the withdrawals are constant and the compressibility is constant; inflow from outside (edge water) is counted as a negative withdrawal the storage capacity of the barrier itself is small; the flow is single-phase

Pressure measurements in producing reservoirs: barriers and accuracy →

Steady-state step and time constant of the barrier

\begin{aligned} \Delta_{\text{ss}} &= (r_2 - r_1)\,\tau = \frac{q_2V_1 - q_1V_2}{T\,(V_1 + V_2)}, \\ r_i &= \frac{q_i}{c_t V_i}, \qquad \tau = \frac{c_t\,V_1V_2}{(V_1 + V_2)\,T} \end{aligned}

metric oilfield, as above

Holds when: constant withdrawals and equal initial potentials of the units the compressibility is not in the steady-state step: it cancels in the right-hand form

Pressure measurements in producing reservoirs: barriers and accuracy →

Potential profile in a tight unit that flows into its neighbour

\begin{aligned} \Phi(\zeta) - \Phi_1 &= \Delta\Phi_{\max}\left[\frac{2\zeta}{h} - \left(\frac{\zeta}{h}\right)^2\right], \\ \Delta\Phi_{\max} &= \frac{\mu\,u\,h}{2\,k_z}, \qquad u = \frac{q}{A}\cdot\frac{V_2}{V_1 + V_2} \end{aligned}

Holds when: the top of the tight unit is impermeable, the flow in it is vertical, and no fluid flows to the wells along the tight unit itself common pseudo-steady-state regime: both units lose pressure at the same rate (the time criterion is given below) single-phase flow, and $k_z$ is the same over the thickness of the unit

Pressure measurements in producing reservoirs: barriers and accuracy →

When the profile of the tight unit settles

t_{Dh} = \frac{k_z\,t}{\phi\,\mu\,c_t\,h^2} > 2

Holds when: the withdrawal from the permeable unit is roughly constant; all of it goes through that unit, and the crossflow from the tight unit builds up from zero the limit holds for any ratio of the pore volumes of the units. If a constant rate were taken from the tight unit itself from the start, $t_{Dh} > 0.5$ would be enough (see the advanced section)

Pressure measurements in producing reservoirs: barriers and accuracy →

Lag of the unit behind the barrier during a pulse

\tau_{\text{loc}} = \frac{\phi_1 h_1\,c_t\,\mu\,b}{k_b},\qquad \Delta p_2 - \Delta p_1 \approx \tau_{\text{loc}}\,\frac{\dd(\Delta p_2)}{\dd t}

Holds when: the well that creates the pulse is open only to unit 2, and unit 1 receives the disturbance only through the barrier lateral flow inside unit 1 is ignored; this overestimates the lag if unit 1 is permeable across the area $\Delta p_2$ is the pressure of unit 2 at the observation well, including the crossflow. The barrier feeds fluid into unit 2 over the whole area that the pulse has reached. The leakage factor is $B = \sqrt{k_2h_2b/k_b}$ (the distance over which the shale can feed unit 2; $k_2$ and $h_2$ are the permeability and thickness of unit 2). If $B$ is smaller than the distance to the well that creates the pulse, the line-source solution for unit 2 alone strongly overestimates its response (see the example below). The line-source solution is the standard formula for the pressure around a well in an infinite reservoir the approximate equality holds for an almost linear change of the pressure of unit 2 and for times much longer than $\tau_{\text{loc}}$ ; the exact relation is $\Delta p_2 - \Delta p_1 = \tau_{\text{loc}}\,\dd(\Delta p_1)/\dd t$

Pressure measurements in producing reservoirs: barriers and accuracy →

Potentials of the two units in time

\begin{aligned} \bar\Phi(t) &= \Phi_0 - \frac{(q_1 + q_2)\,t}{c_t\,(V_1 + V_2)}, \qquad \Delta(t) = \Delta_{\text{ss}}\left(1 - e^{-t/\tau}\right), \\ \Phi_1 &= \bar\Phi + \frac{V_2}{V_1 + V_2}\,\Delta, \qquad \Phi_2 = \bar\Phi - \frac{V_1}{V_1 + V_2}\,\Delta \end{aligned}

metric oilfield, as in the main section

Holds when: constant withdrawals, equal initial potentials, and the other assumptions of the two-unit model.

Pressure measurements in producing reservoirs: barriers and accuracy →

Response of two units with crossflow to a change in the rate of a well in unit 2

\bar p_2(s) = \frac{2\cdot 0.933\,q\mu}{k_2h_2\,s}\,K_0\!\left(\frac{r}{B}\sqrt{\tau_2 s + \frac{\tau_{\text{loc}}\,s}{1 + \tau_{\text{loc}}\,s}}\right),\qquad \bar p_1(s) = \frac{\bar p_2(s)}{1 + \tau_{\text{loc}}\,s}

metric oilfield: MPa, m³/d, mD, m, mPa·s, days

Holds when: unit 1 does not carry flow across the area (lateral flow in it is ignored), the storage capacity of the shale is small, and the formation is uniform across the area and unbounded the pressures in time are found by numerical inversion of the Laplace transform (turning the result in $s$ back into time; Stehfest algorithm) without crossflow ( $k_b \to 0$ ) the formula turns into the line-source solution (Line-source solution and the pressure function)

Pressure measurements in producing reservoirs: barriers and accuracy →

Pressure gradient with vertical flow of two phases

\frac{\dd p}{\dd z} = g\bar\rho + \frac{u_t}{\lambda_t},\qquad \bar\rho = \frac{\lambda_1\rho_1 + \lambda_2\rho_2}{\lambda_1 + \lambda_2},\qquad \lambda_j = \frac{k_z\,k_{rj}}{\mu_j}

Holds when: one-dimensional vertical flow in a uniform unit; the phases are incompressible over the measured interval the pressures of the phases are equal, so capillary pressure is ignored (the correction is in the section on capillary pressure) the formula is local: the mobilities are taken at the saturation of the point in question. If the saturation changes with depth, the gradient changes with it, and the profile bends

Two-phase flow and reservoir description from formation-tester data →

Density from the gradient in countercurrent flow of gas and oil

\bar\rho = \frac{\rho_o + M\rho_g}{1 + M},\qquad M = \frac{k_{rg}/\mu_g}{k_{ro}/\mu_o}

densities in any identical units; the gradient is $g\bar\rho$

Holds when: there is no total vertical flow: the gas rises and the oil drains down at the same volumetric rate the gas–oil capillary pressure is negligible (the interfacial tension between the two fluids is low); the water is immobile at irreducible saturation (the water that stays in the pores and cannot be displaced)

Two-phase flow and reservoir description from formation-tester data →

Gradient of the oil pressure in a two-phase zone with the capillary term

G_{\text{oil}} = \frac{\dd p_o}{\dd z} = G + f_w\,\frac{\dd p_c}{\dd z},\qquad f_w = \frac{\lambda_w}{\lambda_t}

SI (in practice kPa/m)

Holds when: a water-wet rock, with continuous oil in the zone; the capillary pressure curve is the imbibition curve (the curve for water entering the rock) if the zone is being flooded (imbibition profile) the other assumptions of the two-phase gradient; the derivation is in the advanced section

Two-phase flow and reservoir description from formation-tester data →

Shift of the contact if the water line is set to the hydrostatic gradient

\delta z^* = -\frac{G_{\text{visc}}\,L}{G_w - G_o}

any consistent units (kPa/m and m)

Holds when: the line with the set gradient passes through the middle point of the water readings, and the true oil and water lines are straight. Then the formula is exact. With the measured, steeper water gradient in the denominator, it would underestimate the shift the oil line is found from its own points, without a shift

Two-phase flow and reservoir description from formation-tester data →

Water that has entered the accumulation, from the rise of the contact

W_e - W_pB_w \approx A\,\phi\,(1 - S_{wi} - S_{or})\,\Delta z

SI (m, m², m³)

Holds when: the water rises evenly over the whole area $A$ : bottom water, vertical saturation equilibrium. With cones near wells or inflow from the flanks, the rise in one well is not representative the swept zone stays at residual oil saturation; the change in the shape of the transition zone and the compressibility of the pores and the water are ignored the initial and the current FWL are both found from pressure, with common depth matching of the wells

Two-phase flow and reservoir description from formation-tester data →

Material balance of a tank in the multi-tank model

C_j\,\frac{\dd p_j}{\dd t} = -q_j + \sum_k T_{jk}\,(p_k - p_j),\qquad C_j = c_t\,V_{p,j}

metric oilfield: MPa, m³, m³/d, days

Holds when: the pressure inside a tank is the same: it equalises much faster than between tanks single-phase undersaturated oil or water, with constant compressibility and viscosity; below the bubble point pressure the storage capacity depends on pressure, and the equations are nonlinear (see the advanced section) the crossflow through a link is proportional to the pressure difference of the tanks: the flow through it is steady-state, and the storage capacity of the link itself is ignored

Lumped multi-tank material balance model →

Rate of a well in a small block behind a fault

\begin{aligned} q(t) &= q_\infty + (q_0 - q_\infty)\,\e^{-t/\tau},\qquad \tau = \frac{C}{J + T}, \\ q_0 &= J\,(p_{\text{nb}} - p_{wf}),\qquad q_\infty = \frac{J\,T}{J + T}\,(p_{\text{nb}} - p_{wf}) \end{aligned}

metric oilfield

Holds when: the block is a tank; the pressure of the neighbouring block is constant; at the start the pressure of the block equals that of its neighbour the bottomhole pressure and the productivity are constant; the flow is single-phase, above the bubble point pressure

Lumped multi-tank material balance model →

Time constants of three tanks in a chain

\begin{aligned} &\lambda^2 - s_1\lambda + s_2 = 0,\qquad \tau = 1/\lambda, \\ &s_1 = \frac{T_{AB}}{C_A} + \frac{T_{AB} + T_{BC}}{C_B} + \frac{T_{BC}}{C_C},\qquad s_2 = \frac{T_{AB}\,T_{BC}\,(C_A + C_B + C_C)}{C_A\,C_B\,C_C} \end{aligned}

metric oilfield

Holds when: a chain A–B–C with no direct A–C link; the storage capacities and transmissibilities are constant.

Lumped multi-tank material balance model →

Symbols

Symbol	Meaning and context
$\bar z$	mean depth of the valid points, mPressure measurement with the wireline formation tester
$\Delta z$	vertical distance between the probes, mSupercharging and permeability from formation-tester pretests constant error of the depths of the well: by how much the recorded depths are less than the true ones, mForced-gradient technique and multi-well analysis rise of the current FWL relative to the initial one, found from pressure, mTwo-phase flow and reservoir description from formation-tester data
$\nabla h$	gradient of the potentiometric surface (of the heads), m/kmPerched and tilted contacts, trapped water
$\bar d,\ s$	mean and standard deviation of the logarithms, —Comparing formation-tester, core and well-test data
$\bar z,\ \bar p$	mean depth and mean pressure of the chosen reliable points, m, PaForced-gradient technique and multi-well analysis
$(k/\mu)_d$	drawdown mobility of the pretest (how we find it), mD/(mPa·s)Comparing formation-tester, core and well-test data
$(kh)_o$	flow capacity to oil from the slope of the Horner line (formula), mD·mComparing formation-tester, core and well-test data
$\delta z^*$	shift of the FWL that we find (plus is downward), mSupercharging and permeability from formation-tester pretests shift of the depth of the FWL of a well (plus is downward) caused by the shifts of its lines in pressure, mForced-gradient technique and multi-well analysis shift of the contact that we find; plus means downward, as in the lesson Forced-gradient technique and multi-well analysis. With upward flow ( $G_{\text{visc}} > 0$ ) it is negative: the contact is found above the true one. With downward flow it is found below, mTwo-phase flow and reservoir description from formation-tester data
$\bar z^,\ \sigma_{\bar z^}$	weighted mean level and its standard error, mForced-gradient technique and multi-well analysis
$\dd p/\dd z$	pressure gradient with depth, as given by the pretests, Pa/mPressure measurements in producing reservoirs: barriers and accuracy
$\nabla\Phi_w$	areal gradient of the datum pressure of water: its change per unit horizontal distance, in the direction of the greatest change, kPa/kmPerched and tilted contacts, trapped water
$0.809$	$\ln(4/\gamma)$ , where $\gamma = 1.781$ , as in the logarithmic approximation of the line-source solution, —Supercharging and permeability from formation-tester pretests
$\bar p_2,\ \bar p_1$	Laplace transforms of the pressure changes of units 2 and 1 at a distance $r$ from the well, MPa·dPressure measurements in producing reservoirs: barriers and accuracy
$\Delta p_2,\ \Delta p_1$	pressure changes of the unit with the working well and of the unit behind the barrier, PaPressure measurements in producing reservoirs: barriers and accuracy
$A$	anisotropy $k_h/k_v$ of the rock at the probe, —Comparing formation-tester, core and well-test data area of the oil–water contact under which the bottom water rises, m²Two-phase flow and reservoir description from formation-tester data
$a_x,\ a_y$	curvature of the top along the axes, m/km²Perched and tilted contacts, trapped water
$b$	thickness of the barrier (from the log), mPressure measurements in producing reservoirs: barriers and accuracy
$B$	leakage factor $\sqrt{k_2h_2b/k_b}$ , mPressure measurements in producing reservoirs: barriers and accuracy
$B_w$	formation volume factor of water: the volume in the formation per unit volume at standard conditions; for waters without gas it is usually 1.00–1.05, m³/m³Pressure gradients, fluid contacts and capillary equilibrium
$b,\ k_b$	thickness and vertical permeability of the barrier, m, m²Pressure measurements in producing reservoirs: barriers and accuracy
$C$	flow shape factor: 0.5 for flow to a sphere surrounded by rock on all sides, 1 for flow to a hemisphere at an impermeable plane; for a probe on the wall of a borehole of usual diameter, about 0.67, —Pressure measurement with the wireline formation tester storage capacity of the block, $c_tV_p$ , m³/MPaLumped multi-tank material balance model
$C_A,\ C_B,\ C_C$	storage capacities of the tanks, m³/MPaLumped multi-tank material balance model
$C_j$	storage capacity (capacitance) of the tank: the volume that it gives up when its pressure falls by 1 MPa, m³/MPaLumped multi-tank material balance model
$c_t$	total compressibility of the rock and the fluids, 1/MPaPressure measurements in producing reservoirs: barriers and accuracy
$c_t,\ V_{p,j}$	total compressibility of the rock and the fluids, and the pore volume of the tank, 1/MPa, m³Lumped multi-tank material balance model
$c,\ n$	constants of the relation, found by least squares from pairs of samples in logarithms, —Comparing formation-tester, core and well-test data
$\dd p_c/\dd z$	change of the capillary pressure with depth: $(\dd p_c/\dd S_w)(\dd S_w/\dd z)$ . In a zone where there is more water towards the bottom, it is negative, Pa/mTwo-phase flow and reservoir description from formation-tester data
$\Delta p_d$	steady drawdown: the formation pressure minus the pressure at the end of the draw, PaPressure measurement with the wireline formation tester
$d_i$	decimal logarithm of their ratio: 0 means agreement, 0.301 means twice as large, −0.301 means half as large, —Comparing formation-tester, core and well-test data
$F$	the systematic multiplier that we need to notice (for example 1.5), —Comparing formation-tester, core and well-test data
$f_A$	anisotropy factor: the drawdown permeability divided by the horizontal one, —Comparing formation-tester, core and well-test data
$f_w$	share of water in the total mobility, —Two-phase flow and reservoir description from formation-tester data
$\delta p_{\text{FWL}}$	shift of the pressure at its FWL, PaForced-gradient technique and multi-well analysis
$g$	acceleration of gravity, 9.81, m/s²Pressure gradients, fluid contacts and capillary equilibrium
$G$	pressure gradient of the fluid in the formation, Pa/mPressure measurement with the wireline formation tester pressure gradient: the slope of the line, Pa/mPressure measurement with the wireline formation tester gradient without capillary pressure, the formula of the section on the two-phase gradient, Pa/mTwo-phase flow and reservoir description from formation-tester data
$G_a,\ G_b$	gradients of the light and heavy phases, Pa/mPressure gradients, fluid contacts and capillary equilibrium
$G_{\text{app}},\ G$	apparent and true gradients from the points of one zone, Pa/mSupercharging and permeability from formation-tester pretests
$G_o$	oil gradient, Pa/mPressure gradients, fluid contacts and capillary equilibrium
$G_o,\ G_w$	pressure gradients of oil and water, $\rho_o g$ and $\rho_w g$ , Pa/mPressure gradients, fluid contacts and capillary equilibrium gradients of oil and water, Pa/mForced-gradient technique and multi-well analysis pressure gradients of oil and water, $\rho g$ , MPa/mPerched and tilted contacts, trapped water
$G_{\text{oil}}$	gradient of the oil pressure, which is what the pretests show in a water-wet rock, Pa/mTwo-phase flow and reservoir description from formation-tester data
$G_{\text{visc}}$	viscous component, an estimate of $\mu u_z/k_z$ , Pa/mPressure measurements in producing reservoirs: barriers and accuracy viscous component of the water gradient over the interval of its points, Pa/mTwo-phase flow and reservoir description from formation-tester data
$G_w$	pressure gradient of water in the formation, Pa/mPressure gradients, fluid contacts and capillary equilibrium
$G_w - G_o$	difference between the hydrostatic gradient that we set for the water and the gradient of the oil, Pa/mTwo-phase flow and reservoir description from formation-tester data
$G_w,\ G_o$	gradients of water and oil, MPa/mPerched and tilted contacts, trapped water pressure gradients of water and oil, kPa/mPerched and tilted contacts, trapped water
$G,\ s_G$	gradient from the pretests of the unit, and its standard error, Pa/mPressure measurements in producing reservoirs: barriers and accuracy
$h$	thickness of the tight unit, mPressure measurements in producing reservoirs: barriers and accuracy
$H$	height of the accumulation at the crest, $z_1 - z_{\text{crest}}$ , mPerched and tilted contacts, trapped water
$H_{\text{eff}}$	greatest height of the accumulation, at its shifted centre, mPerched and tilted contacts, trapped water
$h(x, y)$	height of the accumulation from the top to the contact; the accumulation is where $h > 0$ , mPerched and tilted contacts, trapped water
$J$	productivity index of the well, m³/(d·MPa)Lumped multi-tank material balance model
$k$	permeability of the rock at the probe for the fluid that flows in, m²Pressure measurement with the wireline formation tester
$K$	probe constant from the mobility formula (39.02 for the classic standard probe), mD·MPa·s/(mPa·s·cm³)Pressure measurement with the wireline formation tester
$K_0$	modified Bessel function of the second kind of order zero, —Pressure measurements in producing reservoirs: barriers and accuracy
$k_2,\ h_2$	permeability and thickness of unit 2, mD, mPressure measurements in producing reservoirs: barriers and accuracy
$k_a$	permeability of the sample to gas in a standard analysis (low confining stress, dry sample), mDComparing formation-tester, core and well-test data
$k_b$	permeability of the same sample to formation water under the reservoir effective stress, mDComparing formation-tester, core and well-test data vertical permeability of the barrier, m²Pressure measurements in producing reservoirs: barriers and accuracy
$k_f$	permeability of the rock to the filtrate: in a water-bearing layer the permeability, in an oil-bearing layer the effective permeability of the flushed zone to the filtrate, m²Supercharging and permeability from formation-tester pretests
$k_{\text{pt},i},\ k_{\text{core},i}$	permeabilities of point $i$ from the pretest (after the corrections) and from the core in the window, mDComparing formation-tester, core and well-test data
$k_{rf}'$	end-point relative permeability of the filtrate in the flushed zone: $k_{rw}'$ for a water-based filtrate in the oil-bearing part, 1 in the water-bearing part, $k_{ro}'$ for a hydrocarbon filtrate, —Comparing formation-tester, core and well-test data
$k_{rg},\ k_{ro}$	relative permeabilities of gas and oil at the current gas saturation, —Two-phase flow and reservoir description from formation-tester data
$k_{ro}'$	end-point relative permeability to oil at irreducible water, —Comparing formation-tester, core and well-test data
$k_s$	spherical permeability $(k_h^2 k_v)^{1/3}$ , m²Supercharging and permeability from formation-tester pretests
$k_z$	effective vertical permeability to this fluid, over the measured interval and its surroundings; if the other phase is at residual saturation, the phase permeability, m²Pressure measurements in producing reservoirs: barriers and accuracy effective vertical permeability of the tight unit, m²Pressure measurements in producing reservoirs: barriers and accuracy
$k_z,\ k_{rj},\ \mu_j$	vertical permeability, relative permeability (the share of the permeability that the phase can use) and viscosity of phase $j$ , m², —, Pa·sTwo-phase flow and reservoir description from formation-tester data
$K(m)$	complete elliptic integral of the first kind with the parameter $m = 1 - 1/A$ , —Comparing formation-tester, core and well-test data
$k/\mu$	mobility, mD/(mPa·s)Pressure measurement with the wireline formation tester
$L$	vertical distance from the mean depth of the water points to the contact, mTwo-phase flow and reservoir description from formation-tester data
$m$	slope of the line per decimal logarithmic cycle (per tenfold change on the logarithmic axis), MPaComparing formation-tester, core and well-test data
$M$	ratio of the mobilities of gas and oil at the current gas saturation, —Two-phase flow and reservoir description from formation-tester data
$m_s$	slope of the spherical line, Pa·s^½Supercharging and permeability from formation-tester pretests the same slope as in the spherical build-up: $q\mu\sqrt{\phi\mu c_t}/(4\pi^{3/2}k_s^{3/2})$ , Pa·s^½Supercharging and permeability from formation-tester pretests
$n$	number of points, —Comparing formation-tester, core and well-test data number of independent pairs “pretest and core”, —Comparing formation-tester, core and well-test data number of these points, —Forced-gradient technique and multi-well analysis
$\bar z_o,\ \bar p_o$	mean depth and pressure of the points of the oil line, m, PaPressure gradients, fluid contacts and capillary equilibrium
$\Delta p_{\text{obs}}$	pressure drop at the observation probe, PaSupercharging and permeability from formation-tester pretests
$\Delta p_{\text{over}}$	overbalance: the mud pressure minus the formation pressure, PaSupercharging and permeability from formation-tester pretests
$p$	pressure at the gauge, MPaPressure measurement with the wireline formation tester
$p^*$	the pressure that the build-up approaches ( $f_s \to 0$ ): the pressure at the wall, PaSupercharging and permeability from formation-tester pretests
$p_a(z_0),\ p_b(z_0)$	pressures on the lines of the light ( $a$ ) and heavy ( $b$ ) phases at the reference depth, PaPressure gradients, fluid contacts and capillary equilibrium
$p_{\text{b}}$	bubble point pressure of the oil from the sample, PaPressure gradients, fluid contacts and capillary equilibrium
$p_j$	average pressure of tank $j$ , brought to one datum depth (a chosen reference depth), MPaLumped multi-tank material balance model
$p_{\text{meas}}(z)$	stabilised pretest pressure at depth $z$ , PaPressure gradients, fluid contacts and capillary equilibrium
$p_{\text{nb}}$	pressure of the neighbouring block: that block is large, or its pressure is held by injection, MPaLumped multi-tank material balance model
$p_{wf}$	bottomhole pressure of the well, MPaLumped multi-tank material balance model
$p(\Delta t)$	pressure at the probe a time $\Delta t$ after the piston stops, PaSupercharging and permeability from formation-tester pretests
$\Delta p_{\text{perch}}$	by how much the pressure of perched water is above the pressure of the water of the main formation at the same depth, MPaPerched and tilted contacts, trapped water
$q$	volumetric rate of the draw by the piston, m³/sPressure measurement with the wireline formation tester rate of the draw by the piston, m³/sSupercharging and permeability from formation-tester pretests change in the rate of the well in unit 2; for a shut-in injection well, its former injection rate at reservoir conditions, m³/dPressure measurements in producing reservoirs: barriers and accuracy
$q_1,\ q_2$	net withdrawal from the units (production minus injection) at reservoir conditions, m³/dPressure measurements in producing reservoirs: barriers and accuracy
$q_{12}$	crossflow from unit 1 to unit 2 through the barrier, m³/dPressure measurements in producing reservoirs: barriers and accuracy
$q_j$	net withdrawal of the wells of the tank (production minus injection) at reservoir conditions, m³/dLumped multi-tank material balance model
$q_{\text{pist}}$	rate of the draw by the piston; zero during the build-up, cm³/sPressure measurement with the wireline formation tester
$q,\ A$	net withdrawal of the site (all of it through the permeable unit) and its area, m³/s, m²Pressure measurements in producing reservoirs: barriers and accuracy
$q,\ B,\ \mu$	oil rate before shut-in at standard conditions, formation volume factor, and oil viscosity at reservoir conditions, m³/d, m³/m³, mPa·sComparing formation-tester, core and well-test data
$q,\ q_0,\ q_\infty$	rate of the well at reservoir conditions: current, initial and limiting, m³/dLumped multi-tank material balance model
$q,\ \mu,\ \phi,\ c_t,\ k_h,\ k_v$	pump rate, viscosity, porosity, compressibility, and horizontal and vertical permeabilities, m³/s, Pa·s, —, 1/Pa, m², m²Supercharging and permeability from formation-tester pretests
$r$	distance from the well that creates the pulse to the observation well, mPressure measurements in producing reservoirs: barriers and accuracy
$r_1,\ r_2$	own depletion rates: how fast the potential of each unit would fall without crossflow, MPa/dPressure measurements in producing reservoirs: barriers and accuracy
$R_{\text{mc}}$	resistance of the mudcake per unit area of the wall, Pa·s/mSupercharging and permeability from formation-tester pretests
$r_{pe}$	effective radius of the probe: the radius of a spherical sink surface that gives the same flow as the round opening of the probe, mPressure measurement with the wireline formation tester
$R_{\text{res}}$	resistance of the rock per unit area of the wall: the excess pressure divided by the filtration rate $u$ (next section), Pa·s/mSupercharging and permeability from formation-tester pretests
$r_w$	radius of the hole, mSupercharging and permeability from formation-tester pretests
$s$	root-mean-square deviation of the points from the line, $\sqrt{\sum e_i^2/(n - 2)}$ , where $e_i$ is the deviation of a point from the line and $n$ is the number of points (two parameters of the line have already been fitted), PaPressure measurement with the wireline formation tester standard deviation of the logarithms of the ratios, —Comparing formation-tester, core and well-test data scatter of a valid point about the line: the repeatability and the depth errors, known from other lines of the survey or from repeat pretests, PaForced-gradient technique and multi-well analysis Laplace transform variable, 1/dPressure measurements in producing reservoirs: barriers and accuracy
$\Delta p_s$	supercharging: the excess of the pressure at the wall $p_s$ over the formation pressure $p_{\text{res}}$ , PaSupercharging and permeability from formation-tester pretests
$S$	tool storage: the volume of liquid in the line and the chamber multiplied by its compressibility, cm³/MPaPressure measurement with the wireline formation tester
$s_1$	trace of the matrix $\mathbf C^{-1}\mathbf K$ (the sum of the numbers on its diagonal), which is the sum of the eigenvalues, 1/dLumped multi-tank material balance model
$s_2$	the sum of its principal minors of second order (small determinants cut out along the diagonal), which is the product of the non-zero eigenvalues, 1/d²Lumped multi-tank material balance model
$s_G$	standard error of the gradient, Pa/mPressure measurement with the wireline formation tester
$S_{wi},\ S_{or}$	irreducible water saturation and residual oil saturation of the swept zone, —Two-phase flow and reservoir description from formation-tester data
$s,\ n,\ \bar z$	scatter of the points about the line, number of points and their mean depth (each line has its own), Pa, —, mPressure gradients, fluid contacts and capillary equilibrium
$\Delta p_{s,o},\ \Delta p_{s,w}$	supercharging of the oil and water lines at the depth where they cross, PaSupercharging and permeability from formation-tester pretests
$t$	Student's coefficient for $n - 1$ degrees of freedom (about 2 for large $n$ ), —Comparing formation-tester, core and well-test data time since the start of withdrawal, sPressure measurements in producing reservoirs: barriers and accuracy time, dLumped multi-tank material balance model
$T$	transmissibility of the barrier, $0.0853\,k_bA/(\mu b)$ , where $A$ is its area, m³/(d·MPa)Pressure measurements in producing reservoirs: barriers and accuracy transmissibility of the fault between the block and its neighbour, m³/(d·MPa)Lumped multi-tank material balance model
$T_{AB},\ T_{BC}$	transmissibilities of the links; there is no A–C link, m³/(d·MPa)Lumped multi-tank material balance model
$T_d$	duration of dynamic filtration before circulation stops, sSupercharging and permeability from formation-tester pretests
$t_{Dh}$	dimensionless time of the vertical drainage of the unit, —Pressure measurements in producing reservoirs: barriers and accuracy
$T_{jk}$	transmissibility of the link between tanks $j$ and $k$ : the crossflow per 1 MPa of pressure difference, $T_{jk} = T_{kj}$ . For a section of a fault it is $0.0853\,(k_f/l_f)\,A_f/\mu$ , m³/(d·MPa)Lumped multi-tank material balance model
$t_p$	duration of the draw, sSupercharging and permeability from formation-tester pretests
$t_s$	time of static filtration up to the measurement, sSupercharging and permeability from formation-tester pretests
$t_x,\ t_y$	components of the tilt of the contact, positive if the contact goes down along the axis, m/kmPerched and tilted contacts, trapped water
$u$	crossflow velocity at the boundary between the units: the share of the withdrawal that comes from the tight unit, divided by the area, m/sPressure measurements in producing reservoirs: barriers and accuracy
$u_1,\ u_2$	filtration rates through unit area of the wall during circulation and after it stops (the volume of filtrate per unit time per unit area), m/sSupercharging and permeability from formation-tester pretests
$u_t$	total vertical Darcy velocity $u_1 + u_2$ , positive for upward flow, m/sTwo-phase flow and reservoir description from formation-tester data
$u_z$	vertical Darcy velocity (the flow rate through a unit of horizontal area); positive for upward flow, m/sPressure measurements in producing reservoirs: barriers and accuracy crossflow velocity through the barrier: the flow rate divided by the area of the barrier, m/sPressure measurements in producing reservoirs: barriers and accuracy
$V$	rock volume of the accumulation, km²·m = million m³, million m³Perched and tilted contacts, trapped water
$V_1,\ V_2$	pore volumes of the units in the area drained by the wells of the site, m³Pressure measurements in producing reservoirs: barriers and accuracy pore volumes of the permeable and the tight unit, m³Pressure measurements in producing reservoirs: barriers and accuracy
$W_e$	cumulative water influx from the aquifer at reservoir conditions (water influx), m³Two-phase flow and reservoir description from formation-tester data
$W_pB_w$	produced water at reservoir conditions, m³Two-phase flow and reservoir description from formation-tester data
$x_0,\ y_0$	shift of the centre of the accumulation from the crest, kmPerched and tilted contacts, trapped water
$z$	number of standard errors; about 2 for 95%, —Comparing formation-tester, core and well-test data true vertical depth, measured downwards, mPressure gradients, fluid contacts and capillary equilibrium true vertical depth, measured downward, mPressure measurements in producing reservoirs: barriers and accuracy
$z^*$	depth of the crossing: the FWL for an oil–water pair, the GOC for a gas–oil pair, the GWC for a gas–water pair, mPressure gradients, fluid contacts and capillary equilibrium
$z^*_{\text{main}}$	FWL of the block that is open to the main aquifer, mPerched and tilted contacts, trapped water
$z^*_{\text{perch}}$	the perched FWL, mPerched and tilted contacts, trapped water
$z^,\ z^_{\text{A}},\ z^*_{\text{B}}$	FWL: the depth where the oil line crosses the water line; in wells A and B, mPerched and tilted contacts, trapped water
$z^*,\ p_{\text{FWL}}$	depth of the free water level and the pressure at it (the pressures of the phases are equal), m, PaPressure gradients, fluid contacts and capillary equilibrium
$z_0$	a reference depth that is convenient for the calculation, usually near the expected contact, mPressure gradients, fluid contacts and capillary equilibrium
$z_i$	depths of the points of the line, mPressure gradients, fluid contacts and capillary equilibrium
$z_i^*,\ \sigma_i$	FWL level of well $i$ and its standard error (from its own lines), mForced-gradient technique and multi-well analysis
$z_i,\ p_i$	true vertical depth and formation pressure of the $i$ -th valid measurement, m, PaPressure measurement with the wireline formation tester
$z_{\text{OWC}}$	oil–water contact from logs, $h_d = p_d/(G_w - G_o)$ above the FWL, mPressure gradients, fluid contacts and capillary equilibrium
$z_{\text{ref}}$	depth of the datum, mPerched and tilted contacts, trapped water
$\beta$	tilt angle of the contact; in practice $\tan\beta$ is given in metres per kilometre, m/kmPerched and tilted contacts, trapped water
$\delta$	constant error of the gauge in this survey, PaForced-gradient technique and multi-well analysis
$\Delta,\ \Delta_{\text{ss}},\ \tau$	potential difference, its steady-state value and the time constant (main section), MPa, MPa, dPressure measurements in producing reservoirs: barriers and accuracy
$\Delta(\Delta p_s)/\Delta z$	change of the supercharging with depth over the interval of the points (the mean slope), Pa/mSupercharging and permeability from formation-tester pretests
$\delta_o,\ \delta_w$	deviations of the oil and water points of a well from the oil and water lines of the field, at the recorded depths, PaForced-gradient technique and multi-well analysis shifts of the oil and water lines in pressure (the formula above without the depth error), PaForced-gradient technique and multi-well analysis
$\Delta_o,\ \Delta_w$	real differences of pressure in the oil and in the water at this well: a separate system, supercharging, depletion (a pressure drop caused by production) by the date of the measurement, PaForced-gradient technique and multi-well analysis
$\Delta_{\text{ss}}$	potential difference $\Phi_1 - \Phi_2$ after the units have reached a common pseudo-steady-state regime (both losing pressure at the same rate), MPaPressure measurements in producing reservoirs: barriers and accuracy
$\delta\Phi_b$	potential step: the potential of the unit that the crossflow comes from, minus the potential of the other unit, PaPressure measurements in producing reservoirs: barriers and accuracy
$\zeta$	height above the base of the tight unit, that is above the boundary with the permeable unit, mPressure measurements in producing reservoirs: barriers and accuracy
$\eta_v$	vertical hydraulic diffusivity (how fast a pressure change spreads in the vertical direction), m²/sSupercharging and permeability from formation-tester pretests
$\lambda$	non-zero eigenvalues of $\mathbf C^{-1}\mathbf K$ , the inverse time constants, 1/dLumped multi-tank material balance model
$\lambda_j,\ \lambda_t$	vertical mobility of phase $j$ , and total mobility $\lambda_1 + \lambda_2$ , m²/(Pa·s)Two-phase flow and reservoir description from formation-tester data
$\mu$	viscosity of the fluid that flows in, usually the mud filtrate, Pa·sPressure measurement with the wireline formation tester fluid viscosity at reservoir conditions, Pa·sPressure measurements in producing reservoirs: barriers and accuracy viscosity of the fluid that flows across, Pa·sPressure measurements in producing reservoirs: barriers and accuracy fluid viscosity, mPa·sPressure measurements in producing reservoirs: barriers and accuracy
$\mu,\ \phi,\ c_t$	viscosity of the fluid that flows in (usually the filtrate), and porosity and total compressibility of the rock at the hole, Pa·s, —, 1/PaSupercharging and permeability from formation-tester pretests
$\mu_f$	viscosity of the filtrate at reservoir temperature, mPa·sComparing formation-tester, core and well-test data
$\mu_f,\ h_{mc},\ k_{mc}$	viscosity of the filtrate, and thickness and permeability of the cake, Pa·s, m, m²Supercharging and permeability from formation-tester pretests
$\mu_f,\ \phi,\ c_t$	viscosity of the filtrate, and porosity and total compressibility of the rock at the hole, Pa·s, —, 1/PaSupercharging and permeability from formation-tester pretests
$\mu_g,\ \mu_o$	viscosities of gas and oil at reservoir conditions, mPa·sTwo-phase flow and reservoir description from formation-tester data
$\rho$	fluid density at reservoir conditions, kg/m³Pressure measurements in producing reservoirs: barriers and accuracy
$\bar\rho$	density weighted by the mobilities of the phases, which is the density shown by the gradient, kg/m³Two-phase flow and reservoir description from formation-tester data density from the pressure gradient in the countercurrent zone, g/cm³Two-phase flow and reservoir description from formation-tester data
$\rho,\ \sigma_\rho$	density of the fluid in the formation from the sample, and its standard error, kg/m³Forced-gradient technique and multi-well analysis fluid density in the formation from the analysis of a sample, and its standard error, kg/m³Pressure measurements in producing reservoirs: barriers and accuracy
$\rho_1,\ \rho_2$	densities of the phases at reservoir conditions, kg/m³Two-phase flow and reservoir description from formation-tester data
$\rho_o,\ \rho_g$	densities of oil and gas at reservoir conditions, g/cm³Two-phase flow and reservoir description from formation-tester data
$\rho_w,\ \rho_o$	densities of water and oil in the formation, g/cm³Perched and tilted contacts, trapped water
$\rho_{w,\text{sc}}$	density of a water sample at standard (laboratory) conditions; from salinity — Formation water and formation temperature, kg/m³Pressure gradients, fluid contacts and capillary equilibrium
$\sigma(z)$	standard error of the pressure on the line at depth $z$ , PaForced-gradient technique and multi-well analysis
$\sigma_a(z^),\ \sigma_b(z^)$	standard errors of the pressure on each line at the depth of the crossing, PaPressure gradients, fluid contacts and capillary equilibrium
$\sigma_{G_{\text{visc}}}$	its standard error, Pa/mPressure measurements in producing reservoirs: barriers and accuracy
$\sigma_{\text{gauge}}$	repeatability of the gauge at a test point (together with thermal effects), PaPressure measurement with the wireline formation tester
$\sigma_p$	random error of the formation pressure, tied to its depth, PaPressure measurement with the wireline formation tester
$\sigma_{p_{\text{b}}}$	error of the bubble point pressure (laboratory and sampling), PaPressure gradients, fluid contacts and capillary equilibrium
$\sigma_z$	random error of the true vertical depth, mPressure measurement with the wireline formation tester
$\sigma_{z^*}$	standard error of the depth of the crossing, mPressure gradients, fluid contacts and capillary equilibrium
$\sigma_{z_{\text{GOC}}}$	standard error of the GOC that we find (the error of the oil line itself is much smaller), mPressure gradients, fluid contacts and capillary equilibrium
$\tau$	time constant: it sets how fast the pressure approaches its steady value, sPressure measurement with the wireline formation tester time constant of the barrier: the step grows as $1 - e^{-t/\tau}$ and reaches 95% of its steady-state value after $3\tau$ , dPressure measurements in producing reservoirs: barriers and accuracy time constant of the decline of the rate, dLumped multi-tank material balance model
$\tau_2,\ \tau_{\text{loc}}$	local time constants of units 2 and 1, $\phi_ih_ic_t\mu b/(0.0853\,k_b)$ , dPressure measurements in producing reservoirs: barriers and accuracy
$\tau_{\text{loc}}$	local time constant: how long unit 1 behind the barrier takes to catch up with the pressure change of unit 2 near the observation well, sPressure measurements in producing reservoirs: barriers and accuracy
$\phi$	porosity of the rocks through which the contact has risen, —Two-phase flow and reservoir description from formation-tester data
$\bar\Phi$	potential of the site averaged over the pore volume, MPaPressure measurements in producing reservoirs: barriers and accuracy
$\phi,\ c_t,\ h$	porosity, total compressibility and thickness of the tight unit, —, 1/Pa, mPressure measurements in producing reservoirs: barriers and accuracy
$\Phi_0$	initial potential, common to both units, MPaPressure measurements in producing reservoirs: barriers and accuracy
$\Phi_1$	potential of the permeable unit (it is in vertical equilibrium), PaPressure measurements in producing reservoirs: barriers and accuracy
$\phi_1,\ h_1$	porosity and thickness of unit 1 (its storage capacity per unit area is $\phi_1h_1c_t$ ), —, mPressure measurements in producing reservoirs: barriers and accuracy
$\Phi_1,\ \Phi_2$	potentials of units 1 and 2, brought to one datum, MPaPressure measurements in producing reservoirs: barriers and accuracy
$\Phi_o$	oil pressure reduced to $z_{\text{ref}}$ with the oil gradient; common for connected oil, MPaPerched and tilted contacts, trapped water
$\Phi_w,\ \Phi_{w,\text{A}},\ \Phi_{w,\text{B}}$	water pressure reduced to $z_{\text{ref}}$ with the water gradient; in wells A and B, MPaPerched and tilted contacts, trapped water
$\chi^2$	a measure of consistency; it is compared with the limit for $n - 1$ degrees of freedom (for four degrees of freedom, 9.49), —Forced-gradient technique and multi-well analysis