Equations: Wellbore storage and type curves

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

Storage coefficient of a wellbore filled with one fluid

C = \frac{\Delta V_{wb}}{\Delta p_w} = c_{wb}\,V_{wb}

Holds when: The wellbore is entirely filled with one fluid, with no free liquid surface in it. The compressibility is taken as constant. The pressure change is small compared with the pressure itself (for gas this is a rough assumption). The pressure changes by the same amount along the whole wellbore: the weight of the fluid column and the friction losses stay as they were. In reality the fluid that has entered makes the column heavier, and $C$ is smaller. In a wellbore with liquid it is 1–4% smaller. In the gas well of the table below it is about 15% smaller. In an oil well with free gas in the tubing (the pipe through which the fluid flows up) it is almost half as large (example below). We ignore the elasticity of the pipes (their stretching under pressure) and the change of temperature.

Wellbore storage →

Volume balance in the wellbore

(q_{sf} - q)\,B = C\,\frac{\dd p_w}{\dd t}

Holds when: The coefficient $C$ is constant: the contents of the wellbore and their compressibility do not change. We treat the wellbore as a single tank: the pressure changes by the same amount everywhere in it. The rate at the wellhead is given (constant or zero), and the sandface rate is the unknown.

Wellbore storage →

Dimensionless wellbore storage coefficient

C_D = \frac{C}{2\pi\,\phi\,c_t\,h\,r_w^2}

metric oilfield units or SI: no constant is needed if $C$ and $c_t$ are given per the same unit of pressure

Holds when: The same as for the coefficient $C$ . $\phi$ , $c_t$ and $h$ refer to the reservoir, not to the wellbore.

Wellbore storage →

Section of pure wellbore storage

\Delta p = \frac{qB}{C}\,t

Holds when: The inflow from the reservoir has not changed yet: this is the section of pure wellbore storage. The coefficient $C$ is constant. Before start-up or shut-in, the pressure in the well changed far more slowly than after it.

Wellbore storage →

End of the unit-slope section

\frac{t_D}{C_D} \approx \alpha\,\ln\left(3\alpha\,C_D\,\e^{2S}\right)

dimensionless

Holds when: An approximate formula for $C_D\e^{2S} > 10^3$ . A constant $C$ and a homogeneous reservoir. For $C_D$ from $10^2$ to $10^4$ and $S$ from 0 to 10, it differs by no more than 15% from the calculation with the lesson's model.

Wellbore storage →

Time constant of the wellbore

\tau = \frac{\mu\,C\,p_D}{2\pi kh},\qquad p_D = \tfrac12\left(\ln t_D + 0.809\right) + S

Holds when: A rough model: the reservoir and the skin are replaced by a constant resistance, though in reality $p_D$ slowly grows with time. It is good for an estimate of the order of magnitude and for the first two or three $\tau$ . The “tail” of a real transition is longer than the exponential.

Wellbore storage →

Estimate of the start of the semilog straight line

t_D \approx (60 + 3.5\,S)\,C_D

dimensionless

Holds when: A guide, not an exact limit: it comes from a comparison of calculated curves. It applies for $C_D\e^{2S} > 10^3$ . A constant $C$ ; until this time the reservoir behaves as an infinite one. For a build-up: if the shut-in time is much shorter than the producing time of the well.

Wellbore storage →

A well with wellbore storage and skin: the Laplace transform

\bar p_{wD}(s) = \frac{s\,\bar p_D^{\,*}(s) + S}{s\left\{1 + C_D\,s\left[s\,\bar p_D^{\,*}(s) + S\right]\right\}}

dimensionless

Holds when: $C_D$ and $S$ are constant; the rate at the wellhead is constant from start-up. The formula holds for any solution $\bar p_D^{\,*}$ , including that of a bounded reservoir: the wellbore and the skin change only the condition at the bottom of the well.

Wellbore storage →

The link between the plot of the record and the type-curve plot, in logarithms

\lg p_D = \lg\Delta p + \lg\frac{2\pi kh}{qB\mu},\qquad \lg\frac{t_D}{C_D} = \lg t + \lg\frac{2\pi kh}{\mu C}

SI; the forms below are in oilfield units

Holds when: A start-up from a state in which the pressure in the reservoir has equalised, or a shut-in after production at a constant rate. For a pressure build-up curve, the shut-in time is much shorter than the producing time of the well. The record obeys the model of the type curves: a homogeneous infinite reservoir, a constant $C$ and a constant skin.

Type-curve matching →

The parameter of the type curves and the two asymptotes of the family

C_De^{2S} = \frac{C}{2\pi\phi c_t h\left(r_we^{-S}\right)^2};\qquad p_D = \frac{t_D}{C_D}\ \text{(wellbore)},\qquad p_D = \frac12\left[\ln\frac{t_D}{C_D} + 0.809 + \ln\left(C_De^{2S}\right)\right]\ \text{(reservoir)}

dimensionless ( $C$ and $c_t$ are given per the same unit of pressure); natural logarithm

Holds when: A homogeneous infinite reservoir; constant $C$ and $S$ . The first asymptote is the section of pure wellbore storage. The second is radial flow, which starts after the transition section (how long wellbore storage lasts). Replacing the skin with an effective radius is exact for radial flow and almost exact in the transition section, if $C_D$ is not too small (advanced section).

Type-curve matching →

Pressure match: the flow capacity

kh = \frac{qB\mu}{2\pi}\left(\frac{p_D}{\Delta p}\right)_M

Holds when: The record is matched to the curve as a whole, including the radial-flow section. The rate is constant. For a build-up, the shut-in time is small compared with the producing time.

Type-curve matching →

Time match: the wellbore storage coefficient

C = \frac{2\pi kh}{\mu}\left(\frac{\Delta t}{t_D/C_D}\right)_M

Holds when: $kh$ was found from the pressure match. Time is counted exactly from the moment of start-up or shut-in.

Type-curve matching →

Dimensionless wellbore storage coefficient and skin from the parameter of the curve

C_D = \frac{C}{2\pi\phi c_t h r_w^2},\qquad S = \frac12\ln\frac{\left(C_De^{2S}\right)_M}{C_D}

metric oilfield units or SI: $C$ and $c_t$ are given per the same unit of pressure; natural logarithm

Holds when: $\phi$ , $c_t$ , $h$ and $r_w$ are known independently: $C_D$ depends on them, and so does the skin. The skin that we find refers to the wellbore radius that we put into $C_D$ .

Type-curve matching →

Time match on the curves

p_D

against

t_D

k = \phi\mu c_t r_w^2\left(\frac{t_D}{\Delta t}\right)_M

Holds when: The curves $p_D(t_D)$ are calculated for a known $C_D$ . The porosity, the compressibility and the wellbore radius are known exactly.

Type-curve matching →

A measure of the difference between the record and the type curve

F(a_p, a_t, X) = \sum_{i=1}^{n}\left[\ln\left(a_p\,\Delta p_i\right) - \ln p_D\!\left(a_t\,\Delta t_i;\ X\right)\right]^2

dimensionless

Holds when: All the measurements are equally reliable. When the noise of the gauge is known, we weight the terms. The model is chosen in advance. The fit does not check that it is right.

Type-curve matching →

The logarithmic derivative of pressure

\Delta p' = \frac{\dd\,\Delta p}{\dd\ln t} = t\,\frac{\dd\,\Delta p}{\dd t}

any system of units: $\Delta p'$ is expressed in the same units as the pressure

Holds when: $\Delta p$ is counted from the pressure at the moment of start-up or shut-in, and $t$ from that moment. The logarithm is natural. If the axis is built in base-10 logarithms, the slope per cycle is $\ln 10 = 2.303$ times larger than $\Delta p'$ . For a build-up after a short producing time of the well, we take the derivative with respect to the equivalent time (Effect of producing time on build-up analysis).

Diagnosis with the pressure derivative →

The derivative of a power-law regime

\Delta p = A\,t^{\,n}\quad\Rightarrow\quad \Delta p' = n\,A\,t^{\,n} = n\,\Delta p,\qquad \lg\Delta p' = \lg\Delta p + \lg n

any consistent system of units

Holds when: The regime lasts long enough for a straight section to form on the plot. If a constant is added to the power term (for example, the skin pressure drop), the plot of $\Delta p$ is curved, but the derivative is not: the derivative of a constant is zero.

Diagnosis with the pressure derivative →

The dimensionless derivative and its value in radial flow

p_D' = \frac{\dd p_D}{\dd\ln t_D} = \frac{2\pi kh}{qB\mu}\,\Delta p',\qquad p_D' = 0.5\ \ \text{(radial flow)}

SI; the forms below are in oilfield units

Holds when: A constant rate after start-up (for a build-up: a constant rate before shut-in, and a shut-in time much shorter than the producing time). The equality $p_D' = 0.5$ holds for radial flow in a homogeneous reservoir, where $p_D = \tfrac12(\ln t_D + 0.809) + S$ .

Diagnosis with the pressure derivative →

Flow capacity from the level of the derivative plateau

kh = \frac{qB\mu}{4\pi\,m_e}

Holds when: The derivative is horizontal for at least half a logarithmic cycle. We average its level over the points of this section. A constant rate. For a build-up, the shut-in time is much shorter than the producing time of the well. The plateau refers to radial flow through the whole thickness of the reservoir. With partial penetration, an early plateau can refer to only part of the thickness (Partial penetration and well deviation).

Diagnosis with the pressure derivative →

Skin factor from the ratio of the pressure to the derivative plateau

S = \frac12\left[\frac{\Delta p(t)}{m_e} - \ln\frac{kt}{\phi\mu c_tr_w^2} - 0.809\right]

SI; natural logarithm

Holds when: The point $t$ lies on the plateau: wellbore storage is over, and the boundaries have not yet had an effect. For a build-up, $\Delta p = p_{ws} - p_{wf}(\Delta t = 0)$ , and the shut-in time is much shorter than the producing time. The skin refers to the wellbore radius $r_w$ that we put into the formula.

Diagnosis with the pressure derivative →

The derivative in the wellbore-storage section

\Delta p' = t\,\frac{\dd\,\Delta p}{\dd t} = \Delta p = \frac{qB}{C}\,t,\qquad p_D' = p_D = \frac{t_D}{C_D}

SI; in metric oilfield units $\Delta p = qBt/(24C)$ : m³/d, m³/MPa, h, MPa

Holds when: The inflow from the reservoir has not changed yet: this is the section of pure wellbore storage. The coefficient $C$ is constant.

Diagnosis with the pressure derivative →

The derivative at a straight boundary

\Delta p' = m_e\left[1 \pm \exp\left(-\frac{\phi\mu c_tL^2}{kt}\right)\right],\qquad p_D' = \frac12\left[1 \pm \e^{-L_D^2/t_D}\right]

Holds when: A straight boundary through the whole thickness of a homogeneous reservoir; there are no other boundaries. Wellbore storage is over, and the logarithmic approximation holds for the well itself. The image is a line source: $L$ is much larger than the wellbore radius.

Diagnosis with the pressure derivative →

The derivative in a closed area (pseudo-steady-state flow)

\Delta p' = t\,\frac{\dd\,\Delta p}{\dd t} = \frac{qB}{c_tV_p}\,t,\qquad p_D' = 2\pi\,t_{DA}

SI; in metric oilfield units $\Delta p' = qBt/(24\,c_tV_p)$ : m³/d, h, 1/MPa, m³, MPa

Holds when: A closed area with no inflow across the boundaries; the disturbance has reached all the boundaries. A constant rate; a single-phase fluid with a constant compressibility.

Diagnosis with the pressure derivative →

The derivative in linear and spherical flow

\Delta p = A_l\sqrt{t} + \Delta p_s\ \Rightarrow\ \Delta p' = \tfrac12\,A_l\sqrt{t};\qquad \Delta p = \Delta p_\infty - \frac{A_s}{\sqrt t}\ \Rightarrow\ \Delta p' = \frac{A_s}{2\sqrt t}

any consistent system of units

Holds when: Linear flow: the flow is along parallel lines towards the plane of a fracture or along a channel, and the disturbance has not reached the ends of the fracture or other boundaries. Spherical flow: a homogeneous reservoir around a small inflow interval, and the disturbance has not yet reached the top and the bottom of the reservoir.

Diagnosis with the pressure derivative →

The derivative from three readings

\Delta p_i' = \frac{s_l\,\Delta X_r + s_r\,\Delta X_l}{\Delta X_l + \Delta X_r},\qquad s_l = \frac{\Delta p_i - \Delta p_l}{\Delta X_l},\quad s_r = \frac{\Delta p_r - \Delta p_i}{\Delta X_r}

$\Delta p'$ and the slopes are in units of pressure; $\Delta X$ has no units

Holds when: The readings are ordered in time, and the pressure is counted from the moment of start-up or shut-in. The interval is the same for the whole record. In the first and the last $L$ cycles of the record there is no neighbour on one side. The derivative is not calculated there, or it is calculated with a shortened interval and is less reliable.

Diagnosis with the pressure derivative →

Build-up from the drawdown response

\Delta p_{BU}(\Delta t) = \Delta p_{DD}(\Delta t) - \Big[\Delta p_{DD}(t_p + \Delta t) - \Delta p_{DD}(t_p)\Big]

any consistent system of units

Holds when: The problem is linear: the fluid is single-phase and slightly compressible, and the reservoir and fluid properties do not depend on pressure. Before the start of production the pressure in the reservoir is uniform, and before shut-in the surface rate was constant. $\Delta p_{DD}$ is the response of the same well, with the same skin and wellbore storage, shut in at the same point. Then the equality is exact, not approximate.

Effect of producing time on build-up analysis →

Local slope of the build-up with respect to the logarithm of the shut-in time

\Delta p' = \frac{\dd\,\Delta p_{BU}}{\dd\ln\Delta t} = m_e\,\frac{t_p}{t_p + \Delta t}

any consistent system of units

Holds when: Radial flow in an infinite reservoir at the times $\Delta t$ and $t_p + \Delta t$ : wellbore storage is over and the boundaries have no effect. The rate is constant before shut-in.

Effect of producing time on build-up analysis →

Equivalent time

\Delta p_{BU}(\Delta t) = \Delta p_{DD}(\Delta t_e),\qquad \Delta t_e = \frac{t_p\,\Delta t}{t_p + \Delta t}

any consistent system; $\Delta t_e$ is in the same units as $t_p$ and $\Delta t$

Holds when: Radial flow in an infinite reservoir at the times $t_p$ and $t_p + \Delta t$ : before shut-in the flow had reached the radial regime, and the disturbance had not reached the boundaries. At the time $\Delta t$ the flow is also radial, or else $\Delta t \ll t_p$ . In that case $\Delta t_e \approx \Delta t$ , and the equality holds during wellbore storage as well. The rate is constant before shut-in.

Effect of producing time on build-up analysis →

Skin from the straight line against equivalent time

S = 1.151\left[\frac{\Delta p_{1\text{h}}}{m} - \lg\frac{k}{\phi\,\mu\,c_t\,r_w^2} + 2.098\right]

metric oilfield units: pressure in MPa, $k$ in mD, $\mu$ in mPa·s, $c_t$ in 1/MPa, $r_w$ in m, time in hours

Holds when: The conditions of equivalent time hold: before shut-in the flow had reached the radial regime. We read $\Delta p_{1\text{h}}$ on the straight line, not from a measured point. If $t_p$ is less than an hour, the point $\Delta t_e$ = 1 h lies on the continuation of the line beyond $t_p$ .

Effect of producing time on build-up analysis →

Derivative with respect to equivalent time

\Delta p'_e = \frac{\dd\,\Delta p_{BU}}{\dd\ln\Delta t_e} = \frac{t_p + \Delta t}{t_p}\cdot\frac{\dd\,\Delta p_{BU}}{\dd\ln\Delta t}

any consistent system; the derivatives are in units of pressure

Holds when: $t_p$ is the producing time at a constant rate before shut-in. With a varying rate, we take the derivative with respect to the superposition function of the whole history (Equivalent producing time and synthetic flowing time). The logarithm is natural. The relation follows from $\dd\ln\Delta t_e/\dd\ln\Delta t = t_p/(t_p + \Delta t)$ .

Effect of producing time on build-up analysis →

Shut-in duration for a given equivalent time

\Delta t = \frac{t_p\,T}{t_p - T},\qquad T < t_p

any identical units of time

Holds when: The reservoir is infinite and the flow is radial. The formula is the inverse of $\Delta t_e(\Delta t)$ . We choose $T$ according to what we need to see: for example, one cycle of the derivative plateau after wellbore storage has ended.

Effect of producing time on build-up analysis →

Storage coefficient of a closed wellbore with free gas

\frac{1}{C} = \frac{1}{C_g} + \frac{\rho g}{A_t},\qquad C_g = \sum_i \frac{V_{g,i}}{p_i} + c_lV_l

Holds when: The wellhead is closed, and the tubing is separated from the annulus by a packer. The gas is ideal, and the temperature of each portion of gas is constant. The liquid is almost incompressible. The gas stays at its own depth or is collected in a pocket under the wellhead. We ignore friction in a shut-in well.

Nonideal wellbore storage, well clean-up and changing skin →

Instantaneous wellbore storage coefficient from the afterflow

C(\Delta t) = \frac{q_{sf}\,B}{\dd p_{ws}/\dd\Delta t}

Holds when: The well is shut in at the wellhead, and the whole inflow goes to fill the wellbore. The flowmeter is between the reservoir and the gauge, and it measures the whole inflow.

Nonideal wellbore storage, well clean-up and changing skin →

Pressure in a closed wellbore after the gas and the liquid have separated

p_c = \frac{\sum_i p_i V_{g,i}}{\sum_i V_{g,i}},\qquad p_{wsc} = p_c + \rho g h_l,\qquad A \approx p_{wsc} - p_{wf}

SI; the pressures are absolute

Holds when: The wellbore is isolated from the reservoir. This is an upper estimate for a well that manages to take part of the liquid back. The gas is ideal and does not dissolve in the oil while the phases separate, and its temperature does not change. We ignore the weight of the gas pocket and the friction (the weight of the pocket would raise the estimate by tenths of a megapascal).

Nonideal wellbore storage, well clean-up and changing skin →

Wellbore balance with the redistribution pressure

q_{sf}B = C\left(\frac{\dd p_{ws}}{\dd\Delta t} - \frac{\dd p_\phi}{\dd\Delta t}\right)

Holds when: Shut-in at the wellhead; the rate at the wellhead after shut-in is zero. $C$ is constant, and everything that changes the behaviour of the wellbore is collected in the function $p_\phi$ . $p_\phi$ is a given function of time that does not depend on the pressure in the reservoir: the model is empirical.

Nonideal wellbore storage, well clean-up and changing skin →

Two empirical forms of the redistribution pressure

p_\phi = A\left(1 - \e^{-\Delta t/t_\phi}\right)\quad\text{or}\quad p_\phi = A\,\operatorname{erf}\frac{\Delta t}{t_\phi}

any consistent units: $A$ in units of pressure, $t_\phi$ in units of time

Holds when: The forms were chosen to fit observations, and were not derived from the physics of the wellbore. We find $A$ and $t_\phi$ by fitting the model to the record. They tell us only roughly about the wellbore (the amplitude is of the order of the estimate for an isolated wellbore, and the time is of the order of the rise time of the bubbles).

Nonideal wellbore storage, well clean-up and changing skin →

Apparent coefficient at the start of the record

\frac{1}{C_a} = \frac{1}{C} + \frac{24A}{qB\,t_\phi}\quad\text{(exponential form)},\qquad \frac{1}{C_a} = \frac{1}{C} + \frac{48A}{\sqrt\pi\,qB\,t_\phi}\quad\text{(error function)}

metric oilfield: m³/MPa, MPa, h, m³/d

Holds when: The start of the record, while the inflow is still almost equal to the rate before shut-in. $t_\phi$ is much larger than the time of the early readings.

Nonideal wellbore storage, well clean-up and changing skin →

Wellbore storage coefficient for a rising level under a gas pocket

\frac{1}{C} = \frac{p_g}{V_g} + \frac{\rho g}{A},\qquad p_gV_g = \text{const}

SI; in metric oilfield units the second term is $10^{-6}\cdot9.81\,\rho/A$ MPa/m³

Holds when: The gas above the level is ideal, and its mass and temperature do not change (no gas comes out of the oil). The liquid is incompressible, and there is no friction.

Nonideal wellbore storage, well clean-up and changing skin →

The largest distortion of the derivative plateau after a short shut-in

\varepsilon_{\max} = \frac{\tau}{\left(\sqrt{t_f + \tau} + \sqrt{t_f}\right)^2} \approx \frac{\tau}{4\,t_f}

dimensionless; $\tau$ and $t_f$ in the same units

Holds when: Before the short shut-in, the well produced for a long time at the same rate, and it returned to it again. Around the well there is radial flow; the wellbore storage after the short shut-in and the restart is over.

Nonideal wellbore storage, well clean-up and changing skin →

The derivative of a drawdown test with a changing skin

\Delta p' = \frac{\dd\,\Delta p}{\dd\ln t} = m_e\left(1 + 2t\,\frac{\dd S}{\dd t}\right)

any consistent system; $t\,\dd S/\dd t$ has no units

Holds when: A constant rate, radial flow in a homogeneous reservoir, and wellbore storage is over. The skin is a thin layer, and the pressure drop across it is proportional to the rate at that moment.

Nonideal wellbore storage, well clean-up and changing skin →

The skin at the time t from a drawdown point

S(t) = \frac12\left[\frac{p^* - p_{wf}(t)}{m_e} - \ln\frac{k\,t}{\phi\mu c_tr_w^2} - 0.809\right]

SI; natural logarithm

Holds when: Before production the pressure in the reservoir was equalised and the rate is constant; at the time $t$ wellbore storage is over. The permeability did not change during production: only the near-wellbore zone changes. The same formula holds also with boundaries, if $p^*$ is taken from the radial-flow line of the build-up.

Nonideal wellbore storage, well clean-up and changing skin →

Symbols

Symbol	Meaning and context
$\Delta p$	change of the bottomhole pressure since start-up or shut-in: $p_i - p_{wf}$ after start-up, $p_{ws} - p_{wf}(\Delta t = 0)$ after shut-in, PaWellbore storage change of the bottomhole pressure since start-up ( $p_i - p_{wf}$ ) or shut-in ( $p_{ws} - p_{wf}(\Delta t = 0)$ ), PaType-curve matching change of the bottomhole pressure since start-up or shut-in: $p_i - p_{wf}$ after start-up, $p_{ws} - p_{wf}(\Delta t = 0)$ after shut-in; always positive, PaDiagnosis with the pressure derivative
$\ln t$	natural logarithm of time; the derivative does not depend on the units of time, because changing them only shifts $\ln t$ by a constant, —Diagnosis with the pressure derivative
$\Delta t$	time after shut-in, sEffect of producing time on build-up analysis required shut-in duration, hEffect of producing time on build-up analysis
$\Delta p,\ \Delta p'$	change of pressure and its logarithmic derivative, PaDiagnosis with the pressure derivative
$\Delta p'$	logarithmic derivative: the slope of the plot of $\Delta p$ against $\ln t$ at the given point, Pa (the same units as $\Delta p$ )Diagnosis with the pressure derivative logarithmic derivative of the build-up with respect to $\ln\Delta t$ — the slope of the MDH plot per unit of natural logarithm, PaEffect of producing time on build-up analysis logarithmic derivative of the pressure drop $p_i - p_{wf}$ , PaNonideal wellbore storage, well clean-up and changing skin
$\Delta p'_e$	logarithmic derivative of the build-up with respect to equivalent time, PaEffect of producing time on build-up analysis
$\left(\Delta t/(t_D/C_D)\right)_M$	time match: the time of the record that coincided with the dimensionless time of the match point, divided by this dimensionless time, sType-curve matching
$\left(C_De^{2S}\right)_M$	parameter of the curve on which the record lies, —Type-curve matching
$\left(p_D/\Delta p\right)_M$	pressure match: the dimensionless pressure of the match point divided by the change of pressure that coincided with it, 1/PaType-curve matching
$\left(t_D/\Delta t\right)_M$	dimensionless time of the match point divided by the time of the record that coincided with it, 1/sType-curve matching
$\Delta p(t)$	change of pressure at the time $t$ , taken on the plateau, PaDiagnosis with the pressure derivative
$\pm$	plus: a sealing boundary (a fault); minus: a constant-pressure boundary, —Diagnosis with the pressure derivative
$\Delta p_\infty$	the value to which the pressure drop tends in spherical flow, PaDiagnosis with the pressure derivative
$\Delta p_{1\text{h}}$	pressure rise on the straight line (or its continuation) at $\Delta t_e$ = 1 h, measured from $p_{wf}(\Delta t = 0)$ , MPaEffect of producing time on build-up analysis
$A$	constant of the regime, which depends on the properties of the reservoir and the well, Pa/sⁿDiagnosis with the pressure derivative amplitude of the redistribution pressure: by how much the separating of gas and liquid could raise the pressure over and above the inflow, PaNonideal wellbore storage, well clean-up and changing skin amplitude: the value to which $p_\phi$ tends; above zero the pressure rises (gas redistribution, humps), below zero the wellbore storage decreases, PaNonideal wellbore storage, well clean-up and changing skin area of the cross-section in which the level rises (the annulus or the tubing), m²Nonideal wellbore storage, well clean-up and changing skin
$A_l,\ A_s$	constants of the linear and the spherical regimes: they contain the permeability, the storativity of the reservoir (its ability to store fluid by elastic expansion) and the size of the fracture or of the opened interval, Pa/s^½, Pa·s^½Diagnosis with the pressure derivative
$a_p,\ a_t$	shifts along the pressure and time axes: $a_p$ is the pressure match, $a_t$ is the inverse of the time match $[\Delta t/(t_D/C_D)]_M$ , 1/MPa, 1/hType-curve matching
$A_t$	area of the cross-section through which the column rises: the inside cross-section of the tubing, m²Nonideal wellbore storage, well clean-up and changing skin
$A,\ t_\phi$	amplitude and time of the redistribution, MPa, hNonideal wellbore storage, well clean-up and changing skin
$B$	formation volume factor of the fluid (its volume in the reservoir per unit volume at the surface), m³/m³Wellbore storage formation volume factor of the fluid, m³/m³Wellbore storage
$B,\ \Delta t$	formation volume factor; time since shut-in, m³/m³, sNonideal wellbore storage, well clean-up and changing skin
$\Delta p_{BU},\ \Delta p_{DD}$	pressure rise after shut-in and pressure drop after the start of production, PaEffect of producing time on build-up analysis
$\Delta p_{BU}(\Delta t)$	rise in bottomhole pressure a time $\Delta t$ after shut-in: $p_{ws}(\Delta t) - p_{wf}(\Delta t = 0)$ , PaEffect of producing time on build-up analysis
$C$	wellbore storage coefficient: the volume of fluid that the wellbore gives out or takes in when the bottomhole pressure changes by one unit, m³/PaWellbore storage wellbore storage coefficient, m³/PaWellbore storage wellbore storage coefficient from the time match, m³/MPaType-curve matching wellbore storage coefficient at the depth of the gauge, m³/PaNonideal wellbore storage, well clean-up and changing skin wellbore storage coefficient after the redistribution is over, m³/PaNonideal wellbore storage, well clean-up and changing skin coefficient after the redistribution, m³/MPaNonideal wellbore storage, well clean-up and changing skin
$C_a$	the coefficient that the early unit slope would give, m³/MPaNonideal wellbore storage, well clean-up and changing skin
$C_D$	dimensionless wellbore storage coefficient, —Wellbore storage
$C_De^{2S}$	parameter of the type curve, —Type-curve matching
$C_D,\ S$	dimensionless wellbore storage coefficient and skin factor, —Wellbore storage
$C_g$	the part of the coefficient that comes from the compression of the contents: the volume that the gas and the liquid free up when the pressure rises by one unit (for an ideal gas the compressibility is $1/p$ ), m³/PaNonideal wellbore storage, well clean-up and changing skin
$c_l,\ V_l$	compressibility and volume of the liquid in the wellbore, 1/Pa, m³Nonideal wellbore storage, well clean-up and changing skin
$c_t$	total compressibility of the reservoir and the fluids in it, 1/MPaWellbore storage total compressibility, 1/PaDiagnosis with the pressure derivative
$c_{wb}$	compressibility of the fluid in the wellbore, 1/PaWellbore storage
$C(\Delta t)$	wellbore storage coefficient at the time $\Delta t$ after shut-in, m³/PaNonideal wellbore storage, well clean-up and changing skin
$\bar p_D^{\,*}(s)$	Laplace transform of the solution at a constant rate at the bottom of the well, without wellbore storage and skin: $K_0(\sqrt{s})/s$ for a line source, $K_0(\sqrt{s})/(s^{3/2}K_1(\sqrt{s}))$ for a well of finite radius, —Wellbore storage
$\dd\,\Delta p_{BU}/\dd\ln\Delta t$	ordinary logarithmic derivative of the build-up with respect to the time after shut-in, $\Delta p'$ , PaEffect of producing time on build-up analysis
$\Delta p_{DD}(t)$	drop in bottomhole pressure a time $t$ after the start of production at a constant rate $q$ from a reservoir at pressure $p_i$ : $p_i - p_{wf}(t)$ , PaEffect of producing time on build-up analysis
$\Delta t_e$	equivalent time: the time after the start of production at which the pressure drop on the drawdown would equal the pressure rise on the build-up a time $\Delta t$ after shut-in, sEffect of producing time on build-up analysis
$\operatorname{erf}$	the error function (a smooth S-shaped curve), $\operatorname{erf}x = \frac{2}{\sqrt\pi}\int_0^x \e^{-u^2}\dd u$ ; at $x = 1$ it equals 0.843, —Nonideal wellbore storage, well clean-up and changing skin
$g$	acceleration of gravity, 9.81, m/s²Nonideal wellbore storage, well clean-up and changing skin
$h$	net thickness of the reservoir, mWellbore storage
$h_l$	height of the liquid column above the gauge after the gas and liquid have separated, mNonideal wellbore storage, well clean-up and changing skin
$i$	the reading at which we calculate the derivative, —Diagnosis with the pressure derivative
$\Delta p_i,\ \Delta t_i$	the measurements: change of pressure and time, MPa, hType-curve matching
$k$	permeability, m²Type-curve matching permeability from the build-up, m²Nonideal wellbore storage, well clean-up and changing skin
$K_0,\ K_1$	modified Bessel functions of the second kind, —Wellbore storage
$k,\ h$	permeability and thickness of the reservoir, m², mType-curve matching
$k,\ \phi,\ \mu,\ c_t,\ r_w$	permeability (from the plateau), porosity, viscosity, total compressibility, wellbore radius, m², —, Pa·s, 1/Pa, mDiagnosis with the pressure derivative permeability (from the slope), porosity, viscosity, total compressibility, wellbore radius, mD, —, mPa·s, 1/MPa, mEffect of producing time on build-up analysis
$kh$	flow capacity, m³Type-curve matching
$kh,\ \mu$	flow capacity (from the pressure match) and viscosity, m³, Pa·sType-curve matching
$L$	distance from the well to the boundary; the image is at the distance $2L$ , mDiagnosis with the pressure derivative differentiation interval, in fractions of a decimal logarithmic cycle, —Diagnosis with the pressure derivative
$L_D$	dimensionless distance $L/r_w$ , —Diagnosis with the pressure derivative
$l,\ r$	the nearest readings to the left and to the right that are at least $L$ decimal cycles from $i$ : $\lg(t_i/t_l) \ge L$ , $\lg(t_r/t_i) \ge L$ , —Diagnosis with the pressure derivative
$\Delta X_l,\ \Delta X_r$	distances in the natural logarithm of time: $\ln(t_i/t_l)$ and $\ln(t_r/t_i)$ , —Diagnosis with the pressure derivative
$m$	slope of the straight line against $\lg\Delta t_e$ per decimal cycle — the same as that of the Horner line, MPaEffect of producing time on build-up analysis
$m_e$	level of the derivative plateau; it is also the slope of the semilog straight line per unit of the natural logarithm, $m_e = m/\ln 10$ , PaDiagnosis with the pressure derivative level of the derivative plateau, PaDiagnosis with the pressure derivative plateau level, $qB\mu/(4\pi kh)$ , PaDiagnosis with the pressure derivative level of the derivative plateau: $qB\mu/(4\pi kh)$ , PaEffect of producing time on build-up analysis plateau level of the derivative, $qB\mu/(4\pi kh)$ , PaNonideal wellbore storage, well clean-up and changing skin plateau level from the build-up, $m/\ln 10$ , PaNonideal wellbore storage, well clean-up and changing skin
$n$	number of measurements, —Type-curve matching exponent: 1 for wellbore storage or a closed area, ½ for linear flow, —Diagnosis with the pressure derivative
$p^*$	the extension of the Horner line from the build-up; for a new well, the initial reservoir pressure, PaNonideal wellbore storage, well clean-up and changing skin
$p_c$	pressure of the gas pocket after the gas and liquid have separated, PaNonideal wellbore storage, well clean-up and changing skin
$p_D$	dimensionless pressure drop of a well without wellbore storage at the time we look at; usually between 5 and 15, —Wellbore storage dimensionless change of pressure, $2\pi kh\,\Delta p/(qB\mu)$ , —Type-curve matching dimensionless change of pressure in the well itself, $2\pi kh\,\Delta p/(qB\mu)$ , with skin and wellbore storage, as in the lessons Line-source solution and the pressure function and Type-curve matching, —Diagnosis with the pressure derivative
$p_D,\ t_D,\ C_D$	in dimensionless form: the change of the bottomhole pressure $2\pi kh\,\Delta p/(qB\mu)$ , the time $kt/(\phi\mu c_tr_w^2)$ and the wellbore storage coefficient; $k$ , $h$ , $\mu$ are the permeability, the thickness of the reservoir and the viscosity, —Wellbore storage
$p_D,\ t_D/C_D$	dimensionless change of pressure and dimensionless time, —Type-curve matching
$p_D'$	dimensionless derivative, —Diagnosis with the pressure derivative
$p_D(\cdot;\ X)$	the type curve with the parameter $X$ , —Type-curve matching
$p_g,\ V_g$	absolute pressure and volume of the gas above the level, Pa, m³Nonideal wellbore storage, well clean-up and changing skin
$p_i,\ V_{g,i}$	mean absolute pressure and volume of free gas in the $i$ -th interval of the wellbore at the moment of shut-in, Pa, m³Nonideal wellbore storage, well clean-up and changing skin
$p_w$	bottomhole pressure, PaWellbore storage
$p_{wf}$	bottomhole pressure at the moment of shut-in, PaNonideal wellbore storage, well clean-up and changing skin
$p_{wf}(t)$	bottomhole pressure at the time $t$ of production at a constant rate, PaNonideal wellbore storage, well clean-up and changing skin
$p_{ws}$	pressure in the wellbore at the depth of the gauge, PaNonideal wellbore storage, well clean-up and changing skin
$p_{wsc}$	pressure at the depth of the gauge in the isolated wellbore after the gas and liquid have separated, PaNonideal wellbore storage, well clean-up and changing skin
$p_\phi$	redistribution pressure: how much the pressure in an isolated wellbore would have risen by the time $\Delta t$ ; $p_\phi(0) = 0$ , PaNonideal wellbore storage, well clean-up and changing skin
$q$	rate at the wellhead at standard conditions, m³/sWellbore storage change of the rate at the wellhead: the rate after start-up or the rate before shut-in, at standard conditions, m³/sWellbore storage
$q_{sf}$	sandface flow rate: the inflow from the reservoir into the wellbore, converted to standard conditions, m³/sWellbore storage sandface flow rate: the afterflow at standard conditions, m³/sNonideal wellbore storage, well clean-up and changing skin inflow from the reservoir into the wellbore after shut-in, at standard conditions, m³/sNonideal wellbore storage, well clean-up and changing skin
$q,\ B$	change of the rate at the wellhead at standard conditions, and formation volume factor, m³/s, m³/m³Diagnosis with the pressure derivative rate at standard conditions and formation volume factor, m³/s, m³/m³Diagnosis with the pressure derivative rate before shut-in at standard conditions and formation volume factor, m³/d, m³/m³Nonideal wellbore storage, well clean-up and changing skin
$q,\ B,\ \mu$	change of the rate at the wellhead at standard conditions, formation volume factor (reservoir volume per unit of surface volume), viscosity, m³/s, m³/m³, Pa·sType-curve matching rate at standard conditions (after start-up or before shut-in), formation volume factor, viscosity, m³/s, m³/m³, Pa·sType-curve matching change of the rate at standard conditions, formation volume factor (reservoir volume per unit of surface volume), viscosity, m³/s, m³/m³, Pa·sDiagnosis with the pressure derivative
$r_w$	wellbore radius, mWellbore storage
$r_w,\ S$	wellbore radius and skin factor; $r_we^{-S}$ is the effective radius, m, —Type-curve matching
$s$	variable of the Laplace transform, which corresponds to the dimensionless time $t_D$ , —Wellbore storage
$\Delta p_s$	a constant pressure drop, for example across the skin; the derivative does not depend on it, PaDiagnosis with the pressure derivative
$S$	skin factor, —Wellbore storage
$s_l,\ s_r$	slopes of the chords (the straight lines between neighbouring points) to the left and to the right, PaDiagnosis with the pressure derivative
$S(t)$	skin at the time $t$ , —Nonideal wellbore storage, well clean-up and changing skin
$t$	time, sWellbore storage time since start-up or shut-in (after shut-in, $\Delta t$ ), sWellbore storage time since start-up or shut-in (for a build-up, $\Delta t$ ), sType-curve matching time since start-up or shut-in, sDiagnosis with the pressure derivative time since the start of production at a constant rate, sNonideal wellbore storage, well clean-up and changing skin
$T$	equivalent time up to which the build-up must reproduce the drawdown, hEffect of producing time on build-up analysis
$t_D$	dimensionless time of the start of the line, $kt/(\phi\mu c_tr_w^2)$ , —Wellbore storage dimensionless time $kt/(\phi\mu c_tr_w^2)$ ; the derivative with respect to $\ln t_D$ equals the derivative with respect to $\ln t$ , because $t_D$ and $t$ differ by a constant factor, —Diagnosis with the pressure derivative
$t_D,\ S$	dimensionless time and skin factor, —Wellbore storage
$t_D/C_D$	dimensionless time divided by the dimensionless wellbore storage coefficient; equal to $0.02232\,kh\,t/(\mu C)$ in metric oilfield units, —Wellbore storage dimensionless time divided by the dimensionless wellbore storage coefficient: $2\pi kh\,t/(\mu C)$ , —Type-curve matching dimensionless time divided by the dimensionless wellbore storage coefficient, —Diagnosis with the pressure derivative
$t_{DA}$	dimensionless time related to the area of the region $A$ : $kt/(\phi\mu c_tA)$ , —Diagnosis with the pressure derivative
$t_f$	time that the well produces after the short shut-in before the valve closes, hNonideal wellbore storage, well clean-up and changing skin
$t_p$	producing time at the rate $q$ before shut-in, sEffect of producing time on build-up analysis producing time before shut-in, hEffect of producing time on build-up analysis
$t_p,\ \Delta t$	producing time and time after shut-in, sEffect of producing time on build-up analysis
$t_\phi$	redistribution time; for the exponential form, 63% of the change happens in this time, sNonideal wellbore storage, well clean-up and changing skin
$t,\ C$	in the dimensional forms: the time of the start of the line and the wellbore storage coefficient, h; m³/MPa or bbl/psiWellbore storage
$V_{g,i},\ p_i$	volume of free gas and its absolute pressure in the $i$ -th interval of the wellbore, m³, PaNonideal wellbore storage, well clean-up and changing skin
$V_p$	pore volume of the closed area, m³Diagnosis with the pressure derivative
$V_{wb}$	volume of the wellbore that is connected with the reservoir: from the closing valve to the reservoir, m³Wellbore storage
$\Delta p_w$	change of the bottomhole pressure, PaWellbore storage
$\Delta V_{wb}$	the volume by which the contents of the wellbore expand or shrink (the index $wb$ stands for wellbore), m³Wellbore storage
$\bar p_{wD}(s)$	Laplace transform of the dimensionless pressure in a well with wellbore storage and skin, —Wellbore storage
$\dd p_{ws}/\dd\Delta t$	the rate of growth of the bottomhole pressure: the derivative with respect to time, not to its logarithm: $\Delta p'/\Delta t$ , Pa/sNonideal wellbore storage, well clean-up and changing skin
$X$	parameter of the curve $C_De^{2S}$ , —Type-curve matching
$\alpha$	allowed relative deviation of the pressure from the unit-slope line: 0.01, 0.05 or 0.10, —Wellbore storage
$\varepsilon_{\max}$	the largest relative lowering of the build-up derivative compared with the radial-flow level, —Nonideal wellbore storage, well clean-up and changing skin
$\mu,\ k,\ h$	viscosity, permeability, thickness of the reservoir, Pa·s, m², mWellbore storage in the dimensional forms: viscosity of the fluid, permeability and thickness of the reservoir, mPa·s or cP; mD; m or ftWellbore storage
$\rho$	density of the liquid in the wellbore, kg/m³Nonideal wellbore storage, well clean-up and changing skin density of the liquid, kg/m³Nonideal wellbore storage, well clean-up and changing skin
$\rho,\ g$	density of the liquid and acceleration of gravity, kg/m³, m/s²Nonideal wellbore storage, well clean-up and changing skin
$\tau$	time constant: in this time the difference between the sandface rate and the wellhead rate falls by a factor of about $\e \approx 2.7$ , sWellbore storage duration of the short shut-in at the wellhead for running the valve, hNonideal wellbore storage, well clean-up and changing skin
$\phi$	porosity of the reservoir, fractionWellbore storage
$\phi,\ \mu,\ c_t,\ k,\ t$	porosity, viscosity, total compressibility, permeability, time, —, Pa·s, 1/Pa, m², sDiagnosis with the pressure derivative
$\phi,\ \mu,\ c_t,\ r_w$	porosity, viscosity, total compressibility, wellbore radius, —, Pa·s, 1/Pa, mType-curve matching
$\phi,\ c_t,\ h$	porosity, total compressibility, thickness of the reservoir, —, 1/MPa, mType-curve matching
$\phi,\ c_t,\ h,\ r_w$	porosity, total compressibility, thickness of the reservoir, wellbore radius, —, 1/MPa, m, mType-curve matching