Equations: Boundaries and late-time behaviour

The equations of the “Boundaries and late-time behaviour” lessons in “Well testing and pressure transient analysis”, with their conditions and units, and what each symbol means in them.

A well near a sealing fault: dimensionless form

p_D = \underbrace{\tfrac12\left(\ln t_D + 0.809\right) + S}_{\text{well}} + \underbrace{\tfrac12\,E_1\!\left(\frac{L_D^2}{t_D}\right)}_{\text{image}},\qquad L_D = \frac{L}{r_w}

no units; natural logarithm

Holds when: The fault is a straight sealing boundary through the whole thickness of a homogeneous reservoir, and the disturbance has not reached any other boundaries. For the well itself the logarithmic approximation holds ( $t_D > 25$ ), and wellbore storage is over. The image is a line source at a distance $2L$ . The argument of $E_1$ is $(2L_D)^2/(4t_D) = L_D^2/t_D$ .

A sealing fault near the well: the method of images →

The first and second semilog straight lines

\begin{aligned}\text{first:}\quad &\Delta p = m_e\left[\ln t_D + 0.809 + 2S\right]\\ \text{second:}\quad &\Delta p = 2m_e\left[\ln t_D + 0.809 + S - \ln\frac{2L}{r_w}\right]\end{aligned}

any consistent system of units; natural logarithm

Holds when: The first line holds while $\eta t/L^2 < 0.2$ , once wellbore storage is over. The second line holds at $\eta t/L^2 \gg 1$ . Its slope differs from $2m$ by less than 5% only after $\eta t/L^2 \approx 10$ . The skin enters the second line once, because only the real well has a pressure drop across the skin.

A sealing fault near the well: the method of images →

Distance to the fault from the intersection time of the lines

t_x = \frac{\gamma\,\phi\mu c_tL^2}{k}\quad\Rightarrow\quad L = \sqrt{\frac{k\,t_x}{\gamma\,\phi\mu c_t}},\qquad \gamma = 1.781

Holds when: Both lines are identified reliably, and the second is drawn through points where the slope has really doubled. $\phi$ and $c_t$ are known independently. Their error passes into $L$ as a square root.

A sealing fault near the well: the method of images →

The derivative step at a sealing fault, and the distance from a single point on it

\Delta p' = m_e\left[1 + \exp\left(-\frac{L^2}{\eta t}\right)\right]\quad\Rightarrow\quad L = \sqrt{\eta\,t\,\ln\frac{m_e}{\Delta p' - m_e}}

Holds when: The reservoir is homogeneous and there is one straight sealing fault. Wellbore storage is over, and $m_e$ and $k$ are known from the first plateau. We take points in the middle of the step, where $\Delta p'/m_e$ is between 1.2 and 1.7. At the start and at the end of the step, small errors in the derivative change $L$ greatly.

A sealing fault near the well: the method of images →

Hemiradial flow: flow capacity and skin for a known distance to the fault

kh = \frac{qB\mu}{2\pi\,\Delta p_2'},\qquad S = \frac{\Delta p(t)}{\Delta p_2'} - \ln\frac{kt}{\phi\mu c_tr_w^2} - 0.809 + \ln\frac{2L}{r_w}

SI; natural logarithm

Holds when: The first plateau is hidden by wellbore storage, and the doubled plateau has formed. The distance $L$ is known independently, because we cannot find it from such a record. The reservoir is homogeneous, and the fault is straight and sealing.

A sealing fault near the well: the method of images →

Build-up of a well near a sealing fault

p_i - p_{ws} = m_e\left[\ln\frac{t_p + \Delta t}{\Delta t} + E_1\!\left(\frac{L^2}{\eta\,(t_p + \Delta t)}\right) - E_1\!\left(\frac{L^2}{\eta\,\Delta t}\right)\right]

SI; natural logarithm

Holds when: Before start-up the pressure in the reservoir was $p_i$ . The rate stays constant during $t_p$ . For the well itself the logarithmic approximation holds, and wellbore storage is over. There are no other boundaries.

A sealing fault near the well: the method of images →

Distance to the fault from the intersection of the Horner lines

L = \sqrt{\frac{\eta\,\Delta t_{ex}}{\gamma}},\qquad \Delta t_{ex} = \frac{t_p\,\Delta t_x}{t_p + \Delta t_x}

Holds when: The fault was felt long before shut-in: $L^2 \ll \eta t_p$ . Then $E_1\big(L^2/(\eta t_p)\big)$ can be replaced by a logarithm. Both Horner lines have fully formed. A second Horner line that has not fully formed gives too small a value of $L$ , as in a drawdown test.

A sealing fault near the well: the method of images →

Largest deviation of the build-up from the line, and the maximum distance at which a fault can be detected

\delta p_{\max} = \frac{qB\,t_p}{4\pi\e\,\phi c_th\,L^2}\quad\Rightarrow\quad L_{\max} = \sqrt{\frac{qB\,t_p}{4\pi\e\,\phi c_th\,\delta p}}

Holds when: The fault was not felt during production: $\eta t_p \ll L^2$ . In practice the formula already works for $L^2 > 3\eta t_p$ . The record lasts at least $L^2/\eta$ after shut-in. The reservoir is homogeneous. There are no other disturbances, such as neighbouring wells in operation or gauge drift (a slow creep of the gauge reading).

A sealing fault near the well: the method of images →

The pressure derivative of a well with several sealing boundaries

\Delta p' = m_e\left[1 + \sum_{j} \exp\left(-\frac{d_j^2}{4\eta t}\right)\right],\qquad m_e = \frac{qB\mu}{4\pi kh},\quad \eta = \frac{k}{\phi\mu c_t}

SI (any consistent system)

Holds when: The boundaries are straight sealing planes through the whole thickness of a homogeneous reservoir. Images describe exactly a channel and a wedge with an angle $180^\circ/n$ . Wellbore storage is over. The images are line sources ( $d_j$ is much larger than the wellbore radius). The rate is constant. For a build-up we add up the same terms by superposition, and we allow for the producing time (Pressure build-up analysis).

Several boundaries: parallel faults and wedge-shaped blocks →

A well in a channel: linear flow

p_D = 2\sqrt{\pi t_{DW}} + S + S_{ch},\qquad t_{DW} = \frac{kt}{\phi\mu c_tW^2}\quad\Leftrightarrow\quad \Delta p = m_L\sqrt t + \frac{qB\mu}{2\pi kh}\,(S + S_{ch})

Holds when: The walls are parallel sealing boundaries through the whole thickness of a homogeneous reservoir; along the channel the reservoir is unbounded. The disturbance has reached both walls: the derivative lies on a line with slope ½ (by the table below, at $\eta t/W^2$ from 0.1 to 0.4, depending on the position of the well). A constant rate; wellbore storage is over.

Several boundaries: parallel faults and wedge-shaped blocks →

Geometric skin of a channel

S_{ch} = \ln\frac{W}{2\pi r_w} - \ln\sin\frac{\pi L_1}{W}

any equal units of length

Holds when: Steady linear flow; the well is a line source in a homogeneous channel (a pseudo-skin of the geometry, and not damage of the reservoir).

Several boundaries: parallel faults and wedge-shaped blocks →

The channel width from the slope of the line Δp against √t

W = \frac{qB}{h\,m_L}\sqrt{\frac{\mu}{\pi k\phi c_t}}

Holds when: The line is drawn through the points where the derivative already has a slope of ½. The permeability was found independently: from the radial flow of the same record or from other data. $\phi$ , $c_t$ and $h$ are known: their error enters $W$ as $1/\sqrt{\phi c_t}$ and $1/h$ .

Several boundaries: parallel faults and wedge-shaped blocks →

A build-up in a channel in the section of linear flow

p_i - p_{ws} = m_L\left(\sqrt{t_p + \Delta t} - \sqrt{\Delta t}\right)

SI (metric oilfield units: MPa, h, $m_L$ in MPa/√h)

Holds when: A constant rate during $t_p$ ; before start-up the pressure in the reservoir was $p_i$ . Both $t_p$ and $\Delta t$ are large enough for both terms of the superposition to be in the linear-flow section ( $\eta\Delta t/W^2$ not less than 0.1–0.4 by the table above).

Several boundaries: parallel faults and wedge-shaped blocks →

The late level of the derivative in a wedge, and the angle from it

\Delta p'_{\infty} = \frac{360^\circ}{\theta}\,m_e\quad\Rightarrow\quad \theta = 360^\circ\,\frac{m_e}{\Delta p'_{\infty}}

any equal units of pressure

Holds when: Two straight sealing faults through the whole thickness of a homogeneous reservoir meet at one point (in plan view). The disturbance has reached no other boundaries. The level has really been reached: on the way to it, the derivative is still lower, and the angle from it comes out too large. The angle is less than 180°: the well is inside an “acute” (less than 90°) or “obtuse” (more than 90°) angle, not beyond a bend of a fault.

Several boundaries: parallel faults and wedge-shaped blocks →

The approach of the derivative to the late level of a wedge, and the angle from two late points

\frac{\Delta p'}{m_e} \approx R_\infty\left(1 - \frac{a}{t}\right),\quad a = \frac{d^2}{2\eta};\qquad a = \frac{r_1 - r_2}{r_1/t_2 - r_2/t_1},\quad R_\infty = \frac{r_2}{1 - a/t_2}

SI; $t$ and $a$ in the same units of time

Holds when: The shortfall is small, no more than 15–20%. At a larger one, the next terms of the expansion matter, and the estimate of $R_\infty$ comes out too low and the angle too large. The derivative from measurements has no noticeable noise: the difference $r_2 - r_1$ is small, and its error strongly affects the result. The disturbance has reached no other boundaries than the two faults.

Several boundaries: parallel faults and wedge-shaped blocks →

Pressure in a well with images, in dimensionless form

p_{wD} = \tfrac12\left(\ln t_D + 0.809\right) + S + \tfrac12\sum_j E_1\!\left(\frac{\beta_j^2}{4\tau}\right),\qquad \tau = \frac{\eta t}{L^2} = \frac{t_D}{L_D^2},\quad \beta_j = \frac{d_j}{L}

dimensionless

Holds when: the boundaries are straight and sealing, through the whole thickness of a homogeneous reservoir; the images are line sources ( $d_j \gg r_w$ ) the rate is constant, wellbore storage is over, $t_D > 25$ for images that go off to infinity (channels), we add terms until they become negligible

Semi-infinite systems: theory and test analysis →

Product solution: the derivative of a rectangular system

\frac{\Delta p'}{m_e} = F_x(t)\,F_y(t),\qquad F_x = \sum_i \exp\!\left(-\frac{\xi_i^2}{4\eta t}\right),\quad F_y = \sum_l \exp\!\left(-\frac{\zeta_l^2}{4\eta t}\right)

SI (any consistent system)

Holds when: all the boundaries are straight sealing planes, parallel to $x$ or $y$ , through the whole thickness of a homogeneous reservoir the well is a line source, wellbore storage is over, and the rate is constant

Semi-infinite systems: theory and test analysis →

A well in a channel closed at one end: late linear flow

\Delta p = 2m_L\sqrt t + \frac{qB\mu}{2\pi kh}\,(S + S_{ch}) - \frac{qB\mu}{khW}\,L_3,\qquad \Delta p' = m_L\sqrt t

Holds when: the walls and the closed end are sealing planes through the whole thickness of a homogeneous reservoir; in the other direction the channel is unbounded the closed end is farther away than the width of the channel ( $L_3 > W$ ), and $\eta t \gg L_3^2$ , so the closed end has long been “felt”; wellbore storage is over, and the rate is constant

Semi-infinite systems: theory and test analysis →

Distance to the closed end of the channel: from a point of the derivative and from the intersection of the lines of Δp against √t

L_3 = \sqrt{\eta t\,\ln\frac{1}{r - 1}},\quad r = \frac{\Delta p'(t)}{\tfrac12 m_L\sqrt t};\qquad L_3 = \sqrt{\frac{\eta\,t_x}{\pi}}

Holds when: the record contains the linear flow of the open channel. We draw the line $\tfrac12m_L\sqrt t$ along it on the derivative, and the line with slope $m_L$ on the plot of $\Delta p$ against $\sqrt t$ the closed end is farther away than the width of the channel; the permeability is known from the radial-flow plateau the second formula holds if the late line has really been reached. If we draw a line with slope $2m_L$ through the last reading at time $t_{end}$ , it gives $L_x = L_3 - L_3^2/\sqrt{\pi\eta t_{end}}$ (the correction is in the example)

Semi-infinite systems: theory and test analysis →

A well at a sealing boundary as a well with a doubled rate

p_{wD} = 2\,p_D^{*}(t_D) - \ln\frac{2\delta}{r_w} + S

dimensionless

Holds when: the boundary is straight and sealing; $\delta$ is small compared with the other dimensions of the system the time is longer than a few $\delta^2/\eta$ : the image at distance $2\delta$ has already merged with the well (the derivation is in the advanced section)

Semi-infinite systems: theory and test analysis →

Build-up of a well in a system with images

p_i - p_{ws} = m_e\left[\ln\frac{t_p + \Delta t}{\Delta t} + \sum_j\left(E_1\!\left(\frac{d_j^2}{4\eta\,(t_p + \Delta t)}\right) - E_1\!\left(\frac{d_j^2}{4\eta\,\Delta t}\right)\right)\right]

SI; natural logarithm

Holds when: before start-up the pressure in the formation was $p_i$ ; the rate is constant during $t_p$ ; for the well itself the logarithmic approximation holds, and wellbore storage is over for channels the sum is infinite but converges: distant images give equal terms at $t_p + \Delta t$ and at $\Delta t$ , and the difference vanishes

Semi-infinite systems: theory and test analysis →

Largest trace of the closed end of a channel on a build-up after a short producing time

\delta p_{\max} = \frac{\e^{-1/2}}{2\sqrt{2\pi}}\cdot\frac{qB\,t_p}{\phi c_t h\,WL_3} = 0.121\,\frac{qB\,t_p}{\phi c_t h\,WL_3},\qquad \Delta t_{\max} \approx \frac{2L_3^2}{\eta}

Holds when: during the producing time the closed end has hardly been felt: $\eta t_p \ll L_3^2$ (in practice it is enough that $L_3^2 > 3\eta t_p$ ) the closed end is farther away than the width of the channel, and by the time $\Delta t_{\max}$ the channel has long been in linear flow a homogeneous formation; no gauge drift and no interference from neighbouring wells

Semi-infinite systems: theory and test analysis →

Flow through the fault zone

u = \frac{k_f}{\mu}\cdot\frac{p_1 - p_2}{l_f}

SI (any consistent system)

Holds when: the zone is thin compared with the distance to the well, and it stores no fluid single-phase flow; the zone is the same along the whole length and thickness of the fault on both sides there is a formation with the same properties and thickness

Partially communicating faults →

Dimensionless fault transmissibility

\alpha = \frac{k_f/l_f}{k/L} = \frac{k_f\,L}{k\,l_f}

dimensionless; $k_f$ and $k$ , and $l_f$ and $L$ , in the same units

Holds when: the thickness of the formation is the same on both sides of the fault (otherwise the numerator and the denominator contain their own $kh$ ) the fluid has the same properties on both sides

Partially communicating faults →

A well near a partially communicating fault: dimensionless pressure

p_D = \tfrac12\left(\ln t_D + 0.809\right) + S + \underbrace{\tfrac12 E_1\!\left(\frac{1}{t_{DL}}\right)}_{\text{image}} - \underbrace{\alpha\int_0^\infty \e^{-2\alpha\xi}\,E_1\!\left(\frac{(2 + \xi)^2}{4t_{DL}}\right)\dd\xi}_{\text{leak}}

dimensionless; natural logarithm

Holds when: the fault is a straight semipermeable barrier through the whole thickness of a homogeneous formation that is the same on both sides; there are no other boundaries for the well itself the logarithmic approximation holds, and wellbore storage is over the image and the leak are line sources ( $L$ is much larger than the well radius)

Partially communicating faults →

Pressure derivative near a partially communicating fault

\frac{\Delta p'}{m_e} = 1 + \e^{-1/t_{DL}}\left[1 - 2\sqrt{\pi}\,\alpha\sqrt{t_{DL}}\;\mathrm{erfcx}\left(2\alpha\sqrt{t_{DL}} + \frac{1}{\sqrt{t_{DL}}}\right)\right]

dimensionless

Holds when: the same as for the pressure; wellbore storage is over.

Partially communicating faults →

The late stretch: the rest of the derivative and the apparent fault skin

\frac{\Delta p'}{m_e} - 1 \approx \frac{1 + 4\alpha}{8\alpha^2\,t_{DL}},\qquad \Delta p = m_e\left[\ln t_D + 0.809 + 2S + 2S_f\right],\quad S_f = \e^{4\alpha}E_1(4\alpha)

the remainder and $S_f$ are dimensionless; $\Delta p$ and $m_e$ are in the same pressure units; natural logarithm

Holds when: $t_{DL} \gg 1/\alpha^2$ and $t_{DL} \gg 1$ : the remainder of the derivative is small; the formula for the remainder is approximate (it overstates it by a few per cent) the disturbance has not reached any other boundaries

Partially communicating faults →

Transmissibility of a section of the fault

T_f = \frac{k_f}{l_f}\,A,\qquad q_r = \frac{T_f}{\mu}\,(p_1 - p_2)

Holds when: single-phase flow; the zone is the same over the area $A$ the jump across the zone is smaller than the difference of the average pressures in the blocks: on the way to the fault and away from it, some pressure drop is also lost in the blocks themselves

Partially communicating faults →

Fault transmissibility multiplier

F = \frac{\Delta x/k}{\Delta x/k + l_f/k_f} = \frac{\alpha\,\Delta x}{\alpha\,\Delta x + L}

dimensionless; $\Delta x$ and $L$ in the same units

Holds when: neighbouring cells have the same size and permeability; for different cells, we add the half-resistances $\Delta x_i/(2k_i)$ separately the width of the zone is small compared with the cell

Partially communicating faults →

Symbols

Symbol	Meaning and context
$\Delta p$	drop in bottomhole pressure $p_i - p_{wf}$ , PaA sealing fault near the well: the method of images drop of the bottomhole pressure, $p_i - p_{wf}$ , PaSemi-infinite systems: theory and test analysis
$\delta p$	smallest deviation that we can tell apart reliably (it depends on the resolution and the noise of the gauge), PaA sealing fault near the well: the method of images
$\Delta x$	size of the cells across the fault (equal cells), mPartially communicating faults
$\Delta p'$	logarithmic derivative at time $t$ on the step, $m_e < \Delta p' < 2m_e$ , PaA sealing fault near the well: the method of images logarithmic derivative of the pressure drop, $\dd\Delta p/\dd\ln t$ , PaSeveral boundaries: parallel faults and wedge-shaped blocks logarithmic derivative of the pressure drop, PaPartially communicating faults
$\Delta p'_{\infty}$	the late level of the derivative, when the disturbance has covered the neighbourhood of the apex of the angle, PaSeveral boundaries: parallel faults and wedge-shaped blocks
$\Delta p'(t)$	logarithmic derivative of the pressure at time $t$ , PaSemi-infinite systems: theory and test analysis
$\Delta p(t)$	pressure drop at time $t$ on the doubled plateau, PaA sealing fault near the well: the method of images
$\Delta p_2'$	level of the doubled plateau of the derivative, $2m_e = qB\mu/(2\pi kh)$ , PaA sealing fault near the well: the method of images
$a$	approach constant: at the time $t$ the shortfall to the late level is the share $a/t$ , sSeveral boundaries: parallel faults and wedge-shaped blocks
$A$	area of the section: its length along the fault (the strike) times the thickness of the formation that is in contact across the fault, m²Partially communicating faults
$d$	distance from the well to the apex of the angle, mSeveral boundaries: parallel faults and wedge-shaped blocks
$d_j$	distance from the well to the $j$ -th image, mSeveral boundaries: parallel faults and wedge-shaped blocks distances to the images, mSemi-infinite systems: theory and test analysis
$d_j,\ \beta_j$	distance from the well to image number $j$ , and the same distance in units of $L$ , m, —Semi-infinite systems: theory and test analysis
$\e$	base of natural logarithms, 2.718, —A sealing fault near the well: the method of images
$E_1$	exponential integral (the E₁ function), —A sealing fault near the well: the method of images exponential integral (Line-source solution and the pressure function), —Semi-infinite systems: theory and test analysis exponential integral (function E₁), —Partially communicating faults exponential integral, —Partially communicating faults
$\mathrm{erfcx}$	scaled complementary error function, $\e^{x^2}\mathrm{erfc}(x)$ ; for large $x$ it is close to $1/(x\sqrt\pi)$ , —Partially communicating faults
$\Delta t_{ex}$	the matching equivalent time, which corrects the shut-in time for the length of production (Effect of producing time on build-up analysis), sA sealing fault near the well: the method of images
$F$	multiplier of the transmissibility between the cells on opposite sides of the fault, from 0 to 1, —Partially communicating faults
$F_x,\ F_y$	direction factors: sums over the one-dimensional images along $x$ and along $y$ , including the well itself (shift 0), —Semi-infinite systems: theory and test analysis
$k$	permeability from the derivative plateau (radial flow), m²Several boundaries: parallel faults and wedge-shaped blocks permeability of the formation, mDPartially communicating faults permeability of the cells (the same as for the formation in the test), mDPartially communicating faults
$k_f,\ l_f$	permeability and width of the fault zone (measured across the fault), m², mPartially communicating faults permeability and width of the fault zone, mD, mPartially communicating faults
$k,\ \phi,\ \mu,\ c_t$	permeability (from the first line), porosity, viscosity, total compressibility, m², —, Pa·s, 1/PaA sealing fault near the well: the method of images
$k,\ \phi,\ \mu,\ c_t,\ r_w$	permeability (from the doubled plateau), porosity, viscosity, total compressibility, wellbore radius, m², —, Pa·s, 1/Pa, mA sealing fault near the well: the method of images
$L$	distance to the fault, mA sealing fault near the well: the method of images distance to the fault, known from geology or seismic, mA sealing fault near the well: the method of images distance from the well to the fault, mPartially communicating faults
$L_1$	distance from the well to one of the walls (the formula is symmetric: $L_1$ and $W - L_1$ give the same), mSeveral boundaries: parallel faults and wedge-shaped blocks
$L,\ t$	distance to the fault and time since start-up, m, sA sealing fault near the well: the method of images
$L,\ L_D$	distance from the well to the fault, and the same distance in wellbore radii, m, —A sealing fault near the well: the method of images characteristic length of the system (for example, the distance to the nearest boundary), and the same length in well radii, m, —Semi-infinite systems: theory and test analysis
$L,\ r_w$	distance to the fault and wellbore radius, mA sealing fault near the well: the method of images
$m_e$	slope of the first line per unit of natural logarithm, $qB\mu/(4\pi kh)$ , PaA sealing fault near the well: the method of images level of the first plateau of the derivative, $qB\mu/(4\pi kh)$ , PaA sealing fault near the well: the method of images level of the derivative plateau in radial flow, PaSeveral boundaries: parallel faults and wedge-shaped blocks plateau level of radial flow, $qB\mu/(4\pi kh)$ , PaSeveral boundaries: parallel faults and wedge-shaped blocks derivative plateau in radial flow, $qB\mu/(4\pi kh)$ , PaSemi-infinite systems: theory and test analysis level of the radial-flow plateau, $qB\mu/(4\pi kh)$ , PaPartially communicating faults
$m_e,\ \eta$	level of the derivative plateau, and hydraulic diffusivity, Pa, m²/sSemi-infinite systems: theory and test analysis
$m_e,\ \eta,\ L$	level of the derivative plateau, hydraulic diffusivity, distance to the fault, Pa, m²/s, mA sealing fault near the well: the method of images
$m_L$	slope of the line $\Delta p$ against $\sqrt t$ , $\dfrac{qB}{hW}\sqrt{\dfrac{\mu}{\pi k\phi c_t}}$ , Pa/√sSeveral boundaries: parallel faults and wedge-shaped blocks slope of the line of the pressure drop in the coordinates $\Delta p$ against $\sqrt t$ in the section of linear flow, Pa/√sSeveral boundaries: parallel faults and wedge-shaped blocks slope of linear flow, the same as while the well produces, Pa/√sSeveral boundaries: parallel faults and wedge-shaped blocks slope of the line of $\Delta p$ against $\sqrt t$ for an open channel of the same width, $\dfrac{qB}{hW}\sqrt{\dfrac{\mu}{\pi k\phi c_t}}$ , Pa/√sSemi-infinite systems: theory and test analysis
$\delta p_{\max}$	largest deviation of the build-up pressure from the Horner line of an infinite reservoir (one with no boundaries), PaA sealing fault near the well: the method of images largest lag of the build-up pressure behind the build-up of the same channel without the closed end, PaSemi-infinite systems: theory and test analysis
$\Delta t_{\max}$	time after shut-in at which the lag is largest, sSemi-infinite systems: theory and test analysis
$p_1 - p_2$	pressure jump across the fault zone itself, PaPartially communicating faults
$p_1,\ p_2$	pressure just on each side of the zone, PaPartially communicating faults
$p_D$	dimensionless drop in bottomhole pressure $2\pi kh\,\Delta p/(qB\mu)$ , with skin — as in the lesson Line-source solution and the pressure function, —A sealing fault near the well: the method of images dimensionless pressure drop at the bottom, $2\pi kh\,\Delta p/(qB\mu)$ , —Several boundaries: parallel faults and wedge-shaped blocks dimensionless drop of the bottomhole pressure $2\pi kh\,\Delta p/(qB\mu)$ , with skin, —Partially communicating faults
$p_D^{*}(t_D)$	dimensionless solution without skin for the same system without the nearest boundary (all the other boundaries and their images stay), —Semi-infinite systems: theory and test analysis
$p_i$	pressure in the reservoir before start-up, PaA sealing fault near the well: the method of images
$p_{wD}$	dimensionless drop of the bottomhole pressure, $2\pi kh\,\Delta p/(qB\mu)$ , —Semi-infinite systems: theory and test analysis dimensionless drop of the bottomhole pressure of the well at the boundary, —Semi-infinite systems: theory and test analysis
$p_{ws}$	bottomhole pressure a time $\Delta t$ after shut-in, PaA sealing fault near the well: the method of images bottomhole pressure at the time $\Delta t$ after shut-in, PaSeveral boundaries: parallel faults and wedge-shaped blocks bottomhole pressure $\Delta t$ after shut-in, PaSemi-infinite systems: theory and test analysis
$q_r$	flow across the section at reservoir conditions, m³/sPartially communicating faults
$q,\ B,\ h,\ \mu,\ \phi,\ c_t$	as in the formula of linear flow, m³/s, m³/m³, m, Pa·s, —, 1/PaSeveral boundaries: parallel faults and wedge-shaped blocks
$q,\ B,\ \mu,\ k,\ h$	rate at standard conditions, formation volume factor, viscosity, permeability, thickness, m³/s, m³/m³, Pa·s, m², mSemi-infinite systems: theory and test analysis
$q,\ B,\ \mu,\ k,\ h,\ \phi,\ c_t$	rate, formation volume factor (reservoir volume per unit of surface volume), viscosity, permeability, thickness, porosity, total compressibility, m³/s, m³/m³, Pa·s, m², m, —, 1/PaSeveral boundaries: parallel faults and wedge-shaped blocks
$q,\ B,\ t_p$	rate before shut-in, formation volume factor (reservoir volume per unit of surface volume), producing time, m³/s, m³/m³, sA sealing fault near the well: the method of images
$qB\,t_p$	volume produced during the producing time, at reservoir conditions, m³Semi-infinite systems: theory and test analysis
$r$	ratio of the derivative at time $t$ to the continuation of the linear-flow line of the open channel, $1 < r < 2$ , —Semi-infinite systems: theory and test analysis
$R_\infty$	the late level in units of $m_e$ , $360^\circ/\theta$ , —Several boundaries: parallel faults and wedge-shaped blocks
$r_1,\ r_2$	the ratios $\Delta p'/m_e$ at the times $t_1 < t_2$ on the way to the level, —Several boundaries: parallel faults and wedge-shaped blocks
$r_w$	wellbore radius, mSeveral boundaries: parallel faults and wedge-shaped blocks
$S$	skin factor of the real well; the image has no skin, —A sealing fault near the well: the method of images skin of the well: it enters once, because it refers to the real rate, —Semi-infinite systems: theory and test analysis
$S_{ch}$	the extra pressure drop (in units of $qB\mu/(2\pi kh)$ ) for the change from linear flow along the channel to radial flow at the well, —Several boundaries: parallel faults and wedge-shaped blocks
$S_f$	apparent fault skin: the extra pressure drop on the late line, in units of $2m_e$ , —Partially communicating faults
$S,\ S_{ch}$	skin of the well and geometric skin of the channel (formula below), —Several boundaries: parallel faults and wedge-shaped blocks skin of the well, and the geometric skin of the channel $\ln\frac{W}{2\pi r_w} - \ln\sin\frac{\pi L_1}{W}$ (channel geometric skin), —Semi-infinite systems: theory and test analysis
$t$	time since start-up, sSeveral boundaries: parallel faults and wedge-shaped blocks
$t_D$	dimensionless time $kt/(\phi\mu c_tr_w^2)$ , —A sealing fault near the well: the method of images
$t_D,\ S$	dimensionless time and skin factor, —A sealing fault near the well: the method of images dimensionless time $kt/(\phi\mu c_tr_w^2)$ , and the skin (a number that shows how much the zone around the well helps or hinders flow), —Semi-infinite systems: theory and test analysis
$t_D,\ t_{DL}$	dimensionless time $kt/(\phi\mu c_tr_w^2)$ , and the same time based on $L^2$ : $t_{DL} = kt/(\phi\mu c_tL^2) = \eta t/L^2$ , —Partially communicating faults
$t_D,\ t_{DL},\ S$	dimensionless time based on the well radius and based on the distance to the fault, and the skin of the well, —Partially communicating faults
$t_{DL}$	dimensionless time $\eta t/L^2$ , —Partially communicating faults
$t_{DW}$	dimensionless time based on the square of the channel width, —Several boundaries: parallel faults and wedge-shaped blocks
$T_f$	transmissibility of the section of the fault: the geometric part, without viscosity, like the transmissibility between cells $T_{ij}$ in the lesson Pressure equation for single-phase flow, m³Partially communicating faults
$t_p,\ \Delta t$	producing time before shut-in and time after shut-in, sA sealing fault near the well: the method of images producing time before shut-in and time since shut-in, sSeveral boundaries: parallel faults and wedge-shaped blocks producing time at constant rate, and time after shut-in, sSemi-infinite systems: theory and test analysis
$t_x$	intersection time of the extensions of the first and second lines, sA sealing fault near the well: the method of images time at which the line of the open channel (slope $m_L$ ) and the late line (slope $2m_L$ ) intersect on the plot of $\Delta p$ against $\sqrt t$ , sSemi-infinite systems: theory and test analysis
$u$	Darcy velocity through the fault: the flow rate at reservoir conditions per unit area of the zone, m/sPartially communicating faults
$W$	channel width: the distance between the walls, $W = L_1 + L_2$ , mSeveral boundaries: parallel faults and wedge-shaped blocks
$W,\ L_3$	width of the channel and distance from the well to the closed end, mSemi-infinite systems: theory and test analysis width of the channel and distance to the closed end, mSemi-infinite systems: theory and test analysis
$\Delta t_x$	shut-in time at which the extensions of the first and second Horner lines meet, sA sealing fault near the well: the method of images
$\alpha$	dimensionless fault transmissibility, —Partially communicating faults
$\alpha,\ L$	dimensionless fault transmissibility, and the distance to which it refers, —, mPartially communicating faults
$\gamma$	the exponential of Euler's constant, —A sealing fault near the well: the method of images
$\delta$	distance from the axis of the well to the nearest boundary, mSemi-infinite systems: theory and test analysis
$\eta$	hydraulic diffusivity $k/(\phi\mu c_t)$ , m²/sA sealing fault near the well: the method of images hydraulic diffusivity (how fast a pressure change spreads through the rock), m²/sSeveral boundaries: parallel faults and wedge-shaped blocks hydraulic diffusivity, m²/sSeveral boundaries: parallel faults and wedge-shaped blocks hydraulic diffusivity $k/(\phi\mu c_t)$ (how fast a pressure change spreads through the rock), m²/sSemi-infinite systems: theory and test analysis
$\eta,\ \gamma$	hydraulic diffusivity and $\gamma = 1.781$ , m²/s, —A sealing fault near the well: the method of images
$\theta$	angle between the faults on the side of the well, degreesSeveral boundaries: parallel faults and wedge-shaped blocks
$\mu$	fluid viscosity, Pa·sPartially communicating faults
$\xi$	distance behind the image in units of $L$ (the variable of integration), —Partially communicating faults
$\xi_i,\ \zeta_l$	shifts of the one-dimensional images along $x$ and along $y$ , mSemi-infinite systems: theory and test analysis
$\tau$	dimensionless time based on $L^2$ ( $\eta$ is the hydraulic diffusivity: how fast a pressure change spreads through the rock), —Semi-infinite systems: theory and test analysis
$\phi,\ c_t,\ h$	porosity, total compressibility, reservoir thickness, —, 1/Pa, mA sealing fault near the well: the method of images porosity, total compressibility, thickness, —, 1/Pa, mSemi-infinite systems: theory and test analysis