Equations: Bounded drainage area

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

The pseudo-steady-state stretch of a drawdown test

p_{wf}(t) = p_{int} - m^*t,\qquad m^* = \frac{qB}{c_tV_p} = \frac{qB}{c_t\,\phi hA}

any consistent units (SI or, for example, m³/d, 1/MPa, m³ and MPa/d)

Holds when: a closed area with no inflow through the boundaries a constant rate since start-up a single-phase slightly compressible fluid, and $c_t$ is constant pseudo-steady-state flow has started (we check it on the derivative)

Reservoir limit test →

Intercept of the depletion line on the pressure axis

p_{int} = p_i - \frac{qB\mu}{4\pi kh}\left(\ln\frac{4A}{\gamma C_A r_w^2} + 2S\right)

SI; $\ln$ is the natural logarithm

Holds when: the same as for the depletion line; a homogeneous formation, and the well penetrates the whole thickness.

Reservoir limit test →

Shape factor from two lines

C_A = \frac{4\pi\,m_e}{m^*\,t_1}\,\exp\frac{p_{int} - p_1}{m_e}

any consistent units: $m^*$ is in units of pressure per the unit of time in which $t_1$ is given

Holds when: both lines are drawn from one record with a constant rate a homogeneous formation; radial flow and pseudo-steady-state flow are really visible in the record

Reservoir limit test →

How the slope in ordinary coordinates approaches its limit

\frac{|\dd p_{wf}/\dd t|}{m^*} - 1 \approx 2\cos^2\frac{\pi x_w}{L}\;\exp\left(-\frac{\pi^2\eta\,t}{L^2}\right)

SI; in metric units $\eta = 3.553\cdot10^{-3}\,k/(\phi\mu c_t)$ m²/h for $t$ in hours

Holds when: a closed rectangular area, a homogeneous formation a late transition period: the faster “modes” have already decayed the well is not at the middle of the length: for a well at the middle of the length, on the long axis, this mode does not appear, and the equalisation goes four times faster

Reservoir limit test →

Estimate of the time to reach pseudo-steady-state flow

t_{pss} \approx \frac{L^2}{\pi^2\eta}\,\ln\left(200\cos^2\frac{\pi x_w}{L}\right)

metric: $L$ and $x_w$ in m, $\eta$ in m²/h, $t_{pss}$ in hours

Holds when: a closed rectangular area; the well is clearly shifted from the middle of the length ( $\cos^2(\pi x_w/L)$ is not small) for a well at the middle of the length, we use the table of $t_{DA}$ of the previous section.

Reservoir limit test →

Lower estimate of the connected pore volume

V_p \gtrsim \pi r_i^2 h\phi,\qquad r_i = 0.119\sqrt{\frac{k\,t}{\phi\mu c_t}}

metric oilfield units: $k$ in mD, $t$ in h, $\mu$ in mPa·s, $c_t$ in 1/MPa; $r_i$ in m

Holds when: there are no signs of boundaries in the whole record: the derivative is on the plateau an estimate of the order of magnitude: different definitions of the radius of investigation differ by tens of per cent

Reservoir limit test →

Rate-normalised pressure and material balance time

\frac{p_i - p_{wf}(t)}{q(t)} = b + \frac{B}{c_tV_p}\,t_{mb},\qquad t_{mb} = \frac{Q(t)}{q(t)}

metric: for $q$ in m³/d, $Q$ in m³ and $c_t$ in 1/MPa, the time $t_{mb}$ is in days, and the slope is in MPa/(m³/d) per day

Holds when: a closed area, pseudo-steady-state flow the rate changes slowly compared with the time of pressure equalisation; we do not use the points right after a sharp change of the rate (tens of hours) a single-phase fluid, and $B$ and $c_t$ are constant

Reservoir limit test →

A build-up in a closed area as the sum of two rates

p_i - p_{ws}(\Delta t) = \frac{qB\mu}{2\pi kh}\Big[p_D(t_p + \Delta t) - p_D(\Delta t)\Big]

SI (any consistent system of units)

Holds when: a single-phase slightly compressible fluid, a homogeneous formation, a closed area a constant rate before shut-in (for a changing rate, the equivalent time $N_p/q$ ) we ignore wellbore storage (the well acting as a small tank): the well is shut in at the bottom, or the early points are not used

Build-up in a bounded reservoir and average reservoir pressure →

The pressure to which a build-up in a closed area levels off

p_{ws}(\Delta t \to \infty) = p_i - 2\pi t_{pDA}\,\frac{qB\mu}{2\pi kh} = p_i - \frac{qB\,t_p}{c_t V_p} = \bar p

SI; in metric units the pressure drop equals $qBt_p/(24\,c_tV_p)$ for $q$ in m³/d, $t_p$ in hours and $c_t$ in 1/MPa

Holds when: $\Delta t_{DA} = k\Delta t/(\phi\mu c_tA)$ is larger than the time to reach pseudo-steady-state flow for the given shape of the area the area is closed, the neighbouring wells do not work, and the fluid is single-phase

Build-up in a bounded reservoir and average reservoir pressure →

The dimensionless MBH correction

p_{DMBH}(t_{pDA}) = \frac{4\pi kh}{qB\mu}\,(p^* - \bar p) = \frac{2.303\,(p^* - \bar p)}{m}

SI; the second form is right in any units, if $p^*$ , $\bar p$ and $m$ are in the same pressure units

Holds when: the build-up has a radial-flow line (we confirm it with the pressure derivative) a homogeneous formation, a single-phase fluid, a closed area, and one well in it (or the area is a part of a block, separated from the neighbouring wells by dividing lines of flow, the lines between the drainage areas of neighbouring wells) the rate was constant before shut-in; on the stretch of the line $\Delta t \ll t_p$

Build-up in a bounded reservoir and average reservoir pressure →

The MBH function through the solution for a producing well

p_{DMBH}(t_{pDA}) = 4\pi t_{pDA} - 2\Big[p_D(t_{pD}) - \tfrac12\big(\ln t_{pD} + 0.809\big)\Big]

dimensionless; $\ln$ is the natural logarithm

Holds when: the same as for the MBH function; $p_D$ is calculated for a specific shape of the area and position of the well.

Build-up in a bounded reservoir and average reservoir pressure →

Average pressure after production in pseudo-steady-state flow

\bar p = p^* - m\,\lg\left(C_A\,t_{pDA}\right)

any consistent units: $p^*$ , $\bar p$ and $m$ in the same pressure units

Holds when: before shut-in the well had reached pseudo-steady-state flow: $t_{pDA}$ is larger than the time from which the MBH function coincides with $\ln(C_At_{pDA})$ (the table above) $p^*$ and $t_{pDA}$ are calculated with the same producing time

Build-up in a bounded reservoir and average reservoir pressure →

Dietz time

\Delta t_{\bar p} = \frac{\phi\mu c_t A}{k\,C_A}

Holds when: before shut-in the well worked in pseudo-steady-state flow, and the producing time is much larger than $\Delta t_{\bar p}$ the MDH line is drawn through the points of radial flow

Build-up in a bounded reservoir and average reservoir pressure →

The local slope of a Horner plot after production in pseudo-steady-state flow

\frac{\text{local slope}}{m} = \left(1 + \frac{\Delta t}{t_p'}\right)\left(1 - 4\pi\,\Delta t_{DA}\right)

dimensionless; $\Delta t$ and $t_p'$ in the same units

Holds when: before shut-in the well worked in pseudo-steady-state flow after shut-in there is still radial flow around the well: the disturbance from the shut-in has not reached the boundaries

Build-up in a bounded reservoir and average reservoir pressure →

A late build-up in a closed area

\bar p - p_{ws}(\Delta t) \approx a\,\e^{-\lambda\,\Delta t},\qquad \lambda \approx \frac{\pi^2\eta}{L^2}

SI; in metric units $\eta = 3.553\cdot10^{-3}\,k/(\phi\mu c_t)$ m²/h, and $\lambda$ comes out in 1/h

Holds when: a late period: the other, faster “modes” have already decayed (for a rectangle this is $\Delta t_{DA}$ of about 0.2–0.5 and more) the formula for $\lambda$ is for a rectangular area and a well that is not at the middle of its length. At the middle, this mode does not appear, and the decay is faster. For a well at the centre of a circle, $\lambda = 14.7\,\eta/r_e^2$ one well in a closed area, and the neighbours do not work

Build-up in a bounded reservoir and average reservoir pressure →

Average pressure of a closed area and cumulative production

\bar p = p_i - \frac{B}{c_t V_p}\,N_p

any consistent units (for example, m³, 1/MPa and MPa)

Holds when: a closed area with no water inflow; single-phase oil above the bubble point pressure (the pressure below which gas comes out of the oil); $B$ and $c_t$ are constant there is one well in the area, or $N_p$ and $V_p$ refer to the whole closed block

Build-up in a bounded reservoir and average reservoir pressure →

A build-up after several flow periods in the Horner form

p_{ws}(\Delta t) = p^* - m\,X_{BU}(\Delta t),\qquad X_{BU} = \sum_{j=1}^{n}\frac{q_j - q_{j-1}}{q_n}\,\lg\frac{\theta_j + \Delta t}{\Delta t}

metric oilfield units: pressure and $m$ in MPa, $q$ in m³/d, $\mu$ in mPa·s, $k$ in mD, $h$ in m; time in any equal units

Holds when: radial flow in an infinite formation over the whole history and the build-up: the disturbance has not reached any boundaries a linear problem (the effects of separate events add up): a single-phase slightly compressible fluid, a constant skin wellbore storage (the well acting as a small tank) is over

Equivalent producing time and synthetic flowing time →

The superposition derivative

\Delta p'_X = -\frac{\dd\,\Delta p}{\dd\,(X_{BU}\ln 10)} = \frac{\Delta p'}{\displaystyle\sum_{j=1}^{n}\frac{q_j - q_{j-1}}{q_n}\,\frac{\theta_j}{\theta_j + \Delta t}}

any consistent system; derivatives in units of pressure

Holds when: the conditions of the previous equation; the minus sign is there because $X_{BU}$ decreases as $\Delta t$ grows.

Equivalent producing time and synthetic flowing time →

The Horner equivalent producing time

t_p = \frac{N_p}{q_n}

$N_p$ in m³, $q_n$ in m³/d, $t_p$ in days; for the Horner ratio in hours we multiply by 24

Holds when: the last rate was held long enough compared with the duration of the build-up (an estimate below) the history went on in an infinitely acting formation; after a long production in a bounded area we use the synthetic time (the sections below)

Equivalent producing time and synthetic flowing time →

The error of the slope with one producing time

R(\Delta t) = \frac{t_p + \Delta t}{t_p}\,\sum_{j=1}^{n}\frac{q_j - q_{j-1}}{q_n}\,\frac{\theta_j}{\theta_j + \Delta t}

dimensionless; times in the same units

Holds when: radial flow in an infinite formation; wellbore storage is over.

Equivalent producing time and synthetic flowing time →

Synthetic flowing time

t_{sDA} = \frac{k\,t_s}{\phi\mu c_tA} = \frac{1}{C_A}\,\exp\frac{p_e - \bar p}{m_e}

SI; in metric oilfield units $t_{sDA} = 3.553\cdot10^{-3}\,kt_s/(\phi\mu c_tA)$ : mD, h, mPa·s, 1/MPa, m²

Holds when: before shut-in the well had reached pseudo-steady-state flow we analyse the build-up by the radial-flow line, while the shut-in has not yet been felt at the boundaries the ratio $(p_e - \bar p)/m_e$ depends only on the shape of the area, the position of the well and the chosen point of the boundary. We calculate it with a model of the area (the table below)

Equivalent producing time and synthetic flowing time →

A build-up as the sum of the trend and the response to the shut-in

p_{ws}(\Delta t) = p_{tr}(\Delta t) + \frac{qB\mu}{2\pi kh}\Big[p_D(\Delta t_D) + S\Big]

Holds when: the problem is linear: a single-phase slightly compressible fluid, and the properties of the formation do not depend on the pressure the rates of the neighbours do not change during the build-up, and if they changed earlier, their history is in the trend we leave out wellbore storage (the well acting as a small tank): the well is shut in at the bottom, or we do not use the early points

Testing in a developed reservoir: Slider's method →

The pressure decline rate of a block in pseudo-steady-state flow

m^* = -\frac{\dd p}{\dd t} = \frac{\sum_j q_jB}{c_tV_p}

Holds when: the block is closed, the fluid is single-phase, and the rates have been constant so long that the transient effects in the whole block have died out there is no water influx (water flowing in from outside) and no gas cap (free gas above the oil)

Testing in a developed reservoir: Slider's method →

Slider's correction: the reconstructed drawdown test

\Delta p_{DD}(\Delta t) = p_{ws}(\Delta t) - p_{wf}(\Delta t = 0) + m^*\Delta t

any consistent units: pressure in MPa, $m^*$ in MPa/h and $\Delta t$ in hours

Holds when: the trend is linear: the block is in pseudo-steady-state flow, and the rates of the neighbours did not change either before the build-up or during it $m^*$ is found from the record before shut-in or from the material balance of the block (the section below)

Testing in a developed reservoir: Slider's method →

The correction for the trend as a fraction of the plateau level

\frac{m^*\Delta t}{m_e} = 4\pi\,\Delta t_{DA_w},\qquad \Delta t_{DA_w} = \frac{k\,\Delta t}{\phi\mu c_tA_w},\qquad A_w = A\,\frac{q}{\sum_j q_j}

dimensionless; in metric oilfield units $\Delta t_{DA_w} = 3.553\cdot10^{-3}\,k\,\Delta t/(\phi\mu c_tA_w)$ for $k$ in mD, $\Delta t$ in hours, $\mu$ in mPa·s, $c_t$ in 1/MPa, $A_w$ in m²

Holds when: the block is in pseudo-steady-state flow, and all the wells work at constant rates in one homogeneous formation.

Testing in a developed reservoir: Slider's method →

The derivative of a raw build-up in a developed accumulation

\Delta p_{BU}' = \Delta p_{DD}' - m^*\Delta t

any consistent units

Holds when: a linear trend.

Testing in a developed reservoir: Slider's method →

Symbols

Symbol	Meaning and context
$\bar p$	average pressure of the area at the moment of shut-in, PaBuild-up in a bounded reservoir and average reservoir pressure
$\Delta t_{\bar p}$	time on the continuation of the MDH line at which the line passes through $\bar p$ , sBuild-up in a bounded reservoir and average reservoir pressure
$\Delta p$	pressure rise after shut-in: $p_{ws}(\Delta t) - p_{wf}(\Delta t = 0)$ , MPaEquivalent producing time and synthetic flowing time
$\Delta t$	time after shut-in, sTesting in a developed reservoir: Slider's method
$\Delta p'$	ordinary logarithmic derivative of the build-up against $\ln\Delta t$ , MPaEquivalent producing time and synthetic flowing time
$\Delta p'_X$	derivative of the pressure rise against the superposition time function, reduced to the natural logarithm, MPaEquivalent producing time and synthetic flowing time
$\tfrac12(\ln t_{pD} + 0.809)$	the same for an infinite formation (Line-source solution and the pressure function), —Build-up in a bounded reservoir and average reservoir pressure
$a$	a constant (it depends on the shape of the area and the production history), PaBuild-up in a bounded reservoir and average reservoir pressure
$A$	area of the drainage region, m²Build-up in a bounded reservoir and average reservoir pressure
$A_w$	drainage area of the well under test: the share of the area of the block that is proportional to its rate, m²Testing in a developed reservoir: Slider's method
$A,\ k,\ \phi,\ \mu,\ c_t$	area of the region, permeability, porosity, viscosity, total compressibility, m², m², —, Pa·s, 1/PaEquivalent producing time and synthetic flowing time
$A,\ q,\ \sum_j q_j$	area of the block, rate of the well under test, and the sum of the rates of all the wells of the block, m², m³/s, m³/sTesting in a developed reservoir: Slider's method
$A,\ C_A$	area of the region and shape factor, m², —Build-up in a bounded reservoir and average reservoir pressure
$A,\ r_w,\ S$	area of the region, well radius, skin, m², m, —Reservoir limit test
$b$	drawdown per unit rate: $0.933\,\frac{B\mu}{kh}\left(\ln\frac{4A}{\gamma C_Ar_w^2} + 2S\right)$ for $k$ in mD, $\mu$ in mPa·s, $h$ in m; $0.933 = 1/(4\pi\cdot0.0853)$ is the constant of the slope per unit of natural logarithm (where the constants come from), MPa/(m³/d)Reservoir limit test
$B$	oil formation volume factor, m³/m³Testing in a developed reservoir: Slider's method
$B,\ c_t$	oil formation volume factor and total compressibility, m³/m³, 1/MPaBuild-up in a bounded reservoir and average reservoir pressure
$B,\ c_t,\ V_p$	formation volume factor, total compressibility, pore volume, m³/m³, 1/MPa, m³Reservoir limit test
$\Delta p_{BU}'$	logarithmic derivative of the pressure rise after shut-in, $\dd\Delta p_{BU}/\dd\ln\Delta t$ , PaTesting in a developed reservoir: Slider's method
$C_A$	shape factor of the area with the given position of the well, —Reservoir limit test
$c_t$	total compressibility of the rocks and fluids, 1/PaReservoir limit test
$c_t,\ V_p$	total compressibility and the pore volume of the whole connected block, 1/Pa, m³Testing in a developed reservoir: Slider's method
$\Delta t_{DA}$	shut-in time referred to the area: $k\Delta t/(\phi\mu c_tA)$ , —Build-up in a bounded reservoir and average reservoir pressure
$\Delta t_{DA_w}$	time after shut-in referred to the area of the drainage region of the well, —Testing in a developed reservoir: Slider's method
$\Delta p_{DD}'$	logarithmic derivative of the reconstructed drawdown test, PaTesting in a developed reservoir: Slider's method
$\Delta p_{DD}(\Delta t)$	reconstructed pressure drop: the drawdown test of the well under test, counted from the moment of shut-in, PaTesting in a developed reservoir: Slider's method
$k,\ h$	permeability and net thickness, m², mBuild-up in a bounded reservoir and average reservoir pressure
$k,\ \phi,\ \mu,\ c_t$	permeability, porosity, viscosity, total compressibility, m², —, Pa·s, 1/PaBuild-up in a bounded reservoir and average reservoir pressure
$k,\ \phi,\ \mu,\ c_t,\ h$	permeability, porosity, viscosity, total compressibility, thickness, mD, —, mPa·s, 1/MPa, mReservoir limit test
$L$	length of the area (the long side), mReservoir limit test length of the rectangular area (the long side), mBuild-up in a bounded reservoir and average reservoir pressure
$L,\ x_w,\ \eta$	as in the previous formula, m, m, m²/hReservoir limit test
$m$	slope of the Horner line per decimal cycle (positive), PaBuild-up in a bounded reservoir and average reservoir pressure slope of radial flow, MPaBuild-up in a bounded reservoir and average reservoir pressure slope of the line per unit of $X_{BU}$ : $2.149\,q_nB\mu/(kh)$ , MPaEquivalent producing time and synthetic flowing time
$m^*$	pressure decline rate on the line (positive), Pa/sReservoir limit test limiting rate of the pressure decline, $qB/(c_tV_p)$ , Pa/sReservoir limit test pressure decline rate in the block (the slope of the trend, taken with the opposite sign), Pa/sTesting in a developed reservoir: Slider's method pressure decline rate before shut-in (the trend); positive if the pressure was falling, Pa/sTesting in a developed reservoir: Slider's method
$m^*\Delta t$	derivative of the correction for the trend against $\ln\Delta t$ , PaTesting in a developed reservoir: Slider's method
$m^*,\ p_{int}$	slope and intercept of the line in ordinary coordinates, MPa/h, MPaReservoir limit test
$m_e$	slope of the semilog line per unit of natural logarithm, $m/2.303$ , MPaReservoir limit test slope of radial flow per unit of natural logarithm, $qB\mu/(4\pi kh)$ , PaEquivalent producing time and synthetic flowing time level of the derivative plateau of the well under test, $qB\mu/(4\pi kh)$ , PaTesting in a developed reservoir: Slider's method
$N_p$	cumulative oil production at standard conditions at the moment of shut-in, m³Build-up in a bounded reservoir and average reservoir pressure cumulative production by the moment of shut-in, at standard conditions, m³Equivalent producing time and synthetic flowing time
$p^*$	pressure on the continuation of the Horner line, drawn with the actual producing time $t_p$ , at a Horner ratio of 1, PaBuild-up in a bounded reservoir and average reservoir pressure the continuation of the line to $X_{BU} = 0$ , that is, to an infinitely long shut-in; in a new formation it equals the initial pressure, MPaEquivalent producing time and synthetic flowing time
$p^*,\ m$	continuation of the Horner line at the ratio 1, and its slope per decimal cycle, MPaBuild-up in a bounded reservoir and average reservoir pressure
$p_1$	pressure on the semilog line (or its continuation) at time $t_1$ , MPaReservoir limit test
$p_D(\Delta t_D)$	dimensionless pressure drop at the wall of a well that was started at a constant rate in the same formation, with its boundaries but without the neighbours; without skin, —Testing in a developed reservoir: Slider's method
$p_D(t_{pD})$	pressure function of a well in a closed area at a constant rate (without skin), at the moment of shut-in, —Build-up in a bounded reservoir and average reservoir pressure
$p_D(t)$	dimensionless pressure drop at the wall of a well that produces at a constant rate for a time $t$ in a closed area, without skin (the skin enters both terms in the same way and cancels), —Build-up in a bounded reservoir and average reservoir pressure
$p_e - \bar p$	excess of the pressure at the chosen point of the boundary over the average pressure in pseudo-steady-state flow, PaEquivalent producing time and synthetic flowing time
$p_i$	initial reservoir pressure, PaBuild-up in a bounded reservoir and average reservoir pressure
$p_{int}$	intercept of the line on the pressure axis, that is, its continuation to $t = 0$ , PaReservoir limit test continuation of the line in ordinary coordinates to $t = 0$ , PaReservoir limit test
$p_{tr}(\Delta t)$	the trend: the bottomhole pressure that the well would have had at the same moment if it had kept producing at the rate $q$ ; at the moment of shut-in it equals $p_{wf}(\Delta t = 0)$ , PaTesting in a developed reservoir: Slider's method
$p_{wf}(\Delta t = 0)$	bottomhole pressure at the moment of shut-in, PaTesting in a developed reservoir: Slider's method
$p_{ws}(\Delta t)$	bottomhole pressure a time $\Delta t$ after shut-in, PaBuild-up in a bounded reservoir and average reservoir pressure pressure in the shut-in well a time $\Delta t$ after shut-in, PaTesting in a developed reservoir: Slider's method the measured pressure after shut-in, PaTesting in a developed reservoir: Slider's method
$q_j$	rate of flow period number $j$ at standard conditions; $q_0 = 0$ , and $q_n$ is the last rate before shut-in, m³/dEquivalent producing time and synthetic flowing time rates of all the wells of the block at standard conditions; if the block has injection wells, their injection, converted to reservoir conditions, is subtracted from the sum $\sum q_jB$ , m³/sTesting in a developed reservoir: Slider's method
$q_n$	the last steady rate before shut-in, m³/dEquivalent producing time and synthetic flowing time
$q,\ B$	rate at standard conditions and formation volume factor; $qB$ is the rate at reservoir conditions, m³/s, m³/m³Reservoir limit test
$q,\ B,\ \mu$	rate at standard conditions, formation volume factor, viscosity, m³/s, m³/m³, Pa·sBuild-up in a bounded reservoir and average reservoir pressure
$q,\ B,\ \mu,\ k,\ h$	as in the inflow formula, m³/s, m³/m³, Pa·s, m², mReservoir limit test rate before shut-in, formation volume factor, viscosity, permeability, thickness, m³/s, m³/m³, Pa·s, m², mBuild-up in a bounded reservoir and average reservoir pressure rate before shut-in at standard conditions, formation volume factor, viscosity, permeability, thickness, m³/s, m³/m³, Pa·s, m², mTesting in a developed reservoir: Slider's method
$q(t)$	rate at the moment of the reading, at standard conditions, m³/dReservoir limit test
$Q(t)$	cumulative production up to the moment of the reading, m³Reservoir limit test
$r_i$	radius of investigation at the end of the record, mReservoir limit test
$R(\Delta t)$	ratio of the local slope of the Horner plot with the time $t_p$ to the correct slope $m$ , —Equivalent producing time and synthetic flowing time
$S$	skin of the well under test, —Testing in a developed reservoir: Slider's method
$t$	duration of production at a constant rate, hReservoir limit test
$t_{mb}$	material balance time, dReservoir limit test
$t_p$	producing time at the rate $q$ before shut-in, sBuild-up in a bounded reservoir and average reservoir pressure equivalent producing time that we put into the Horner function, dEquivalent producing time and synthetic flowing time producing time put into the Horner function, hEquivalent producing time and synthetic flowing time
$t_p'$	producing time put into the Horner function (not necessarily the actual one), hBuild-up in a bounded reservoir and average reservoir pressure
$t_{pD}$	producing time in dimensionless form, $kt_p/(\phi\mu c_tr_w^2)$ , —Build-up in a bounded reservoir and average reservoir pressure
$t_{pDA}$	producing time referred to the area: $kt_p/(\phi\mu c_tA)$ , —Build-up in a bounded reservoir and average reservoir pressure producing time referred to the area, —Build-up in a bounded reservoir and average reservoir pressure
$t_{pss}$	time from which the slope in ordinary coordinates differs from the limit by less than 1%, hReservoir limit test
$t_s$	synthetic flowing time, the time that we put into the Horner function, sEquivalent producing time and synthetic flowing time
$t_{sDA}$	the same, referred to the area of the region, —Equivalent producing time and synthetic flowing time
$V_p$	pore volume of the area connected to the well, m³Build-up in a bounded reservoir and average reservoir pressure
$V_p = \phi hA$	pore volume connected to the well, m³Reservoir limit test pore volume of the area, m³Build-up in a bounded reservoir and average reservoir pressure
$X_{BU}$	superposition time function for a build-up: the sum of the decimal logarithms of the Horner ratios for each rate change, with weights $w_j = (q_j - q_{j-1})/q_n$ , —Equivalent producing time and synthetic flowing time
$x_w$	distance from the well to the nearer short side, mReservoir limit test
$\gamma$	1.781, the exponential of Euler's constant, —Reservoir limit test
$\eta = k/(\phi\mu c_t)$	hydraulic diffusivity of the formation (how fast a pressure change spreads through the rock), m²/sReservoir limit test hydraulic diffusivity of the formation, m²/sBuild-up in a bounded reservoir and average reservoir pressure
$\theta_j$	time from the start of flow period number $j$ to shut-in: $t_n - t_{j-1}$ , where $t_n$ is the moment of shut-in, counted from the first start-up, hEquivalent producing time and synthetic flowing time
$\theta_j,\ q_j$	age of rate change number $j$ at the moment of shut-in, and the rates of the flow periods, h, m³/dEquivalent producing time and synthetic flowing time
$\lambda$	decay exponent, 1/sBuild-up in a bounded reservoir and average reservoir pressure