Equations: Aquifer influx

The equations of the “Aquifer influx” lessons in “Reservoir engineering”, with their conditions and units, and what each symbol means in them.

Elastic store of an aquifer

W_{e,\max} = (c_w + c_f)\,W\,\Delta p,\qquad W = \pi\,(r_a^2 - r_o^2)\,h\,\phi\,f

consistent units: m³, MPa and 1/MPa

Holds when: the aquifer is full of water, and the compressibilities are constant the geometry is radial, a sector of a ring with the angle $\theta$ ; for a strip-shaped aquifer of width $w$ and length $L$ the volume is $wLh\phi$ the pressure drop has had time to reach all parts of the aquifer

Aquifer models: water influx →

Influx from a small aquifer

W_e = (c_w + c_f)\,W\,(p_i - p)

m³, MPa, 1/MPa (or barrels, psi and 1/psi)

Holds when: the pressure throughout the aquifer equals the pressure at the contact: the aquifer equalises faster than the pressure in the accumulation changes there is no recharge from outside the pressure at the contact is taken as equal to the average pressure of the accumulation

Aquifer models: water influx →

Dimensionless time of the aquifer and the time to reach its boundary

t_D = \frac{\eta\,t}{r_o^2} = \frac{k\,t}{\phi\,\mu_w\,c_t\,r_o^2},\qquad t_a \approx \frac{(r_a - r_o)^2}{4\eta}

Holds when: a uniform aquifer of constant thickness, with radial geometry $t_a$ is an order-of-magnitude estimate: a noticeable pressure drop at the boundary starts earlier, and the equalisation of the whole aquifer comes several times later

Aquifer models: water influx →

Steady-state model of water influx

\frac{\dd W_e}{\dd t} = J_a\,(p_i - p),\qquad W_e = J_a\int_0^t (p_i - p)\dd t

m³ at reservoir conditions, days, MPa; in oilfield units, barrels per day per psi

Holds when: the pressure at the outer boundary of the aquifer is constant and equal to the initial pressure the flow in the aquifer is steady: the transient period is short compared with the development history the pressure at the contact is taken as equal to the average pressure of the accumulation

Aquifer models: water influx →

Productivity index of a radial aquifer (steady flow)

J_a = \frac{2\pi\,f\,k\,h}{\mu_w\,\ln(r_a/r_o)}

Holds when: a radial aquifer of constant thickness and permeability the pressure at the radius $r_a$ is constant; the flow is steady and single-phase

Aquifer models: water influx →

Pseudo-steady-state aquifer model

\frac{\dd W_e}{\dd t} = J\,(\bar p_a - p),\qquad W_e = (c_w + c_f)\,W\,(p_i - \bar p_a)

m³ at reservoir conditions, days, MPa

Holds when: the aquifer is closed; after a transient period the pressure in it falls everywhere at the same rate the pressure at the contact is taken as equal to the average pressure of the accumulation

Aquifer models: water influx →

Influx for a constant pressure drop at the contact (unsteady-state model)

W_e = U\,\Delta p\,W_D(t_D),\qquad U = 2\pi\,f\,\phi\,c_t\,h\,r_o^2

m³, MPa, 1/MPa, m

Holds when: a uniform radial aquifer; the pressure at the contact fell by $\Delta p$ in one step and does not change afterwards for a changing pressure the history is split into steps and the responses are added (Superposition principle)

Aquifer models: water influx →

Response of a pseudo-steady-state aquifer to a constant pressure drop

\frac{\dd W_e}{\dd t} = J\left[\Delta p - \frac{W_e}{(c_w + c_f)W}\right]\;\Rightarrow\; W_e = (c_w + c_f)\,W\,\Delta p\left(1 - \e^{-t/\tau}\right),\quad \tau = \frac{(c_w + c_f)\,W}{J}

m³, days, MPa

Holds when: the pressure drop at the contact is constant; at $t = 0$ the aquifer is untouched: $W_e = 0$

Aquifer models: water influx →

The van Everdingen–Hurst problem in dimensionless form

\frac{1}{r_D}\frac{\partial}{\partial r_D}\!\left(r_D\,\frac{\partial P}{\partial r_D}\right) = \frac{\partial P}{\partial t_D},\qquad P(1, t_D) = 1,\quad P(r_D, 0) = 0,\quad \left.\frac{\partial P}{\partial r_D}\right|_{r_D = r_{aD}} = 0

dimensionless form

Holds when: a uniform aquifer of constant thickness, full of water; the compressibilities are small and constant, so the equation is linear the flow is radial, with no crossflow through the top and bottom the outer boundary is impermeable; for an infinite aquifer the condition on it is replaced by $P \to 0$ as $r_D \to \infty$

Unsteady-state water influx: the van Everdingen–Hurst method →

Influx for a constant pressure drop at the contact

W_e = U\,\Delta p\,W_D(t_D,\,r_{aD}),\qquad U = 2\pi f\phi c_t h r_o^2,\qquad t_D = \frac{k\,t}{\phi\,\mu_w c_t\,r_o^2}

$W_e$ and $U$ in one system of volume and pressure (m³ and MPa); $t_D$ in SI

Holds when: the problem above: a uniform radial aquifer, with the contact pressure falling in one step and not changing afterwards the pressure at the contact is taken as equal to the average pressure of the accumulation, as in the material balance the aquifer is full of water, $c_t = c_w + c_f$ , and the compressibilities are constant

Unsteady-state water influx: the van Everdingen–Hurst method →

Influx as the expansion of the pressure-drop funnel

W_e = c_t\int_{V_p}(p_i - p)\dd V_p = U\,\Delta p\int_1^{r_{aD}} P\,r_D\dd r_D,\qquad W_D = \int_1^{r_{aD}} P\,r_D\dd r_D

m³, MPa, 1/MPa

Holds when: there is only water in the aquifer; $c_t$ is constant; there is no recharge from outside

Unsteady-state water influx: the van Everdingen–Hurst method →

Influx for a stepwise pressure drop at the contact

W_e(t_n) = U\sum_{j=0}^{n-1}\Delta p_j\,W_D\big(t_{Dn} - t_{Dj}\big)

m³, MPa

Holds when: the conditions of linearity: the properties of the water and the aquifer do not depend on pressure the contact pressure is constant between the moments of the steps; how well the steps replace a smooth history depends on the choice of the drops and of the step (the next section) all the steps are counted from one initial state: before development the pressure everywhere in the aquifer equals $p_i$

Unsteady-state water influx: the van Everdingen–Hurst method →

Step drops from the average pressures of the intervals

\Delta p_0 = \frac{p_0 - p_1}{2},\qquad \Delta p_j = \frac{p_{j-1} - p_{j+1}}{2}\quad (j \ge 1),\qquad \sum_{j=0}^{n-1}\Delta p_j = p_0 - \frac{p_{n-1} + p_n}{2}

MPa

Holds when: if the intervals between measurements differ, the formula is the same, since the stepped curve still follows the average pressures of the intervals, but the longest intervals bring the largest error the pressure in an interval changes smoothly, without sharp jumps (the start-up or shutdown of many wells)

Unsteady-state water influx: the van Everdingen–Hurst method →

Rate of influx and dimensionless rate

q_w = \frac{2\pi f k h\,r_o}{\mu_w}\left.\frac{\partial p}{\partial r}\right|_{r_o} = \frac{2\pi f k h\,\Delta p}{\mu_w}\,q_D(t_D),\qquad q_D = -\left.\frac{\partial P}{\partial r_D}\right|_{r_D = 1}

Holds when: a constant pressure drop $\Delta p$ at the contact from the moment $t = 0$

Unsteady-state water influx: the van Everdingen–Hurst method →

From the rate of influx to the cumulative influx

W_e = \int_0^t q_w\dd t = \frac{2\pi fkh\,\Delta p}{\mu_w}\cdot\frac{\phi\mu_wc_tr_o^2}{k}\int_0^{t_D}q_D\dd t_D = 2\pi f\phi c_thr_o^2\,\Delta p\,W_D

Holds when: the same as for the problem in dimensionless form

Unsteady-state water influx: the van Everdingen–Hurst method →

Laplace transform of the dimensionless influx

\bar W_D(s) = \frac{K_1(\sqrt s)\,I_1(r_{aD}\sqrt s) - I_1(\sqrt s)\,K_1(r_{aD}\sqrt s)}{s^{3/2}\big[K_0(\sqrt s)\,I_1(r_{aD}\sqrt s) + I_0(\sqrt s)\,K_1(r_{aD}\sqrt s)\big]}\;\xrightarrow{\;r_{aD}\to\infty\;}\;\frac{K_1(\sqrt s)}{s^{3/2}\,K_0(\sqrt s)}

dimensionless

Holds when: a closed outer boundary at $r_{aD}$ ; on the right is the infinite aquifer

Unsteady-state water influx: the van Everdingen–Hurst method →

Influx from a linear aquifer

W_e = U_L\,\Delta p\,W_D(t_D),\qquad U_L = w\,h\,L\,\phi\,c_t,\qquad t_D = \frac{k\,t}{\phi\,\mu_wc_t\,L^2};\qquad W_D = 2\sqrt{t_D/\pi}\ \ \text{(infinite strip)}

m³, MPa, 1/MPa, m; $t_D$ in SI (the metric form has the multiplier $3.553\cdot10^{-3}$ for time in hours)

Holds when: a uniform strip of constant cross-section; the flow is perpendicular to the contact for an infinite strip $L$ cancels: $W_e = 2wh\phi c_t\Delta p\sqrt{\eta t/\pi}$

Unsteady-state water influx: the van Everdingen–Hurst method →

Influx of the unsteady-state aquifer model

W_e(t_n) = U\,w_n,\qquad w_n = \sum_{j=0}^{n-1}\Delta p_j\,W_D\big(t_{Dn} - t_{Dj};\ r_{aD}\big),\qquad t_D = \frac{k\,t}{\phi\,\mu_wc_t\,r_o^2}

m³, MPa; $t_D$ in SI, or with the multiplier $3.553\cdot10^{-3}$ for $k$ in mD, $t$ in hours, $\mu_w$ in mPa·s and $c_t$ in 1/MPa

Holds when: a uniform radial aquifer; the pressure at the contact equals the average pressure of the accumulation the function $w$ is calculated with short steps, or exactly; otherwise the error of the stepped representation of the pressure passes into the fitted parameters

History matching of aquifer parameters →

Apparent aquifer constant

U_n = \frac{W_{e,n}}{w_n} = \frac{F_n - N E_{t,n}}{w_n}

m³/MPa

Holds when: the oil in place is known independently; there is no gas cap, or its size is known ( $E_t$ includes $mE_g$ ) the early dates, where the pressure drop is about the same size as the error of its measurement, get a smaller weight

History matching of aquifer parameters →

Havlena–Odeh line with an unsteady-state influx function

\frac{F}{E_t} = N + U\,\frac{w}{E_t},\qquad E_t = E_o + mE_g + E_{fw}

m³, MPa, m³/MPa

Holds when: there is no gas cap, or its size $m$ is known the early points, where $E_t$ is small, are unreliable and must not decide the slope

History matching of aquifer parameters →

Measure of agreement with the pressure history

M = \frac1n\sum_{k=1}^{n}\left(\frac{p_k^{\text{meas}} - p_k^{\text{calc}}}{\sigma}\right)^{2},\qquad M \le 1\ \Longleftrightarrow\ \text{RMS} = \sqrt{\frac1n\sum_{k}\big(p_k^{\text{meas}} - p_k^{\text{calc}}\big)^2} \le \sigma

$M$ is dimensionless; pressures in MPa

Holds when: the measurement errors are independent and equal in size each candidate aquifer is fitted so that $M$ is the smallest for its $r_{aD}$ and permeability; the constant $U$ is then fitted automatically

History matching of aquifer parameters →

The pseudo-steady-state aquifer model in the notation of the Fetkovich method

\frac{\dd W_e}{\dd t} = J\,\big(\bar p_a - p\big),\qquad W_e = c_tW\,\big(p_i - \bar p_a\big) = W_{ei}\left(1 - \frac{\bar p_a}{p_i}\right),\qquad W_{ei} = c_tW\,p_i

m³, days, MPa

Holds when: The aquifer is closed. After the transient period it drains so that the pressure falls at the same rate everywhere in it (the pseudo-steady-state regime). Compressibility and permeability are constant. The pressure at the contact equals the average pressure of the accumulation.

Simplified water influx models: Fetkovich and Carter–Tracy →

The Fetkovich step

\Delta W_{e,n} = c_tW\,\big(\bar p_{a,n-1} - \bar p_n\big)\left(1 - \e^{-\Delta t_n/\tau}\right),\qquad \bar p_{a,n} = p_i - \frac{W_{e,n-1} + \Delta W_{e,n}}{c_tW},\qquad \bar p_n = \frac{p_{n-1} + p_n}{2}

m³, days, MPa

Holds when: Within the step, the pressure at the contact is replaced by its average value. The pseudo-steady-state aquifer model (the equations above).

Simplified water influx models: Fetkovich and Carter–Tracy →

Productivity index of the aquifer for the Fetkovich method

J_{\text{rad}} = \frac{2\pi f\,k\,h}{\mu_w\left(\ln r_{aD} - \dfrac34\right)},\qquad J_{\text{lin}} = \frac{3\,k\,w\,h}{\mu_w\,L}

Holds when: Pseudo-steady-state regime: the pressure difference that drives the flow is the average aquifer pressure minus the pressure at the contact. The formula for a radial aquifer is correct for $r_{aD} \gg 1$ . For small $r_{aD}$ use the exact expression for a ring (see the advanced section).

Simplified water influx models: Fetkovich and Carter–Tracy →

Influence function of a radial aquifer

p_D(t_D) = \frac{2\pi f k h\,\big(p_i - p_o(t)\big)}{q_w\,\mu_w};\qquad p_D \approx 2\sqrt{t_D/\pi}\ \ (t_D \ll 1),\qquad p_D \approx \tfrac12\big(\ln t_D + 0.809\big)\ \ \text{(infinite)},\qquad p_D \approx \frac{2t_D}{r_{aD}^2 - 1} + \ln r_{aD} - \frac34\ \ \text{(closed, late)}

SI; a dimensionless function

Holds when: A uniform radial aquifer. The influx rate is constant from the moment $t = 0$ . There is no skin at the contact. The logarithmic form holds for $t_D$ above about 25, while the aquifer still behaves as an infinite one. The last form holds after the pressure drop has reached the closed boundary, for $r_{aD} \gg 1$ (for small $r_{aD}$ , use the exact expression for a ring instead of $\ln r_{aD} - 3/4$ ).

Simplified water influx models: Fetkovich and Carter–Tracy →

The Carter–Tracy step

W_{e,n} = W_{e,n-1} + \big(t_{Dn} - t_{D,n-1}\big)\,\frac{U\,\Delta p_n - W_{e,n-1}\,p_D'(t_{Dn})}{p_D(t_{Dn}) - t_{D,n-1}\,p_D'(t_{Dn})},\qquad \Delta p_n = p_i - p_n

m³, MPa; time is dimensionless

Holds when: The influx rate is constant within the step. The whole history before the step is replaced by an equivalent one: an instant influx at the initial moment plus the constant rate of the current step from the very beginning (see the advanced section). The influence function matches the aquifer: a closed one with its own $r_{aD}$ , or an infinite one.

Simplified water influx models: Fetkovich and Carter–Tracy →

Pressure at the contact as a convolution of the influx history

U\,\big(p_i - p(t_D)\big) = \int_0^{t_D}\frac{\dd W_e}{\dd\lambda}\;p_D'(t_D - \lambda)\dd\lambda

m³ (both sides)

Holds when: The problem is linear. Before the start of development the pressure in the aquifer equals $p_i$ everywhere. An instant influx of volume $V$ at the moment $\lambda$ gives the response $V\,p_D'(t_D - \lambda)$ , the derivative of the response to a constant rate.

Simplified water influx models: Fetkovich and Carter–Tracy →

Pressure decline from the equivalent influx history

U\,\Delta p_n = a_n\,p_D(t_{Dn}) + \big(W_{e,n-1} - a_n\,t_{D,n-1}\big)\,p_D'(t_{Dn})

m³ (both sides)

Holds when: The equivalent history instead of the real one; the influence function is for the given aquifer.

Simplified water influx models: Fetkovich and Carter–Tracy →

The limit of the Carter–Tracy formula for very short steps

\big(p_D - t_D\,p_D'\big)\,\frac{\dd W_e}{\dd t_D} + p_D'\,W_e = U\,\big(p_i - p(t_D)\big)

m³ (both sides); time is dimensionless

Holds when: The steps tend to zero. The aquifer and the influence function are the same as in the formula of the method.

Simplified water influx models: Fetkovich and Carter–Tracy →

Dimensionless resistance of a ring-shaped aquifer in the pseudo-steady-state regime

b(r_{aD}) = \frac{4\,r_{aD}^4\ln r_{aD} - 3\,r_{aD}^4 + 4\,r_{aD}^2 - 1}{4\,\big(r_{aD}^2 - 1\big)^2}\;\xrightarrow{\;r_{aD}\gg1\;}\;\ln r_{aD} - \frac34,\qquad J = \frac{2\pi f k h}{\mu_w\,b(r_{aD})}

dimensionless; $J$ in SI (in metric oilfield units $2\pi \to 0.536$ )

Holds when: A uniform ring. The outer boundary is impermeable. The flow has settled into the pseudo-steady-state regime.

Simplified water influx models: Fetkovich and Carter–Tracy →

The Fetkovich step when the pressure at the contact changes linearly

\Delta W_{e,n} = c_tW\left[\big(\bar p_{a,n-1} - p_{n-1}\big)\big(1 - \e^{-\Delta t_n/\tau}\big) + \big(p_{n-1} - p_n\big)\left(1 - \frac{\tau}{\Delta t_n}\big(1 - \e^{-\Delta t_n/\tau}\big)\right)\right]

m³, MPa, days

Holds when: The pseudo-steady-state aquifer model. The pressure at the contact changes along a straight line inside the step.

Simplified water influx models: Fetkovich and Carter–Tracy →

Symbols

Symbol	Meaning and context
$\Delta p$	fall in pressure from the initial value, $p_i - p$ , MPaAquifer models: water influx pressure drop at the contact, constant from the moment $t = 0$ , MPaAquifer models: water influx
$\bar p_a$	average pressure in the aquifer, MPaAquifer models: water influx
$a_n$	constant influx rate of the current step, m³ per unit of $t_D$ Simplified water influx models: Fetkovich and Carter–Tracy
$\bar p_{a,n-1}$	average aquifer pressure at the start of the step, MPaSimplified water influx models: Fetkovich and Carter–Tracy
$\bar p_{a,n-1},\ \bar p_{a,n}$	average aquifer pressure at the start and at the end of the step; $\bar p_{a,0} = p_i$ , MPaSimplified water influx models: Fetkovich and Carter–Tracy
$b$	dimensionless difference between the average aquifer pressure and the pressure at the contact, in units of $q_w\mu_w/(2\pi fkh)$ , —Simplified water influx models: Fetkovich and Carter–Tracy
$c_t$	total compressibility of the aquifer, $c_w + c_f$ , 1/PaAquifer models: water influx
$c_tW$	storage capacity of the aquifer (the water it gives up per megapascal of pressure drop): the compressibility $c_w + c_f$ times its pore volume $W$ , m³/MPaSimplified water influx models: Fetkovich and Carter–Tracy
$c_w,\ c_f$	compressibility of the water and of the pore volume of the aquifer (the fraction by which a volume changes for a unit change in pressure), 1/MPaAquifer models: water influx
$c_w + c_f$	compressibility of the water and the pores of the aquifer, 1/MPaAquifer models: water influx
$\bar W_D(s)$	Laplace transform of $W_D(t_D)$ , —Unsteady-state water influx: the van Everdingen–Hurst method
$\Delta W_{e,n}$	influx during step $n$ , m³Simplified water influx models: Fetkovich and Carter–Tracy
$\dd W_e/\dd t$	rate of water influx at reservoir conditions, m³/dAquifer models: water influx
$f$	fraction of the perimeter of the accumulation that the aquifer touches, $\theta/360°$ , —Aquifer models: water influx fraction of the perimeter of the accumulation that the aquifer touches, —Aquifer models: water influx
$F_n,\ E_{t,n}$	underground withdrawal (the reservoir volume of the fluids produced) and expansion per 1 m³ of oil in place on the same date, m³, m³/m³History matching of aquifer parameters
$f,\ k,\ h,\ \mu_w$	fraction of the perimeter, permeability, thickness of the aquifer, viscosity of water, —, m², m, Pa·sSimplified water influx models: Fetkovich and Carter–Tracy
$f,\ \phi,\ c_t,\ h,\ r_o$	fraction of the perimeter, porosity, compressibility $c_w + c_f$ and thickness of the aquifer, radius of the accumulation, —, —, 1/MPa, m, mAquifer models: water influx
$F/E_t$	ordinate: the apparent oil in place, m³History matching of aquifer parameters
$h,\ \phi$	thickness and porosity of the aquifer, m, —Aquifer models: water influx
$I_\nu,\ K_\nu$	modified Bessel functions of the first and second kind of order $\nu$ , —Unsteady-state water influx: the van Everdingen–Hurst method
$\Delta p_j$	drop of the $j$ -th step, applied at time $t_j$ ; the sum of the drops is the pressure drop from the initial value, MPaUnsteady-state water influx: the van Everdingen–Hurst method drop of the step applied at time $t_j$ , MPaUnsteady-state water influx: the van Everdingen–Hurst method pressure steps at the contact from the average pressures of the intervals (or the exact integral for a pressure that changes along a straight line between the measurements), MPaHistory matching of aquifer parameters
$J$	productivity index of the aquifer for pseudo-steady-state flow, m³/(d·MPa)Aquifer models: water influx productivity index of the aquifer in pseudo-steady-state flow (the inflow for each unit of pressure difference), m³/(d·MPa)Simplified water influx models: Fetkovich and Carter–Tracy
$J_a$	aquifer productivity index (the influx constant), m³/(d·MPa)Aquifer models: water influx productivity index of the aquifer, m³/(s·Pa)Aquifer models: water influx
$J_{\text{lin}}$	linear aquifer of width $w$ along the contact and length $L$ , closed at the far end, m³/(s·Pa)Simplified water influx models: Fetkovich and Carter–Tracy
$J_{\text{rad}}$	radial aquifer on a fraction $f$ of the perimeter, closed at the radius $r_a$ ; $r_{aD} = r_a/r_o$ , m³/(s·Pa)Simplified water influx models: Fetkovich and Carter–Tracy
$k,\ h$	permeability and thickness of the aquifer, m², mAquifer models: water influx
$k,\ h,\ \mu_w$	permeability and thickness of the aquifer, viscosity of water, m², m, Pa·sSimplified water influx models: Fetkovich and Carter–Tracy
$k,\ h,\ \mu_w,\ f$	permeability and thickness of the aquifer, viscosity of the water, fraction of the perimeter, m², m, Pa·s, —Unsteady-state water influx: the van Everdingen–Hurst method
$k,\ \mu_w$	permeability of the aquifer and viscosity of the reservoir water, m², Pa·sUnsteady-state water influx: the van Everdingen–Hurst method
$k,\ \phi$	permeability and porosity of the aquifer, m², —Aquifer models: water influx
$M$	misfit function (a number that shows how far the calculation is from the measurements), —History matching of aquifer parameters
$\bar p_n$	average pressure at the contact during the step: the mean of the readings at its two ends, MPaSimplified water influx models: Fetkovich and Carter–Tracy
$\Delta t_n$	length of the step, dSimplified water influx models: Fetkovich and Carter–Tracy
$\Delta p_n$	total pressure decline at the contact from the initial pressure to the end of the step (not the drop over one step), MPaSimplified water influx models: Fetkovich and Carter–Tracy total pressure decline at the contact by the end of the step, MPaSimplified water influx models: Fetkovich and Carter–Tracy
$N$	oil in place: known, m³History matching of aquifer parameters intercept on the vertical axis: the oil in place, m³History matching of aquifer parameters
$p$	pressure at the contact (the average pressure of the accumulation), MPaSimplified water influx models: Fetkovich and Carter–Tracy
$P$	fraction of the pressure drop at the contact reached at a point of the aquifer, $(p_i - p)/\Delta p$ , —Unsteady-state water influx: the van Everdingen–Hurst method
$p_0 = p_i$	initial pressure at the contact, MPaUnsteady-state water influx: the van Everdingen–Hurst method
$p_D,\ p_D'$	influence function and its derivative with respect to $t_D$ at the end of the step, —Simplified water influx models: Fetkovich and Carter–Tracy influence function and its derivative at the moment $t_D$ , —Simplified water influx models: Fetkovich and Carter–Tracy
$p_D'$	derivative of the influence function, —Simplified water influx models: Fetkovich and Carter–Tracy
$p_i$	initial pressure (of the aquifer and the accumulation), MPaSimplified water influx models: Fetkovich and Carter–Tracy
$p_i - p$	pressure drop at a point of the aquifer, MPaUnsteady-state water influx: the van Everdingen–Hurst method
$p_i - p(t_D)$	pressure decline at the contact, MPaSimplified water influx models: Fetkovich and Carter–Tracy
$p_i,\ p$	initial and current pressure at the contact, MPaAquifer models: water influx initial pressure and pressure at the contact, MPaAquifer models: water influx
$p_j$	pressure at the contact, measured (or averaged) at time $t_j$ , MPaUnsteady-state water influx: the van Everdingen–Hurst method
$p_k^{\text{meas}},\ p_k^{\text{calc}}$	measured and calculated pressure at the contact on date $k$ , MPaHistory matching of aquifer parameters
$p_{n-1},\ p_n$	pressure at the contact at the start and at the end of the step, MPaSimplified water influx models: Fetkovich and Carter–Tracy
$p_o(t)$	pressure at the contact for a constant influx rate, PaSimplified water influx models: Fetkovich and Carter–Tracy
$P,\ r_D$	fraction of the pressure drop and the dimensionless distance, as in the problem above, —Unsteady-state water influx: the van Everdingen–Hurst method
$q_D$	dimensionless rate of influx, —Unsteady-state water influx: the van Everdingen–Hurst method
$q_w$	rate of water influx at reservoir conditions, m³/sUnsteady-state water influx: the van Everdingen–Hurst method constant water influx rate at reservoir conditions, m³/sSimplified water influx models: Fetkovich and Carter–Tracy
$r_a$	radius at which the pressure is constant (the recharge boundary), mAquifer models: water influx
$r_{aD}$	outer radius of the aquifer in radii of the accumulation, $r_a/r_o$ , —Unsteady-state water influx: the van Everdingen–Hurst method ratio of the outer radius of the aquifer to the radius of the accumulation, —Simplified water influx models: Fetkovich and Carter–Tracy
$r_D$	distance from the centre of the accumulation in radii of the accumulation, $r/r_o$ , —Unsteady-state water influx: the van Everdingen–Hurst method
$r_o$	radius of the accumulation, mAquifer models: water influx radius of the accumulation (the initial contact with the water), mUnsteady-state water influx: the van Everdingen–Hurst method
$r_o,\ r_a$	radius of the accumulation (the initial contact with the water) and the outer radius of the aquifer, mAquifer models: water influx radius of the accumulation and outer radius of the aquifer, mAquifer models: water influx
$\text{RMS}$	root-mean-square (RMS) deviation of the calculation from the measurements, MPaHistory matching of aquifer parameters
$s$	variable of the transform (conjugate to $t_D$ ), —Unsteady-state water influx: the van Everdingen–Hurst method
$t$	time since the pressure at the contact started to fall, sAquifer models: water influx time, dAquifer models: water influx time since the pressure drop, sUnsteady-state water influx: the van Everdingen–Hurst method
$t_a$	time in which the radius of investigation $\sqrt{4\eta t}$ (how far the pressure drop has spread) goes from the contact to the outer boundary, sAquifer models: water influx
$t_D$	dimensionless time of the aquifer, —Aquifer models: water influx dimensionless time of the aquifer, $kt/(\phi\mu_wc_tr_o^2)$ , —Unsteady-state water influx: the van Everdingen–Hurst method dimensionless time, referred to the length of the strip, —Unsteady-state water influx: the van Everdingen–Hurst method dimensionless time of the aquifer, as in the van Everdingen–Hurst method, —Simplified water influx models: Fetkovich and Carter–Tracy
$t_{Dn} - t_{Dj}$	dimensionless time that has passed since the $j$ -th step, —Unsteady-state water influx: the van Everdingen–Hurst method
$t_{Dn},\ t_{D,n-1}$	dimensionless time at the end of the current and the previous step, —Simplified water influx models: Fetkovich and Carter–Tracy
$U$	aquifer constant, m³/MPaAquifer models: water influx slope: the aquifer constant, m³/MPaHistory matching of aquifer parameters aquifer constant, $2\pi f\phi c_thr_o^2$ , m³/MPaSimplified water influx models: Fetkovich and Carter–Tracy
$U_L$	constant of the linear aquifer: its elastic store per 1 MPa, m³/MPaUnsteady-state water influx: the van Everdingen–Hurst method
$U_n$	apparent constant for date $n$ , m³/MPaHistory matching of aquifer parameters
$U,\ W_D$	aquifer constant and dimensionless influx, as above, m³/MPa, —Unsteady-state water influx: the van Everdingen–Hurst method
$V_p$	pore volume of the aquifer; $\dd V_p = 2\pi f h\phi\,r\dd r$ , m³Unsteady-state water influx: the van Everdingen–Hurst method
$W$	pore volume of the aquifer, which is also the initial volume of water in it (often written $W_i$ in the literature), m³Aquifer models: water influx pore volume of the aquifer, m³Aquifer models: water influx
$W_D$	dimensionless influx: a function of the dimensionless time $t_D$ and of the ratio $r_{aD} = r_a/r_o$ , —Aquifer models: water influx dimensionless influx: a function of $t_D$ and $r_{aD} = r_a/r_o$ , —Unsteady-state water influx: the van Everdingen–Hurst method
$W_D,\ r_{aD},\ t_D$	dimensionless influx, radius ratio and dimensionless time of the aquifer, —History matching of aquifer parameters
$W_D = \int_0^{t_D}q_D\dd t_D$	dimensionless influx, —Unsteady-state water influx: the van Everdingen–Hurst method
$W_e$	cumulative water influx at reservoir conditions, m³Aquifer models: water influx cumulative influx, m³Aquifer models: water influx cumulative water influx at reservoir conditions by time $t$ , m³Unsteady-state water influx: the van Everdingen–Hurst method cumulative influx by the method, m³Simplified water influx models: Fetkovich and Carter–Tracy
$W_{e,\max}$	largest influx for a pressure drop of $\Delta p$ throughout the aquifer, m³Aquifer models: water influx
$W_{e,n},\ W_{e,n-1}$	cumulative influx at the end of the current and the previous step, m³Simplified water influx models: Fetkovich and Carter–Tracy
$W_{e,n-1}$	cumulative influx at the start of the step, m³Simplified water influx models: Fetkovich and Carter–Tracy
$W_{e,n-1} - a_n\,t_{D,n-1}$	instant influx of the equivalent history at the initial moment, m³Simplified water influx models: Fetkovich and Carter–Tracy
$W_e(\lambda)$	cumulative influx up to the dimensionless moment $\lambda$ , m³Simplified water influx models: Fetkovich and Carter–Tracy
$W_e(t_n)$	cumulative influx by time $t_n$ , m³Unsteady-state water influx: the van Everdingen–Hurst method cumulative water influx by date $t_n$ , at reservoir conditions, m³History matching of aquifer parameters
$W_{ei}$	maximum encroachable water, m³Simplified water influx models: Fetkovich and Carter–Tracy
$w_n$	influx function by date $t_n$ , MPaHistory matching of aquifer parameters influx function of the chosen aquifer model, MPaHistory matching of aquifer parameters
$w,\ h,\ L$	width of the strip along the contact, thickness, and length of the aquifer from the contact to the far boundary, mUnsteady-state water influx: the van Everdingen–Hurst method
$W,\ c_w + c_f$	pore volume and compressibility of the aquifer, m³, 1/MPaAquifer models: water influx
$W,\ c_w + c_f,\ \Delta p$	pore volume and compressibility of the aquifer, and the pressure drop at the contact, m³, 1/MPa, MPaAquifer models: water influx
$w/E_t$	abscissa: the influx function divided by the expansion, MPaHistory matching of aquifer parameters
$\eta$	hydraulic diffusivity of the aquifer, $k/(\phi\mu_wc_t)$ , m²/sAquifer models: water influx
$\mu_w$	viscosity of the reservoir water, Pa·sAquifer models: water influx
$\sigma$	error in finding the average pressure (standard deviation), MPaHistory matching of aquifer parameters
$\tau$	time constant of the aquifer, dAquifer models: water influx time constant of the aquifer, $c_tW/J$ , dSimplified water influx models: Fetkovich and Carter–Tracy
$\tau,\ \Delta t_n$	time constant of the aquifer and length of the step, dSimplified water influx models: Fetkovich and Carter–Tracy
$\phi,\ c_t,\ h$	porosity, total compressibility $c_w + c_f$ and thickness of the aquifer, —, 1/MPa (1/Pa in $t_D$ ), mUnsteady-state water influx: the van Everdingen–Hurst method
$\phi,\ c_t,\ r_o$	porosity and total compressibility of the aquifer, radius of the accumulation, —, 1/Pa, mUnsteady-state water influx: the van Everdingen–Hurst method