Equations: Flow in porous media and transient inflow

The equations of the “Flow in porous media and transient inflow” lessons in “Reservoir engineering”, with their conditions and units, and what each symbol means in them.

Total compressibility of the rocks and fluids

c_t = S_o c_o + S_w c_w + S_g c_g + c_f

any consistent units, for example 1/MPa or 1/psi ( $1\ \text{MPa}^{-1} = 6.895\cdot10^{-3}\ \text{psi}^{-1}$ )

Holds when: The compressibilities are measured at reservoir temperature and change little over the pressure range considered. The saturations do not change noticeably over the time considered. For oil below the bubble point pressure, $c_o$ also includes the release of gas. The compressibility then rises sharply, and such cases are described with multiphase models.

Flow regimes →

Diffusivity equation for radial flow

\frac{1}{r}\frac{\partial}{\partial r}\!\left(r\,\frac{\partial p}{\partial r}\right) = \frac{1}{\eta}\,\frac{\partial p}{\partial t},\qquad \eta = \frac{k}{\phi\mu c_t}

Holds when: The reservoir is uniform and isotropic, of constant thickness, with horizontal radial flow. There is one mobile fluid with a small, constant compressibility ( $c_t\,\Delta p \ll 1$ ), and the viscosity does not depend on pressure. Darcy's law applies. Small terms with the square of the pressure gradient are neglected (the derivation is in the advanced section).

Flow regimes →

Time for the disturbance to reach the boundary

t_b \approx \frac{\phi\mu c_t\,r_e^2}{4k}

Holds when: The reservoir is uniform, the well is at the centre of a circular area, and the rate is constant after the start. For elongated areas and for wells shifted towards a boundary, the disturbance reaches the nearest boundary sooner, but the regime in the whole area sets in later (Pseudo-steady-state flow and the shape factor).

Flow regimes →

Rate of pressure decline in a closed area

\frac{\dd \bar p}{\dd t} = -\frac{qB}{c_t V_p}

consistent units: m³/d, 1/MPa and m³ give MPa/d

Holds when: The area is closed, with no inflow across the boundaries, and the rate is constant. The fluid is single-phase with a constant compressibility.

Flow regimes →

Diffusivity equation for linear flow

\frac{\partial^2 p}{\partial x^2} = \frac{\phi\mu c_t}{k}\,\frac{\partial p}{\partial t}

Holds when: The reservoir is uniform, and $k$ and $\mu$ do not depend on pressure. The compressibility is small and constant. This holds for liquids, but not for gas.

Flow regimes →

Dimensionless radius, time and pressure

r_D = \frac{r}{r_w},\qquad t_D = \frac{k\,t}{\phi\mu c_t r_w^2},\qquad p_D = \frac{2\pi kh\,(p_i - p)}{qB\mu}

SI; in metric oilfield units $t_D = 3.553\cdot10^{-3}\,kt/(\phi\mu c_t r_w^2)$ with $t$ in hours

Holds when: The conditions of the diffusivity equation; a constant rate.

Flow regimes →

Diffusivity equation in dimensionless form

\frac{1}{r_D}\frac{\partial}{\partial r_D}\!\left(r_D\,\frac{\partial p_D}{\partial r_D}\right) = \frac{\partial p_D}{\partial t_D}

dimensionless

Holds when: The same as for the dimensional equation. At the well, $r_D\,\partial p_D/\partial r_D = -1$ at a constant rate.

Flow regimes →

Darcy's law in vector form

\mathbf{u} = -\frac{k}{\mu}\left(\nabla p - \rho g\,\nabla D\right),\qquad \nabla p = \left(\frac{\partial p}{\partial x},\ \frac{\partial p}{\partial y},\ \frac{\partial p}{\partial z}\right)

Holds when: The flow of one fluid is slow (laminar). The irreducible water does not move, and $k$ is the permeability to the mobile fluid. The rock is isotropic: the permeability is the same in all directions (the general case is below). Inertial forces are neglected. For high-rate gas wells an inertial resistance is added to the viscous one (Inflow at high rates: non-Darcy flow).

Diffusivity equation →

Darcy's law with a permeability tensor

\mathbf{u} = -\frac{1}{\mu}\,\mathbf{K}\,\nabla p,\qquad \mathbf{K} = \begin{pmatrix} k_{xx} & k_{xy} & k_{xz}\\ k_{yx} & k_{yy} & k_{yz}\\ k_{zx} & k_{zy} & k_{zz}\end{pmatrix}

Holds when: Gravity is left out; it is added in the same way as in the isotropic case. The tensor is symmetric ( $k_{xy} = k_{yx}$ and so on), so by turning the axes it can always be brought to a diagonal form with three principal permeabilities.

Diffusivity equation →

Continuity equation and Darcy's law for radial flow

\frac{1}{r}\frac{\partial}{\partial r}\big(r\,\rho\,u\big) = \frac{\partial(\rho\phi)}{\partial t},\qquad u = \frac{k}{\mu}\,\frac{\partial p}{\partial r}

Holds when: The flow is radial and horizontal, in a reservoir of constant thickness, and the well penetrates the whole thickness. There is one mobile fluid, and Darcy's law applies. The velocity $u$ is positive when the fluid flows towards the well. Pressure grows with distance, so there is a plus sign before $\partial p/\partial r$ .

Diffusivity equation →

Diffusivity equation for radial flow: the linear form and its assumptions

\frac{\partial^2 p}{\partial r^2} + \frac{1}{r}\frac{\partial p}{\partial r} = \frac{1}{r}\frac{\partial}{\partial r}\!\left(r\,\frac{\partial p}{\partial r}\right) = \frac{1}{\eta}\,\frac{\partial p}{\partial t},\qquad \eta = \frac{k}{\phi\mu c_t}

Holds when: The reservoir is uniform and isotropic, of constant thickness. Porosity and permeability do not depend on pressure. The flow is horizontal and radial: the well penetrates the whole thickness, and gravity is not taken into account. There is one mobile fluid, and Darcy's law applies (flow without inertial losses). The flow is isothermal (at constant temperature), and the viscosity is constant. The compressibility is small and constant: $c\,\Delta p \ll 1$ .

Diffusivity equation →

The condition of a constant rate at the wall of the well

\left.r\,\frac{\partial p}{\partial r}\right|_{r = r_w} = \frac{qB\mu}{2\pi kh}

Holds when: The sandface rate (the rate at the bottom of the well) is constant from the start. The effect of the volume of the wellbore is not taken into account (Wellbore storage). The inflow is uniform over the whole thickness of the reservoir.

Diffusivity equation →

Dimensionless variables: radius, time and pressure

r_D = \frac{r}{r_w},\qquad t_D = \frac{\eta\,t}{r_w^2} = \frac{k\,t}{\phi\mu c_t r_w^2},\qquad p_D = \frac{2\pi kh\,(p_i - p)}{qB\mu}

Holds when: The conditions of the diffusivity equation hold. The rate is constant, and the initial pressure is the same throughout the reservoir. $p_D$ is positive when the pressure falls. For an injection well the sign is the opposite.

Diffusivity equation →

The dimensionless problem of a well at a constant rate

\begin{aligned} &\frac{1}{r_D}\frac{\partial}{\partial r_D}\!\left(r_D\,\frac{\partial p_D}{\partial r_D}\right) = \frac{\partial p_D}{\partial t_D}\\[4pt] &p_D = 0\ \text{at}\ t_D = 0,\qquad r_D\,\frac{\partial p_D}{\partial r_D} = -1\ \text{at}\ r_D = 1\end{aligned}

dimensionless

Holds when: The same as for the dimensional equation; a constant rate. The third condition is set at the outer boundary: $p_D \to 0$ as $r_D \to \infty$ , or $\partial p_D/\partial r_D = 0$ , or $p_D = 0$ at $r_D = r_{eD}$ , where $r_{eD} = r_e/r_w$ .

Diffusivity equation →

The exact pressure drop through the drop from the linear equation

\Delta p = -\frac{1}{c}\ln\left(1 - c\,\Delta p_{\text{lin}}\right) \approx \Delta p_{\text{lin}}\left(1 + \frac{c\,\Delta p_{\text{lin}}}{2}\right)

any consistent units (the product $c\,\Delta p$ has no dimension)

Holds when: The same as for the density equation; the approximate form holds for $c\,\Delta p_{\text{lin}} \ll 1$ .

Diffusivity equation →

Diffusivity equation in three dimensions

\nabla\cdot\left(\frac{\mathbf{K}}{\mu}\,\nabla p\right) = \phi c_t\,\frac{\partial p}{\partial t}\quad\Rightarrow\quad k_x\frac{\partial^2 p}{\partial x^2} + k_y\frac{\partial^2 p}{\partial y^2} + k_z\frac{\partial^2 p}{\partial z^2} = \phi\mu c_t\,\frac{\partial p}{\partial t}

Holds when: The equation is linearised. Gravity is not taken into account (or the pressure is replaced by the pressure brought to the datum depth). The second form is for a uniform reservoir with the axes along the principal directions of permeability.

Diffusivity equation →

Similarity variable (Boltzmann variable)

x = \frac{r^2}{4\eta t} = \frac{\phi\mu c_t\,r^2}{4kt} = \left(\frac{r}{r_i}\right)^{2},\qquad r_i = \sqrt{4\eta t}

Holds when: An unbounded homogeneous reservoir, a line source and a constant rate from start-up.

Line-source solution and the pressure function →

Line-source solution

p_i - p(r, t) = \frac{qB\mu}{4\pi kh}\,E_1\!\left(\frac{\phi\mu c_t\,r^2}{4kt}\right),\qquad E_1(x) = -\mathrm{Ei}(-x) = \int_x^{\infty}\frac{\e^{-u}}{u}\,\dd u

Holds when: An unbounded homogeneous isotropic (the same permeability in every direction) reservoir of constant thickness. Before start-up the pressure is $p_i$ everywhere. A constant rate from start-up. The well is replaced by a line source and penetrates the full thickness. One slightly compressible fluid (these are the conditions of the diffusivity equation: Diffusivity equation). There is no skin and no wellbore storage (the effect of the volume of fluid in the well itself). The formula describes the pressure in the reservoir, not in the well itself.

Line-source solution and the pressure function →

Series for the exponential integral

E_1(x) = -\ln x - 0.5772 + x - \frac{x^2}{2\cdot2!} + \frac{x^3}{3\cdot3!} - \dots

no units

Holds when: The series converges for any $x > 0$ , but it is convenient only for $x \le 1$ . For larger $x$ , use tables or a continued fraction (advanced section).

Line-source solution and the pressure function →

Logarithmic approximation of the line-source solution

p_i - p(r, t) = \frac{qB\mu}{4\pi kh}\left[\ln\frac{kt}{\phi\mu c_t r^2} + 0.809\right]

SI; natural logarithm

Holds when: The conditions of the line-source solution. $x < 0.01$ , that is, $kt/(\phi\mu c_tr^2) > 25$ : the error is then less than 0.25%. At $kt/(\phi\mu c_tr^2) = 5$ the error is about 2%.

Line-source solution and the pressure function →

Pressure function of a well in an infinite reservoir

p_D(t_D) = \frac{2\pi kh\,(p_i - p_{wf})}{qB\mu} = \frac12\left(\ln t_D + 0.809\right) + S,\qquad t_D = \frac{kt}{\phi\mu c_t r_w^2}

SI; in metric oilfield units $t_D = 3.553\cdot10^{-3}\,kt/(\phi\mu c_tr_w^2)$ with $t$ in hours

Holds when: An unbounded reservoir and a constant rate from start-up. $t_D > 25$ (error about 2%) or $t_D > 100$ (about 0.5%): for an ordinary well that is only seconds. The skin is an extra pressure drop in a thin zone near the wellbore. It appears together with the flow. The effect of wellbore storage (the fluid volume in the well itself) has ended (Wellbore storage).

Line-source solution and the pressure function →

Rate at constant bottomhole pressure

q(t) = \frac{2\pi kh\,(p_i - p_{wf})}{B\mu}\,q_D(t_D),\qquad q_D(t_D) \approx \frac{1}{p_D(t_D)} = \frac{2}{\ln t_D + 0.809}

SI; in metric oilfield units the factor $2\pi$ is replaced by the constant 0.536

Holds when: An unbounded reservoir, a constant bottomhole pressure from start-up, and no skin. With skin, $q_D \approx 1/(p_D + S)$ . The approximation $q_D \approx 1/p_D$ overestimates the rate: by about 2% at $t_D = 10^4$ and by less than 1% at $t_D > 10^6$ .

Line-source solution and the pressure function →

Linearity of the diffusivity equation

\nabla^2 \Delta p = \frac{\phi\mu c_t}{k}\,\frac{\partial\,\Delta p}{\partial t}\quad\Rightarrow\quad \Delta p = a_1\,\Delta p_1 + a_2\,\Delta p_2\ \ \text{— also a solution}

SI (any consistent system of units)

Holds when: The fluid is single-phase and slightly compressible. The porosity, permeability, viscosity and compressibility do not depend on pressure. Darcy's law applies: the filtration velocity is proportional to the pressure gradient, with no inertial losses. The conditions at the wells and at the boundaries are linear too: given rates, constant pressures or no-flow boundaries. Before the events begin, the pressure in the whole reservoir is the same, $p = p_i$ . If the reservoir was already disturbed, the solution that describes this earlier disturbance is added to the sum.

Superposition principle →

Adding in space: the pressure at a point from several wells

p_i - p(x, y, t) = \sum_{j=1}^{N} \frac{q_jB\mu}{4\pi kh}\,E_1\!\left(\frac{\phi\mu c_t\,r_j^2}{4k\,(t - t_j)}\right)

Holds when: The reservoir is uniform and unbounded; each well works at a constant rate from the moment it starts. The linearity conditions of the previous section hold. For a producing well, at the point of its own wellbore ( $r = r_w$ ), the function is replaced by a logarithm and the skin of that well is added. The skins of the other wells are not in the sum.

Superposition principle →

Adding in time: the bottomhole pressure for a stepwise change of the rate

p_i - p_{wf}(t) = \frac{B\mu}{2\pi kh}\sum_{j=1}^{n}\left(q_j - q_{j-1}\right)\left[p_D\!\left(\frac{k\,(t - t_{j-1})}{\phi\mu c_t r_w^2}\right) + S\right]

Holds when: The linearity conditions hold. The skin does not depend on the rate and does not change with time. The sandface rate (the rate at the bottom of the well) changes instantly: the effect of the wellbore is not taken into account. It is taken into account by putting the solution with wellbore storage and skin in place of $p_D + S$ (Wellbore storage). In an infinite reservoir the logarithmic form of $p_D$ is correct if more than a few seconds have passed after each rate change. In a bounded reservoir we take the solution for a bounded reservoir.

Superposition principle →

Pressure build-up after work at several rates

p_i - p_{ws}(\Delta t) = \frac{B\mu}{2\pi kh}\left[\sum_{j=1}^{n}\left(q_j - q_{j-1}\right)p_D\big(t_n - t_{j-1} + \Delta t\big) - q_n\,p_D(\Delta t)\right]

Holds when: The conditions of adding in time hold; the effect of the wellbore after the shut-in is not taken into account. For a single rate step ( $n = 1$ , $t_1 = t_p$ ) the formula coincides with the one from which the Horner equation is derived.

Superposition principle →

The superposition time function and the straight line of a multi-rate record

\frac{p_i - p_{wf}(t)}{q_n} = m'\,X_n(t) + b',\qquad X_n(t) = \sum_{j=1}^{n}\frac{q_j - q_{j-1}}{q_n}\,\lg\left(t - t_{j-1}\right)

metric oilfield: pressure in MPa, rate in m³/d, time in h

Holds when: The reservoir is infinite and uniform. After each rate change, enough time has passed for the effect of the wellbore to end. The rates and the moments of the rate changes are known exactly. Before the first start the pressure in the reservoir had evened out and equals $p_i$ . The skin is constant and does not depend on the rate.

Superposition principle →

A well near a sealing fault

p_i - p_{wf} = \frac{qB\mu}{4\pi kh}\left[\ln\frac{kt}{\phi\mu c_t r_w^2} + 0.809 + 2S + E_1\!\left(\frac{\phi\mu c_t\,(2L)^2}{4kt}\right)\right]

SI; the logarithm is natural

Holds when: The fault is a straight sealing boundary through the whole thickness of the reservoir. The disturbance has not reached any other boundary. The reservoir is uniform. Beyond the fault we imagine the same reservoir continuing. The skin is added only for the real well. The image has the same rate history. For a changing rate, both are split into the same steps.

Superposition principle →

A well near a constant-pressure boundary: the steady state

p_i - p_{wf} \to \frac{qB\mu}{2\pi kh}\left[\ln\frac{2L}{r_w} + S\right]\quad \text{for}\ t \gg \frac{\phi\mu c_t\,(2L)^2}{4k}

SI; in metric oilfield units the factor $2\pi$ is replaced by the constant 0.536

Holds when: The boundary is a straight line with the constant pressure $p_i$ through the whole thickness of the reservoir; there are no other boundaries. The time is much longer than the time in which the disturbance reaches the image.

Superposition principle →

The Duhamel integral (convolution) for a changing rate

p_i - p_{wf}(t) = \int_0^t \Delta p_u(t - \tau)\,\dd q(\tau)

SI (any consistent system of units)

Holds when: The problem is linear; before the start the pressure in the reservoir has evened out. For a stepwise history the integral turns into the sum of the section on adding in time.

Superposition principle →

Symbols

Symbol	Meaning and context
$\bar p$	pore-volume-averaged pressure in the drainage area, MPaFlow regimes
$\nabla p$	pressure gradient, Pa/mDiffusivity equation
$\Delta p$	exact pressure fall $p_i - p$ at a given point at a given moment, PaDiffusivity equation pressure drop relative to the initial pressure, $p_i - p(x, y, t)$ , at the point of the reservoir with coordinates $x$ , $y$ at the time $t$ , PaSuperposition principle
$\dd q(\tau)$	increase of the rate at the moment $\tau$ , including the jump from zero at the start, m³/sSuperposition principle
$\nabla^2$	Laplace operator over the coordinates in the plane of the reservoir, $\partial^2/\partial x^2 + \partial^2/\partial y^2$ (the sum of the second derivatives, a measure of the curvature of the pressure profile), 1/m²Superposition principle
$0.5772$	Euler's constant, —Line-source solution and the pressure function
$0.809$	$\ln(4/\gamma)$ , where $\gamma = 1.781$ , —Line-source solution and the pressure function
$\Delta p_1,\ \Delta p_2$	two solutions of the same equation, for example from two wells or from two rate steps of one well, PaSuperposition principle
$a_1,\ a_2$	constant factors (ratios of rates), —Superposition principle
$B$	formation volume factor of the fluid, m³/m³Flow regimes formation volume factor of the fluid (at the initial reservoir pressure), m³/m³Diffusivity equation
$B,\ \mu,\ k,\ h,\ \phi,\ c_t$	formation volume factor, viscosity, permeability, thickness, porosity, total compressibility, m³/m³, Pa·s, m², m, —, 1/PaSuperposition principle
$B,\ \mu,\ k,\ h,\ \phi,\ c_t,\ r_w$	formation volume factor, viscosity, permeability, thickness, porosity, total compressibility, radius of the well, m³/m³, Pa·s, m², m, —, 1/Pa, mSuperposition principle
$b'$	value of $(p_i - p_{wf})/q_n$ on the line at $X_n = 0$ , MPa/(m³/d)Superposition principle
$c$	compressibility of the mobile fluid, 1/PaDiffusivity equation
$c_f$	pore (rock) compressibility: the relative decrease of the pore volume when the pore pressure falls by one unit, 1/MPaFlow regimes
$c_o,\ c_w,\ c_g$	compressibility of oil, water and gas, $c = -\frac{1}{V}\frac{\dd V}{\dd p}$ , 1/MPaFlow regimes
$c_t$	total compressibility, per unit of pore volume, 1/MPaFlow regimes total compressibility, 1/PaFlow regimes
$D$	depth, counted downward; $\nabla D$ is a unit vector pointing down, mDiffusivity equation
$E_1$	exponential integral (a standard mathematical function that describes how the pressure drop depends on the distance from a well and on time), —Superposition principle exponential integral: the contribution of the image, —Superposition principle
$E_1(x)$	exponential integral, —Line-source solution and the pressure function
$g$	acceleration due to gravity, 9.81, m/s²Diffusivity equation
$k$	permeability of the reservoir, m²Flow regimes permeability (here the same in all directions), m²Diffusivity equation permeability of the reservoir to the mobile fluid, m²Diffusivity equation
$\mathbf{K}$	permeability tensor, m²Diffusivity equation
$k_x,\ k_y,\ k_z$	principal permeabilities (the coordinate axes point along the principal directions), m²Diffusivity equation
$k_{xx},\ k_{yy},\ k_{zz}$	diagonal components: the link between flow along an axis and the gradient along the same axis, m²Diffusivity equation
$k_{xy},\ k_{xz},\ k_{yz}$	off-diagonal components: flow along one axis caused by the gradient along another, m²Diffusivity equation
$k,\ h,\ B,\ \mu$	permeability, thickness, formation volume factor, viscosity, m², m, m³/m³, Pa·sLine-source solution and the pressure function
$k,\ h,\ B,\ \mu,\ \phi,\ c_t,\ r_w,\ S$	in the forms below: permeability, thickness, formation volume factor, viscosity, porosity, total compressibility, radius of the well, skin, mD, m, m³/m³, mPa·s, —, 1/MPa, m, —Superposition principle
$k,\ h,\ \mu$	permeability, thickness of the reservoir, viscosity, m², m, Pa·sDiffusivity equation
$k,\ h,\ \phi,\ \mu,\ c_t,\ \eta$	permeability, thickness, porosity, viscosity, total compressibility, hydraulic diffusivity, m², m, —, Pa·s, 1/Pa, m²/sDiffusivity equation
$k,\ h,\ \phi,\ c_t$	permeability, thickness, porosity, total compressibility, m², m, —, 1/PaLine-source solution and the pressure function
$k,\ \mu,\ p$	permeability, viscosity, pressure, m², Pa·s, PaDiffusivity equation
$L$	distance from the well to the fault; $2L$ is the distance to the image, mSuperposition principle distance from the well to the constant-pressure boundary, mSuperposition principle
$\Delta p_{\text{lin}}$	the same from the linear diffusivity equation, PaDiffusivity equation
$m'$	slope of the line: the change of $(p_i - p_{wf})/q_n$ when $X_n$ grows by one (per log cycle), MPa/(m³/d)Superposition principle
$n$	number of the current rate step: $t_{n-1} < t$ , —Superposition principle
$N$	number of wells, —Superposition principle
$p_D$	dimensionless pressure at the wall of the well at a constant rate, without skin; in an infinite reservoir $\tfrac12(\ln t_D + 0.809)$ , —Superposition principle
$p_D(\cdot)$	pressure function of the well without skin; the argument is the time converted to dimensionless time, —Superposition principle
$p_D(t_D)$	dimensionless drop in bottomhole pressure, —Line-source solution and the pressure function pressure function of the well at constant rate, —Line-source solution and the pressure function
$p_i$	initial reservoir pressure, PaFlow regimes initial reservoir pressure (the pressure before the first start), PaSuperposition principle initial pressure, which is also the pressure at the boundary, PaSuperposition principle
$p_i - p_{wf}$	constant difference between the initial reservoir pressure and the bottomhole pressure, PaLine-source solution and the pressure function
$p_i,\ q,\ B,\ \mu$	initial pressure, rate, formation volume factor, viscosity, Pa, m³/s, m³/m³, Pa·sLine-source solution and the pressure function
$p_i,\ B,\ \mu,\ k,\ h$	initial pressure, formation volume factor, viscosity, permeability, thickness, Pa, m³/m³, Pa·s, m², mSuperposition principle
$p_i,\ p_{wf}$	the pressure before the first start, and the bottomhole pressure, MPaSuperposition principle
$p_{wf}$	bottomhole flowing pressure of the producing well a time $t$ after start-up, PaLine-source solution and the pressure function
$p_{wf},\ p_i$	bottomhole and initial pressure, PaSuperposition principle
$p_{wf}(t)$	bottomhole pressure at the time $t$ , PaSuperposition principle
$p_{ws}(\Delta t)$	bottomhole pressure the time $\Delta t$ after the shut-in, PaSuperposition principle
$p(r, t)$	pressure at the distance $r$ from the well axis at the time $t$ , PaFlow regimes pressure in the reservoir at distance $r$ from the well axis, a time $t$ after start-up, PaLine-source solution and the pressure function pressure in the reservoir at distance $r$ , a time $t$ after start-up, PaLine-source solution and the pressure function
$p(x, t)$	pressure, PaFlow regimes
$p(x, y, t)$	pressure at a point of the reservoir at the time $t$ , PaSuperposition principle
$p(x, y, z, t)$	pressure, PaDiffusivity equation
$q$	rate at standard conditions, m³/dFlow regimes rate at standard conditions; $qB$ is the rate at reservoir conditions, m³/sDiffusivity equation
$q_D(t_D)$	dimensionless rate, —Line-source solution and the pressure function
$q_j$	rate of well $j$ at standard conditions; negative for an injection well, m³/sSuperposition principle rate in step $j$ at standard conditions; $q_0 = 0$ , m³/sSuperposition principle
$q_j,\ t_{j-1}$	rates of the steps and the moments they begin, $q_0 = 0$ , $t_0 = 0$ , m³/s, sSuperposition principle
$q_n$	rate before the shut-in, m³/sSuperposition principle rate of the current, $n$ -th step at standard conditions, m³/dSuperposition principle
$q,\ B$	rate at standard conditions and formation volume factor, m³/s, m³/m³Flow regimes
$q,\ B,\ \mu,\ k,\ h$	rate, formation volume factor, viscosity, permeability, thickness, m³/s, m³/m³, Pa·s, m², mSuperposition principle
$q,\ B,\ \mu,\ k,\ h,\ \phi,\ c_t$	as in the line-source solution, m³/s, m³/m³, Pa·s, m², m, —, 1/PaLine-source solution and the pressure function
$q,\ B,\ \mu,\ k,\ h,\ r_w,\ S$	rate, formation volume factor, viscosity, permeability, thickness, radius of the well, skin, m³/s, m³/m³, Pa·s, m², m, m, —Superposition principle
$q(t)$	rate at standard conditions a time $t$ after start-up, m³/sLine-source solution and the pressure function
$r_D,\ t_D,\ p_D$	dimensionless distance, time and pressure drop, —Flow regimes dimensionless distance, time and pressure drop, defined above, —Flow regimes
$r_e$	distance to the boundary; for a circular area, its radius, mFlow regimes
$r_i$	radius of investigation at time $t$ (how far the disturbance has spread), mLine-source solution and the pressure function
$r_j$	distance from well $j$ to the point, mSuperposition principle
$r_w$	radius of the well, mFlow regimes wellbore radius, mLine-source solution and the pressure function
$r,\ t$	distance from the well axis, and time, m, sDiffusivity equation distance from the well axis and time since start-up, m, sLine-source solution and the pressure function
$S$	skin factor, —Line-source solution and the pressure function
$S_o,\ S_w,\ S_g$	oil, water and gas saturation, fractionFlow regimes
$t_b$	time from the start of the well to the arrival of the disturbance at the boundary, sFlow regimes
$t_D$	dimensionless time based on the wellbore radius, —Line-source solution and the pressure function
$t_j$	moment well $j$ starts; while $t \le t_j$ , its term is zero, sSuperposition principle
$t_{j-1}$	moment step $j$ begins; $t_0 = 0$ , sSuperposition principle
$t_n$	moment of the shut-in (the end of the $n$ -th, last rate step), sSuperposition principle
$\mathbf{u}$	filtration velocity vector: the flow rate through a unit of area perpendicular to the flow, m/sDiffusivity equation
$u$	filtration velocity towards the well: the flow rate through a unit area of the cylindrical surface, m/sDiffusivity equation
$\mathbf{u},\ \mu,\ \nabla p$	filtration velocity vector, viscosity, pressure gradient, m/s, Pa·s, Pa/mDiffusivity equation
$\Delta p_u(t)$	pressure drop at the bottom of the well when the well starts at a unit constant rate (together with the skin and, if needed, the wellbore), Pa/(m³/s)Superposition principle
$V_p$	pore volume of the drainage area, m³Flow regimes
$x$	dimensionless variable on which the solution depends, —Line-source solution and the pressure function argument, $x > 0$ , —Line-source solution and the pressure function
$X_n(t)$	superposition time function: a weighted sum of the base-10 logarithms ( $\lg$ ) of the times elapsed since each rate change, —Superposition principle
$\eta$	hydraulic diffusivity, m²/sFlow regimes hydraulic diffusivity; in Russian-language literature it is also written $\kappa$ or $\chi$ , m²/sDiffusivity equation hydraulic diffusivity $k/(\phi\mu c_t)$ , m²/sLine-source solution and the pressure function
$\mu$	viscosity of the fluid, Pa·sFlow regimes
$\mu,\ k,\ h$	fluid viscosity, permeability and reservoir thickness, Pa·s, m², mLine-source solution and the pressure function
$\rho$	density of the fluid, kg/m³Diffusivity equation density of the fluid at reservoir conditions, kg/m³Diffusivity equation
$\tau$	the moment to which the increase of the rate refers, sSuperposition principle
$\phi$	porosity, fractionFlow regimes
$\phi,\ \mu,\ c_t$	porosity, viscosity, total compressibility, —, Pa·s, 1/PaDiffusivity equation
$\phi,\ \mu,\ c_t,\ k$	porosity, viscosity, total compressibility, permeability, —, Pa·s, 1/Pa, m²Flow regimes
$\phi,\ c_t$	porosity and total compressibility, —, 1/PaLine-source solution and the pressure function porosity and total compressibility (in the condition on time), —, 1/PaSuperposition principle
$\phi,\ c_t,\ r_w,\ S$	porosity, total compressibility, radius of the well, skin, —, 1/Pa, m, —Superposition principle