Equations: Transient testing: fundamentals

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

Rate of pressure decline in a closed area

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

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

Holds when: The area is closed (nothing flows in across the boundaries), and the rate is constant. Pseudo-steady state has already been reached: the disturbance has reached all the boundaries. The fluid is single-phase and slightly compressible, with constant compressibility.

Well testing: purposes and types of test →

Radius of investigation

r_i = \sqrt{\frac{4kt}{\phi\mu c_t}}

Holds when: The reservoir is homogeneous, and the disturbance has not yet reached the boundaries. The criterion $kt/(\phi\mu c_t r_i^2) = 1/4$ is a convention. If we choose a different level for what counts as a “noticeable” disturbance, the constant changes by tens of per cent. In practice, whether we can detect a disturbance also depends on the resolution of the gauge (the smallest change it can show).

Well testing: purposes and types of test →

Pressure drawdown in an infinite reservoir (logarithmic approximation)

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

SI; the logarithm is natural

Holds when: The rate is constant from start-up, and wellbore storage is ignored. The disturbance has not reached the boundaries, and the dimensionless time $kt/(\phi\mu c_t r_w^2)$ is greater than about 25. The reservoir is homogeneous, and the fluid is single-phase and slightly compressible.

Well testing: purposes and types of test →

Bottomhole pressure in pseudo-steady-state flow

p_i - p_{wf} = \frac{qB\,t}{c_t V_p} + \frac{q}{J},\qquad J = \frac{2\pi kh}{B\mu\left(\ln\dfrac{r_e}{r_w} - \dfrac34 + S\right)}

Holds when: The area is circular and closed, with the well at the centre. The time is longer than the time needed to reach pseudo-steady state.

Well testing: purposes and types of test →

Conversion of a dimensionless solution to pressure drop and time

\Delta p = \frac{qB\mu}{2\pi kh}\,p_D,\qquad t = \frac{\phi\mu c_t r_w^2}{k}\,t_D

Holds when: The conditions of the diffusivity equation hold: the fluid is single-phase and slightly compressible, and the properties of the reservoir and the fluid are constant (Diffusivity equation). $\Delta p$ is positive for production. For an injection well the sign is opposite.

Dimensionless well solutions: skin, radius of investigation and units →

Flow rate through a circle in the line-source solution

\frac{q(r, t)}{q} = \exp\!\left(-\frac{r^2}{4\eta t}\right) = \exp\!\left(-\frac{r_D^2}{4t_D}\right)

no units (a ratio of rates)

Holds when: This is the line-source solution: the reservoir is infinite and homogeneous, and the rate is constant from start-up (the derivation is in the advanced section).

Dimensionless well solutions: skin, radius of investigation and units →

Two ways of accounting for skin in the solution for a well

p_{wD}(t_D) = p_D(t_D) + S\qquad\text{or}\qquad p_{wD}(t_D) = p_D\!\left(t_D\,\e^{2S}\right)

no units

Holds when: The first form is the skin of an infinitely thin zone that follows the rate instantly. The second form is a well without skin, of radius $r_w' = r_w\e^{-S}$ , in the same reservoir. The pressure is still referred to the $k$ and $h$ of the reservoir. Wellbore storage is not included. It can be added to either form (Wellbore storage).

Dimensionless well solutions: skin, radius of investigation and units →

Rate of pressure decline at a point in the reservoir

\frac{\partial\,\Delta p}{\partial t} = \frac{qB\mu}{4\pi kh}\cdot\frac1t\,\exp\!\left(-\frac{r^2}{4\eta t}\right)

Holds when: This is the line-source solution: the reservoir is infinite, and the rate is constant.

Dimensionless well solutions: skin, radius of investigation and units →

Share of the withdrawn volume supplied by the reservoir inside radius r

\frac{V(r)}{V} = 1 - \e^{-X} + X\,E_1(X),\qquad X = \frac{r^2}{4\eta t} = \left(\frac{r}{r_i}\right)^2

no units

Holds when: This is the line-source solution: the reservoir is infinite, and the rate is constant from start-up (the derivation is in the advanced section).

Dimensionless well solutions: skin, radius of investigation and units →

The radius of drainage in transient flow

r_d = \sqrt{\frac{4\eta t}{\gamma}} = 0.749\,r_i,\qquad p_i - p_{wf} = \frac{qB\mu}{2\pi kh}\left(\ln\frac{r_d}{r_w} + S\right)

SI; the logarithm is natural

Holds when: The rate is constant, and the reservoir is infinite and homogeneous. $t_D > 25$ , so the logarithmic approximation is valid at the wall of the well.

Dimensionless well solutions: skin, radius of investigation and units →

Transient productivity index

J_t = \frac{q}{p_i - p_{wf}(t)} = \frac{2\pi kh}{B\mu\left[\ln(r_d/r_w) + S\right]}

Holds when: The flow is transient: the disturbance has not reached the boundaries. We measure the drawdown from the initial pressure $p_i$ , because the average pressure in an infinite reservoir is not defined. If the drawdown is held constant, the rate falls by the same formula, with a small correction (the rate at constant pressure).

Dimensionless well solutions: skin, radius of investigation and units →

Bottomhole pressure at a constant rate in an infinite reservoir

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

SI; natural logarithm

Holds when: The rate is constant from the moment of start-up; before start-up the pressure throughout the reservoir equals $p_i$ . The reservoir behaves as an infinite one: the disturbance has not reached the boundaries. Wellbore storage is over. The condition $kt/(\phi\mu c_tr_w^2) > 25$ is met within seconds. The reservoir is homogeneous, there is a single slightly compressible fluid, and the skin does not change with time.

Pressure drawdown analysis →

Flow capacity from the slope of the semilog straight line

kh = \frac{qB\mu\,\ln 10}{4\pi m}

Holds when: The slope is measured on the section of infinite-acting radial flow. The rate is constant. If the thickness of the producing part of the reservoir is not known, we report the result as $kh$ .

Pressure drawdown analysis →

Skin factor from a drawdown test

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

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

Holds when: We read $p_{1\text{h}}$ on the line and do not take it from the measurements, because wellbore storage may not be over an hour after start-up. The well stood idle long enough before start-up for the pressure in the reservoir to equalise. Otherwise we need superposition with the previous history (adding up the effects of earlier rate changes) in place of $p_i$ . $m$ and $k$ are found from the same straight section.

Pressure drawdown analysis →

Flow capacity and skin from a plot of pressure against the natural logarithm of time

kh = 0.933\,\frac{qB\mu}{m_e},\qquad S = \frac12\left[\frac{p_i - p_{1\text{h}}}{m_e} - \ln\frac{k}{\phi\mu c_t r_w^2} + 4.831\right]

metric oilfield units; time in hours

Holds when: The same as for the formulas with the base-10 logarithm.

Pressure drawdown analysis →

Pressure build-up as the superposition 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: The problem is linear: one slightly compressible fluid, and the properties of the reservoir and the fluid do not depend on the pressure. The rate was constant before shut-in. The inflow from the reservoir stops at once at shut-in (we ignore wellbore storage).

Pressure build-up analysis →

The Horner equation

p_{ws} = p_i - \frac{qB\mu}{4\pi kh}\,\ln\frac{t_p + \Delta t}{\Delta t}

SI (any consistent system of units)

Holds when: The reservoir behaves as an infinite one both while the well produces and during the shut-in: the disturbance has not reached the boundaries. The rate before shut-in is constant. The reservoir is homogeneous, and there is one slightly compressible fluid. Wellbore storage is over: the inflow from the reservoir into the wellbore has stopped after shut-in.

Pressure build-up analysis →

Flow capacity from the slope of the Horner line

kh = \frac{qB\mu\,\ln 10}{4\pi m}

Holds when: The slope is measured on the section of radial flow. $h$ is the thickness of the producing part of the reservoir. If it is not known, the result stays $kh$ .

Pressure build-up analysis →

Skin factor from a build-up

S = 1.151\left[\frac{p_{1\text{h}} - p_{wf}(\Delta t = 0)}{m} - \lg\frac{k}{\phi\,\mu\,c_t\,r_w^2} + 2.098\right]

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

Holds when: The producing time $t_p$ is much longer than 1 h. In the general case we add the term $\lg[(t_p + 1)/t_p]$ to the brackets. Both the production and the shut-in take place as radial flow in an infinite reservoir. We read $p_{1\text{h}}$ on the line, even if the measurement itself at 1 h is still distorted by wellbore storage.

Pressure build-up analysis →

The MDH line

p_{ws} = p_{1\text{h}} + m\,\lg\Delta t

metric oilfield units (pressure in MPa, time in hours)

Holds when: $\Delta t \ll t_p$ over the whole section we use. If the shut-in time is no more than about a tenth of the producing time, the slope is distorted by no more than a few per cent. The other conditions are the same as for the Horner method.

Pressure build-up analysis →

Radius of the “probe” in a build-up

r_p = \sqrt{\frac{4k\,\Delta t_e}{\gamma\,\phi\mu c_t}},\qquad \Delta t_e = \frac{t_p\,\Delta t}{t_p + \Delta t}

Holds when: The reservoir is infinite. At the point $r_p$ the logarithmic approximation holds: the disturbance passed it long ago. Wellbore storage is over. The skin changes the pressure only in the well itself, and it does not enter the formula.

Pressure build-up analysis →

Symbols

Symbol	Meaning and context
$\Delta p$	pressure decline $p_i - p$ at the point considered, PaDimensionless well solutions: skin, radius of investigation and units pressure decline $p_i - p$ at the distance $r$ from the well, PaDimensionless well solutions: skin, radius of investigation and units
$\Delta t$	time since shut-in, hPressure build-up analysis
$(t_p + \Delta t)/\Delta t$	the Horner time ratio; it has no units, —Pressure build-up analysis
$\dd p/\dd t$	rate of pressure decline — the same throughout the area and in the well, MPa/dWell testing: purposes and types of test
$B$	formation volume factor of the fluid (its volume in the reservoir per unit volume at standard conditions), m³/m³Well testing: purposes and types of test
$c_t$	total compressibility of rock and fluids, 1/MPaWell testing: purposes and types of test total compressibility, 1/MPaPressure drawdown analysis total compressibility of the reservoir and the fluids, 1/MPaPressure build-up analysis
$\Delta t_e$	equivalent time (Effect of producing time on build-up analysis); for $\Delta t \ll t_p$ it is almost equal to $\Delta t$ , sPressure build-up analysis
$E_1(X)$	exponential integral, —Dimensionless well solutions: skin, radius of investigation and units
$\e^{2S}$	the ratio $(r_w/r_w')^2$ : the dimensionless time of a well with the effective radius is $t_D\e^{2S}$ , —Dimensionless well solutions: skin, radius of investigation and units
$J$	productivity index with respect to the average pressure of the area, m³/(s·Pa)Well testing: purposes and types of test
$J_t$	productivity index at time $t$ , with respect to the initial pressure, m³/(s·Pa)Dimensionless well solutions: skin, radius of investigation and units
$k$	permeability of the reservoir, m²Well testing: purposes and types of test permeability found from the slope, mDPressure drawdown analysis
$k,\ h$	permeability and thickness of the reservoir, m², mWell testing: purposes and types of test permeability and net thickness of the reservoir, m², mPressure drawdown analysis
$k,\ h,\ B,\ \mu,\ r_w,\ S$	permeability, thickness, formation volume factor, viscosity, well radius, skin, m², m, m³/m³, Pa·s, m, —Dimensionless well solutions: skin, radius of investigation and units
$k,\ h,\ \mu$	permeability, reservoir thickness, viscosity, m², m, Pa·sDimensionless well solutions: skin, radius of investigation and units
$k,\ h,\ \phi,\ c_t,\ r_w$	permeability, thickness, porosity, total compressibility, wellbore radius, mD, m, —, 1/MPa, mPressure drawdown analysis
$k,\ \phi,\ \mu,\ c_t$	permeability, porosity, viscosity, total compressibility, m², —, Pa·s, 1/PaPressure build-up analysis
$kh$	flow capacity: the product of permeability and the thickness of the producing part of the reservoir, m³Pressure drawdown analysis flow capacity: the product of the permeability and the thickness of the producing part of the reservoir, m³Pressure build-up analysis
$m$	pressure drop over one decimal cycle of time on the line, PaPressure drawdown analysis pressure drop per decimal cycle of time on the line, MPaPressure drawdown analysis slope of the line per decimal logarithmic cycle, PaPressure build-up analysis slope of the line per logarithmic cycle, MPaPressure build-up analysis slope per logarithmic cycle, the same as for the Horner line, MPaPressure build-up analysis
$m_e$	pressure drop on the line when $\ln t$ increases by one, that is, when time grows by a factor of 2.718, MPaPressure drawdown analysis
$m,\ p_{1\text{h}}$	in the oilfield forms: the pressure drop per decimal cycle of time and the pressure on the line at $t$ = 1 h, MPa or psiPressure drawdown analysis
$p_{1\text{h}}$	pressure on the line (or its continuation) at $t$ = 1 h, MPaPressure drawdown analysis pressure on the line at $t$ = 1 h ( $\ln t = 0$ ), MPaPressure drawdown analysis pressure on the Horner line (or its extension) at $\Delta t$ = 1 h, MPaPressure build-up analysis pressure on the MDH line at $\Delta t$ = 1 h, MPaPressure build-up analysis
$p_D,\ t_D$	dimensionless pressure and time read from the solution or the type curve, —Dimensionless well solutions: skin, radius of investigation and units
$p_D(t_D)$	solution for a finite-radius well without skin, at the wall, —Dimensionless well solutions: skin, radius of investigation and units
$p_D(t)$	dimensionless pressure drop at the wall of a well that has produced at a constant rate for a time $t$ (a drop without units: the real drop divided by $qB\mu/(2\pi kh)$ ), with the skin included (Line-source solution and the pressure function), —Pressure build-up analysis
$p_i$	initial reservoir pressure, PaWell testing: purposes and types of test reservoir pressure before start-up (initial reservoir pressure), PaPressure drawdown analysis reservoir pressure before the well is started up, MPaPressure drawdown analysis
$p_i,\ q,\ B,\ \mu$	pressure before start-up, rate, formation volume factor, viscosity, MPa, m³/d, m³/m³, mPa·sPressure drawdown analysis
$p_i,\ p_{wf}$	initial reservoir pressure and bottomhole flowing pressure, PaDimensionless well solutions: skin, radius of investigation and units
$p_{wD}$	dimensionless decline of the bottomhole pressure of a well with skin, —Dimensionless well solutions: skin, radius of investigation and units
$p_{wf}$	bottomhole flowing pressure at time $t$ after start-up, PaWell testing: purposes and types of test bottomhole pressure at time $t$ after start-up, PaPressure drawdown analysis
$p_{wf}(\Delta t = 0)$	bottomhole pressure at the moment of shut-in, MPaPressure build-up analysis
$p_{wf}(t)$	bottomhole flowing pressure at time $t$ at a constant rate, PaDimensionless well solutions: skin, radius of investigation and units
$p_{ws}$	bottomhole pressure at the time $\Delta t$ after shut-in, PaPressure build-up analysis
$p_{ws}(\Delta t)$	bottomhole pressure at the time $\Delta t$ after shut-in, PaPressure build-up analysis
$q$	flow rate at standard conditions, m³/dWell testing: purposes and types of test well rate at standard conditions, m³/sDimensionless well solutions: skin, radius of investigation and units
$q,\ B$	rate at standard conditions and formation volume factor; $qB$ is the rate at reservoir conditions, m³/s, m³/m³Dimensionless well solutions: skin, radius of investigation and units
$q,\ B,\ \mu$	flow rate at standard conditions, formation volume factor, viscosity, m³/s, m³/m³, Pa·sWell testing: purposes and types of test rate at standard conditions, formation volume factor, viscosity, m³/s, m³/m³, Pa·sPressure drawdown analysis rate before shut-in at standard conditions, formation volume factor, viscosity, m³/s, m³/m³, Pa·sPressure build-up analysis
$q,\ B,\ \mu,\ k,\ h$	rate at standard conditions, formation volume factor, viscosity, permeability, thickness, m³/s, m³/m³, Pa·s, m², mDimensionless well solutions: skin, radius of investigation and units rate before shut-in, formation volume factor, viscosity, permeability, thickness, m³/s, m³/m³, Pa·s, m², mPressure build-up analysis
$q,\ B,\ \mu,\ k,\ h,\ S$	rate at standard conditions, formation volume factor, viscosity, permeability, thickness, skin, m³/s, m³/m³, Pa·s, m², m, —Dimensionless well solutions: skin, radius of investigation and units
$q(r, t)$	flow rate of fluid through a cylindrical surface of radius $r$ at time $t$ , reduced to standard conditions, m³/sDimensionless well solutions: skin, radius of investigation and units
$r_d$	radius of drainage at time $t$ , mDimensionless well solutions: skin, radius of investigation and units
$r_D,\ t_D$	dimensionless distance and time, —Dimensionless well solutions: skin, radius of investigation and units
$r_e$	radius of the circular drainage area, mWell testing: purposes and types of test
$r_i$	radius to which a noticeable pressure disturbance has travelled in time $t$ , mWell testing: purposes and types of test radius of investigation $\sqrt{4\eta t}$ , mDimensionless well solutions: skin, radius of investigation and units
$r_p$	distance from the well at which the pressure at the moment of shut-in was equal to the reading $p_{ws}(\Delta t)$ , mPressure build-up analysis
$r_w$	wellbore radius, mPressure drawdown analysis
$S$	skin factor, —Well testing: purposes and types of test
$t$	time since the rate change, sWell testing: purposes and types of test time after start-up, sWell testing: purposes and types of test time since start-up of the well, sDimensionless well solutions: skin, radius of investigation and units time since start-up, sDimensionless well solutions: skin, radius of investigation and units
$t_p,\ \Delta t$	producing time before shut-in and time since shut-in, sPressure build-up analysis producing time before shut-in and time since shut-in (in any equal units), sPressure build-up analysis
$V$	the whole withdrawn volume $qBt$ , m³Dimensionless well solutions: skin, radius of investigation and units
$V_p$	pore volume of the drainage area, m³Well testing: purposes and types of test pore volume of the area, $\pi r_e^2 h\phi$ , m³Well testing: purposes and types of test
$V(r)$	volume of fluid (at reservoir conditions) supplied by elastic expansion inside a circle of radius $r$ , m³Dimensionless well solutions: skin, radius of investigation and units
$\gamma$	$\e^{0.5772} = 1.781$ , —Dimensionless well solutions: skin, radius of investigation and units the exponential of Euler's constant, 1.781, —Pressure build-up analysis
$\eta$	hydraulic diffusivity $k/(\phi\mu c_t)$ , m²/sDimensionless well solutions: skin, radius of investigation and units
$\mu$	fluid viscosity, Pa·sWell testing: purposes and types of test viscosity of the fluid in the reservoir, mPa·sPressure drawdown analysis
$\phi$	porosity, fractionWell testing: purposes and types of test
$\phi,\ c_t,\ r_w$	porosity, total compressibility, well radius, —, 1/Pa, mWell testing: purposes and types of test porosity, total compressibility, wellbore radius, —, 1/Pa, mPressure drawdown analysis