Content

Definition

Mathematical model of multiphase wellbore flow predicts the temperature, pressure and flow speed distribution along the wellbore trajectory with account for:

tubing head pressure which is set by gathering system or injection pump
wellbore design
pump characterisits
fluid friction with tubing /casing walls
interfacial phase slippage
heat exchange between wellbore fluid and surrounding rocks

Flow Model

(1)

$\begin{array}{l}\displaystyle (\rho \,c_{pt})_p \frac{\partial T}{\partial t} - \ \phi \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t} + \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \epsilon_\alpha \ \mathbf{u}_\alpha \bigg) \nabla P + \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \nabla T - \nabla (\lambda_t \nabla T) = \frac{\delta E_H}{ \delta V \delta t}\end{array}$

The disambiguation fo the properties in the above equation is brought in The list of dynamic flow properties and model parameters.

Equations

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define the continuity of the fluid components flow or equivalently represent the mass conservation of each mass component

$\begin{array}{l}\{ m_W, \ m_O, \ m_G \}\end{array}$ during its transportation in space.

Equations

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define the motion dynamics of each phase, represented as linear correlation between phase flow speed

$\begin{array}{l}\bar u_\alpha\end{array}$ and partial pressure gradient of this phase

$\begin{array}{l}\bar \nabla P_\alpha\end{array}$ .

Equation (1) defines the heat flow continuity or equivalently represents heat conservation due to heat conduction and convection with account for adiabatic and Joule–Thomson throttling effect.

The term $\begin{array}{l}\frac{\delta E_H}{ \delta V \delta t}\end{array}$ defines the speed of change of heat energy $\begin{array}{l}E_H\end{array}$ volumetric density due to the inflow from formation into the wellbore.

The term $\begin{array}{l}\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \bar \nabla T\end{array}$ represents heat convection defined by the wellbore mass flow.

The term $\begin{array}{l}\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \epsilon_\alpha \ \mathbf{u}_\alpha \bigg) \bar \nabla P\end{array}$ represents the heating/cooling effect of the multiphase flow through the porous media. This effect is the most significant with light oils and gases.

The term $\begin{array}{l}\ \phi \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t}\end{array}$ represents the heating/cooling effect of the fast adiabatic pressure change. This usually takes effect in and around the wellbore during the first minutes or hours after changing the well flow regime (as a consequence of choke/pump operation). This effect is absent in stationary flow and negligible during the quasi-stationary flow and usually not modeled in conventional monthly-based flow simulations.

The set

Stationary Flow Model

Stationary wellbore flow is defined as the flow with constant pressure and temperature: $\begin{array}{l}\frac{\partial T}{\partial t} = 0\end{array}$ and $\begin{array}{l}\frac{\partial P}{\partial t} = 0\end{array}$ .

This happens during the long-term (usually hours & days & weeks) production/injection or long-term (usually hours & days & weeks) shut-in.

The temperature dynamic equation (1) is going to be:

(2)	$\begin{array}{l}\displaystyle \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \nabla T = \frac{\delta E_H}{ \delta V \delta t}\end{array}$

and its discrete computational scheme will be:

(3)

$\begin{array}{l}\displaystyle \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha^{k-1} \ c_{p \alpha}^{k-1} \ q_\alpha^{k-1} \bigg) T^{k-1} - \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha^k \ c_{p \alpha}^k \ q_\alpha^k \bigg) T^k = \sum_{a = \{w,o,g \}} \rho_\alpha^k \ c_{p \alpha}^k \ (q_\alpha^{k-1} - q_\alpha^k) \, (T_r^k + \epsilon_\alpha^k \delta p^k )\end{array}$

where $\begin{array}{l}\delta p^k = p_e^k - p_{wf}^k\end{array}$ is drawdown, $\begin{array}{l}p_e^k\end{array}$ – formation pressure in $\begin{array}{l}k-\end{array}$ th grid layer, $\begin{array}{l}p_{wf}^k\end{array}$ – bottom-hole pressure across $\begin{array}{l}k-\end{array}$ th grid layer, $\begin{array}{l}T_r^k\end{array}$ – remote reservoir temperature of $\begin{array}{l}k-\end{array}$ th grid layer.

The $\begin{array}{l}l-\end{array}$ axis is pointing downward along hole with $\begin{array}{l}(k-1)-\end{array}$ th grid layer sitting above the $\begin{array}{l}k-\end{array}$ th grid layer.

If the flowrate is not vanishing during the stationary lift ( $\begin{array}{l}\sum_{a = \{w,o,g \}} |q_\alpha^{k-1}| > 0\end{array}$ ) then $\begin{array}{l}T^{k-1}\end{array}$ can be calculated iteratively from previous values of the wellbore temperature $\begin{array}{l}T^k\end{array}$ as:

(4)

$\begin{array}{l}\displaystyle T^{k-1} = \frac{\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha^k \ c_{p \alpha}^k \ q_\alpha^k \bigg) T^k + \sum_{a = \{w,o,g \}} \rho_\alpha^k \ c_{p \alpha}^k \ (q_\alpha^{k-1} - q_\alpha^k) \, (T_r^k + \epsilon_\alpha^k \delta p^k )}{\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha^{k-1} \ c_{p \alpha}^{k-1} \ q_\alpha^{k-1} \bigg) }\end{array}$

Deduction

The wellbore fluid velocity $\begin{array}{l}u_\alpha\end{array}$ can be expressed thorugh the volumetric flow profile $\begin{array}{l}q_\alpha\end{array}$ and tubing/casing cross-section area $\begin{array}{l}\pi r_f^2\end{array}$ as:

(5)	$\begin{array}{l}\displaystyle u_\alpha = \frac{q_\alpha}{\pi r_f^2}\end{array}$

so that

(6)	$\begin{array}{l}\displaystyle \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \nabla T = \frac{\delta E_H}{ \delta V \delta t}\end{array}$

References

Beggs, H. D. and Brill, J. P.: "A Study of Two-Phase Flow in Inclined Pipes," J. Pet. Tech., May (1973), 607-617

The list of dynamic flow properties and model parameters

$\begin{array}{l}(t,x,y,z)\end{array}$	time and space corrdinates , $\begin{array}{l}z\end{array}$ -axis is orientated towards the Earth centre, $\begin{array}{l}(x,y)\end{array}$ define transversal plane to the $\begin{array}{l}z\end{array}$ -axis
$\begin{array}{l}\mathbf{r} = (x, \ y, \ z)\end{array}$	position vector at which the flow equations are set
$\begin{array}{l}q_{mW} = \frac{d m_W}{dt}\end{array}$	speed of water-component mass change in wellbore draining points
$\begin{array}{l}q_{mO} = \frac{d m_O}{dt}\end{array}$	speed of oil-component mass change in wellbore draining points
$\begin{array}{l}q_{mG} = \frac{d m_G}{dt}\end{array}$	speed of gas-component mass change in wellbore draining points
$\begin{array}{l}q_W = \frac{1}{\rho_W^{\LARGE \circ}} \frac{d m_W}{dt} = \frac{d V_{Ww}^{\LARGE \circ}}{dt} = \frac{1}{B_w} q_w\end{array}$	volumetric water-component flow rate in wellbore draining points recalculated to standard surface conditions
$\begin{array}{l}q_O = \frac{1}{\rho_O^{\LARGE \circ}} \frac{d m_O}{dt} = \frac{d V_{Oo}^{\LARGE \circ}}{dt} + \frac{d V_{Og}^{\LARGE \circ}}{dt} = \frac{1}{B_o} q_o + \frac{R_v}{B_g} q_g\end{array}$	volumetric oil-component flow rate in wellbore draining points recalculated to standard surface conditions
$\begin{array}{l}q_G = \frac{1}{\rho_G^{\LARGE \circ}} \frac{d m_G}{dt} = \frac{d V_{Gg}^{\LARGE \circ}}{dt} + \frac{d V_{Go}^{\LARGE \circ}}{dt} = \frac{1}{B_g} q_g + \frac{R_s}{B_o} q_o\end{array}$	volumetric gas-component flow rate in wellbore draining points recalculated to standard surface conditions
$\begin{array}{l}q_w = \frac{d V_w}{dt}\end{array}$	volumetric water-phase flow rate in wellbore draining points
$\begin{array}{l}q_o = \frac{d V_o}{dt}\end{array}$	volumetric oil-phase flow rate in wellbore draining points
$\begin{array}{l}q_g = \frac{d V_g}{dt}\end{array}$	volumetric gas-phase flow rate in wellbore draining points
$\begin{array}{l}q^S_W =\frac{dV_{Ww}^S}{dt}\end{array}$	total well volumetric water-component flow rate
$\begin{array}{l}q^S_O = \frac{d (V_{Oo}^S + V_{Og}^S )}{dt}\end{array}$	total well volumetric oil-component flow rate
$\begin{array}{l}q^S_G = \frac{d (V_{Gg}^S + V_{Go}^S )}{dt}\end{array}$	total well volumetric gas-component flow rate
$\begin{array}{l}q^S_L = q^S_W + q^S_O\end{array}$	total well volumetric liquid-component flow rate
$\begin{array}{l}P_w = P_w (t, \vec r)\end{array}$	water-phase pressure pressure distribution and dynamics
$\begin{array}{l}P_o = P_o (t, \vec r)\end{array}$	oil-phase pressure pressure distribution and dynamics
$\begin{array}{l}P_g = P_g (t, \vec r)\end{array}$	gas-phase pressure pressure distribution and dynamics
$\begin{array}{l}\vec u_w = \vec u_w (t, \vec r)\end{array}$	water-phase flow speed distribution and dynamics
$\begin{array}{l}\vec u_o = \vec u_o (t, \vec r)\end{array}$	oil-phase flow speed distribution and dynamics
$\begin{array}{l}\vec u_g = \vec u_g (t, \vec r)\end{array}$	gas-phase flow speed distribution and dynamics
$\begin{array}{l}P_{cow} = P_{cow} (s_w)\end{array}$	capillary pressure at the oil-water phase contact as function of water saturation
$\begin{array}{l}P_{cog} = P_{cog} (s_ g)\end{array}$	capillary pressure at the oil-gas phase contact as function of gas saturation
$\begin{array}{l}k_{rw} = k_{rw}(s_w, \ s_g)\end{array}$	relative formation permeability to water flow as function of water and gas saturation
$\begin{array}{l}k_{ro} = k_{ro}(s_w, \ s_g)\end{array}$	relative formation permeability to oil flow as function of water and gas saturation
$\begin{array}{l}k_{rg} = k_{rg}(s_w, \ s_g)\end{array}$	relative formation permeability to gas flow as function of water and gas saturation
$\begin{array}{l}\phi = \phi(P)\end{array}$	porosity as function of formation pressure
$\begin{array}{l}k_a = k_a(P)\end{array}$	absolute formation permeability to air
$\begin{array}{l}\vec g = (0, \ 0, \ g)\end{array}$	gravitational acceleration vector
$\begin{array}{l}g = 9.81 \ m/s^2\end{array}$	gravitational acceleration constant
$\begin{array}{l}\rho_\alpha(P,T)\end{array}$	mass density of $\begin{array}{l}\alpha\end{array}$ -phase fluid
$\begin{array}{l}\mu_\alpha(P,T)\end{array}$	viscosity of $\begin{array}{l}\alpha\end{array}$ -phase fluid
$\begin{array}{l}\lambda_t(P,T,s_w, s_o, s_g)\end{array}$	effective thermal conductivity of the rocks with account for multiphase fluid saturation
$\begin{array}{l}\lambda_r(P,T)\end{array}$	rock matrix thermal conductivity
$\begin{array}{l}\lambda_\alpha(P,T)\end{array}$	thermal conductivity of $\begin{array}{l}\alpha\end{array}$ -phase fluid
$\begin{array}{l}\rho_r(P,T)\end{array}$	rock matrix mass density
$\begin{array}{l}\eta_{s \alpha}(P,T)\end{array}$	differential adiabatic coefficient of $\begin{array}{l}\alpha\end{array}$ -phase fluid
$\begin{array}{l}c_{pr}(P,T)\end{array}$	specific isobaric heat capacity of the rock matrix
$\begin{array}{l}c_{p\alpha}(P,T)\end{array}$	specific isobaric heat capacity of $\begin{array}{l}\alpha\end{array}$ -phase fluid
$\begin{array}{l}\epsilon_\alpha (P, T)\end{array}$	differential Joule–Thomson coefficient of $\begin{array}{l}\alpha\end{array}$ -phase fluid дифференциальный коэффициент Джоуля-Томсона фазы $\begin{array}{l}\alpha\end{array}$

Page tree

Wellbore Multiphase Flow

Definition

Flow Model

Stationary Flow Model

References

The list of dynamic flow properties and model parameters