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

Mathematical Model

The wellbore flow is modeled with 1D-model with $\begin{array}{l}l-\end{array}$ axis aligned with well trajectory.

(1)

$\begin{array}{l}\displaystyle (\rho \,c_{pt})_p \frac{\partial T}{\partial t} - \sum_\alpha \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t} + \sum_\alpha \rho_\alpha \ c_{p \alpha} \ u_\alpha \frac{\partial T}{\partial l} \ = \ \frac{1}{ \pi r_f^2} \ \sum_\alpha \rho_\alpha \ c_{p \alpha} \frac{\partial T_\alpha q_\alpha}{\partial l}\end{array}$

where $\begin{array}{l}\alpha = \{w,o,g \}\end{array}$ .

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

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

The term $\begin{array}{l}\sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ u_\alpha \frac{\partial T}{\partial l}\end{array}$ represents heat convection defined by the wellbore mass flow.

The term $\begin{array}{l}\sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t}\end{array}$ 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.

Derivation

(2)

$\begin{array}{l}\displaystyle (\rho \,c_{pt})_p \frac{\partial T}{\partial t} - \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t} + \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ u_\alpha \frac{\partial T}{\partial l} \ = \ \frac{\delta E_H}{ \delta V \delta t}\end{array}$

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.

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 $\begin{array}{l}\frac{\partial P_\alpha}{\partial t} = 0\end{array}$ .

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:

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

and its discrete computational scheme will be:

(4)

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

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:

(5)

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

Deduction

(6)

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

The 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:

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

so that

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

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

Mathematical Model

Stationary Flow Model

References

The list of dynamic flow properties and model parameters