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) | (\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} |
The disambiguation fo 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 \{ m_W, \ m_O, \ m_G \} during its transportation in space.Equations
– define the motion dynamics of each phase, represented as linear correlation between phase flow speed \bar u_\alpha and partial pressure gradient of this phase \bar \nabla P_\alpha .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 \frac{\delta E_H}{ \delta V \delta t} defines the speed of change of heat energy E_H volumetric density due to the inflow from formation into the wellbore.
The term
\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \bar \nabla T represents heat convection defined by the wellbore mass flow.
The term \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \epsilon_\alpha \ \mathbf{u}_\alpha \bigg) \bar \nabla P 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 \ \phi \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t} 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: \frac{\partial T}{\partial t} = 0 and \frac{\partial P}{\partial t} = 0 .
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) | \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} |
and its discrete computational scheme will be:
(3) | \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 ) |
where \delta p^k = p_e^k - p_{wf}^k is drawdown, p_e^k – formation pressure in k-th grid layer, p_{wf}^k – bottom-hole pressure across k-th grid layer, T_r^k – remote reservoir temperature of k-th grid layer.
The l-axis is pointing downward along hole with (k-1)-th grid layer sitting above the k-th grid layer.
If the flowrate is not vanishing during the stationary lift ( \sum_{a = \{w,o,g \}} |q_\alpha^{k-1}| > 0) then T^{k-1} can be calculated iteratively from previous values of the wellbore temperature T^k as:
(4) | 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) } |
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
(t,x,y,z) | time and space corrdinates , z -axis is orientated towards the Earth centre, (x,y) define transversal plane to the z -axis |
\mathbf{r} = (x, \ y, \ z) | position vector at which the flow equations are set |
q_{mW} = \frac{d m_W}{dt} | speed of water-component mass change in wellbore draining points |
q_{mO} = \frac{d m_O}{dt} | speed of oil-component mass change in wellbore draining points |
q_{mG} = \frac{d m_G}{dt} | speed of gas-component mass change in wellbore draining points |
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 | volumetric water-component flow rate in wellbore draining points recalculated to standard surface conditions |
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 | volumetric oil-component flow rate in wellbore draining points recalculated to standard surface conditions |
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 | volumetric gas-component flow rate in wellbore draining points recalculated to standard surface conditions |
q_w = \frac{d V_w}{dt} | volumetric water-phase flow rate in wellbore draining points |
q_o = \frac{d V_o}{dt} | volumetric oil-phase flow rate in wellbore draining points |
q_g = \frac{d V_g}{dt} | volumetric gas-phase flow rate in wellbore draining points |
q^S_W =\frac{dV_{Ww}^S}{dt} | total well volumetric water-component flow rate |
q^S_O = \frac{d (V_{Oo}^S + V_{Og}^S )}{dt} | total well volumetric oil-component flow rate |
q^S_G = \frac{d (V_{Gg}^S + V_{Go}^S )}{dt} | total well volumetric gas-component flow rate |
q^S_L = q^S_W + q^S_O | total well volumetric liquid-component flow rate |
P_w = P_w (t, \vec r) | water-phase pressure pressure distribution and dynamics |
P_o = P_o (t, \vec r) | oil-phase pressure pressure distribution and dynamics |
P_g = P_g (t, \vec r) | gas-phase pressure pressure distribution and dynamics |
\vec u_w = \vec u_w (t, \vec r) | water-phase flow speed distribution and dynamics |
\vec u_o = \vec u_o (t, \vec r) | oil-phase flow speed distribution and dynamics |
\vec u_g = \vec u_g (t, \vec r) | gas-phase flow speed distribution and dynamics |
P_{cow} = P_{cow} (s_w) | capillary pressure at the oil-water phase contact as function of water saturation |
P_{cog} = P_{cog} (s_ g) | capillary pressure at the oil-gas phase contact as function of gas saturation |
k_{rw} = k_{rw}(s_w, \ s_g) | relative formation permeability to water flow as function of water and gas saturation |
k_{ro} = k_{ro}(s_w, \ s_g) | relative formation permeability to oil flow as function of water and gas saturation |
k_{rg} = k_{rg}(s_w, \ s_g) | relative formation permeability to gas flow as function of water and gas saturation |
\phi = \phi(P) | porosity as function of formation pressure |
k_a = k_a(P) | absolute formation permeability to air |
\vec g = (0, \ 0, \ g) | gravitational acceleration vector |
g = 9.81 \ m/s^2 | gravitational acceleration constant |
\rho_\alpha(P,T) | mass density of \alpha-phase fluid |
\mu_\alpha(P,T) | viscosity of
\alpha-phase fluid |
\lambda_t(P,T,s_w, s_o, s_g) | effective thermal conductivity of the rocks with account for multiphase fluid saturation |
\lambda_r(P,T) | rock matrix thermal conductivity |
\lambda_\alpha(P,T) | thermal conductivity of \alpha-phase fluid |
\rho_r(P,T) | rock matrix mass density |
\eta_{s \alpha}(P,T) | differential adiabatic coefficient of \alpha-phase fluid |
c_{pr}(P,T) | specific isobaric heat capacity of the rock matrix |
c_{p\alpha}(P,T) | specific isobaric heat capacity of \alpha-phase fluid |
\epsilon_\alpha (P, T) | differential Joule–Thomson coefficient of \alpha-phase fluid дифференциальный коэффициент Джоуля-Томсона фазы \alpha |