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Motivation
Assume the well is producing
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While going down to depth
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Wellbore temperature
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The volume shares
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It is now possible to simulate the stationary multiphase wellbore flow and link the surface flow conditions (pressure, temperature and rates) to downhole flow conditions at any depth.
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The most adequate and practical model can be built for a stationary fluid flow at hydrodynamic and thermodynamic equilibrium.
If flow is changing slowly over time the same solutions can be used with inputs as time functions which is called quasi-stationary flow conditions.
This model finds important applications in industry.
If well production has been changed abruptly (for example opening or closing the flow) there will be a transition period towards the new quasi-stationary flow conditions (usually minutes or few hours).
The transition period for pressure and temperature is different and wellbore temperature takes much longer time to stabilize than pressure.
Unlike single-phase flow (Wellbore Water Flow and Wellbore Gas Flow) the complexity of the multiphase flow during transition is so high and unstable that building a simple mathematical model does not make a practical sense.
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 controled by gathering system or injection pump
- wellbore design (pipe diameters, pipe materials and inter-pipe annular fillings)
- fluid friction with tubing /casing walls
- interfacial phase slippage
- heat exchange between wellbore fluid and surrounding rocks via complex well design
Consider a 3-phase water-oil-gas flow:
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\gamma_\alpha = \frac{q_\alpha}{q_t} |
where
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q_t = \sum_\alpha q_\alpha = q_w + q_o + g_g |
In multiphase wellbore flow each phase occupies its own area
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This area can be connected into a single piece of cross-sectional area (like in case of slug or annular flow) or dispersed into a number of connected spots (like in case of bubbly flow).
A share of total pipe cross-section area occupied by moving
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s_\alpha = \frac{A_\alpha}{A} |
so that a sum of all in-situ hold-ups is subject to natural constraint:
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\sum_\alpha s_\alpha = s_w + s_o + s_g = 1 |
When word hold-up is used alone it usually means in-situ hold-up and should not be confused with input hold-up or no-slip hold-up which should be better called volumetric flow fraction.
The actual average cross-sectional velocity of moving
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u_\alpha = \frac{q_\alpha}{A_\alpha} |
where
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The superficial velocity of
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u_{s \alpha} = \frac{q_\alpha}{A}= s_\alpha \cdot u_\alpha |
The multiphase mixture velocity is defined as total flow volume normalized by the total cross-sectional area:
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u_m = \frac{1}{A} \sum_\alpha q_\alpha = \sum_\alpha u_{s \alpha} = \sum_\alpha s_\alpha \cdot u_\alpha |
The difference between velocities of
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u_{\alpha_1 \alpha_2} = u_{\alpha_1} - u_{\alpha_2} |
The multiphase fluid density
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\rho_m = \sum_\alpha s_\alpha \rho_\alpha |
where
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The two-phase gas-liquid model is defined in the following terms:
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u_m = u_{s g} + u_{s l} = s_g u_g + (1-s_g) u_l |
The two-phase oil-water model is defined in the following terms:
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u_m = u_{s o} + u_{s w} = s_o u_o + (1-s_o-s_g) u_w |
The 3-phase water-oil-gas model is usually built as a superposition of gas-liquid model and then oil-water model:
Input & Output
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Application
Activity | Input | Output | ||||||||||
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1 | WPA – Well Performance Analysis | Optimizing the lift performance based on the IPR vs VLP models |
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2 | DM – Dynamic Modelling | Relating production rates at separator to bottom-hole pressure with VLP |
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3 | PRT – Pressure Testing | Adjust gauge pressure to formation datum |
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4 | PLT – Production Logging | Interpretation of production logs |
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5 | RFP – Reservoir Flow Profiling | Interpretation of reservoir flow logs |
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Mathematical Model
Homogeneous Model
The multiphase homogeneous wellbore flow in model assumes that fluid is at stationary hydrodynamic and thermodynamic equilibrium is defined by the following set of 1D equations: and there is no slip between phases.
This means that all have the same temperature:
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A(l) \sum_\alpha \rho_\alpha u_\alphaT_w = T_o = \sumT_\alpha \rho_{s\alpha} q_{s\alpha} = \rm constg = T |
same pressure:
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P_w = P_o = P_g = T |
same velocity:
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\sum_\alpha \rho_\alpha u_w = u_o = u_\alpha \frac{\partial u_\alphag = u_m |
and all dynamic fluid parameters are constant in time:
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\frac{\partial T}{\partial lt} = 0, -\quad \frac{dp\partial P}{dl} + \rho_m \, g \, \sin \theta -\partial t} = 0, \quad \frac{\partial fu_m }{\, \rho_m \, u_m^2 \, }{2 d}partial t} = 0 |
The model is defined by the following set of 1D equations:
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\sum_\alphaA(l) \rho_m u_m = \rho_\alpha \ c{sm} q_{p \alpha}s} = \rm const |
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\frac{dp}{dl} = - \rho_m u_mu_\alpha \frac{\partial Tu_m}{\partial l} \ = + \rho_m \, g \, \frac{1}{A} cos \theta + \frac{ \sumf_m \alpha, \rho_m \, u_m^2 \, }{2 d} |
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\sum_\alpha \big( \rho_\alpha \, c_{p \, \alpha} T_\alpha \big) \ q_m \ \frac{\partial q_\alphaT}{\partial l} |
It carries the original reservoir temperature with heating/cooling effect from reservoir-flow throttling and well-reservoir contact throttling:
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T_\alpha = T_r + \epsilon \ = \ \sum_\alpha \rho_\,alpha \delta P =c_{p \alpha} T_r +\alpha \epsilon_\alpha \, (P_e - P_{wf}) |
where
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time and space corrdinates ,
-axis is orientated towards the Earth centre, LaTeX Math Inline body z
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measured depth along wellbore trajectory
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differential Joule–Thomson coefficient of
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differential adiabatic coefficient of
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Darci flow friction coefficient at fluid velocity
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cross-sectional area
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volumetric flow rate
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in-situ velocity of
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gravitational acceleration constant
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effective thermal conductivity of the rocks with account for multiphase fluid saturation
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kinematic viscosity of
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thermal conductivity of
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dynamic viscosity of
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rock matrix mass density
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specific isobaric heat capacity of
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specific isobaric heat capacity of the rock matrix
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The right side of equation
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T_\alpha = T_r + \epsilon_\alpha \, \delta p = T_r + \epsilon_\alpha \, (p_e - p) |
The discrete computational scheme for
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\bigg( \sum_\alpha \rho_\alpha^{k-1} \ c_{p \alpha}^{k-1} \ q_\alpha^{k-1} \bigg) T^{k-1} - \bigg( \sum_\alpha \rho_\alpha^k \ c_{p \alpha}^k \ q_\alpha^k \bigg) T^k
= \sum_\alpha \rho_\alpha^k \ c_{p \alpha}^k \ (q_\alpha^{k-1} - q_\alpha^k) \, (T_r^k + \epsilon_\alpha^k \delta p^k ) |
where
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The
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If the flowrate is not vanishing during the stationary lift (
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T^{k-1} = \frac{\bigg( \sum_\alpha \rho_\alpha^k \ c_{p \alpha}^k \ q_\alpha^k \bigg) T^k + \sum_\alpha \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_\alpha \rho_\alpha^{k-1} \ c_{p \alpha}^{k-1} \ q_\alpha^{k-1} \bigg) }
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Slippage Model
The multiphase slippage wellbore flow model assumes that fluid is at hydrodynamic and thermodynamic equilibrium and there is a slip between phases so that phases may be moving with different flow speeds
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The model is defined by the following set of 1D equations:
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The discrete computational scheme for
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\bigg( \sum_\alpha \rho_\alpha^{k-1} \ c_{p \alpha}^{k-1} \ q_\alpha^{k-1} \bigg) T^{k-1} - \bigg( alpha u_\alpha = \sum_\alpha \rho_{s\alpha} q_{s\alpha} = \rm const |
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\sum_\alpha \rho_\alpha^kalpha u_\alpha c_{p \frac{\partial u_\alpha}^k {\ q_\alpha^k \bigg) T^k partial l} = - \sum_\alphafrac{dp}{dl} + \rho_m \alpha^k, g \ c_{p, \cos \alpha}^ktheta \+ (q_\alpha^frac{k-1} - q_\alpha^k) f_m \, (T\rho_r^km +\, \epsilonu_\alpha^km^2 \delta, p^k ) |
where
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The
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If the flowrate is not vanishing during the stationary lift (
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T^{k-1} = \frac{\bigg( \sum_\alpha \rho_\alpha^k \ c_{p \alpha}^k \ q_\alpha^k \bigg) T^k + \sum_\alpha \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_\alpha \rho_\alpha^{k-1} \ c_{p \alpha}^{k-1} \ q_\alpha^{k-1} \bigg) }
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title | Derivation |
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(\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} |
Equation
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The term
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The multiphase wellbore flow in hydrodynamic and thermodynamic equilibrium is defined by the following set of 1D equations:
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\frac{\partial (\rho_m A)}{\partial t} + \frac{\partial}{\partial l} \bigg( A \, \sum_\alpha \rho_\alpha \, u_\alpha \bigg) = 0 |
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\sum_\alpha \rho_\alpha \bigg[ \frac{\partial u_\alpha}{\partial t} + u_\alpha \frac{\partial u_\alpha}{\partial l} - \nu_\alpha \Delta u_\alpha\bigg] = - \frac{dp}{dl} + \rho_m \, g \, \sin \theta - \frac{ f_m \, \rho_m \, u_m^2 \, }{2 d} |
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(\rho \,c_p)_m \frac{\partial T}{\partial t}
- \bigg( \sum_\alpha \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha}\bigg) \ \frac{\partial p}{\partial t}
+ \bigg( \sum_\alpha \rho_\alpha \ c_{p \alpha} \ u_\alpha \bigg) \frac{\partial T}{\partial l}
\ = \ \frac{1}{A} \ \sum_\alpha \rho_\alpha \ c_{p \alpha} T_\alpha \frac{\partial q_\alpha}{\partial l} |
Equations
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The model is set in 1D-model with
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The disambiguation of the properties in the above equation is brought in The list of dynamic flow properties and model parameters.
Equation
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Equation
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The term
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This usually takes effect in the wellbore during the first minutes or hours after changing the well flow regime (as a consequence of choke/pump operation).
The term
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(\rho \,c_p)_m = \sum_\alpha \rho_\alpha c_\alpha s_\alpha |
Stationary wellbore flow is defined as the flow with constant pressure and temperature:
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This happens during the long-term (usually hours & days & weeks) production/injection or long-term (usually hours & days & weeks) shut-in.
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(\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 wellbore fluid velocity
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u_\alpha = \frac{q_\alpha}{\pi r_f^2} |
so that
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\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \nabla T
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References
Beggs, H. D. and Brill, J. P.: "A Study of Two-Phase Flow in Inclined Pipes," J. Pet. Tech., May (1973), 607-617}{2 d} |
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\sum_\alpha \rho_\alpha \ c_{p \alpha} \ u_\alpha \frac{\partial T}{\partial l}
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It carries the original reservoir temperature with heating/cooling effect from reservoir-flow throttling and well-reservoir contact throttling:
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T_\alpha = T_r + \epsilon_\alpha \, \delta P = T_r + \epsilon_\alpha \, (P_e - P_{wf}) |
see Nomenclature below.
References
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Nomenclature
| position vector at which the flow equations are set | ||||||||||||||||||||||
| time and space corrdinates ,
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| measured depth along wellbore trajectory
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| indicates a mixture of fluid phases | ||||||||||||||||||||||
| water, oil, gas phase indicator |
| differential Joule–Thomson coefficient of
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| pressure at separator |
| differential adiabatic coefficient of
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| temperature at separator |
| Darcy friction factor at fluid velocity
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| wellbore fluid pressure |
| cross-sectional average pipe flow diameter | ||||||||||||||||||||
| wellbore fluid temperature |
| cross-sectional area
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| volumetric flow rate
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| wellbore trajectory inclination to horizon | ||||||||||||||||||||
| in-situ velocity of
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| effective thermal conductivity of the rocks with account for multiphase fluid saturation | ||||||||||||||||||||
| cross-sectional average fluid density |
| rock matrix thermal conductivity | ||||||||||||||||||||
| kinematic viscosity of
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| dynamic viscosity of
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| specific isobaric heat capacity of
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| specific isobaric heat capacity of the rock matrix |