Motivation

Assume the well is producing 

While going down to depth 

Wellbore temperature 

The volume shares 

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.

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: 

The 

\gamma_\alpha = \frac{q_\alpha}{q_t}

where 

q_t = \sum_\alpha q_\alpha = q_w + q_o + g_g

In multiphase wellbore flow each phase occupies its own area 

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 

s_\alpha = \frac{A_\alpha}{A}

so that a sum of all in-situ hold-ups is subject to natural constraint:

\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 

u_\alpha = \frac{q_\alpha}{A_\alpha}

where 

The superficial velocity of 

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:

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 

u_{\alpha_1 \alpha_2} = u_{\alpha_1} - u_{\alpha_2}

The multiphase fluid density 

\rho_m =  \sum_\alpha  s_\alpha \rho_\alpha

where 

The two-phase gas-liquid model is defined in the following terms:

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:

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

Application

Mathematical Model

Homogeneous Model

The multiphase homogeneous wellbore flow model assumes that fluid is at stationary hydrodynamic and thermodynamic equilibrium and there is no slip between phases.

This means that all have the same temperature:

T_w = T_o = T_g = T

same pressure:

P_w = P_o = P_g = T

same velocity:

u_w = u_o = u_g = u_m

and all dynamic fluid parameters are constant in time:

\frac{\partial T}{\partial t} = 0, \quad \frac{\partial P}{\partial t} = 0, \quad \frac{\partial u_m}{\partial t} = 0

The model is defined by the following set of 1D equations: 

A(l) \rho_m u_m = \rho_{sm} q_{s} = \rm const

\frac{dp}{dl} = - \rho_m  u_m \frac{\partial u_m}{\partial l}    + \rho_m \, g \, \cos \theta + \frac{ f_m \, \rho_m \, u_m^2 \, }{2 d}

\sum_\alpha \big( \rho_\alpha \, c_{p \, \alpha}  \big) \ q_m \  \frac{\partial T}{\partial l}
 \  =   \    \sum_\alpha \rho_\alpha \ c_{p \alpha} T_\alpha \frac{\partial  q_\alpha}{\partial l}  

The right side of equation 

T_\alpha = T_r + \epsilon_\alpha \, \delta p = T_r + \epsilon_\alpha \, (p_e - p)

Derivation – Homogenuous Wellbore Multiphase Flow

The discrete computational scheme for 

\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 

The 

If the flowrate is not vanishing during the stationary lift (

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) } 

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 

The model is defined by the following set of 1D equations: 

A(l) \sum_\alpha \rho_\alpha u_\alpha = \sum_\alpha \rho_{s\alpha} q_{s\alpha} = \rm const

\sum_\alpha \rho_\alpha  u_\alpha \frac{\partial u_\alpha}{\partial l}   =  - \frac{dp}{dl} + \rho_m \, g \, \cos \theta + \frac{ f_m \, \rho_m \, u_m^2 \, }{2 d}

\sum_\alpha \rho_\alpha \ c_{p \alpha} \ u_\alpha \frac{\partial T}{\partial l}
 \  =   \   \frac{1}{A}  \ \sum_\alpha \rho_\alpha \ c_{p \alpha} T_\alpha \frac{\partial  q_\alpha}{\partial l} 

It carries the original reservoir temperature with heating/cooling effect from reservoir-flow throttling and well-reservoir contact throttling:

T_\alpha = T_r + \epsilon_\alpha \, \delta P = T_r + \epsilon_\alpha \, (P_e - P_{wf})

see Nomenclature below.

References

Nomenclature