Fast Offshore Wells Model (FOWM) In this repo the FOWM, from Diehl et al., 2017 , will be replicated using Julia Programming Language.
Here is a representation of the system available in the work of Diehl et al., 2017
$$\dfrac{dm_{ga}}{dt} = W_{gc} - W_{iV}$$ (1)
$$\dfrac{dm_{gt}}{dt} = W_r\alpha_{gw} + W_{iV}-W_{whg}$$ (2)
$$\dfrac{dm_{lt}}{dt} = W_r(1-\alpha_{gw}) - W_{whl}$$ (3)
$$\dfrac{dm_{gb}}{dt} = (1-E)W_{whg}-W_g$$ (4)
$$\dfrac{dm_{gr}}{dt} = EW_{whg}+W_g-W_{gout}$$ (5)
$$\dfrac{dm_{lr}}{dt} = W_{whl}-W_{lout}$$ (6)
Being:
Variables
Description
$m_{ga}$ Gas mass in the annualar
$m_{gt}$ Gas mass in the tubbing
$m_{lt}$ Liq mass in the tubing
$m_{gb}$ Gas mass in the bubble
$m_{gr}$ Gas mass in the flowline
$m_{lr}$ Liquid mass in the flowline
$W_{iV}$ Gas mass flow from annular to tubing
$W_{r}$ Reservoir to the bottom hole mass flow
$W_{whg}$ Gas mass flow at Xmas Tree
$W_{whl}$ Liq mass flow at Xmas Tree
$W_{gc}$ Gas lift mass flow annular
$W_{g}$ Gas mass flow at the virtual valve
$W_{gout}$ Gas mass flow through topside valve (Choke)
$W_{lout}$ Gas mass flow through topside valve (Choke)
$E$ Mass fraction of gas bypassing the bubble
$\alpha_{gw}$ Gas mass fraction at resorvoir's pressure and temperature
Where:
$$W_{iV} = K_a\sqrt{\rho_{ai}(P_{ai}-P_{tb})}$$ (7)
$$W_{r} = K_r \left [1 - 0.2\dfrac{P_{bh}}{P_{r}} - \left(0.8\dfrac{P_{bh}}{P_r}\right)^2 \right]$$ (8)
$$W_{whg} = K_w\sqrt{\rho_{L}(P_{tt}-P_{rb})}\alpha_{gt}$$ (9)
$$W_{whl} = K_w\sqrt{\rho_{L}(P_{tt}-P_{rb})}(1-\alpha_{gt})$$ (10)
$$W_{g} = C_g(P_{eb} - P_{rb}) $$ (11)
$$W_{gout} = \alpha_{gr} C_{out} z \sqrt{\rho_L(P_{rt}-P_{s})}$$ (12)
$$W_{lout} = \alpha_{lr} C_{out} z \sqrt{\rho_L(P_{rt}-P_{s})}$$ (13)
Being:
Variables
Description
$K_{a}$ Flow coefficient between annular and tubing
$K_{r}$ Resorvoir's flow coefficient
$K_{w}$ Flow coefficient at Xmas Tree
$\rho_{ai}$ Gas density in the annular
$\rho_{L}$ Liquid density (assumed constant)
$\alpha_{gt}$ Gas mass fraction in tubing
$\alpha_{gr}$ Gas mass fraction in the subsea pipeline
$\alpha_{lr}$ Liquid mass fraction in the subsea pipeline
$C_{g}$ Virtual valve flow constant
$C_{out}$ Choke valve constant
$z$ Choke valve opening fraction
$P_{r}$ Reservoir's Pressure
$P_{s}$ Pressure after Choke valve
$P_{rt}$ Pressure at the top of the riser
$P_{rb}$ Pressure at the flowline before the bubble
$P_{tt}$ Pressure at the top of tubing
$P_{eb}$ Bubble Pressure
$P_{ai}$ Pressure in the annular gas injection point to tubing
$P_{tb}$ Pressure in the gas injection point on the tubing side
$P_{bh}$ Pressure in the bottom hole
$$\rho_{ai} = \dfrac{MP_{ai}}{RT}$$ (14)
$$\alpha_{gt} = \dfrac{m_{gt}}{m_{gt}+m_{lt}}$$ (15)
$$\alpha_{gr} = \dfrac{m_{gr}}{m_{gr}+m_{lr}}$$ (16)
$$\alpha_{lr} = 1-\alpha_{gr}$$ (17)
$$P_{ai} = \left(\dfrac{RT}{V_aM} + \dfrac{gL_a}{V_a} \right)m_{ga} $$ (18)
$$P_{tb} = P_{tt} + \rho_{mt}gH_{vgl}$$ (19)
$$P_{bh} = P_{pdg} + \rho_{mres}g(H_t-H_{pdg})$$ (20)
$$P_{pdg} = P_{tb} + \rho_{mres}g(H_{pdg}-H_{vgl})$$ (21)
$$P_{tt} = \dfrac{\rho_{gt}RT}{M}$$ (22)
$$P_{rb} = P_{rt}+\dfrac{(m_{lr}+m_{L,still})gsin(\theta)}{A_{ss}}$$ (23)
$$P_{eb} = \dfrac{m_{gb}RT}{MV_{eb}}$$ (24)
$$P_{rt} = \dfrac{m_{gr}RT}{M\left(\omega_{u}V_{ss}-\dfrac{m_{lr}+m_{L,still}}{\rho_L}\right)}$$ (25)
Being:
Variables
Description
$R$ Universal gas constant
$T$ Average temperature
$M$ Gas molecular weight
$\rho_{mt}$ Mixture density
$\rho_{mres}$ Reservoir's density
$\rho_{gt}$ Gas density
$g$ Gravity acceleration
$V_{a}$ Annular volume
$V_{eb}$ Bubble Volume
$V_{ss}$ Pipe volume downstream virtual valve
$H_{t}$ Vertical length Xmas Tree - Bottom Hole
$H_{pdg}$ Vertical length Xmas Tree - PDG point
$H_{vgl}$ Vertical length Xmas Tree - Gas Lift
$A_{ss}$ Riser cross sectional area
$L_{a}$ Annular length
$m_{L,still}$ Minimum mass of liq in the subsea pipeline
$\omega_{u}$ Bubble Location (assistant parameter)
$\theta$ Average riser inclination
Where:
$$\rho_{mt} = \dfrac{m_{gt}+m_{lt}}{V_t}$$ (26)
$$\rho_{gt} = \dfrac{m_{gt}}{V_{gt}}$$ (27)
$$V_{gt} = V_t - \dfrac{m_{lt}}{\rho_L}$$ (28)
$$A_{ss} = \dfrac{\pi D_{ss}^2}{4}$$ (29)
$$V_{ss} = \dfrac{\pi D_{ss}^2L_r}{4}+\dfrac{\pi D_{ss}^2L_{fl}}{4}$$ (30)
$$V_a = \dfrac{\pi D_{a}^2L_a}{4}$$ (31)
$$V_t = \dfrac{\pi D_{t}^2L_t}{4}$$ (32)
Being:
Variables
Description
$V_t$ Volume of tubing
$V_{gt}$ Gas volume in the tubing
$L_r$ Riser length
$L_{fl}$ Flowline length
$L_t$ Tubing length
$D_a$ Annular diameter
$D_t$ Tubing diameter
$D_{ss}$ Subsea pipeline diameter
Variables
Value
Unit
$\rho_L$ $900$ $kg/m^3$
$P_r$ $225$ $bar$
$P_s$ $10$ $bar$
$\alpha_{gw}$ $0.0188$ $-$
$\rho_{mres}$ $892$ $kg/m^3$
$M$ $18$ $kg/kmol$
$T$ $298$ $K$
$L_{r}$ $1569$ $m$
$L_{fl}$ $2928$ $m$
$L_{t}$ $1639$ $m$
$L_{a}$ $1118$ $m$
$H_{t}$ $1279$ $m$
$H_{pdg}$ $1117$ $m$
$H_{vgl}$ $916$ $m$
$D_{ss}$ $0.15$ $m$
$D_{t}$ $0.15$ $m$
$D_{a}$ $0.14$ $m$
$m_{L,still}$ $6.222\times 10^{+1}$ kg
$C_{g}$ $1.137\times 10^{-3}$ $-$
$C_{out}$ $2.039\times 10^{-3}$ $-$
$V_{eb}$ $6.098\times 10^{+1}$ $m^3$
$E$ $1.545\times 10^{-1}$ $-$
$K_{w}$ $6.876\times 10^{-4}$ $-$
$K_{a}$ $2.293\times 10^{-5}$ $-$
$K_{r}$ $1.269\times 10^{+2}$ $-$
$\omega_{u}$ $2.780\times 10^{0}$ $-$
