Fast Offshore Wells Model (FOWM)

In this repo the FOWM, from Diehl et al., 2017, will be replicated using Julia Programming Language.

System Representation

Here is a representation of the system available in the work of Diehl et al., 2017

Model

$$\dfrac{dm_{ga}}{dt} = W_{gc} - W_{iV}$$ (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

Where:

$$\rho_{ai} = \dfrac{MP_{ai}}{RT}$$ (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

Simulation

Parameters

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
$C_{g}$	$1.137\times 10^{-3}$	$-$
$C_{out}$	$2.039\times 10^{-3}$	$-$
$V_{eb}$	$6.098\times 10^{+1}$	$m^3$
$E$	$1.545\times 10^{-1}$	$-$
$K_{w}$	$6.876\times 10^{-4}$	$-$
$K_{a}$	$2.293\times 10^{-5}$	$-$
$K_{r}$	$1.269\times 10^{+2}$	$-$
$\omega_{u}$	$2.780\times 10^{0}$	$-$