List of Symbols

A area, ft²

A_b total effective bellows area, in.²

A_eng net cross-sectional area of engine piston, in.²

A_fb total firebox surface area, ft²

A′_i inner area of tubing sleeve, in.²

A′_o outer area of tubing sleeve, in.²

A_p valve seat area, gross plunger cross-sectional area, or inner area of packer, in.²

A_pump net cross-sectional area of pump piston, in.²

A_r cross-sectional area of rods, in.²

A_t tubing inner cross-sectional area, in.²

^oAPI API gravity of stock tank oil

B formation volume factor of fluid, rb/stb

b constant 1.5×10⁻⁵ in SI units

B_o formation volume factor of oil, rb/stb

B_w formation volume factor of water, rb/bbl

C_A drainage area shape factor

C_a weight fraction of acid in the acid solution

C_c choke flow coefficient

C_D choke discharge coefficient

C_g correction factor for gas-specific gravity

C_i productivity coefficient of lateral i

C_l clearance, fraction

C_m mineral content, volume fraction

C_s structure unbalance, lb_f

C_t correction factor for operating temperature

c_t total compressibility, psi⁻¹

C_p specific heat of gas at constant pressure, lb_f-ft/lb_m-R

${\overline{C}}_p$ specific heat under constant pressure evaluated at cooler

C_wi water content of inlet gas, lb_m H₂O/MMscf

D outer diameter, in., or depth, ft, or non-Darcy flow coefficient, d/Mscf, or molecular diffusion coefficient, m²/s

d diameter, in.

d₁ upstream pipe diameter, in.

d₂ choke diameter, in.

d_b barrel inside diameter, in.

D_ci inner diameter of casing, in.

d_f fractal dimension constant 1.6

D_h hydraulic diameter, in.

D_H hydraulic diameter, ft

D_i inner diameter of tubing, in.

D_o outer diameter, in.

d_p plunger outside diameter, in.

D_pump minimum pump depth, ft

D_r length of rod string, ft

E rotor/stator eccentricity, in., or Young's modulus, psi

E_v volumetric efficiency, fraction

e_v correction factor

e_p efficiency

F_b axial load, lb_f

F_CD fracture conductivity, dimensionless

F_F fanning friction factor

F_gs modified Foss and Gaul slippage factor

f_hi flow performance function of the vertical section of lateral i

f_Li inflow performance function of the horizontal section of lateral i

f_M Darcy-Wiesbach (Moody) friction factor

F_pump pump friction-induced pressure loss, psia

f_Ri flow performance function of the curvic section of lateral i

f_sl slug factor, 0.5 to 0.6

G shear modulus, psia

g gravitational acceleration, 32.17 ft/s²

G_b pressure gradient below the pump, psi/ft

g_c unit conversion factor, $32.17{\mathrm{lb}}_{\mathrm{m}}-{\text{ft}/\text{1b}}_{\mathrm{f}}-{\mathrm{s}}^2$

G_fd design unloading gradient, psi/ft

G_i initial gas-in-place, scf

G_p cumulative gas production, scf

$G_p^1$ cumulative gas production per stb of oil at the beginning of the interval, scf

G_s static (dead liquid) gradient, psi/ft

G₂ mass flux at downstream, lb_m/ft²/sec

GLR_fm formation oil GLR, scf/stb

GLR_inj injection GLR, scf/stb

GLR_min minimum required GLR for plunger lift, scf/bbl

GLR_opt,o optimum GLR at operating flow rate, scf/stb

GOR producing gas-oil ratio, scf/stb

GWR glycol to water ratio, gal TEG/lb_m H₂O

H depth to the average fluid level in the annulus, ft, or dimensionless head

h reservoir thickness, ft, or pumping head, ft

h_f fracture height, ft

HP required input power, hp

Hp_MM required theoretical compression power, hp/MMcfd

H_t total heat load on reboiler, Btu/h

Δh depth increment, ft

ΔHp_m mechanical power losses, hp

∇_hi pressure gradient in the vertical section of lateral i, psi/ft

J productivity of fractured well, stb/d-psi

J_i productivity index of lateral i.

J_o productivity of nonfractured well, stb/d-psi

K empirical factor, or characteristic length for gas flow in tubing, ft

k permeability of undamaged formation, md, or specific heat ratio

k_f fracture permeability, md

k_H the average horizontal permeability, md

k_h the average horizontal permeability, md

k_i liquid/vapor equilibrium ratio of compound i

k_p a constant

k_ro the relative permeability to oil

k_V vertical permeability, md

L length, ft, or tubing inner capacity, ft/bbl

L_g length of gas distribution line, mile

L_N net lift, ft

L_p length of plunger, in.

M total mass associated with 1 stb of oil

M₂ mass flow rate at downstream, lb_m/sec

MW_a molecular weight of acid

MW_m molecular weight of mineral

N pump speed, spm, or rotary speed, rpm

n number of layers, or polytropic exponent for gas

N_Ac acid capillary number, dimensionless

N_Cmax maximum number of cycles per day

n_G number of lb-mole of gas

N_i initial oil in place in the well drainage area, stb

n_i productivity exponent of lateral i

n_L number of mole of fluid in the liquid phase

N_max maximum pump speed, spm

n_p number of pitches of stator

$N_p^1$ cumulative oil production per stb of oil in place at the beginning of the interval

$N_{p,n}^f$ forcasted annual cumulative production of fractured well for year n

$N_{p,n}^{nf}$ predicted annual cumulative production of nonfractured well for year n

$N_{p,n}^{no}$ predicted annual cumulative production of non-optimized well for year n

$N_{p,n}^{op}$ forcasted annual cumulative production of optimized system for year n

N_Re Reunolds number

N_s number of compression stages required

N_st number of separation stages – 1

n_V number of mole of fluid in the vapor phase

N_w number of wells

ΔN_p,n predicted annual incremental cumulative production for year n

P pressure, lb/ft²

p pressure, psia

p_b base pressure, psia

p_bd formation breakdown pressure, psia

P_c casing pressure, psig

p_c critical pressure, psia, or required casing pressure, psia, or the collapse pressure with no axial load, psia

p_cc the collapse pressure corrected for axial load, psia

P_cd2 design injection pressure at valve 2, psig

P_Cmin required minimum casing pressure, psia

p_c,s casing pressure at surface, psia

p_c,v casing pressure at valve depth, psia

P_d pressure in the dome, psig

p_d final discharge pressure, psia

p_eng,d engine discharge pressure, psia

p_eng,i pressure at engine inlet, psia

p_f frictional pressure loss in the power fluid injection tubing, psi

P_h hydraulic power, hp

p_h hydrostatic pressure of the power fluid at pump depth, psia

p_hf wellhead flowing pressure, psia

p_hfi flowing pressure at the top of lateral i, psia

p_L pressure at the inlet of gas distribution line, psia

p_i initial reservoir pressure, psia, or pressure in tubing, psia, or pressure at stage i, psia p_kd1 kick-off pressure opposite the first valve, psia

p_kfi flowing pressure at the kick-out-point of lateral i, psia

p_L pressure at the inlet of the gas distribution line, psia

P_lf flowing liquid gradient, psi/bbl slug

P_lh hydrostatic liquid gradient, psi/bbl slug

p_Lmax maximum line pressure, psia

p_o pressure in the annulus, psia

p_out output pressure of the compression station, psia

P_p W_p/A_t, psia

p_p pore pressure, psi

p_pc pseudocritical pressure, psia

p_pump,i pump intake pressure, psia

p_pump,d pump discharge pressure, psia

P_r pitch length of rotor, ft

p_r pseudoreduced pressure

P_s pitch length of stator, ft, or shaft power, ft–lb_f/sec

p_s surface operating pressure, psia, or suction pressure, psia, or stock-tank pressure, psia

p_sc standard pressure, 14.7 psia

p_sh slug hydrostatic pressure, psia

p_si surface injection pressure, psia

p_suction suction pressure of pump, psia

P_t tubing pressure, psia

p_tf flowing tubing head pressure, psig

p_up pressure upstream the choke, psia

P_vc valve closing pressure, psig

P_vo valve opening pressure, psig

p_wh upstream (wellhead) pressure, psia

p_wf flowing bottom hole pressure, psia

p_wfi the average flowing bottom-lateral pressure in lateral i, psia

p_wfo dynamic bottom hole pressure because of cross-flow between, psia

$p_{wf}^c$ critical bottom hole pressure maintained during the production decline, psia

p_up upstream pressure at choke, psia

P₁ pressure at point 1 or inlet, lb_f/ft²

P₂ pressure at point 2 or outlet, lb_f/ft²

p₁ upstream/inlet/suction pressure, psia

p₂ downstream/outlet/discharge pressure, psia

$\overline{p}$ average reservoir pressure, psia

${\overline{p}}_f$ reservoir pressure in a future time, psia

${\overline{p}}_0$ average reservoir pressure at decline time zero, psia

${\overline{p}}_t$ average reservoir pressure at decline time t, psia

ΔP pressure drop, lb_f/ft²

Δp pressure increment, psi

δ_p head rating developed into an elementary cavity, psi

Δp_f frictional pressure drop, psia

Δp_h hydrostatic pressure drop, psia

Δp_{i avg} the average pressure change in the tubing, psi

Δp_{o avg} the average pressure change in the annulus, psi

Δp_sf safety pressure margin, 200 to 500 psi

Δp_v pressure differential across the operating valve (orifice), psi

Q volumetric flow rate

q volumetric flow rate

Q_c pump displacement, bbl/day

q_eng flow rate of power fluid, bbl/day

Q_G gas production rate, Mscf/day

q_G glycol circulation rate, gal/hr

q_g gas production rate, scf/day

q_g,inj the lift gas injection rate (scf/day) available to the well

q_gM gas flow rate, Mscf/d

q_g,total total output gas flow rate of the compression station, scf/day

q_h injection rate per unit thickness of formation, m³/sec-m

q_i flow rate from/into layer i, or pumping rate, bpm

q_i,max maximum injection rate, bbl/min

q_L liquid capacity, bbl/day

Q_o oil production rate, bbl/day

q_o oil production rate, bbl/day

q_pump flow rate of the produced fluid in the pump, bbl/day

Q_s leak rate, bbl/day, or solid production rate, ft³/day

q_s gas capacity of contactor for standard gas (0.7 specific gravity) at standard temperature (100°F), MMscfd, or sand production rate, ft³/day

q_sc gas flow rate, Mscf/day

q_st gas capacity at standard conditions, MMscfd

q_total total liquid flow rate, bbl/day

Q_w water production rate, bbl/day

q_w water production rate, bbl/d

q_wh flow rate at wellhead, stb/day

R producing gas-liquid ratio, Mcf/bbl, or dimensionless nozzle area, or area ratio A_p/A_b, or the radius of fracture, ft, or gas constant, 10.73 ft³-psia/lbmol-R

r distance between the mass center of counterweights and the crank shaft, ft or cylinder compression ratio

r_a radius of acid treatment, ft

R_c radius of hole curvature, in.

r_e drainage radius, ft

r_eH radius of drainage area, ft

R_p pressure ratio

R_s solution gas-oil ratio, scf/stb

r_w radius of wellbore, ft

r_wh desired radius of wormhole penetration, m

R² A_o/A_i

∇_Ri vertical pressure gradient in the curvic section of lateral i, psi/ft

S skin factor, or choke size, $1/64$ in.

S_A axial stress at any point in the tubing string, psi

S_f specific gravity of fluid in tubing, water=1, or safety factor

S_g specific gravity of gas, air=1

S_o specific gravity of produced oil, fresh water=1

S_s specific gravity of produced solid, fresh water=1

S_t equivalent pressure caused by spring tension, psig

S_w specific gravity of produced water, fresh water=1

T temperature, °R

t temperature, °F, or time, hour, or retention time, min

T_av average temperature, °R

T_avg average temperature in tubing, °F

T_b base temperature, °R, or boiling point, °R

T_c critical temperature, °R

T_ci critical temperature of component i, °R

T_d temperature at valve depth, °R

TF₁ maximum upstroke torque factor

TF₂ maximum downstroke torque factor

T_m mechanical resistant torque, lb_f-ft

t_r retention time ≈ 5.0 min

T_sc standard temperature, 520°R

T_up upstream temperature, °R

T_v viscosity resistant torque, lb_f-ft

T₁ suction temperature of the gas, °R

$\overline{T}$ average temperature, °R

u fluid velocity, ft/s

u_m mixture velocity, ft/s

u_SL superficial velocity of liquid phase, ft/s

u_SG superficial velocity of gas phase, ft/s

V volume of the pipe segment, ft³

v superficial gas velocity based on total cross-sectional area A, ft/s

V_a the required minimum acid volume, ft³

V_fg plunger falling velocity in gas, ft/min

V_fl plunger falling velocity in liquid, ft/min

V_g required gas per cycle, Mscf

V_gas gas volume in standard condition, scf

V_G1 gas specific volume at upstream, ft³/lb_m

V_G2 gas specific volume at downstream, ft³/lb_m

V_h required acid volume per unit thickness of formation, m³/m

V_L specific volume of liquid phase, ft³/mol–lb, or volume of liquid phase in the pipe segment, ft³, or liquid settling volume, bbl, or liquid specific volume at upstream, ft³/lb_m

V_m volume of mixture associated with 1 stb of oil, ft³, or volume of minerals to be removed, ft³

V₀ pump displacement, ft³

V_P initial pore volume, ft³

V_r plunger rising velocity, ft/min

V_res oil volume in reservoir condition, rb

V_s required settling volume in separator, gal

V_slug slug volume, bbl

V_st oil volume in stock tank condition, stb

V_t A_t(D–V_slugL), gas volume in tubing, Mcf

V_Vsc specific volume of vapor phase under standard condition, scf/mol-lb

V₁ inlet velocity of fluid to be compressed, ft/sec

V₂ outlet velocity of compressed fluid, ft/sec

v₁ specific volume at inlet, ft³/lb

v₂ specific volume at outlet, ft³/lb

w fracture width, ft, or theoretical shaft work required to compress the gas, ft-lb_f/lb_m

W_air weight of tubing in air, lb/ft

W_c total weight of counterweights, lb_f

W_f weight of fluid, lb_f

W_fi weight of fluid inside tubing, lb/ft

W_fo weight of fluid displaced by tubing, lb/ft

WOR producing water-oil ratio, bbl/stb

W_p plunger weight, lb_f

W_s mechanical shaft work into the system, ft-lb_f per lb of fluid

w_w fracture width at wellbore, in.

$\overline{w}$ average width, in.

X volumetric dissolving power of acid solution, ft³ mineral/ ft³ solution

x_f fracture half-length, ft

x_i mole fraction of compound i in the liquid phase

x₁ free gas quality at upstream, mass fraction

y_a actual pressure ratio

y_c critical pressure ratio

y_i mole fraction of compound i in the vapor phase

y_L liquid hold up, fraction

Z gas compressibility factor in average tubing condition

Z gas compressibility factor

z_b gas deviation factor at T_b and p_b

z_d gas deviation factor at discharge of cylinder, or gas compressibility factor at valve depth condition

z_s gas deviation factor at suction of the cylinder

z₁ compressibility factor at suction conditions

$\overline{z}$ the average gas compressibility factor

ΔZ elevation increase, ft