A matrix defined by Eq. (D.29)

A(σ) arbitrary function, Chapter 9

a constants in Eq. (A.7)

a_j constants in Eq. (3.13)

a_si stable value for a_i(t_D), Eq. (5.18); constant defined in Eq. (D.26)

B formation volume factor, RB/STB

b length of linear reservoir, ft (m); constants in Eq. (A.7)

b′ intercept, Eq. (7.13)

b₁, b₂ intercept of semilog straight line, Chapter 9

b_i constant in Eq. (3.35)

C wellbore storage, m³/MPa

C_D dimensionless wellbore storage

c compressibility of the fluid, psi⁻¹ (MPa⁻¹)

c_t system total compressibility, psi⁻¹ (MPa⁻¹)

D(σ) arbitrary function, Chapter 9

D_i constant defined by Eq. (3.36); constant defined by Eq. (5.21); eigenvector defined in Eq. (A.2)

d_e weighted average dimensionless diffusivity

de effective dimensionless diffusivity of reservoirs, Eq. (4.14)

de₁ effective dimensionless diffusivity of section 1, Eq. (4.12)

de₂ effective dimensionless diffusivity of section 2, Eq. (4.12)

d_i dimensionless diffusivity of layers i; element of eigenvector

E objective function, defined by Eq. (8.28)

E₁,E₂ expression, Eq. (F.22)

Ei exponential integral function

F kh-weighted dimensionless pressure, Eq. (5.3)

f dimensionless pressure difference between the two layers leakage function behind the casing, dimensionless, Chapters 8 and 9

f_i dimensionless pressure difference, Eq. (5.2), $p_{D1}-p_{D2}$

f_S steady value of f when time is long

G₁,G₂ expression, Eq. (8.43)

G, G₀, G₁, G₂ expression, Eqs. (9.54)–(9.57)

g expression, Eq. (E.26)

g₁,g₂ expression, Eqs. (8.61), (8.63)

h total thickness of all the layers, ft (m)

h_i thickness of the ith layer, ft (m)

I₀, I₁ Bessel function of imaginary argument

i₀ layer number for separating the two sections

K(x) function, Eq. (9.59)

K₀, K₁ Bessel function of imaginary argument

k thickness-weighted permeability, md

k_i permeability for ith layer, md

k_i′ vertical permeability of layer i, md

${\tilde{k}}_i$ semipermeability between layer i and layer $i+1$, md/ft (m)

${\tilde{k}}_{Di}$ dimensionless semipermeability

(kh)_t total (kh) product of reservoir, md-ft (μm² m)

L objective function, Eq. (8.29)

L_i coefficients in Eq. (3.37)

L₁,L₂ expression, Eq. (G.6)

L₃,L₄ expression, Eq. (8.37)

L₁,L₂,…,L₆ expressions, Eqs. (9.70)–(9.75)

L₇,L₈ expressions, Eqs. (H.42), (9.83)

M expression, Eq. (F.12)

M₁,M₂,…,M₆ expressions, Eqs. (9.62)–(9.67)

m slope of straight line, Eq. (7.39)

m′ slope of straight line, Eq. (7.12)

m₁,m₂,m_t slope of semilog straight line

n total number of layers; total number of measured points, Chapter 9

n₁ number of measured points in the early time period, Chapter 9

p pressure, psi (MPa)

p_i pressure of layer i, psi (MPa)

p₀ original reservoir pressure, psi (MPa)

p^o observed pressure, psi (MPa)

Δp pressure change from initial trend, $p_i-p$, psi (MPa)

Δp_D pressure difference, dimensionless, Eq. (7.14)

p_a, p_bi boundary pressure for linear flow, psi (MPa)

p_D kh-weighted reservoir pressure (Eq. 3.4), dimensionless

p_Di dimensionless pressure of layer i, defined after Eq. (5.1)

p_pc critical pressure, psi (MPa)

p_w weighted wellbore pressure, psi (MPa)

p_wi wellbore pressure for ith layer, psi (MPa)

p_wDi dimensionless wellbore pressure of layer i

q total flow rate for the multilayer system, B/D (m³/d)

q₀, $q_{\infty }$ total flow rate for ${\tilde{k}}_1=0$ and ${\tilde{k}}_1=\infty$ respectively, B/D (m³/d)

q_ci area crossflow rate, layer defined by Eq. (5.24)

q_cD area crossflow rate, dimensionless

q_cDi area crossflow rate of layer i, Eq. (4.11)

q_cp peak value of area crossflow rate, B/D (m³/d)

q_cpi value of area crossflow rate, layer i

q_cpi^S steady-state peak value of area crossflow rate, layer i, B/D (m³/d)

${\tilde{q}}_D$ total production of the reservoir, B/D (m³/d)

q_Di dimensionless production rate of layer i

${\tilde{q}}_{Di}$ production rate of layer i, B/D (m³/d)

q_i flow rate for the ith layer, B/D (m³/d)

r radius, ft (m)

r_w wellbore radius, ft (m)

r_we max effective hole-diameter, ft (m)

r_Dd dimensionless drainage radius

r_eD dimensionless reservoir radii

s_i skin factor of layer i

T dimensionless calculation time, $={\displaystyle {\int }_0^{\tau }Q\left(x\right)dx}/Q\left(\tau \right)$; formation temperature, K

T_sc temperature of standard state, K

T_pc critical temperature, K

t time, h

t′ calculation time, $={\displaystyle {\int }_0^{\tau }q\left(x\right)dx}/q\left(t\right)$, h

t_cD dimensionless time of the crosspoint for drawdown

t_D dimensionless time, Eq. (4.6)

t_p production time, h

${\overrightarrow{U}}_i$ Darcy's velocity in layer i, ft/s (m/s)

v $ln\left(1+4t_D\right)$, Eq. (3.40)

v_cD crossflow velocity (Eq. 3.19), dimensionless

v_cDi dimensionless crossflow velocity of layer i, Eq. (4.10)

v_i crossflow velocity per unit area for layer i, ft⁻¹ s⁻¹ (m⁻¹ s⁻¹)

x independent variable

x_i,j element of eigenvector, X_j

X_j eigenvector

W matrix defined in Eq. (7.52)

w_i dimensionless transmissibility of layer i

w_s accumulated w, Eq. (3.15)

$w_{s_1},w_{s_2}$ total dimensionless productivity of sections 1 and 2, Eq. (4.12)

w_si accumulated dimensionless productivity, Eq. (5.19)

Z Z-factor

z Laplace variable or vertical coordinate

Greek Letters

α communication coefficient, Eq. (8.34); leakage coefficient, Chapter 9

$\overline{\alpha }$ average diffusivity, ft/d

α_e weighted average diffusivity, equal to $\overline{\alpha }{d}_e$

α_i diffusivity of layer i, md psi/cp; function of time defined by Eq. (3.13)

α_p,α_t,α_c coefficient for different units, Eqs. (9.28), (9.29), (9.31)

α_H, α_L critical values for determining whether to unite or separate the two layers

α_F,α_C,α_P,α_t unit conversion coefficient

α₁,α₂ expression, Eq. (F.25)

β constant, Eq. (8.57); constant, Eq. (9.92)

β₁,β₂ expression, Eqs. (9.52), (9.53)

β_i constant in Eq. (7.50)

ɛ ratio of the vertical permeability to the horizontal permeability

ϕ_i porosity of layer i, fraction

γ 1.781, Euler's constant

γ, γ₂ dimensionless productivity of Layer 1 and Layer 2 (Section 5.3)

γ_g gas gravity

γ_i constant, defined by Eq. (5.21)

ς expression, Eq. (G.16)

η diffusivity, md-psi/cp (mm²/s); constant, Eq. (I.6)

λ eigenvalue

λ₁,λ₂ eigenvalue, Eq. (9.51); eigenvalue, Eq. (F.23)

μ viscosity of the fluid, cp (Pa s)

υ expression, Eq. (9.58)

θ_j expression, Eq. (8.69)

ρ_i fluid density in layer i, g/mL (kg/m³)

ρ₀ fluid density under pressure p₀, g/mL (kg/m³)

σ correctness factor porosity, fraction; Laplace space argument, Eq. (8.19)

σ_i wall resistance for layer i, ft/md (m/md)

τ dimensionless time

τ_f dimensionless time for the first straight line to end

ω storativity ratio; eigenvalue of equation

ω, ω₂ dimensionless storativity of Layer 1 and Layer 2 (Section 5.3)

ξ expression, Eq. (G.16); Boltzmann variable

Δ_D $\frac{1}{r_D}\frac{\partial }{\partial r_D}\left(r_D\frac{\partial }{\partial r_D}\right)$, dimensionless radial differential operator

Δ_r $\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial }{\partial r}\right)$, radial differential operator

Subscripts

b caused by boundary

c crossflow or crosspoint

d caused by diffusivity

D dimensionless

e effective value

f flow condition

i,j,k layer or section number or eigenvalue number

L leakage

min minimum

p peak value or production condition

r reference value

s shut-in or steady value

t total value

v vertical value

w wellbore

wf wellbore condition for flowing period

ws wellbore condition for shut in

η caused by diffusivity

Superscript

' value at any location of the reservoir

o observation value