Welltest Analysis
Contents
Review of Derivative Analysis Approach
Variable Rate Analysis
Effect of Sealing Faults
Limited Entry Effects
Radial Composite Systems
Gas Cap and Aquifer Support
Vertically Fractured Reservoirs
Contents
Dual Permeability Models
Dual Porosity Models
Layered Reservoirs
Horizontal Wells
Compartmentalized Systems
Case Studies
Review of Derivative Analysis Approach
Reservoir Model
dr
r
dθ
o
k1
k2
h1
h2
Material Balance
Mass In - Mass Out = Mass Accumulation
Principle
qr ρ
r + dr
+ qθ ρ θ + dθ + Qm − qr ρ r − qθ ρ θ
∂ρ
= φ1h1rdrdθ
∂t
∂ (qθ ρ )
∂ ( qr ρ )
qr ρ r +
dr − qr ρ r + qθ ρ θ +
dθ − qθ ρ θ + Qm
∂r
∂θ
∂ρ
= φ1h1rdrd θ
∂t
General Equation
∂ (qθ ρ )
∂ ( qr ρ )
∂ρ
dr +
dθ + Qm = φ1h1rdrdθ
∂r
∂θ
∂t
Darcy’s Law
k1h1rdθ ∂p
qr =
μ
∂r
qθ =
k1h1dr ∂p
μ
∂θ
Cross-Flow
ρk 2 ( p − p )
Qm =
r dr dθ
μ
h2
Compressibility
∂ρ
∂p
= ct ρ
∂t
∂t
Governing Equation
φ1ct μ ∂p
k2
1 ∂ ⎛ ∂p ⎞
(p− p ) =
⎜r ⎟ +
r ∂r ⎝ ∂r ⎠ k1h1h2
k1 ∂t
For k2/k1 = 0:
1 ∂ ⎛ ∂p ⎞ φ1ct μ ∂p
⎜r ⎟ =
r ∂r ⎝ ∂r ⎠
k1 ∂t
Dimensionless Variables
r
rD =
rw
kt
tD =
2
φμct rw
2πkh
( pi − p )
pD =
q s Bμ
1 ∂ ⎛ ∂pD
⎜⎜ rD
rD ∂ rD ⎝
∂ rD
⎞ ∂pD
⎟⎟ =
⎠ ∂tD
Simple Boundaries
I .C. :
tD = 0
B.C.1 :
rD → 0
B.C.2 :
rD → inf
pD = 0
⎛ ∂p D ⎞
⎜⎜ rD
⎟⎟ = −1
lim
∂rD ⎠
rD → 0 ⎝
pD = 0
Line Source Solution:
1 ⎛ rD ⎞
⎟
pD = − Ei⎜⎜ −
2 ⎝ 4t D ⎟⎠
2
2
rD
≤ 0 . 01 ,
For
4tD
1 ⎛ γrD
pD = − ln⎜⎜
2 ⎝ 4t D
2
⎞
⎟
⎟
⎠
In Consistent Units:
⎞
qs Bμ ⎛
kt
⎟
⎜ ln
pw = pi −
+
0
.
80907
2
⎟
4πkh ⎜⎝ φμct rw
⎠
Theoretical Pressure Distribution
Wellbore
pwf
Skin Effect
q
pw
Δ ps
pwf
q s Bμ
Δp s =
S
2πkh
Mechanical
Skin
With Skin Near Wellbore:
⎞
qs Bμ ⎛
kt
⎟
⎜ ln
0
.
80907
2
pw = pi −
+
+
S
2
⎟
4πkh ⎜⎝ φμct rw
⎠
In US Field Units:
pw = pi − m' q(log t + s ')
where
m' = 162.6
μB
kh
k
s ' = log
− 3.23 + 0.869S
2
φμct rw
Semi-Log Plot
p
Radial Flow
Slope = m'q
WBS
log t
Pressure Derivative:
d ( pi − p w )
= m' q
d (log t )
Derivative Diagnostic
p'
Radial Flow
Plateau
WBS
log t
Derivative Approach
Pressure & Pressure Derivative
100
Early
Mid Time
Late Time
10
1
0.1
0.01
0.1
1
Time
10
100
Reservoir Signatures
10
Pressure & Pressure Derivative
Early
Mid Time
Late Time
Closed System
U Fault
Paralell Fault
Vertical Fracture/Horizontal Well
Dual Permeability
Compartmental
Radial Homogeneous
Storage
DD
Radial Composite
BU
Closed
Dual Porosity
System-BU
Partial Penetration
Gas Cap/Aquifer
1
0.01
Angle Fault
Single Fault
0.1
Time
1
Constant Pressure
10