Introducon Shahab Shahab Gerami, Gerami, PhD
Reservoir Information
Predictive Models (forward solution)
Production Analysis Models (backward solution)
Production Forecast
Field Data
(i) Well test data (ii) Production data
Economic Study and Decision Making for the Field Development
Conditions
Objective
Conditions
Objective
Conditions
Objective
Falloff Test:
Interference Test:
Drill Stem Test (DST):
Direct solution
Inverse solution
“Average” permeability in a region Not Permeability at a “fixed radius”
Well Testing Lecture #2: Fundamentals of fluid flow in porous media Shahab Gerami, PhD
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•
The linear, one dimensional, horizontal, one phase, partial differential flow equation for a liquid, assuming constant permeability, viscosity and compressibility for transient or time dependent flow:
•
If the flow reaches a state where it is no longer time dependent, we denote the flow as steady state. The equation then simplifies to: •
The analytical solution of the transient pressure development in the slab is given by:
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fluid S.Gerami
∂ 2 P ⎛ φ µc ⎞ ∂ P =⎜ ⎟ 2 ∂ x k ⎝ ⎠ ∂t ∂ 2 P =0 ∂ x 2 x
Development of Hydraulic Diffusivity Equation for Flow of a Slightly Compressible Oil and Its Solution Subjected to Different Boundary Conditions •
•
Simplifying assumptions
•
Mathematical model
Physical model
–
Choosing an appropriate element
–
Governing equation
•
Mass balance
•
Momentum balance (Darcy’s law)
Equation of state
• –
Initial and Boundary conditions
•
Infinite acting
– Constant rate production – Constant pressure production •
Finite acting
– Constant rate production – Constant pressure production –
Solutions •
• 5
Laplace space solutions
•
Time domain solutions
•
Simplified solutions
Applications (Drawdown (single rate & multi rate), Reservoir limit test, Build up, Superposition (time & space), …), S.Gerami
Simplifying Assumptions
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Hydraulic Diffusivity Equation
1 ⎡ ∂ ⎛ ∂ p ⎞⎤ 1 ∂ p ⎜ r ⎟⎥ = ⎢ r ⎣ ∂r ⎝ ∂r ⎠⎦ η ∂t η =
0.000264 k φµ ct
Hydraulic diffusivity equation determines the velocity at which pressure waves propagate in the reservoir. The more the permeability the faster the pressure wave will propagate.
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Initial And Boundary Conditions Radial Flow In a Circular Reservoir Initial Condition: p = pi , Well production
Constant rate
Flow regime
Infinite acting
t = 0, r ≥ r w
Inner Boundary Condition
µ qBo ⎛ ∂ p ⎞ ⎜ ⎟ =− 2π r w hk ⎝ ∂r ⎠ r
Outer Boundary conditions
( p ) →∞ = r
pi
w
Constant rate
Finite acting (Bounded)
µ qBo ⎛ ∂ p ⎞ ⎜ ⎟ =− 2π r w hk ⎝ ∂r ⎠ r w
Constant pressure
Infinite acting
( p ) = p
wf
( p ) = p
wf
r w
Constant pressure
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Finite acting (Bounded)
r w
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⎛ ∂ p ⎞ =0 ⎜ ⎟ ⎝ ∂r ⎠ r →re
( p ) →∞ = r
pi
⎛ ∂ p ⎞ =0 ⎜ ⎟ ∂ r ⎝ ⎠ r →re
Line-source & Finite-wellbore Solutions •
The solution to differential equations treating the well as a vertical line through a porous medium .The solution is nearly identical to the finitewellbore solution. At very early times, there is a notable difference in the solutions, but the differences disappear soon after a typical well is opened to flow or shut in for a buildup test, and in practice the differences are masked by wellbore storage .
•
The solution to the diffusivity equation that results when the well (inner) boundary condition is treated as a cylinder of finite radius instead of treating the well as a line source.
Line-source: the well has zero radius
µ qBo ⎛ ∂ p ⎞ ⎜ ⎟ =− 2π r w hk ⎝ ∂r ⎠ r →0
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Finite-wellbore
µ qBo ⎛ ∂ p ⎞ ⎜ ⎟ =− 2π r w hk ⎝ ∂r ⎠ r w
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Infinite cylindrical reservoir with line‐source well (approximate solution) Line-source: the well has zero radius
Dimensionless solution µ ct r 2 ⎞ 1 ⎛ ⎟ p D = − E i ⎜ − 948 2 ⎜⎝ k t ⎠⎟
Dimensional solution µ ct r w ⎞ qBµ ⎛ ⎟ E i ⎜ − 948 = pi + 70.6 ⎜ ⎟ kh k t ⎝ ⎠ 2
pwf
∞
− E (− x ) = i
∫u x
19
e −u
⎧
du = ⎨
⎩≈ 0
≈ ln(1.781 x )
for x < 0.02 for x > 10.9
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(error ≈ 0.6%)
Example •
A well is producing only oil at constant rate of 20 STB/D. Data describing the well and formation are summarized below. Calculate the reservoir pressure at radii of 1, 10, and 100 ft after 3 hrs of production.
r w = 0.5
ft
r e = 3,000
ft
h = 150
ft
k = 0.1
md
φ = 0.23 − S wi = 0
21
−
µ = 0.72
cp
Bo = 1.475
RB / STB
ct = 1.5 × 10 −5
psi −1
q = 20
STB / Day
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Flowing Wellbore Pressure Flowing wellbore pressure
Flowing wellbore pressure
3600
3600
3500
3500
3400
3400
µ ct r w ⎞ qBµ ⎛ ⎟ E i ⎜ − 948 ⎜ kh k t ⎠⎟ ⎝ 2
3300
3300
) a i s p3200 (
) a i s p3200 (
f w
f w
p
p
3100
3100
3000
3000
2900
2900
2800 0
pwf = pi + 70.6
200
400 600 time(hr) ∞
− E i (− x ) =
800
2800 0 10
1000
e− u
∫ u du ≈ ln(1.781 x)
1
10
for x < 0.02
2
time(hr)
(error ≈ 0.6%)
x
p wf (t ) = pi −
23
162.6qBo µ ⎡ kh
⎛ k ⎢log⎜ ⎢⎣ ⎜⎝ φµ c r
2
t w
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⎞ ⎟ ⎠
10
⎤
t ⎟ − 3.23⎥
⎥⎦
10
3
Dimensionless transient pressure response of a radial well in infinite reservoir p wD (t D ) =
2t D 2 eD
r
∞
+ ln(r eD ) − 0.75 + 2∑ n=1
e
[
−α n2 t D
J 1 (α n r eD ) 2
2 1
2
p wD (t D ) = 25
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]
α J (α n r eD ) − J 1 (α n ) 2 n
1 2
[ln(t ) + 0.80908 ] D
Skin •
The skin effect, first introduced by van Everdingen and Hurst (1949) defines a steady-state pressure difference around the wellbore.
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Unsteady ‐state radial flow (accounting for the skin factor) for slightly compressible fluids Hawkins (1956) suggested that the permeability in the skin zone, i.e., skin, is uniform and the pressure drop across the zone can be approximated by Darcy’s equation. Hawkins proposed the following approach:
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Application: Accounting for Skin pi − pwf (t ) =
162.6qBo µ ⎡ kh
⎤ ⎛ k ⎞ ⎟ − 3.23 + 0.87 S ⎥ ⎢log (t ) + log⎜ ⎜ φµ ct r w2 ⎟ ⎢⎣ ⎥⎦ ⎝ ⎠
pwf (t ) = a − m log(t )
m=−
162.6 qBo µ
a = pi −
31
kh 162.6qBo µ ⎡ kh
⎤ ⎛ k ⎞ ⎜ ⎟ ⎢log − 3.23 + 0.87 S ⎥ ⎢⎣ ⎜⎝ φµ c r ⎠⎟ ⎥⎦ 2
t w
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Application: Semi-log Pressure Drawdown Data
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Solution Step 1. From Figure 1.34, calculate p1 hr: p1 hr = 954 psi Step 2. Determine the slope of the transient flow line: m = ‐22 psi/cycle Step 3. Calculate the permeability:
Step 4. Solve for the skin factor s
Step 5. Calculate the additional pressure drop: 35 35
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Radius of Investigation
r i =
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kt 948φµ C t
Reservoir‐Limits Test (estimation of reservoir pore volume) pwf = pi −
141.2qBµ ⎡ 0.0005274 k
⎢ ⎢⎣
kh
2
φµ ct r e
⎤ ⎛ r e ⎞ ⎟⎟ − 0.75⎥ ⎥⎦ ⎝ r w ⎠
t + ln⎜⎜
∂ p 0.07447qB = ∂t φ c r V = π r hφ wf
o
2
t e
2
p
e
∂ p 0.234qB = ∂t c V wf
o
t
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p
Bessel Differential Equation
Modified Bessel Differential Equation
Properties of Bessel function d dr D d dr D
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(
)
I 0 r D S =
(
)
(
)
(
)
S I 1 r D S
K 0 r D S = − S K 1 r D S
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Solutions‐ Laplace Domain (Sabet, 1991). Infinite-acting reservoir
p wD (t D ) =
1 2
[ln(t ) + 0.80908] D
Constant rate solution Boundary dominated flow- approximate late time
pwD (t D ) =
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2t D r eD2
+ ln(r eD ) − 0.75
Numerical Inverse Laplace Transformation (Stehfest Algorithm)
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Assignment #2.b A well has flown single phase oil for 10 days at rate of 800 STB/D . Rock and fluid properties are: Rock and fluid properties Bo, RB/STB
1.13
h, ft
50
Pi, Psia
3000
Ct psi-1
2.00E-05
µ,cp
0.5
Φ
0.16
k md(constant)
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r w, ft
0.33
(1) Assume infinite reservoir and calculate pressure at radii of 0.33,10 , 1000, 3160ft and plot the results as pressure vs. logarithm of radius . (2) Estimate the radius of investigation achieved after 10 days flow time, calculate the pressure drop at radius of investigation, is the pressure drop at radius of investigation equal to zero? Explain briefly. (3) Suppose the production rate was 400 STB/Day . Prepare a plot of pressure vs. logarithm of radius after 10 day on the same graph as the plot developed for a rate of 800 STB//Day. Is the radius of investigation calculated from the appropriate equation affected by change in flow rate? What is the effect of increased rate on pressure inside the reservoir? 47
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Assignment #2.d A 12 inch diameter hole has a damaged region 24 inchec tick measured from the wellbore wall. The permeability in this region is one tenth of undamaged region.
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1.
find the skin value?
2.
Find the equivalent wellbore radius that would represent the above skin.
3.
Repeat part a and b for the case where the area around the wellbore was
4.
Compare the results of damaged well with stimulated one.
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Well Testing Lecture #3: Drawdown Analysis Shahab Gerami, PhD
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Dimensionless Hydraulic Diffusivity Equation
∂ 2 p D 1 ∂ p D ∂ p D + = 2 r D ∂r D ∂t D ∂r D 3
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Application: Semi-log Pressure Drawdown Data
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Transient approximate solution
P.S.S approximate solution
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Derivative Analysis: Transient Radial Flow Regime pi − pwf (t ) = ∆ pwf =
162.6qBo µ ⎡ kh
⎤ ⎛ k ⎞ ⎟ − 3.23 + 0.87 S ⎥ ⎢log(t ) + log⎜ ⎜ φµ c r 2 ⎟ ⎢⎣ ⎥⎦ ⎝ ⎠
d ∆ pwf d log t
t w
=
162.6 qBo µ kh
⎛ 162.6qB µ ⎞ ⎛ d ∆ p ⎞ ⎟⎟ ⎟⎟ = 0 × log(t ) + log⎜⎜ d log t kh ⎝ ⎠ ⎝ ⎠
log⎜⎜ 11
wf
o
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Procedure for Derivative Analysis To calculate the pressure derivative curve we need to use the formula of derivative which is: P Drivative = t .
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∆ p ∆t
ti-1
Pi-1
ti
Pi
ti+1
Pi+1
P Drivative( i ) = t i ×
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pi −1 − pi +1 t i +1 − t i −1
Wellbore Storage Due to the finite wellbore volume, the initial production from a well opened at surface is dominated by expansion of the fluids in the wellbore. q Rate
Surface Rate
Sandfacerate Wellbore rate
qsf Vwb
Time
qsf
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In = q sf B ρ
q
Out = qB ρ Accum. =
d (24 ρ wbV wb ) dt
q sf B ρ − qB ρ = 24V wb
d ρ dt
Vwb
We can write
c=
1 d ρ
d ρ wb
ρ dp
dt
ρ B = C st
q sf = q +
Define
C = c wbV wb
Ass ume
ρ wb ≈ ρ R
17
=
d ρ wb dp wb dp
dt
= ρ wb c wb
dp wb dt
qsf
24c wbV wb ⎛ ρ wb ⎞ dp w B
⎜⎜ ⎟⎟ ρ ⎝ R ⎠ dt
q sf = q +
24C wb dp w B
dt
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C D =
0.8936C 2 w
φ c t hr
=
0.8936c wbV wb φ ct hr w2
Van Everdingen and Hurst, 1949 The rate of unloading off/ or storage in, the wellbore per unit pressure difference is constant. This constant is known as the wellbore storage constant.
C s = V ws cws Vws(bbl): Volume of wellbore tubing (and annulus if there is no packer) cws: Compressibility of the wellbore fluid evaluated at the mean wellbore pressure and temperature and not at reservoir condition, as is usually the case.
Dimensionless wellbore storage constant
C sD =
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0.894C s φ h c r w2
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• Agarwal, Al‐Hussainy and Ramey (1970) showed that for all practical purposes, the duration of wellbore storage effects is also given by For negative skin and No skin t wsD > 60C sD For positive skin t wsD = (60 + 3.5S )C sD or t ws =
21
(200,000 + 12,000S )C
s
kh µ
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Partial Penetration Skin The partial penetration skin is used when the perforations of a vertical wellbore do not span the entire net pay of the reservoir. In these situations, the reservoir flow has to flow vertically and the flow lines converge at the perforations.
Flow line
(a) Complete penetration
Flow line
(b) Partial penetration
The convergence of flow lines near the wellbore result in an additional pressure drop; an effect similar to that caused by wellbore damage. Therefore this pressure drop is dealt with as if it was a skin effect and it is labelled as the skin due to partial penetration. This effect is always positive. It is a function of the perforated interval, the distance from the top of the zone to the top of the perforations and the horizontal to vertical permeability ratio (Muskat, 1946; Nisle, 1958; Brons and Marting, 1959; Kirkham, 1959; Odeh, 1968; Seth, 1968; Clegg and Mills, 1969; Kazemi, and Seth, 1969; Gringarten and Ramey, 1975; Streltsova‐Adams, 1978).23 S.Gerami
A radial well in a finite acting reservoir in general displays 3 flow periods. The flow periods are most easily identified from the derivative plot. The initial unit‐slope is indicative of wellbore storage. The initial production is dominated by expansion of the fluids in the wellbore. In general friction‐losses along the wellbore are negligible and the wellbore behaves like a tank. For a slightly compressible fluid as well as for an ideal gas the first order derivative of the pressure will be constant (assuming a constant production), resulting in a unit‐slope derivative.
The hump that follows the unit‐ slope is caused by near‐wellbore impairment, often characterized by a skin factor.
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Superposition Principle Linear diffusivity equation
A mathematical technique based on the property that solutions to linear partial equations can be added to provide yet another solution. This permits constructions of mathematical solutions to situations with complex boundary conditions, especially drawdown and buildup tests, and in settings where flow rates change with time. Mathematically the superposition theorem states that any sum of individual solutions to the diffusivity equation is also a solution to that equation. This concept can be applied to account for the following effects on the transient flow solution:
• Superposition in time – Effects of rate change
• Super position in space – Effects of multiple wells – Effects of the boundary 27
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(1) Effects of Multiple Wells •
37
The superposition concept states that the total pressure drop at any point in the reservoir is the sum of the pressure changes at that point caused by flow in each of the wells in the reservoir. In other words, we simply superimpose one effect upon the other.
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(2) Production schedule for variable rate well q2 q1
•Each well that contribute to the total pressure drawdown will be at the same position in the reservoir. The wells simply will be “ turned on “ at different times.
q3
q
t1
•These wells, in general, will be inside a zone of altered permeability zone.
t2
t
q1 Well#1
q2 – q1 Well#2 t1
t2
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Well#3 q3 – q2
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•
Two questions?!
t p = 24
1.
What is the basis of this equation?
2.
Under what conditions is it applicable?
N p qlast
µ ct r w ⎞ qlast Bµ ⎛ ⎟ E i ⎜ − 948 ⎜ kh k t p ⎠⎟ ⎝ 2
pi − pwf = −70.6
• Answers 1.
The basis for the approximation is intuitive and not rigorous – Clear choice is the most recent rate which is maintained for any significant period. –
2.
The product of effective production time and production rate results in correct cumulative production. Thus, it honors the material balance equation.
The approximation is adequate if the most recent flow rate is maintained long enough. – Guideline: Horner’s approximation is valid when:
∆t
last
∆t
next − to − last
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>2
q D = 0.0002637
47
BO µ q khpi
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Assignment#3.b: A drawdown test is performed in a well close to a sealing fault , the reservoir is otherwise infinite. a‐ Write the equation describes Pwf ( assume well is at distance L from Fault) b‐Use the above equation and describe why at the early time the slop of Pwf vs. Time on a semi‐log plot is 162.6 qµ B , and why at late time the slop doubles. kh
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time hr
Assignment#3.d: Repeat Assignment#3.c using Horner’s approximation. Compare the results with those found in Assignment#3.c solution . Next ignore the variation in rate and analyze the data using constant rate analysis technique. Using the initial rate.
59
q(STB/Day)
time hr
0
5000
200
3.64
4797
121
0.114
4927
145
4.37
4798
119
1.136
4917
143
5.27
4798
118
0.164
4905
142
6.29
4798
117
0.197
4893
141
7.54
4799
116
0.236
4881
140
9.05
4799
114
0.283
4868
138
10.9
4800
113
0.34
4856
137
13
4801
112
0.408
4844
136
15.6
4801
110
0.49
4833
135
18.8
4802
109
0.587
4823
133
22.5
4803
108
0.705
4815
132
27
4803
107
0.846
4809
131
32.4
4804
105
1.02
4804
129
38.9
4805
104
1.22
4801
128
46.7
4806
103
1.46
4799
127
56.1
4807
102
1.75
4798
126
67.3
4807
100
2.11
4797
124
80.7
4808
99
2.53
4797
123
96.9
4809
98
3.03
4797
122
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Pwf
Pwf
q(STB/Day)
Well Testing Lecture #4&5: Buildup Analysis Shahab Gerami, PhD
w w w . p e t r o m a n .i r
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Outline • Buildup Test • Behavior of Static Sandface Pressure Upon Shut‐in of a Well • Buildup as superposition of rates • Horner plot relationship • Detecting Faults from Buildup w w w . p e t r o m a n .i r
•
Agarwal Equivalent Time
• Qualitative Interpretation of Buildup Curves •
Builup during pseudo steady state flow
• Average Reservoir Pressure – Miller‐Dyes‐Hutchinson (MDH) Method – The Matthews–Brons–Hazebroek (MBH) Method – Ramey–Cobb method – Dietz method
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Buildup Test • Drawdown data quality is subject to many operational problems; slugging, turbulence, rate variation, inaccurate rate measurements, instability, unsteady flow, plugging, interruptions, equipment adjustments, etc… w w w . p e t r o m a n .i r
• Buildup is measurement of pressure and time when well is shut‐in. • In high permeability reservoirs the pressure will buildup to a stabilized value quickly, but in tight formations the pressure may continue to buildup for month before stabilization attained.
• • • • •
Buildup must be preceded by flow period. Simplified Analysis assumes constant flow rate for a duration t hours. Shut‐in time, ∆t , measured from end flow. Buildup Analysis treated as superposition of flow and injection. Analysis of buildup data may yield the values of K, S, and the average reservoir pressure.
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Buildup is always preceded by a drawdown and the buildup data are directly affected by this drawdown.
w w w . p e t r o m a n .i r
Methods of analysis: •Horner plot (1951): Infinite acting reservoir •Matthews‐Brons‐Hazebroek (MBH,1954): Extension of Horner plot to finite reservoir. •Miller‐Dyes‐Hutchinson (MDH plot, 1950): Analysis of P.S.S. flow conditions.
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Behavior of Static Sandface Pressure Upon Shut‐in of a Well Reflects “kh”
w w w . p e t r o m a n .i r
Reflects the wellbore storage (afterflow) 5
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Reflects the effects of boundaries.
w w w . p e t r o m a n .i r
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w w w . p e t r o m a n .i r
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•Flowing sandface pressure during drawdown pi − pwf =
w w w . p e t r o m a n .i r
162.6qBo µ ⎡
⎤ ⎛ k ⎞ ⎜ ⎟ ⎢log(t ) + log − 3.23 + 0.87 S ⎥ ⎜ φµ c r 2 ⎟ ⎢⎣ ⎥⎦ ⎝ ⎠
kh
t w
•Shut‐in wellbore pressure: The static sandface pressure is given by the sum of the continuing effect of the drawdown rate, qsc, and the superposed effect of the change in rate(0‐qsc) pi − pws =
162.6qBo µ ⎡
⎤ ⎛ k ⎞ ⎜ ⎟ ⎢log (t + ∆t ) + log − 3.23 + 0.87 S ⎥ + ⎜ φµ c r ⎟ ⎢⎣ ⎥⎦ ⎝ ⎠ 2
kh
t w
162.6(0 − q ) Bo µ ⎡ kh
⎤ ⎛ k ⎞ ⎜ ⎟ ⎢log (∆t ) + log − 3.23 + 0.87 S ⎥ ⎜ φµ c r ⎟ ⎢⎣ ⎥⎦ ⎝ ⎠ 2
t w
Horner plot relationship‐ Infinite acting reservoir
pi − pws (t ) = 8
162.6qBo µ kh S.Gerami
⎛ t + ∆t ⎞ ⎟ t ∆ ⎝ ⎠
log⎜
Horner plot relationship pi − pws (t ) =
162.6qBo µ kh
⎛ t + ∆t ⎞ ⎟ ∆ t ⎝ ⎠
log⎜
⎛ t + ∆t ⎞ ⎟ t ∆ ⎝ ⎠
Horner time = ⎜
w w w . p e t r o m a n .i r
Slope of semilog straight line same as drawdown – used to calculate permeability. m=
9
162.6qBo µ kh
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Buildup test does NOT allow for skin calculation. Skin is obtained from FLOWING pressure before shut-in. pws (t p + ∆t ) − pwf (t p ) =
w w w . p e t r o m a n .i r
p ws (t p + ∆t ) − p wf (t p ) =
162.6qBo µ ⎡
⎤ ⎛ k ⎞ t + ∆t ⎞ ⎟ − 3.23 + 0.87 S ⎥ − 162.6qB µ log⎛ ⎜⎜ ⎟⎟ ⎢log(t ) + log⎜ ⎜ φµ c r ⎟ kh t ∆ ⎢⎣ ⎥ ⎝ ⎠ ⎝ ⎠ ⎦
kh
2
t w
162.6qBo µ ⎡ kh
o
p
⎤ ⎛ t ∆t ⎞ ⎛ ⎞ ⎟ + log⎜ k ⎟ − 3.23 + 0.87S ⎥ ⎢log⎜⎜ ⎜ φµ c r ⎟ ⎢⎣ ⎝ t + ∆t ⎠⎟ ⎥⎦ ⎝ ⎠ p
2
p
t w
∆t = 1 hr
⎧⎪ p − p ⎡ k t − log ⎢ S = 1.151⎨ m ⎪⎩ ⎣⎢ (t + 1)φµ c r 1 hr
p
wf
2
p
10
t w
⎫⎪ ⎤ ⎥ + 3.23⎬ ⎦⎥ ⎭⎪
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p
w w w . p e t r o m a n .i r
⎡ p − p ⎤ ⎛ ⎞ k t ⎜ ⎟ − log + 3.23⎥ S = 1.151⎢ ⎜ ⎟ ( ) 1 φµ + m t c r ⎢⎣ ⎝ ⎠ ⎦⎥ 1 hr
wf
p
2
p
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t w
w w w . p e t r o m a n .i r
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w w w . p e t r o m a n .i r
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Detecting Faults from Buildup
w w w . p e t r o m a n .i r
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w w w . p e t r o m a n .i r
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Agarwal Equivalent Time Log-Log Analysis for drawdown test:
⎛ d ∆ p ⎞ ⎟⎟ vs log(t ) d t log ⎝ ⎠
log⎜⎜
wf
Log-Log Analysis for buildup test ?
pi
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pi − p wf (t )
p wf (t ) p ws (∆t ) − p wf t p + ∆t
p wf t p Measured Me asured pressur e
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Agarwal Equivalent Time
Measurable pressur e diff erence
[ p (∆t ) − p (t )] ws
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Correct pressure difference
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wf
p
[ p (∆t ) − p (t ws
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wf
p
+ ∆t )]
Agarwal Equivalent Time A time at which measurable pressure difference is equal to correct pressure difference.
[ p (∆t ) − p (t )]= [ p (∆t ) − p (t ws
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e
wf
p
ws
wf
p
+ ∆t )]
• ∆te can be determined exactly for infinite acting radial flow, when the log approximation is valid. •Using ∆te in place of ∆t, will allow drawdown type-curves to be used for buildup. This strictly true if only for infinite acting radial flow without wellbore storage. ∆te
= t ∆t/(t + ∆t)
The type curve analysis approach was introduced in the petroleum industry by Agarwal et al. (1970) as a valuable tool when used in conjunction with conventional semilog plots. A type curve is a graphical representation of the theoretical solutions to flow equations.
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Agarwal Equivalent Time ∆te
= t ∆t/(t + ∆t)
Definition of equivalent time illustrates that radius of investigation in a buildup depends on: 1. duration of Drawdown 2. duration of Buildup w w w . p e t r o m a n .i r
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Qualitative Interpretation of Buildup Curves Wellbore storage derivative transients are recognized as a “hump” in early time. The flat derivative portion in late time is easily analyzed as the Horner semilog straight line.
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The level of the second-derivative plateau is twice the value of the level of the first-derivative plateau, and the Horner plot shows the familiar slope-doubling effect.
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Unlike the drawdown pressure transient, this has a unit-slope line in late time that is indicative of pseudosteady-state flow; the buildup pressure derivative drops to zero. The permeability and skin cannot be determined from the Horner plot because no portion of the data exhibits a flat derivative for this example. When transient data resembles example d, the only way to determine the reservoir parameters is with a type curve match.
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Average Reservoir Pressure ●
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material balance studies; ● water influx; ● pressure maintenance projects; ● secondary recovery; ● degree of reservoir connectivity.
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Average Reservoir Pressure
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Figure 1.39: Typical pressure buildup curve for a well in a fini te reservoir reservoir 40
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The Matthews–Brons–Hazebroek (MBH) Method • A methodology for estimating average pressure from buildup tests in bounded drainage regions. w w w . p e t r o m a n .i r
• Theoretical correlations between the extrapolated semilog straight line to the p∗ and current average drainage area pressure p.
• The average pressure in the drainage area of each of each well can be related to p∗ if the if the geometry, shape, and location of the of the well relative to the drainage boundaries are known.
• A set of correction of correction charts for various drainage geometries are developed.
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m=The m=The Horner semilog semilog straight-line plot slope
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Modified Muskat p wD (t D ) =
2t D 2
r eD
∞
+ ln(r eD ) − 0.75 + 2∑ n =1
e
−α n2 t D
(
J 12 α n r eD
α n2 [ J 12 (α n r eD ) − J 12 (α n )]
J 1 (α n r eD )Y 1 (α n ) − J 1 (α n )Y 1 (α n r eD ) = 0 w w w . p e t r o m a n .i r
Approximate Solutions once boundary effect are felt p − p ws = 118.6
qBµ kh
⎛
k ∆t ⎞
⎝
φµ ct r e2 ⎠⎟
exp⎜⎜ − 0.00388
log( p − p ws ) = A + B∆t ⎛ 250 φµ ct r e2 ⎞ ⎛ 750 φµ ct r e2 ⎞ ⎜ ⎟ < ∆t < ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ k k ⎝ ⎠ ⎝ ⎠
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)
⎟
Modified Muskat log( p − pws ) = A + B∆t
⎛ 250 φµ ct r e2 ⎞ ⎛ 750 φµ ct r e2 ⎞ ⎜ ⎟ < ∆t < ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ k k ⎝ ⎠ ⎝ ⎠
1. Assume a value for p-bar w w w . p e t r o m a n .i r
2. Plot log (pavg-pws) versus ∆t 3. Is it a straight line? 4. If the answer is yes, the assumed value is the average reservoir pressure otherwise GO TO 1. log( p − pws )
As su med p avg too high
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∆t
Modified Muskat log( p − pws ) = A + B∆t Advantages 1. It requires no estimate no estimates of reservoir properties when it is used to establish pavg. w w w . p e t r o m a n .i r
2.
It provide satisfactory estimates of pavg for hydraulically fractured wells and layered reservoirs.
Disadvantages 1. It fails when the tested well is not reasonably centered in its drainage area. 2.
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The required shut-in times are frequently impractically long, particularly in low permeability reservoirs.
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⎛ 250 φµ ct r e2 ⎞ ⎛ 750 φµ ct r e2 ⎞ ⎜ ⎟ < ∆t < ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ k k ⎝ ⎠ ⎝ ⎠
∆t
Ramey–Cobb method Ramey and Cobb (1971) proposed that the average pressure in the well drainage area can be read directly fromthe Horner semilog straight line if the following data is available: ●
shape of the well drainage area; ● location of the well within the drainage area; ● size of the drainage area.
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Dietz method Dietz (1965) indicated that if the if the test well has been producing long enough to reach the pseudosteady state before shut‐in, the average pressure can be read directly from the MDH semilog straight straight‐line plot, i.e., pws vs. log(t ), ), at the following shut‐in time:
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Assignment# 4.a
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Well Testing Lecture #6: Hydraulically Fractured Well Shahab Gerami, PhD
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Radial System Flow Regime
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Hydraulic Fracturing •Often newly drilled wells do not flow satisfactorily and stimulation is required. A popular and effective stimulation practice is hydraulic fracturing. The objective of this technique is to provide a greatly increased surface for the reservoir fluid to enter the wellbore. In order for this to be effective the pressure drop along the fracture needs to be small, requiring a high fracture conductivity (defined by the product of fracture width and fracture permeability).
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•A fracture is defined as a single crack initiated from the wellbore by hydraulic fracturing. It should be noted that fractures are different from “fissures,” which are the formation of natural fractures.
• •
Massive hydraulic fracturing (MHF) stimulation treatments are extensively used in tight reservoirs to boost the reservoir performance. A good fractured well surveillance is essential for optimal reservoir exploitation and long‐term strategic plan development.
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Hydraulically Fractured Well Depth >3000 ft: It is believed that the hydraulic fracturing results in the formation of vertical fractures. Depth< 3000 ft: The likelihood is that horizontal fractures will be induced. w w w . p e t r o m a n .i r
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Characterization of Hydraulic Fractures fracture half ‐length x f , ft; ● dimensionless radius r , where r = r / x ; eD eD e f ● fracture height h , which is often assumed equal to the formation thickness, ft; f ● fracture permeability k , md; f ● fracture width w , ft; f ● fracture conductivity F , where F = k w f f C C ●
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•The fracture has a much greater permeability than the formation it penetrates; hence it influences the pressure response of a well test significantly. •The analysis of fractured well tests deals with the identification of well and reservoir variables that would have an impact on future well performance. •The fractured well has unknown geometric features, i.e., x f , w f , hf , and unknown conductivity properties. S.Gerami
Pressure Response in a Hydraulic Fractured Well •The fracture has a much greater permeability than the formation it penetrates; hence it influences the pressure response of a well test significantly. •The following dimensionless groups are used when analyzing pressure transient data in a hydraulically fractured well: w w w . p e t r o m a n .i r
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Hydraulic Fractures Models
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Gringarten et al. (1974) and Cinco and Samaniego (1981), among others, proposed three transient flow models to consider when analyzing transient pressure data from vertically fractured wells. These are: (1) infinite conductivity vertical fractures; • A very high conductivity, which for all practical purposes can be considered as infinite (No significant pressure drop from the tip of the fracture to the wellbore) (2) finite conducvity vercal fractures; • These are very long fractures created by massive hydraulic fracture (MHF). • These types of fractures need large quantities of propping agent to keep them open and, as a result, the fracture permeability k f is reduced as compared to that of the infinite conductivity fractures. • These finite conductivity vertical fractures are characterized by measurable pressure drops in the fracture and, therefore, exhibit unique pressure responses when testing hydraulically fractured wells. (3) uniform flux fractures. • A uniform flux fracture is one in which the reservoir fluid flow rate from the formation into the fracture is uniform along the entire fracture length. • This model is similar to the infinite conductivity vertical fracture in several aspects. The difference between these two systems occurs at the boundary of the fracture. The system is characterized by a variable pressure along the fracture.
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Flow Periods for Vertically Fractured Well •Several flow regimes are observed in fractured wells. One of the responsibilities of the well test analyst is to use the appropriate tools to predict the type of flow regime that may develop in the fracture around the wellbore.
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Hydraulic Fractures Flow Periods
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(1) infinite conductivity vertical fractures; 1. fracture linear flow period; 2. formation linear flow period; 3. infinite‐acting pseudo‐radial flow period. (2) finite conducvity vercal fractures; 1. initially “linear flow within the fracture”; 2. followed by “bilinear flow”; 3. then “linear flow in the formation”; and 4. eventually “infinite acting pseudo‐radial flow.” (3) uniform flux fractures. 1. linear flow; 2. infinite‐acting pseudo‐radial flow. S.Gerami
Fracture Linear Flow
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•The first flow period which occurs in a fractured system. •Dominant production mechanism is the of expansion fluid within the fracture, i.e., there is negligible fluid coming from the formation. •Flow within the fracture and from the fracture to the wellbore during this time period is linear. •The flow in this period can be described by the linear diffusivity equation and is applied to both the fracture linear flow and formation linear flow periods. •The pressure transient test data during the linear flow period can be analyzed with a graph of p vs (time)0.5 •Unfortunately, the fracture linear flow occurs at very early time to be of practical use in well test analysis. •The fracture linear flow exists for fractures with F CD > 300. •The duration of the fracture linear flow period is short, as it often is in finite conductivity fractures with F CD < 300, and care must be taken not to misinterpret the early pressure data. •In some situations the linear flow straight line is not recognized from well test analysis due to the skin effects or wellbore storage effects. •End of fracture linear flow can be estimated from the following relation.
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Bilinear flow •The pressure drop through the fracture is significant for the finite conductivity case and the bilinear flow behavior is observed; however, the infinite conductivity case does not exhibit bilinear flow behavior because the pressure drop in the fracture is negligible. •Two types of linear flow occur simultaneously. w w w . p e t r o m a n .i r
•One flow is a linear incompressible flow within the fracture and the other is a linear compressible flow in the formation. • Most of the fluid which enters the wellbore during this flow period comes from the formation. •Fracture tip effects do not affect well behavior during bilinear flow and, accordingly, it will not be possible to determine the fracture length from the well bilinear flow period data. •The actual value of the fracture conductivity F C can be determined during this flow period.
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Importance of the Identification of the Bilinear Flow Period
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(1)
It will NOT be possible to determine a unique fracture length from the well bilinear flow period data. If this data is used to determine the length of the fracture, it will produce a much smaller fracture length than the actual.
(2)
The actual fracture conductivity k f w f can be determined from the bilinear flow pressure data.
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Estimation Fracture Conductivity
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The fracture tip begins to affect wellbore behavior.
•When the bilinear flow ends, the plot will exhibit curvature which could concave upwards or downwards depending upon the value of the dimensionless fracture conductivity F CD, as shown in Figure 1.72.
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•If the test is not run sufficiently long for bilinear flow to end when F CD > 1. 6, it is not possible to determine the length of the fracture.
•When the dimensionless fracture conductivity F CD < 1. 6, it indicates that the fluid flow in the reservoir has changed from a predominantly one‐dimensional linear flow to a two‐dimensional flow regime. In this particular case, it is not possible to uniquely determine fracture length even if bilinear flow does end during the test.
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End of bilinear flow
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Formation linear flow •At the end of the bilinear flow, there is a transition period after which the fracture tips begin to affect the pressure behavior at the wellbore and a linear flow period might develop. •This linear flow period is exhibited by vertical fractures whose dimensionless conducvity is greater that 300, i.e., F CD > 300. •As in the case of fracture linear flow, the formation linear flow pressure data collected during this period is a function of the fracture length x f and fracture conductivity F C. •The pressure behavior during this linear flow period can be described by the diffusivity equation as expressed in linear form: w w w . p e t r o m a n .i r
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Linear Flow •
Linear flow occurs in some petroleum reservoirs with, long, highly conductive vertical fractures.
•The governing equation for linear flow in x‐direction •Slightly compressible oil w w w . p e t r o m a n .i r
Linear flow
∂ 2 P ⎛ φ µct ⎞ ∂ P =⎜ ⎟ 2 0 . 000264 k ∂ x ⎝ ⎠ ∂t
•Homogeneous reservoir •Isotropic •Constant porosity and permeability
Radial flow
1⎡
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Solution ∂ 2 P ⎛ φ µct ⎞ ∂ P =⎜ ⎟ ∂ x 2 ⎝ 0.000264k ⎠ ∂t Initial Condition: p = pi , w w w . p e t r o m a n .i r
( p ) x→∞ = BCs
p i − p wf
t = 0
pi
µ qBo ⎛ ∂ p ⎞ = − ⎜ ⎟ 4 x f hk ⎝ ∂ x ⎠ x =0
qB ⎛ µ ⎞⎟ ⎜ = 16.26 A f ⎜⎝ k φ ct ⎠⎟
0.5
t
A f = 4hL f qB ⎛ µ ⎞⎟ ⎜ pi − pwf = 4.064 hx f ⎜⎝ k φ ct ⎠⎟
0.5
t
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Difficulties in Test Interpretation •In practice, the (1/2) slope is rarely seen except in fractures with high conductivity.
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•Finite conductivity fracture responses generally enter a transition period after the bilinear flow (the (1/4) slope) and reach the infinite‐acting pseudo‐radial flow regime before ever achieving a (1/2) Slope (linear flow). •For a long duration of wellbore storage effect, the bilinear flow pressure behavior may be masked and data analysis becomes difficult with current interpretation methods.
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Infinite‐acting pseudo‐radial flow • During this period, the flow behavior is similar to the radial reservoir flow with a negative skin effect caused by the fracture. •The traditional semilog and log–log plots of transient pressure data can be used during this period; for example, the drawdown pressure data can be analyzed by using the following Equations:
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Pressure Response in a Hydraulic Fractured Well •In general, a fracture could be classified as an infinite conductivity fracture when the dimensionless fracture conducvity is greater than 300, i.e., F CD >300. Specialized graphs for analysis of the start and end of each flow period: p w w w . p e t r o m a n .i r
p
p
p
vs. (time)0.25 for bilinear flow
vs. (time)0.5 for linear flow
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.Use the following data and perform a completed analysis of the data on appropriate plots.
Well Testing Lecture #8: Well Test Analysis of Gas Reservoirs-Module B Shahab Gerami, PhD
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Flow tests conducted on gas wells 1. Tests designed to yield knowledge of reservoir
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•
Drawdown
•
Buildup
2. Tests designed to measure the deliverability (downhole deliverability)
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•
Back pressure tests
•
Isochronal type tests
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Deliverabilit y Tests
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AOF
Idea behind determination of AOF is to be compare the productivity of wells in the same fields.
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Various deliverability tests of gas well •Flow‐after‐flow (Conventional Back Pressure Test) •Flowing the well at several different flow rates •Each flow rate being continued to pressure stabilization •Isochronal •A series flow tests at different rates for equal periods of time •Alternately closing in the well until a stabilized flow (last flow rate is long enough to achieve stabilization) •Modified isochronal deliverability tests • A series tests at different rates for equal periods of flow‐time and shut‐in times
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Stabilized Flow Equations ; r i > r e The approximate time to stabilization
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Transient Flow Equations; r i < r e
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Analysis of Conventional Backpressure Test
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Summary Conventional Back‐pressure Test
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Example: Conventional Back‐pressure Test
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Analysis of Isochronal Test
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Example: Analysis of
Isochronal Test
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Analysis of Modified Isochronal Test
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Example: Analysis of
Modified Isochronal Test
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The idea behind the isochronal methods: the radius of investigation is independent of q .
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•Collect and utilize all information •logs, drill‐stem tests, •previous deliverability tests conducted on that well, •production history, •fluid composition and temperature, •cores, and •geological studies. •Knowledge of the time required for stabilization (a very important factor in deciding the type of test to be used for determining the deliverability of a well) •from previous tests (such as drill‐stem, deliverability tests, the production characteristics of the well) •When the approximate time to stabilization is not known, it may be estimated from Eq. 19‐3
•Duration equal to about four times this value is recommended for the isochronal periods. •The minimum flow rate •In conducting a multipoint test, the minimum flow rate used should be at least equal to that required lifting the liquids, if any, from the well. It should also be sufficient to 19 S.Gerami maintain a wellhead temperature above the hydrate point.
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