Rate Transient Analysis

Theory/Software Course
Introduction to Well Performance
Analysis
 Traditional

- Production rate only

- Using historical trends to predict future

- Empirical (curve fitting)

- Based on analogy

- Deliverables:
- Production forecast
- Recoverable Reserves under current
conditions
 Modern

- Rates AND Flowing Pressures

- Based on physics, not empirical

- Reservoir signal extraction and characterization

- Deliverables:
- OGIP / OOIP and Reserves
- Permeability and skin
- Drainage area and shape
- Production optimization screening
- Infill potential
 Recommended Approach

- Use BOTH Traditional and Modern together

- Production Data Analysis should include a

comparison of multiple methods

- No single method always works

- Production data is varied in frequency, quality

and duration
 Modern Production Analysis - Integration of
Knowledge

Modern Production Analysis

Welltest Analysis Empirical Decline
Analysis
- Flow regime
characterization over
- High resolution life of well
early-time
- Characterization
characterization - Estimation of fluids-
of perm and skin
in-place - Estimation of
- Projection
- High resolution reserves when
-Estimation of of recovery
characterization of - Performance based flowing pressure is
contacted constrained
the near-wellbore recovery factor unknown
drainage area by historical
operating
-Point-in-time - Able to analyze
-Estimation of conditions
characterization of transient production
reservoir pressure
wellbore skin data (early-time
production, tight gas
etc)
Theory
 Traditional Decline Curves – J.J. Arps

- Graphical – Curve fitting exercise

- Empirical – No theoretical basis

- Implicitly assumes constant operating conditions

 The Exponential Decline Curve
Unnamed Well Rate vs Time
5.00

4.50

4.00

q = qie − Dit
3.50
fd

3.00
,M
te s
Mc

Slope
2.50
a
sR

Di =
a
G

2.00

1.50

1.00
q
0.50

0.00
2001 2002 2003 2004 2005 2006

Unnamed Well Rate vs Time Rate vs. Cumulative Prod.

Unnamed Well
101
4.50
7
6

Dit
log q = log qi −
5 4.00

q = qi − DiQ
4

3.50
3

2
2.302
Di = Slope
3.00
Gas Rate, MMscfd

Di = 2.302* Slope
G as Rate, MMscfd
2.50
1.0

2.00
7
6
5
1.50
4

3
1.00

2
0.50

0.00
10-1
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50
2001 2002 2003 2004 2005 2006
Gas Cum. Prod., Bscf
 The Hyperbolic Decline Curve
Unnamed Well Rate vs. Cumulative Prod.
4.50

4.00

3.50
qi
q=
(1 + bDit )1/ b
3.00
Gas Rate, MMscfd

2.50
Di b
2.00 D= b q
1.50
qi
1.00

0.50
D = f (t )
0.00
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60
Gas Cum. Prod., Bscf
 Hyperbolic Exponent “b”
Unnamed Well Rate vs. Cumulative Prod.
4.50

4.00

Mild Hyperbolic – b ~ 0
3.50

3.00
M
,Msfd
c

2.50
sR
a te
a

2.00
G

1.50

1.00

0.50

0.00
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60
Gas Cum. Prod., Bscf

NBU 921-22G R a t e v s . C u m u la t iv e P r o d .

3 .2 0

3 .0 0

2 .8 0

2 .6 0

2 .4 0
Strong Hyperbolic – b ~ 1
2 .2 0

2 .0 0
Gas Rate, MMscfd

1 .8 0

1 .6 0

1 .4 0

1 .2 0

1 .0 0

0 .8 0

0 .6 0

0 .4 0

0 .2 0

0 .0 0
0 .0 0 0 .0 5 0 .1 0 0 .1 5 0 .2 0 0 .2 5 0 .3 0 0 .3 5 0 .4 0 0 .4 5 0 .5 0 0 .5 5 0 .6 0 0 .6 5 0 .7 0 0 .7 5 0 .8 0 0 .8 5 0 .9 0 0 .9 5 1 .0 0 1 .0 5
G a s C u m u la t iv e , B s c f
Analytical Solutions
Definition of Compressibility

pi pi-dp
dV

V V

1 ∂V
c=−
V ∂p
Compressibility Defines Material Balance of
a Closed Oil Reservoir (above bubble point)

Δp = pi - p ΔV = Np

V=N

1 Np
c=
N pi − p
Np
p = pi −
ctN
p = pi − mpssNp

Assumptions: 1. c is constant
2. Bo is constant
 Illustration of Pseudo-Steady-State

p1
1
p2
2
p3
pressure

3
pwf1

pwf2 time
Constant Rate q
pwf3

rw Distance re
 Steady-State Inflow Equation

pi

p
p − pwf = qbpss
pressure

141.2Bμ q ⎛ re 3 ⎞
bpss = ⎜ ln − ⎟
kh ⎝ rwa 4 ⎠
pwf

Inflow (Darcy) pressure drop- Constant- Productivity

Index

rw Distance re
The Two Most Important Equations in
Modern Production Analysis

p = pi − mpssNp

p = pwf + qbpss
 Operating Conditions - Simplified

Constant Pressure Constant Rate

= =
Production Welltest

q q

pwf pwf
 Constant Flowing Pressure Solution

- Required: q(t), Npmax and N for constant pwf

- Take derivative of both equations and solve for q

- Integrate to find Np(t), as t goes to infinity Np

goes to Npmax

pi − pwf
mpss
− t
q(t ) = e bpss

bpss
pi − pwf
Np max = = ( pi − pwf ) ctN
mpss
Constant Flowing Pressure Solution – Relate
back to Arps Exponential, Determine N

pi − pwf
qi =
bpss
mpss
Di =
bpss
qi
Np max =
Di
ct ( pi − pwf ) ct ( pi − pwf ) Di
N= =
Np max qi
 Constant Rate Solution

- Required: pwf(t), Npmax and N for constant rate

- Equate left side to right and solve for pi-pwf

- Set pwf = 0 to find Npmax

- Plot pi-pwf versus Np to get N

pi − pwf (t ) = mpssNp + bpssq

pi − bpssq
Np max =
mpss
 Constant Rate – PSS Plot

y = mx + b
Np 141.2Bμ q ⎛ re 3 ⎞
pi − pwf = + ⎜ ln − ⎟
ctN kh ⎝ rwa 4 ⎠

pi − pwf

1
m= = mpss
cN

Np
Constant Rate Solution – Relate back to
Arps Harmonic

- Normalize the PSS equation with q

- Invert the PSS equation

q 1 1
= =
pi − pwf (t ) mpssNp + bpss mpsst + bpss
q
1
q bpss
=
pi − pwf (t ) mpss t + 1
bpss
Plot Constant p and Constant q together

0. 9
Constant rate q/Δp (Harmonic)

0. 8
1
q bpss
=
pi − pwf (t ) mpss t + 1
0. 7

0. 6 bpss
0. 5

0. 4

Constant pressure q/Δp (Exponential)

0. 3

mpss
0. 2
q(t ) 1 − bpss t
= e
0. 1
pi − pwf bpss

0 5 10 15 20 25 30 35 40 45
 Transient Flow

- Early-time OR Low Permeability

- Flow that occurs while a pressure “pulse” is

moving out into an infinite or semi-infinite acting
reservoir

- Like the “fingerprint” of the reservoir

- Contains information about reservoir
properties (permeability, drainage shape)
 Boundary Dominated Flow

- Late-time flow behavior

- Typically dominates long-term production data

- Reservoir is in a state of pseudo-equilibrium –

physics reduces to a mass balance

- Contains information about reservoir pore volume

(OOIP and OGIP)
Transient Flow
 Transient and Boundary Dominated Flow

3600

3400

3200

3000
Cross Section 2800

2600
Transient Well Performance = Boundary Dominated
2400 f(k, skin, time) Well Performance =
2200
f(Volume, PI)
ps

2000

Plan View
 Radius (Region) of Investigation

3600

3400

3200

3000
Cross Section 2800

2600

2400
kt
2200 rinv =
948φμ c
ps

2000

π kt
Ainv =
948φμ c

Plan View
 Transient Equation

Describes radial flow in an infinite acting reservoir

q kh 1
=
( pi − pwf ) 141.2 μ B 1 ⎛ 0.0063kt ⎞
ln ⎜ ⎟ + 0.4045 + s
2 ⎝ φμ ct ⎠
 q(t)’s compared

1. 6

1. 4

1. 2

1 Transient flow: compares to Arps “super

hyperbolic” (b>1)
0. 8

0. 6

0. 4

0. 2

0
0 5 10 15 20 25 30 35 40 45
 Blending of Transient into
Boundary Dominated Flow
3

2. 5
Complete q(t) consists of:
Transient q(t) from t=0 to tpss
2 Depletion equation from t = tpss and higher

1. 5

0. 5

0 5 10 15 20 25 30 35 40 45
 Log-Log plot: Adds a new visual dynamic
Comparison of qD with 1/pD
Cylindrical Reservoir with Vertical Well in Center

1000

Infinite Acting Boundary Dominated

100

10 Constant Rate Solution 0.9

Harmonic
1
qD and 1/p

0.1

0.01

0.001

0.0001
Constant Pressure Solution Exponential

0.00001

0.000001
0.000001 0.0001 0.01 1 100 10000 1000000 100000000 1E+10 1E+12 1E+14

tD
Type Curves
 Type Curve

- Dimensionless model for reservoir / well system

- Log-log plot

- Assumes constant operating conditions

- Valuable tool for interpretation of production and

pressure data
 Type Curve Example - Fetkovich

Fetkovich Typecurve Analysis

1.0

7
6
5
4
Harmonic
q (t ) 1
3
qDd = qDd =
1 + tDd
2 qi
tDd = Dit
qDd
Rate,

10-1
9

7
6
5 Exponential Hyperbolic
4 qDd = e − tDd qDd =
1
3 (1 + btDd )1/ b
2

10-2
2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 2 3 4 5 6 7 8
10-1 1.0 101

tDd Time
 Plotting Fetkovich Type Curves- Example
Time (years) Rate (MMscfd) tDd qDd
Well 1 (exponential) Well 1 Well 2 Well 1 Well 2 Well 1 Well 2
0 2.50 10.00 0.00 0.00 1.00 1.00
qi = 2.5 MMscfd 1 2.26 8.19 0.10 0.20 0.90 0.82
Di = 10 % per year 2 2.05 6.70 0.20 0.40 0.82 0.67
3 1.85 5.49 0.30 0.60 0.74 0.55
4 1.68 4.49 0.40 0.80 0.67 0.45
Well 2 (exponential) 5 1.52 3.68 0.50 1.00 0.61 0.37
6 1.37 3.01 0.60 1.20 0.55 0.30
qi = 10 MMscfd
7 1.24 2.47 0.70 1.40 0.50 0.25
Di = 20 % per year 8 1.12 2.02 0.80 1.60 0.45 0.20
9 1.02 1.65 0.90 1.80 0.41 0.17
10 0.92 1.35 1.00 2.00 0.37 0.14

Raw Data Plot Dimensionless Plot

12.00 1.00
10.00
Rate (MMscfd)

8.00
Well 1 Well 1

qDd
6.00
Well 2 Well 2
4.00
2.00
0.00 0.10
0 5 10 15 0.01 0.10 1.00 10.00
Time (years) tDd
 Fetkovich Typecurve Matching

In most cases, we don’t know what “qi” and “Di” are ahead of time. Thus, qi and Di
are calculated based on the typecurve match (ie. The typecurve is superimposed on
the data set
NBU 921-22G Fetkovich Typecurve Analysis

q (t )
qi = 1.0

qDd 8
7

q6
5

tDd
Di =
4

3
qDd
Rate,

t 2

10-1
9
8
7
6

5
3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2
1.0 101

tDd
Time

Knowing qi and Di, EUR (expected ultimate recovery) can be calculated

 Analytical Model Type Curve
Fetkovich Typecurve Analysis
101

6
4
3
2

1.09

6
Transient Flow
4
3
2
Rate,

qDd 10-19 re/rwa = 10 re/rwa = 100 re/rwa = 10,000

6
4
3
2

10-29

6
4
3
2 Boundary Dominated Flow
Exponential
2 3 4 5 6 7 9 -3 2 3 4 5 678 -2 2 3 4 5 678 -1 2 3 4 5 678 2 3 4 5 678 1 2 3 4 5 67
10-4 10 10
tDd
10
Time
1.0 10
Dimensionless Variable Definitions (Fetkovich)

141.2q μ B ⎡ ⎛ re ⎞ 1 ⎤
qDd = ⎢ ln ⎜ ⎟− ⎥
kh( pi − pwf ) ⎣ ⎝ rwa ⎠ 2 ⎦

0.00634kt
φμ ctrwa 2
tDd =
⎡
1 ⎡ ⎛ re ⎞ 1 ⎤ ⎛ re ⎞
2
⎤
⎢ ln ⎜ ⎟ − ⎥ ⎢⎜ ⎟ − 1⎥
2 ⎣ ⎝ rwa ⎠ 2 ⎦ ⎢⎣⎝ rwa ⎠ ⎥⎦
 Type Curve Matching (Fetkovich)

The Fetkovich analytical typecurves can be used to calculate three paramters:

permeability, skin and reservoir radius

141.2μ B ⎡ ⎛ re ⎞ 1 ⎤ q
k= ⎢ ln ⎜ ⎟− ⎥
h( pi − pwf ) ⎣ ⎝ rwa ⎠ 2 ⎦ qDd match

0.00634k 1 t ⎛ rw ⎞
rwa = s = ln ⎜ ⎟
φμ ct 1 ⎡ ⎛ re ⎞ 1 ⎤ ⎡⎛ re ⎞ 2 ⎤ tDd ⎝ rwa ⎠
⎢ ln ⎜ ⎟ − ⎥ ⎢⎜ ⎟ − 1⎥
2 ⎣ ⎝ rwa ⎠ 2 ⎦ ⎣⎢⎝ rwa ⎠ ⎦⎥ match

141.2 B 0.00634 q t
re = 2
h( pi − pwf ) φ ct qDd match tDd match
 Type Curve Matching - Example
10 Fetkovich Typecurve Analysis
101
8
6
k = f(q/qDd)
4
3 reD = 50 s = f(q/qDd * t/tDd, reD)
2 re = f(q/qDd * t/tDd)
q
1.0
8
6
4 Transient Flow
3
2
Rate,

qDd 10-1
8
6
4 t
3
2

10-2
8
6
4
3
2 Boundary Dominated Flow
Exponential
10-3
2 3 4 5 678 -3 2 3 4 5 678 -2 2 3 4 5 678 -1 2 3 4 5 678 2 3 4 5 678 1 2 3 4 5 6 78
10-4 10 10
tDd
10
Time
1.0 10
What about Variable Rate / Variable Pressure Production?
The Principle of Superposition
Superposition in Time:

1. Divide the production history into a series of constant rate periods

2. The observed pressure response is a result of the additive effect of each rate
change in the history

Example: Two Rate History

q2

q q1
pi − pwf = q1 f (t ) + (q 2 − q1) f (t − t1)
pwf

Effect of (q2-q1)

t1
 The Principle of Superposition - Continued

Two Rate History

pi − pwf = q1 f (t ) + (q 2 − q1) f (t − t1)

N - Rate History
N
pi − pwf = ∑ (qj − qj − 1) f (t − tj − 1)
j =1
f(t) is the Unit Step Response
 Superposition Time

Convert multiple rate history into an equivalent single rate history by re-plotting
data points at their “superposed” times

pi − pwf N
(qj − qj − 1)
=∑ f (t − tj − 1)
qN j =1 qN
 The Principle of Superposition – PSS Case

pi − pwf N
(qj − qj − 1)
=∑ f (t − tj − 1)
qN j =1 qN

pi − pwf t 141.2Bμ ⎛ re 3 ⎞
f (t ) = = + ⎜ ln − ⎟
q ctN kh ⎝ rwa 4 ⎠

pi − pwf 1 N
(qj − qj − 1) 141.2Bμ ⎛ re 3 ⎞
qN
=
ctN
∑
j =1 qN
(t − tj − 1) + ⎜ ln − ⎟
kh ⎝ rwa 4 ⎠
pi − pwf 1 Np 141.2Bμ ⎛ re 3 ⎞
= + ⎜ ln − ⎟
qN ctN qN kh ⎝ rwa 4 ⎠

Superposition Time: Material Balance Time

 Definition of Material Balance Time
(Blasingame et al)

Actual Rate Decline Equivalent Constant Rate

Q
Q

actual material
time (t) balance = Q/q
time (tc)
 Features of Material Balance Time

-MBT is a superposition time function

- MBT converts VARIABLE RATE data into an

EQUIVALENT CONSTANT RATE solution.

- MBT is RIGOROUS for the BOUNDARY

DOMINATED flow regime

- MBT works very well for transient data also, but

is only an approximation (errors can be up to 20%
for linear flow)
MBT Shifts Constant Pressure to Equivalent Constant
Rate
Comparison of qD (Material Balance Time Corrected) with 1/pD
Cylindrical Reservoir with Vertical Well in Center

1000 1.2
Very early time radial flow
Ratio (qD to 1/pD) ~ 90%
100
1
0.97
10

1 Constant Rate Solution 1/pD 0.8

Harmonic

Ratio 1/pD to q
qD and 1/p

0.1
Beginning of "semi-log" radial flow (tD=25) 0.6
Ratio (qD to 1/pD) ~ 97%
0.01

0.001 0.4

0.0001
Constant Pressure Solution qD
0.2
Corrected to Harmonic
0.00001

0.000001 0
0.000001 0.0001 0.01 1 100 10000 1000000 100000000 1E+10 1E+12 1E+14

tD
Corrections for Gas Reservoirs
Corrections Required for Gas Reservoirs

• Gas properties vary with pressure

– Formation Volume Factor
– Compressibility
– Viscosity
Corrections Required for Gas Reservoirs

Depletion Term Reservoir FlowTerm:

Depends on Depends on “B” and
compressibility Viscosity

qt 141.2qBoμ ⎛ re 3 ⎞
pi − pwf = + ⎜ ln − ⎟
coN kh ⎝ rwa 4 ⎠
Darcy’s Law Correction for Gas Reservoirs

Darcy’s Law states : Δp ∝ q

For Gas Flow, this is not true because

viscosity (μ) and Z-factor (Z) vary with pressure

Solution: Pseudo-Pressure

p
pdp
pp = 2 ∫
0
μZ
 Depletion Correction for Gas Reservoirs

Gas properties (compressiblity and viscosity) vary

significantly with pressure
Gas Compressibility

0.012

0.01

0.008
Compressibility (1/psi)

0.006
1
cg ≈
0.004
p

0.002

0
0 1000 2000 3000 4000 5000 6000
Pressure (psi)
 Depletion Correction for Gas Reservoirs:
Pseudo-Time

Solution: Pseudo-Time

ta = (μcg )i ∫
dt
t

0 μc g

μ , c g → Evaluated
pressure
at average reservoir

Not to be confused with welltest pseudo-

time which evaluates properties at well
flowing pressure
 Boundary Dominated Flow Equation for Gas

Constant Rate Case

Pseudo-pressure Pseudo-time

2 pi 1.417e6 * Tq ⎛ re 3 ⎞
Δpp = ppi − ppwf = qta + ⎜ ln − ⎟
( μcgZ )iGi kh ⎝ rwa 4 ⎠

Variable Rate Case

Pseudo-Cumulative Production
Δpp αGpa
= + bpss
q qGi
Overall time function -
Material Balance Pseudo-time

1 t
tc = ∫ qdt
q 0
1 ta
tca = ∫ qdta =
(μcg )i t qdt
q 0 q ∫0 μ c g
Corrected Material Balance Pseudo-time

Overall material balance pseudo-time function (corrected

for variable fluid saturations and formation expansion):

( μct )i t q (t )
tca =
q ∫0 μc t [1 − cf ( pi − p)] dt
Where, Evaluated at average
reservoir pressure
c t = cf + s oco + s wcw + s gcg
 Practice

- Traditional
- Blasingame
- Agarwal – Gardner and NPI
- Flowing p/z analysis
- Transient
- Models and History Matcning
Notes About Drive Mechanism and b Value (from
Arps and Fetkovich)

b value Reservoir Drive Mechanism

0 Single phase liquid expansion (oil above bubble point)

Single phase gas expansion at high pressure
Water or gas breakthrough in an oil well

0.1 - 0.4 Solution gas drive

0.4 - 0.5 Single phase gas expansion

0.5 Effective edge water drive

0.5 - 1.0 Layered reservoirs

>1 Transient (Tight Gas)

 Advantages of Traditional

- Easy and convenient

- No simplifying assumptions are required regarding the

physics of fluid flow. Thus, can be used to model very
complex systems

- Very “Real” indication of well performance

 Limitations of Traditional

- Implicitly assumes constant operating conditions

-Non-unique results, especially for tight gas (transient flow)

- Provides limited information about the reservoir

 Example 1: Decline Overpredicts Reserves
Unnamed Well Rate vs Time
Gas Rate, MMscfd

October November December January February March April

2001 2002

Unnamed Well Rate vs. Cumulative Prod.

4
EUR = 9.5 bcf
Gas Rate, MMscfd

0
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 10.50
Gas Cum. Prod., Bscf
 Example 1 (cont’d)
Flowing Pressure and Rate vs Cumulative Production
Rates
5 1200
4.5
1000
4

Flowing Pressure (psia)

3.5
True EUR does not 800
Rate (MMscfd)

3 exceed 4.5 bcf

Pressures
2.5 600
2 Forecast is not valid
here 400
1.5

1
200
0.5
0 0
0 1 2 3 4 5 6 7 8 9 10
Cumulative Production (bcf)
 Example 2: Decline Underpredicts Reserves
Unnamed Well Rate vs. Cumulative Prod.
8.50

8.00

7.50

7.00

6.50

6.00

5.50
EUR = 3.0 bcf
5.00
Gas Rate, MMscfd

4.50

4.00

3.50

3.00

2.50

2.00

1.50

1.00

0.50

0.00
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20
Gas Cum. Prod., Bscf
 Example 2 (cont’d)
Unnamed Well Flowing Material Balance
0.085
Legend
Decline FMB
0.080

0.075

0.070

0.065

0.060

OGIP = 24 bcf
Normalized Rate, MMscfd/(106psi2/cP)

0.055

0.050

0.045

0.040

0.035

0.030

0.025

0.020

0.015

0.010

0.005 Original Gas In Place

0.000
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Normalized Cumulative Production, Bscf
 Example 2 (cont’d)
Unnamed Well Data Chart
18 Legend 1300
Pressure
17 Actual Gas Data
1200
16

15 1100

14
1000
13
900
12

11
Operating conditions: Low drawdown 800

Increasing back pressure

Pressure, psi
Gas, MMscfd

10
700
9

600
8

7 500

6
400
5

4 300

3
200

2
100
1

0 0
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600 620 640 660 680 700 720
Time, days
 Example 3 – Illustration of Non-Uniqueness

Arps Production Forecast

10

1
Gas Rate (MMscfd)

0.1

Economic Limit = b = 0.25, b = 0.50, b = 0.80,

0.05 MMscfd
EUR = 2.0 bcf EUR = 2.5 bcf EUR = 3.6 bcf

0.01
Dec-00 May-06 Nov-11 May-17 Oct-22 Apr-28 Oct-33
Time
 Blasingame Typecurve Analysis

Blasingame typecurves have identical format to those of Fetkovich.

However, there are three important differences in presentation:

1. Models are based on constant RATE solution instead of

constant pressure

2. Exponential and Hyperbolic stems are absent, only

HARMONIC stem is plotted

3. Rate Integral and Rate Integral - Derivative typecurves

are used (simultaneous typecurve match)

Data plotted on Blasingame typecurves makes use of MODERN

DECLINE ANALYSIS methods:

- NORMALIZED RATE (q/Δp)

- MATERIAL BALANCE TIME / PSEUDO TIME

 Blasingame Typecurve Analysis-
Comparison to Fetkovich

Fetkovich Blasingame

log(q) log(q/Δp)

log(qDd) log(qDd)

log(t) log(tca)

log(tDd) log(tDd)

- Usage of q/Δp and tca allow boundary dominated flow to be

represented by harmonic stem only, regardless of flowing
conditions

- Blasingame harmonic stem offers an ANALYTICAL fluids-in-place

solution

- Transient stems (not shown) are similar to Fetkovich

 Blasingame Typecurve Analysis- Definitions

Typecurves Data - Oil Data - Gas

141.2qβμ ⎡ ⎛ re ⎞ 1 ⎤ q q
q Dd = ⎢ln⎜ rwa ⎟ − 2 ⎥
Normalized Rate khΔP ⎣ ⎝ ⎠ ⎦ ΔP ΔPp

t DA
⎛ q ⎞ 1 q
t
c
⎛ q ⎞ t ca

∫ qDd (t )dt
1
⎟ = ∫ ⎜ ⎟ = 1 q
Rate Integral q Ddi =
t DA
⎜
⎝ ΔP ⎠i tc 0 ΔP
dt
⎜ ΔP ⎟ t ∫ ΔP dt
0 ⎝ p ⎠ i ca 0 p

Rate Integral - q Ddid = t DA

dq Ddi ⎛ q ⎞ ⎛ q ⎞
d⎜ ⎟ tca d ⎜ ⎟
Derivative dt DA ⎛ q ⎞ ⎝ ΔP ⎠ i ⎛ q ⎞ ⎜ ΔP ⎟
⎜ ⎟ = tc ⎜ ⎟ ⎝ p ⎠i
⎝ ΔP ⎠id =
dtc ⎜ ΔP ⎟ dtca
⎝ p ⎠id
 Concept of Rate Integral
(Blasingame et al)

actual rate rate integral

= Q/t

Q
Q

actual actual
time time
 Rate Integral: Like a Cumulative Average

Average rate over time period

“0 to t1”
Average rate over time period
“0 to t2”

t1 t2
Effective way to remove noise
Rate Integral: Definition

⎛ q ⎞ 1 q
tc

⎜⎜ ⎟⎟ = ∫ dt
⎝ Δp ⎠i tc 0 Δp
Typecurve Interpretation Aids: Integrals, Derivatives

Typecurve Most Useful For Drawback Used in Analysis

Integral / Removing the scatter from Dilutes the reservoir Fetkovich,

Cumulative noisy data sets signal Blasingame, NPI

Amplifying the reservoir

Amplifies noise - Agarwal-Gardner,
Derivative signal embedded in
often unusable PTA
production data

Maximizing the strengths

Integral-Derivative Can still be noisy Blasingame, NPI
of Integral and Derivative

Other methods: Data filtering, Moving averages, Wavelet decomposition

Rate Integral and Rate Integral Derivative
(Blasingame et al)

Rate Integral

Rate (Normalized)

Rate Integral Derivative

Blasingame Typecurve Analysis-
Transient Calculations

Oil:

k is obtained from rearranging the definition of

q ⎛ 141 .2 βμ ⎞⎛⎜ ⎛⎜ re ⎞ 1⎞
q Dd = ⎜ ⎟⎜ ln ⎜ ⎟⎟ − ⎟
Δ p ⎝ kh ⎠⎝ ⎝ rwa ⎟
⎠ match 2 ⎠

⎛ q ⎞
⎜ Δp ⎟ ⎛ 141 .2 βμ ⎞⎛⎜ ⎛⎜ re ⎞ 1⎞
k =⎜ ⎟ ⎜ ⎟⎜ ln ⎜ ⎟⎟ − ⎟
⎜ q Dd ⎟ ⎝ ⎠⎝ ⎝ rwa ⎟
h ⎠ match 2 ⎠
⎝ ⎠ match

Solve for rwa from the definition of

0 .006328 kt c
t Dd =
⎛
2 ⎜ ⎛ re ⎞
2
⎞⎛ ⎛ r ⎞ 1⎞
− 1 ⎟⎜ ln ⎜⎜ e
1
φμ c t rwa ⎜⎜ ⎟⎟ ⎟⎟ − ⎟
2 ⎜ ⎝ rwa ⎠ ⎟⎜ ⎝ rwa ⎟
⎠ match 2 ⎠
⎝ match ⎠⎝

⎛ t ⎞
= ⎜ c ⎟ 0 .006328 k
r
wa ⎜t ⎟ ⎛ 2 ⎞⎛ ⎛ ⎞
⎝ Dd ⎠ match 1 ⎜ ⎛⎜ re ⎞⎟ ⎟⎜ r ⎞ 1⎟
φμ c ⎜ − 1 ⎟⎜ ln ⎜ e ⎟ − ⎟
t 2 ⎜⎜ r ⎟ ⎟⎜ ⎜⎝ rwa ⎟ 2⎟
⎝ ⎝ wa ⎠ match
⎠⎝ ⎠ match ⎠
⎛ r ⎞
s = ln ⎜⎜ w ⎟⎟
⎝ rwa ⎠
 Blasingame Typecurve Analysis-
Boundary Dominated Calculations- Oil

Oil-in-Place calculation is based on the harmonic stem of Fetkovich typecurves.

In Blasingame typecurve analysis, qDd and tDd are defined as follows:

qDd =
(q / Δp ) and tDd = Ditc
(q / Δp )i
Recall the Fetkovich definition for the harmonic typecurve and the PSS equation for oil in
harmonic form:
Definition of Harmonic PSS equation for oil in
typecurve 1 harmonic form, using
material balance time
1 q b
q Dd = and =
1 + tDd Δp 1
tc + 1
ctNb
From the above equations:

⎛
⎜
⎜
q ⎞⎟⎟
⎜⎜
Δp ⎟⎟⎠ i ⎛ q ⎞ 1
q
= ⎝
where ⎜ ⎟ = , and Di =
1
Δp 1 + Ditc ⎜ Δp ⎟ b ctNb
⎝ ⎠i
Blasingame Typecurve Analysis- Boundary
Dominated Calculations- Oil (cont’d)

Oil-in-Place (N) is calculated as follows:

Rearranging the equation for Di:

1
N=
ctDib
Now, substitute the definitions of qDd and tDd back into the above equation:

1 1 ⎡ tc ⎤ ⎡ (q / Δp ) ⎤
N= = ⎢ ⎥⎢ ⎥
⎡ tDd ⎤ ⎡ qDd ⎤ ct ⎣ tDd ⎦ ⎣ qDd ⎦
ct ⎢ ⎥ ⎢ ⎥
⎣ tc ⎦ ⎣ (q / Δp ) ⎦

X-axis “match-point from Y-axis “match-point”

typecurve analysis from typecurve analysis
Blasingame Typecurve Analysis- Boundary
Dominated Calculations- Gas
Gas-in-Place calculation is similar to that of oil, with the additional complications of
pseudo-time and pseudo-pressure.

In Blasingame typecurve analysis, qDd and tDd are defined as follows:

q Dd =
(q / Δpp ) and tDd = Ditca
(q / Δpp )i

Recall the Fetkovich definition for the harmonic typecurve and the PSS equation for gas
in harmonic form:
Definition of Harmonic PSS equation for gas in
typecurve harmonic form, using
1 material balance pseudo-
time
1 q b
qDd = and =
1 + tDd Δpp 2 pi
tca + 1
(Zμct )iGib
From the above equations:

⎛
⎜
⎜
q ⎞⎟⎟
⎜⎜
Δp ⎟⎟⎠ i ⎛ q ⎞ 1
q
= ⎝
where ⎜ ⎟ = , and Di =
2 pi
Δp 1 + Ditc ⎜ Δpp ⎟ b
⎝ ⎠i
(Zμct )iGib
Blasingame Typecurve Analysis- Boundary
Dominated Calculations- Gas

Gas-in-Place (Gi) is calculated as follows:

Rearranging the equation for Di:

2 pi
Gi =
Di (Zμct )ib
Now, substitute the definitions of qDd and tDd back into the above equation:

2 pi 2 pi ⎡ tca ⎤ ⎡ (q / Δpp ) ⎤
Gi = =
⎛ tDd ⎞ ⎛ ⎞
(Zμct )i ⎢⎣ tDd ⎥⎦ ⎢⎣ qDd ⎥⎦
⎜ ⎟(Zμct )i ⎜⎜
⎜ q Dd ⎟
⎟
⎝ ( q / Δpp ) ⎠
⎟⎟
⎝ tca ⎠ ⎜

X-axis “match-point from Y-axis “match-point”

typecurve analysis from typecurve analysis
 Agarwal-Gardner Typecurve Analysis

Agarwal and Gardner have developed several different diagnostic

methods, each based on modern decline analysis theory. The AG
typecurves are all derived using the WELLTESTING definitions of
dimensionless rate and time (as opposed to the Fetkovich
definitions). The models are all based on the constant RATE
solution. The methods they present are as follows:

1. Rate vs. Time typecurves (tD and tDA format)

2. Cumulative Production vs. Time typecurves (tD and tDA

format)

3. Rate vs. Cumulative Production typecurves (tDA

format)
- linear format
- logarithmic format
Agarwal-Gardner Typecurve Analysis
Agarwal-Gardner - Rate vs. Time typecurves

Agarwal and Gardner Rate vs. Time typecurves are the same as
conventional drawdown typecurves, but are inverted and plotted in
tDA (time based on area) format.

qD vs tDA

The AG derivative plot is not a rate derivative (as per Blasingame).

Rather, it is an INVERSE PRESSURE DERIVATIVE.

pD(der) = t(dpD/dt) qD(der) = t(dqD/dt)

1/pD(der) = ( t(dpD/dt) ) -1
Agarwal-Gardner - Rate vs. Time typecurves

Comparison to Blasingame typecurves

Rate Integral-
Derivative
Inv. Pressure
Integral-
Derivative

qDd and tDd

plotting format
qD and tDA
plotting fomat
Agarwal-Gardner - Rate vs. Cumulative typecurves

Agarwal and Gardner Rate vs. Cumulative typecurves are different

from conventional typecurves because they are plotted on LINEAR
coordinates.

They are designed to analyze BOUNDARY DOMINATED data only.

Thus, they do not yield estimates of permeability and skin, only
fluid-in-place.

Plot: qD (1/pD) vs QDA

Where (for oil):

141.2 μB q (t )
qD =
kh pi − pwf (t )

1 Q 1 pi − p
QDA = qD * tDA = or alternatively
2π ctN ( pi − pwf ) 2π pi − pwf
Agarwal-Gardner - Rate vs. Cumulative typecurves

Where (for gas):

1.417e6 * T q (t )
qD =
kh ψ i −ψ wf (t )
1 2qtca 1 ψ i −ψ
QDA = qD * tDA = or alternatively
2π (ctμZ )iGi (ψ i −ψ wf ) 2π ψ i − ψ wf
Agarwal-Gardner - Rate vs. Cumulative typecurves

qD vs QDA typecurves always

converge to 1/2π (0.159)
 NPI (Normalized Pressure Integral)

NPI analysis plots a normalized PRESSURE rather than a normalized

RATE. The analysis consists of three sets of typecurves:

1. Normalized pressure vs. tc (material balance time)

2. Pressure integral vs. tc

3. Pressure integral - derivative vs. tc

- Pressure integral methodology was developed by Tom Blasingame;

originally used to interpret drawdown data with a lot of noise. (ie.
conventional pressure derivative contains far too much scatter)

- NPI utilizes a PRESSRE that is normalized using the current RATE.

It also utilizes the concepts of material balance time and pseudo-
time.
 NPI (Normalized Pressure
Integral): Definitions

Typecurves Data - Oil Data - Gas

khΔP ΔP ΔPp
Normalized Pressure PD =
141.2qβμ q q
dPD ⎛ ΔP ⎞ d (ΔPp )
PDd = d ⎜⎜ ⎟⎟
d (ln t DA ) ⎛ ΔP ⎞ ⎛ ΔPp ⎞
⎟⎟ = ⎝ ⎠
Conventional q q
⎜⎜ ⎜⎜ ⎟⎟ =
⎝ q ⎠ d d (ln t c ) ⎝ q ⎠ i d (ln t ca )
Pressure Derivative

⎛ ΔP ⎞ ΔP ⎛ ΔPp ⎞ 1 ca ΔPp
t DA tc t

∫ Pp (t )dt
1 1
Pressure Integral PDi =
t DA 0
⎜⎜ ⎟⎟ =
⎝ q ⎠i tc
∫0 q dt ⎜⎜
⎝ q
⎟⎟ =
⎠i t ca
∫0
q
dt

dPDi ⎛ ΔP ⎞ ⎛ ΔPp ⎞
PDid = t DA d ⎜⎜ ⎟⎟ t ca d ⎜⎜ ⎟⎟
Pressure Integral - dt DA ⎛ ΔP ⎞ ⎝ q ⎠i ⎛ ΔPp ⎞ ⎝ q ⎠i
Derivative ⎜⎜ ⎟⎟ = tc ⎜⎜ ⎟⎟ =
⎝ q ⎠ id dt c ⎝ q ⎠ id dt ca
NPI (Normalized Pressure
Integral): Diagnostics

Transient

Normalized Pressure
Typecruve

Integral - Derivative
Typecurve

Boundary
Dominated
NPI (Normalized Pressure Integral): Calculation
of Parameters- Oil

Oil - Radial

khΔP 0.00634kt c
PD = t DA =
141.2qβμ πφμC t re2

⎛ ⎞
⎜ ⎟
141.2 βμ ⎜ PD ⎟
k=
h ⎜ ΔP ⎟
⎜ ⎟
⎝ q ⎠ match

0.00634k ⎛ tc ⎞
re = ⎜ ⎟
πφμCt ⎜⎝ t DA ⎟⎠ match
re ⎛r ⎞
rwq = S = ln⎜⎜ w ⎟⎟
⎛ re ⎞ ⎝ rwa ⎠
⎜⎜ ⎟⎟
⎝ rwa ⎠ match

⎛ ⎞
⎜ ⎟
0.00634 ⎛ 141.2 S 0 ⎞⎜ PD ⎟ ⎛ tc ⎞
N= ⎜ ⎟ ⎜⎜ ⎟⎟ (MBBIS)
⎜
C t ⎝ 5.615 * 1000 ⎠ ΔP ⎟ ⎝ DA ⎠ match
t
⎜ ⎟
⎝ q ⎠ match
NPI (Normalized Pressure Integral):
Calculation of Parameters- Gas

Gas – Radial

khΔPp 0.00634kt ca
PD = t DA =
1.417Ε6Tq πφμ i C ti re2

⎛ ⎞
⎜ ⎟
1.417Ε6T ⎜ PD ⎟
k= ⎜ ΔPp ⎟
h
⎜⎜ ⎟⎟
⎝ q ⎠ match

0.00634k ⎛ tca ⎞
re = ⎜ ⎟
πφμ i Cti ⎜⎝ t DA ⎟⎠ match

re ⎛r ⎞
rwa = S = ln⎜⎜ w ⎟⎟
⎛ re ⎞ ⎝ rwa ⎠
⎜⎜ ⎟⎟
⎝ rwa ⎠ match
⎛ ⎞
⎜ ⎟
(0.00634)(1.417Ε6)S g PiTsc ⎛ t ca ⎞ ⎜ PD ⎟
G= ⎜⎜ ⎟⎟ ⎜ ⎟ * 10 9 (bcf)
μ i cti z i Psc ⎝ t DA ⎠ match ⎜ ΔPp ⎟⎟
⎜ q
⎝ ⎠ match
 Flowing p/z Method for Gas – Constant Rate

- Mattar L., McNeil, R., "The 'Flowing' Gas

Material Balance", JCPT, Volume 37 #2, 1998
pi
zi Pressure loss due to flow
in reservoir (Darcy’s Law)
is constant with time
pwf
p ⎛ p⎞
zwf = ⎜ ⎟ + constant
z ⎝ z ⎠ wf

Measured at well
Gi
during flow

Gp
 Graphical Flowing p/z Method for Gas –
Variable Rate

pi
zi
Graphical Method
Doesn’t Work!
pwf
zwf

Gi ?

Measured at well
during flow

Gp
 Flowing p/z Method for Gas – Variable Rate

pi
zi Pressure loss due to flow
in reservoir is NOT
constant
pwf
zwf p ⎛ p⎞
= ⎜ ⎟ + qbpss
z ⎝ z ⎠ wf
Unknown

Gi
Measured at well
during flow

Gp
 Variable Rate p/z – Procedure (1)
Unnamed Well Flowing Material Balance
Legend 550
Static P/Z*
P/Z Line
Flowing Pressure
500

450
Step 1: Estimate OGIP and plot
a straight line from pi/zi to OGIP. 400

Include flowing pressures (p/z)wf

Flowing Pressure, psi

350
on plot
300

250

200

150

100

Original Gas In Place 50

0
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70
Cumulative Production, Bscf
 Variable Rate p/z – Procedure (2)
Unnamed Well Flowing Material Balance
Legend
550
Static P/Z*
4.40 P/Z Line
Flowing Pressure
500
Productivity Index
4.00

Step 2: Calculate bpss for each 450

3.60
production point using the
Productivity Index, MMscfd/(106psi2/cP)

3.20 following formula: 400

⎛ p⎞ ⎛ p⎞

Flowing Pressure, psi

350
2.80
⎜ ⎟ − ⎜ ⎟
⎝ z ⎠line ⎝ z ⎠ wf 300
2.40
bpss =
2.00
q 250

1.60 Plot 1/bpss as a function of Gp 200

1.20 150

0.80 100

0.40 Original Gas In Place 50

0.00 0
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70
Cumulative Production, Bscf
 Variable Rate p/z – Procedure (3)
Unnamed Well Flowing Material Balance
Legend
550
Static P/Z*
4.40 P/Z Line
Flowing Pressure
500
Productivity Index
4.00
Step 3: 1/bpss should tend 450
3.60
towards a flat line. Iterate on
OGIP estimates until this
Productivity Index, MMscfd/(106psi2/cP)

400
3.20
happens

Flowing Pressure, psi

350
2.80

300
2.40

250
2.00

200
1.60

1.20 150

0.80 100

0.40 50
Original Gas In Place

0.00 0
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70
Cumulative Production, Bscf
 Variable Rate p/z – Procedure (4)
Unnamed Well Flowing Material Balance
Legend
550
Static P/Z*
4.40 P/Z Line
Flowing P/Z*
500
Flowing Pressure
4.00 Productivity Index
Step 4: Plot p/z points on the p/z 450
3.60
line using the following formula:
Productivity Index, MMscfd/(106psi2/cP)

400
3.20 ⎛ p⎞ ⎛ p⎞
⎜ ⎟ = ⎜ ⎟ + qbpss

P/Z*, Flowing Pressure, psi

2.80 ⎝ z ⎠ data ⎝ z ⎠ wf 350

2.40
“Fine tune” the OGIP estimate 300

250
2.00

200
1.60

1.20 150

0.80 100

1/bpss
0.40 50
Original Gas In Place

0.00 0
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70
Cumulative Production, Bscf
 Transient (tD format) Typecurves

Transient typecurves plot a normalized rate against material

balance time (similar to other methods), but use a dimensionless
time based on WELLBORE RADIUS (welltest definition of
dimensionless time), rather than AREA. The analysis consists of
two sets of typecurves:

1. Normalized rate vs. tc (material balance time)

2. Inverse pressure integral - derivative vs. tc

- Transient typecurves are designed for analyzing EARLY-TIME data

to estimate PERMEABILITY and SKIN. They should not be used (on
their own) for estimating fluid-in-place

- Because of the tD format, the typecurves blend together in the

early-time and diverge during boundary dominated flow (opposite
of tDA and tDd format typecurves)
 Transient versus Boundary Scaling Formats

log(qD) log(qDd)

log(tD) log(tDd)
 Transient (tD format) Typecurves: Definitions

Typecurves Data - Oil Data - Gas

141.2qβμ q q
qD =
Normalized Rate khΔP ΔP ΔPp
−1 −1 −1
⎡ 1 t DA ⎤ ⎛ ΔP ⎞ ⎡ 1 c ΔP ⎤ ⎛ ΔPp ⎞ ⎡ 1 ΔPp ⎤
t t ca
Inverse Pressure 1 / PDi = ⎢ P (
∫ p ⎥⎥
t )dt Inv⎜⎜ ⎟⎟ = ⎢ ∫ dt ⎥ Inv⎜⎜ ⎟⎟ = ⎢ ∫ dt ⎥
Integral
⎣⎢ t DA 0 ⎦ ⎝ q ⎠i ⎢⎣ tc 0 q ⎦⎥ ⎝ q ⎠i ⎢⎣ tca 0
q ⎦⎥

−1 −1 −1
⎡ dP ⎤ ⎡ ⎛ ΔP ⎞ ⎤ ⎡ ⎛ ΔPp ⎞ ⎤
Inverse Presssure 1 / PDid = ⎢t DA Di ⎥ ⎢ d ⎜⎜ ⎟⎟ ⎥ ⎢ tca d ⎜⎜ ⎟⎟ ⎥
⎣ dt DA ⎦ ⎛ ΔP ⎞ ⎝ q ⎠i ⎥ ⎛ ΔPp ⎞ ⎝ q ⎠i ⎥
Integral - Derivative
Inv⎜⎜ ⎟⎟ = ⎢tc Inv⎜⎜ ⎟⎟ =⎢
⎝ q ⎠id ⎢ dtc ⎥ ⎝ q ⎠id ⎢ dtca ⎥
⎢ ⎥ ⎢ ⎥
⎣⎢ ⎦⎥ ⎢⎣ ⎥⎦
Transient (tD format) Typecurves:
Diagnostics (Radial Model)

Transient Transition to Boundary

Dominated occurs at
Inverse Integral - different points for
Derivative Typecurve
different typecurves

Normalized Rate
Typecurve
Transient (tD format) Typecurves:
Finite Conductivity Fracture Model

Increasing Fracture
Conductivity (FCD stems)

Increasing Reservoir
Size (xe/xf stems)
 Transient (tD format) Typecurves:
Calculations (Radial Model)
O il W e lls : Gas Wells:

U s in g th e d e fin itio n o f q D ,
For gas wells, qD is defined as follows:
141 . 2 qB μ
qD =
kh ( p i − p wf )
1.417 E 6TR q
qD =
kh Δpp
p e rm e a b ility is c a lc u la te d a s fo llo w s :

141 . 2 B μ ⎛ q / Δ p ⎞
k = ⎜⎜ q D ⎟⎟ The permeability is calculated from above, as follows:
h ⎝ ⎠ match

F ro m th e d e fin itio n o f t D , 1.417 E 6TR ⎛ q/Δpp ⎞

k= ⎜⎜ ⎟⎟
0 . 00634 kt c
h ⎝ qD ⎠ match
tD =
φμ c t r wa 2

r w a is c a lc u la te d a s fo llo w s : From the definition of tD and k, rwa is calculated as follows

r wa =
0 . 00634 ⎛ 141 . 2 B ⎞ ⎛ q / Δ p
⎟ ⎜⎜
⎞
⎟⎟
⎛ tc ⎞ 0.00634 ⎛ 1.417 E 6TR ⎞⎛ tca ⎞ ⎛ q/Δpp ⎞
φct
⎜
⎝ ⎠ ⎝ qD ⎠
⎜ ⎟
⎝ t D ⎠ match rwa = ⎜ ⎟⎜ ⎟ ⎜⎜ ⎟⎟
h match
φμicti ⎝ h ⎠⎝ tD ⎠ match ⎝ qD ⎠ match
S k in is c a lc u la te d a s fo llo w s :
Skin is calculated as follows:
⎛ rw ⎞
s = ln ⎜ ⎟
⎝ r wa ⎠
⎛ rw ⎞
s = ln⎜ ⎟
⎝ rwa ⎠
Modeling and History Matching
1. Pressure Constrained System:

Constraint Signal
(Input) (Output)

Well Well / Reservoir

Pressure at Production
Model Volumes
Sandface

2. Rate Constrained System:

Constraint Signal
(Input) (Output)

Production Well / Reservoir Well

Volumes Model Pressure at
Sandface
 Modeling and History Matching

Models - Horizontal
Models - Radial
Rectangular reservoir with a horizontal well located anywhere inside.
Rectangular reservoir with a vertical well located anywhere inside.

Models - Fracture

Rectangular reservoir with a vertical infinite conductivity fracture located anywhere inside.
A Systematic and Comprehensive
Method for Analysis
 Modern Production Analysis Methodology

Diagnostics Interpretation and Modeling and Forecasting

Analysis History Matching

- Data Validation - Identifying dominant - Validating interpretation

flow regimes - Reserves
- Reservoir signal - Optimizing solution
- Estimating reservoir - Optimization scenarios
extraction - Enabling additional
characteristics flexibility and complexity
- Identifying important
system parameters
- Qualifying
uncertainty

- Data Chart - Traditional - Analytical Models

- Typecurves - Fetkovich - Numerical Models
- Blasingame
- AG / NPI
- Flowing p/z
- Transient
Practical Diagnostics
 What are diagnostics?
• Qualitative investigation of data
– Pre-analysis, pre-modeling
– Must be quick and simple
• A VITAL component of production data
analysis (and reservoir engineering in
general)
 Illustration- Typical Dataset
Unnam ed Well Data Chart
28
Legend 1600
5.50
26 Pressure 1500
Actual Gas Data

24 5.00 1400

1300
22
4.50
1200
20
4.00 1100
18
1000
Liquid Rates , bbl/d

3.50

Pressure , psi
16
G as , MMcfd

900

14 800
3.00

12 700

2.50 600
10

500
8 2.00
400
6
1.50 300
4
200
1.00
2 100

0 0.50 0
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500 520 540
Tim e, days
“Face Value” Analysis of Data

OGIP = 90 bcf
 Go Back: Diagnostics
Unnam ed Well Data Chart
28
Legend 1600
5.50
26 Pressure 1500
Actual Gas Data

24 5.00 1400

1300
22
4.50
1200
20
4.00 1100
18
1000
Liquid Rates , bbl/d

3.50

Pressure , psi
16
G as , MMcfd

900
Unnam e d We ll Data Chart

14 Legend
800
3.00 Pre s s ure
Actual Gas Data

12 700

2.50 600
10

500
8 2.00
Pressures are not 400
6
1.50 representative of bh 300
4
deliverability 200
1.00
2 100

0 0.50 0
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500 520 540
Tim e, days
 Correct Data Used
UnnamedWell DataChart
7400
Legend
6.00 5.50
Pressure 7200
Actual Gas Data
5.50 Oil Production
5.00 7000
Water Production

5.00 6800
4.50
4.50 6600

4.00
4.00 6400
Liquid Rates , bbl/d

3.50 6200

Pressure , psi
Gas , MMcfd

3.50

6000
3.00 3.00

5800
2.50
2.50
5600
2.00
2.00 5400
OGIP = 19 bcf
1.50
5200
1.50
1.00
5000

1.00
0.50 4800

0.00 0.50 4600

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500 520 540
Time, days
 Diagnostics using Typecurves

Radial Model
Blasingam e Typecurve Match

10-7
8
5
3
2

10-8
Transient
8
5

qDd 32 (concave up) Boundary Dominated

(concave down)
10-9
8
5

Base Model:
3
2

10-10

5
8
- Vertical Well in Center of Circle
3
2
- Homogeneous, Single Layer
10-11
8
5
3
2

4 56 8 -1 2 3 45 79 2 3 45 7 9 1 2 3 4 56 8 2 2 3 4 56 8 3 2 3 4 56 8 4 2 3 4 56 8 5 2 3 4 56 8 6 2 3 45 7 2 3 45 7
10 1.0 10 10
tDd
10 10 10 10 107
 Diagnostics using Typecurves

Material Balance Diagnostics

Radial Model
Blasingam e Typecurve Match

10-7
8
5
3
2 Reservoir With
10-8
8
Pressure Support
5

qDd 32
Dual
10-9 Depl
e
Syst tion
8
5 em
3
2
Infi
n
10-10 Pre ite A
8 ssu ctin
re S g

Vo
5 upp
ort

lu
3

m
et
2

ric
10-11

5
8
Leaky Reservoir
3 (interference)
2

4 56 8 -1 2 3 45 79 2 3 45 7 9 1 2 3 4 56 8 2 2 3 4 56 8 3 2 3 4 56 8 4 2 3 4 56 8 5 2 3 4 56 8 6 2 3 45 7 2 3 45 7
10 1.0 10 10
tDd
10 10 10 10 107
 Diagnostics using Typecurves

Productivity Diagnostics
Radial Model
Blasingam e Typecurve Match

10-7
8
Increasing Damage (difficult to identify)
5
3
2

10-8
8
5
Productivity
qDd 3
2 Shifts (workover,
10-9
8 unreported
5
tubing change)
3 Well Cleaning Up
2

10-10
8 Liquid Loading
5
3
2

10-11
8
5
3
2

4 56 8 -1 2 3 45 79 2 3 45 7 9 1 2 3 4 56 8 2 2 3 4 56 8 3 2 3 4 56 8 4 2 3 4 56 8 5 2 3 4 56 8 6 2 3 45 7 2 3 45 7
10 1.0 10 10
tDd
10 10 10 10 107
 Diagnostics using Typecurves

Transient Flow Diagnostics

Radial Model
Blasingam e Typecurve Match

10-7 Fracture Linear Flow

8
5 (Stimulated)
3
Transitionally
2 Dominated Flow (eg:
10-8
8 Channel or Naturally
5
Damaged Fractured)
qDd 3
2

10-9
8
Radial Flow
5
3
2

10-10
8
5
3
2

10-11
8
5
3
2

4 56 8 -1 2 3 45 79 2 3 45 7 9 1 2 3 4 56 8 2 2 3 4 56 8 3 2 3 4 56 8 4 2 3 4 56 8 5 2 3 4 56 8 6 2 3 45 7 2 3 45 7
tDd
10 1.0 10 10 10 10 10 10 107
 Diagnostics using Typecurves

“Bad Data” Diagnostics

Radial Model
Blasingam e Typecurve Match

10-7
8
Δp in reservoir is too low
5
3
-Tubing size too small ?
2 - Initial pressure too low ?
10-8 - Wellbore correlations
8
5
overestimate pressure loss ?
qDd 3
2

10-9
8
5
Δp in reservoir is too high
3
2 -Tubing size too large ?
10-10
- Initial pressure too high ?
8
- Wellbore correlations
5
3
underestimate pressure loss ?
2

10-11
8
5
3
2

4 56 8 -1 2 3 45 79 2 3 45 7 9 1 2 3 4 56 8 2 2 3 4 56 8 3 2 3 4 56 8 4 2 3 4 56 8 5 2 3 4 56 8 6 2 3 45 7 2 3 45 7
10 1.0 10 10
tDd
10 10 10 10 107
Selected Topics and Examples
Tight Gas
Industry Migration to Tight Gas Reservoirs
Production Analysis – Tight Gas versus Conventional Gas

¾ Analysis methods are no different from that

of high permeability reservoirs

¾ Transient effects tend to be more dominant

– Establishing the region (volume) of
influence is critical

¾ Drainage shape becomes more important

(Transitional effects)

¾ Linear flow is more common

¾ Layer effects are more common

 Tight Gas- Common Geometries
Tight Gas Type Curves

1.00E-05

Infinite acting reservoir

1.00E-06

1.00E-07 1/2
qDd

1.00E-08

1.00E-09 1

Linear flow
1.00E-10
dominated Limited, bounded
drainage area
1.00E-11
1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04
tDd
 Tight Gas Model 1

¾ Extensive, continuous porous media; very low

permeability

1800 psi
Pi = 2000 psi

Pi = 1500 psi
 Infinite Acting System
Tight Gas Type Curves

1.00E-05

1.00E-06

1.00E-07 1/2
qDd

1.00E-08

1.00E-09

1.00E-10

1.00E-11
1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04
tDd
 Example#1 – Infinite Acting System
10 Agarwal Gardner Rate vs Time Typecurve Analysis
10 Agarwal Gardner Rate vs Time Typecurve Analysis 2
2
102
102
6
6 4
4 3
3 2
2
101
101 7
7 5
5
3
3 2
2

Normalized Rate
Normalized Rate

1.09
1.09
6
6 4
4 3
3 2
2
10-1
10-1 7
7 5
5
3
3
2
2
10-2
10-2 7
7 5
5
3
3
2
2 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 -2 2 3 4 5 6 78 -1 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78
2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 -2 2 3 4 5 6 78 -1 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 10-5 10-4 10-3 10 10 1.0 101 102
10-5 10-4 10-3 10 10 1.0 101 102
Material Balance Pseudo Time
Material Balance Pseudo Time

k = 0.08 md k = 0.08 md
xf = 53 ft xf = 53 ft
OGIP = 10 bcf Minimum OGIP = 2.6 bcf
 Tight Gas Model 2

¾ No flow continuity across reservoir- Well only drains

a limited bounded volume

Example: Lenticular Sands

 Bounded Reservoir
Tight Gas Type Curves

1.00E-05

1.00E-06

1.00E-07 1/2
qDd

1.00E-08

- Limited or no flow continuity in reservoir

1.00E-09
- Very small drainage areas 1

- Very large effective fracture lengths

1.00E-10

1.00E-11
1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04
tDd

Commonly observed in practice

 Example #2- Bounded Drainage Areas

10 ROBINSON 11-1 ALT Blasingame Typecurve Analysis

2

- West Louisiana gas field 101

7

9 - 80 acre average spacing 5

- All wells in boundary dominated flow 2

Normalized Rate
1.0
7
8 5

3
2

10-1
7 7
5

3
2
.

6 10-2
2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78
10-3 10-2 10-1 1.0 101 102
Material Balance Pseudo Time
O G IP (b c f)

5 35 120%

30
100%

4 25
80%

Frequency
20

3 60%
15

40%
10
2
20%
5

0 0%
1 10 20 30 40 50 60 70 80 90 100 More

Drainage Area (acres)

Frequency Cumulative %
0
0 100 200 300 400 500 600
xf (feet)
 Tight Gas Model 3

¾ Linear flow dominated system

Example: Naturally fractured, tight reservoir

kx

ky
Infinite Systems versus Linear Flow Systems

Establish
permeability and xf
independently

Establish xf sqrt (k)

product only
 Linear Flow Systems
Tight Gas Type Curves

1.00E-05

1.00E-06

1.00E-07 1/2
qDd

1.00E-08

- Channel and faulted reservoirs

1.00E-09
- Naturally fractured (anisotropic) reservoirs
- Very large effective fracture lengths
- Very difficult to uniquely interpret
1.00E-10

1.00E-11
1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04
tDd

Commonly observed in practice

 Example #3- Linear Flow System
Fracture Model
Blasingame Typecurve Match

5
4

3
k = 1.1 md
xf = 511 ft
2 ye = 5,500 ft
yw = 2,900 ft
10-7

ye
7

yw
4

2
2xf
10-8
9
7

2 3 4 5 6 7 89 1 2 3 4 5 6 7 89 2 2 3 4 5 6 7 8
10 10 103
More Examples
 Example #3- Multiple Layers
Blasingame Typecurve Analysis

3
Multi Layer Model
Well Blasingame Typecurve Match

10-8
2

7
5
4
1.0 3
8
2
Normalized Rate

5
10-9
4 8
6
3
4
3
2

10-1
10-10
9
2 3 4 5 6 7 89 1 2 3 4 5 6 7 89 2 2 3 4 5 6 7 89 3 2 3 4 5 6 78
1.0 10 10 10 104
3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9
10-1 1.0
Material Balance Pseudo Time

- Blasingame typecurve match, using Fracture - Three-Layer Model (one layer with very low
Model permeability) used, late-time match improved
- Pressure support indicated
 Example #4- Shale Gas
Well Agarwal Gardner Rate vs Time Typecurve Analysis
5

4
- Multi-stage fractures, horizontal well
3
- Analyzed as a vertical well in a circle
2

1.0

7
6
Normalized Rate

10-1
9

7
6 k = 0.02 md
5

4
s = -4
3
OGIP = 4.5 bcf
6 7 8 9 -3 2 3 4 5 6 7 8 9 -2 2 3 4 5 6 7 8 9 -1 2 3 4 5 6 7 8 9
10 10 10 1.0
Material Balance Pseudo Time
 Tight Gas: Assessing Reserve
Potential – Recovery Plots

¾ Objectives

¾ Determine incremental reserves that are added as the ROI

expands into the reservoir (only relevant for infinite or
semi-infinite systems)

¾ To establish a practical range of Expected Ultimate

Recovery
 Typical Recovery Profile
Recovery Curves for k = 1 md

10

9 1 md reservoir, unfractured
8 (~10 bcf / section)
7 100% Recovery
6
EUR (bcf)

0
0 1 2 3 4 5 6 7 8 9 10
Original Gas in Place (bcf)
 Typical Recovery Profile
Recovery Curves for k = 1 md

10

9
1 md reservoir, unfractured
8 (~10 bcf / section)
7 100% Recovery
6
EUR (bcf)

3
Actual EUR (qab = 0.05 MMscfd)
2

0
0 1 2 3 4 5 6 7 8 9 10
Original Gas in Place (bcf)

EUR- unlimited time

 Typical Recovery Profile
Recovery Curves for k = 1 md

10

9
1 md reservoir, unfractured
8 (~10 bcf / section)
7 100% Recovery
6
30 Year Limited
EUR (bcf)

3
Actual EUR (qab = 0.05 MMscfd)
2

0
0 1 2 3 4 5 6 7 8 9 10
Original Gas in Place (bcf)

EUR- 30 year EUR- unlimited time

 Typical Recovery Profile
Recovery Curves for k = 1 md

10

9
1 md reservoir, unfractured
8 (~10 bcf / section)
7 100% Recovery
6
30 Year Limited
EUR (bcf)

5
20 Year Limited
4

3
Actual EUR (qab = 0.05 MMscfd)
2

0
0 1 2 3 4 5 6 7 8 9 10
Original Gas in Place (bcf)

EUR- 30 year EUR- 20 year EUR- unlimited time

 Tight Gas Recovery Profile
Recovery Curves for k = 0.02 md

10

9 0.02 md reservoir, fractured

8 (~10 bcf / section)
7 Actual EUR (qab = 0.05 MMscfd)
6
EUR (bcf)

0
0 1 2 3 4 5 6 7 8 9 10
Original Gas in Place (bcf)

EUR- unlimited time

 Tight Gas Recovery Profile
Recovery Curves for k = 0.02 md

10

9 0.02 md reservoir, fractured

8 (~10 bcf / section)
7 Actual EUR (qab = 0.05 MMscfd)
6
EUR (bcf)

4
30 Year
3

0
0 1 2 3 4 5 6 7 8 9 10
Original Gas in Place (bcf)

EUR- 30 year EUR- unlimited time

 Tight Gas Recovery Profile
Recovery Curves for k = 0.02 md

10

9 0.02 md reservoir, fractured

8 (~10 bcf / section)
7 Actual EUR (qab = 0.05 MMscfd)
6
EUR (bcf)

4
30 Year
3 20 Year
2

0
0 1 2 3 4 5 6 7 8 9 10
Original Gas in Place (bcf)

EUR- 30 year EUR- 20 year EUR- unlimited time

 Tight Gas Recovery Profile
Recovery Curves for k = 0.02 md

10

9 0.02 md reservoir, fractured

8 (~10 bcf / section)
7 Actual EUR (qab = 0.05 MMscfd)
6
Max EUR (30 y) = 2 bcf
EUR (bcf)

4
30 Year
3 20 Year
2

0
0 1 2 3 4 5 6 7 8 9 10
Original Gas in Place (bcf)

EUR- 30 year EUR- 20 year EUR- unlimited time

 Example – South Texas, Deep Gas Well
Fracture Model
AGTypecurve Match

10-8

5
4
Sqrt k X xf = 155
3
Min OGIP = 4.2 bcf
2

10-9
9
7
2 3 4 5 6 7 89 2 3 4 5 6 7 89 1 2 3 4 5 6 7 89 2 2 3 4 5 6 78
1.0 10 10 103
 Example – South Texas, Deep Gas Well
Recovery Plot - Linear System

MaximumEUR= 6.7 bcf

6

Recovery period = 30 years

4
sqrt k X xf = 155
pi = 6971 psia
EUR (bc

MinimumEUR= 3.5 bcf

0
0 100 200 300 400 500 600
ROI (acres)
Water Drive Models
 Water Drive (Aquifer) Models:
Models for reservoirs under the influence of active water encroachment can
be categorized as follows:

1. Steady State Models (inaccurate for finite reservoir sizes)

- Schilthuis

2. Pseudo Steady-State Models (geometry independent,

time discretized)
- Fetkovich

3. Single Phase Transient Models (geometry dependent)

- infinite aquifer (linear, radial or layer geometry)
- finite aquifer (linear, radial or layer geometry)

4. Modified Transient Models

- Moving saturation front approximations
- Two phase flow approximations
 Water Drive (Aquifer) Models:
Pseudo Steady-State Models

PSS models (such as that of Fetkovich) use a TRANSFER

COEFFICIENT (similar to a well productivity index) to describe the
PSS rate of water influx into the reservoir, in conjunction with a
MATERIAL BALANCE model that predicts the decline in reservoir
boundary pressure over time.

The Fetkovich model is generally used to determine reservoir fluid-

in-place by history matching the CUMULATIVE PRODUCTION and
AVERAGE RESERVOIR PRESSURE.
 Water Drive (Aquifer) Models:
Pseudo Steady-State Models

Advantages:

- Geometry independent (applicable to aquifers of any shape, size or

connectivity to the reservoir)
- Works well for finite sized aquifers of medium to high mobility
- Computationally efficient

Disadvantages:

- Does not provide a full time solution (transient effects are ignored)
- Does not work well for infinite acting or very low mobility aquifers
 Water Drive (Aquifer) Models:
Pseudo Steady-State Model- Equations

The Fetkovich water influx equation for a finite aquifer is:

Aquifer transfer coefficient

We =
Wei
( pi-p ) ⎛⎜1 − e − Jpit /Wei ⎞⎟ Initial encroachable water
pi ⎝ ⎠
Reservoir boundary pressure

The above equation applies to the water influx due to a constant pressure difference
between aquifer and reservoir. In practice, the reservoir pressure “p” will be declining
with time. Thus, the equation must be discretized as follows:

ΔW e n =
Wei
pi
(
pa n−1- p n )⎛⎜⎝1 − e Jp t /W
− i ei ⎞⎟
⎠
(1)

The average aquifer pressure at the previous timestep (n-1) is evaluated explicitly, as follows:

⎛ n −1
⎞
⎜ ∑ ΔWej ⎟
= pi ⎜⎜1 − ⎟
j =1
pa n−1
Wei ⎟
⎜⎜ ⎟⎟
⎝ ⎠
 Water Drive (Aquifer) Models:
Pseudo Steady-State Model- Equations

Now, we have one equation with two unknowns (water influx “We” and reservoir boundary
pressure “p”)

But there is another equation that relates the average reservoir pressure to the amount of
water influx: the material balance equation for a gas reservoir under water drive.

Cumulative Production
-1
p pi ⎛ Gp ⎞ ⎛ WeBi ⎞
= ⎜1 − ⎟ ⎜1 − ⎟ FVF at initial conditions
z zi ⎝ Gi ⎠ ⎝ Gi ⎠ Gas-in-place

As with the water influx equation, the material balance equation can be discretized in time:

-1
⎛ p⎞ pi ⎛ Gp n ⎞ ⎛ W e n Bi ⎞
⎜ ⎟ = ⎜1 − ⎟ ⎜1 − ⎟ (2)
⎝ ⎠n
z z i ⎝ Gi ⎠ ⎝ Gi ⎠

Equations 1 and 2 are now solved simultaneously at each timestep, to obtain a discretized
reservoir pressure and water influx profile through time.
 Water Drive (Aquifer) Models:
Transient Models
Transient models use the full solution to the hydraulic DIFFUSIVITY
EQUATION to model rates and pressures.

The transient equations can be used to model either FINITE or

INFINITE acting aquifers. There are a number of different transient
models available for analyzing a reservoir under active water drive:

- Radial Composite (edge water drive)

- Linear (edge water drive)
- Layered (bottom water drive)

Advantages:

- Offers full continuous pressure solution in the reservoir

- Includes early time effects

Disadvantages:

- Geometry dependent (only a disadvantage if aquifer properties are

unknown)
- Limited to assumption of single phase flow
- Does not account for water influx
 Water Drive (Aquifer) Typecurves:
Radial Composite Model
Blasingame, AG and NPI dimensionless formats can be used to plot
typecurves for SINGLE PHASE production (oil or gas) from a
reservoir under the influence of an EDGE WATER DRIVE. A
typecurve match using this model can be used to predict

1. Reservoir fluid-in-place

2. Aquifer mobility

- These typecurves are designed to estimate fluid-in-place by

detecting the shift in fluid mobility as the transient passes the
reservoir boundaries, into the aquifer.

- Their usefulness is limited to single phase flow (ie: the transition

from reservoir fluid to aquifer is assumed to be abrupt)
Water Drive (Aquifer) Typecurves: Definitions
Model Type: Radial Composite (two zones);
outer zone is of infinite extent

Reservoir Aquifer

Maq kaq μres

Mobility Ratio (M): M= =
Mres kres μaq
Water Drive (Aquifer) Typecurves: Diagnostics

M=10 (Constant Pressure System (approx))

Decreasing reD value

Increasing Aquifer Mobility (M)

M=0 (Volumetric Depletion)

Water Drive (Aquifer) Typecurves: Diagnostics

M=10 (Constant Pressure System (approx))

Increasing Aquifer Mobility (M)

Decreasing reD value

M=0 (Volumetric Depletion)

 Water Drive (Aquifer) Models:
Modified Transient Models
1. Moving aquifer front (reservoir boundary)

The radial composite model previously discussed can be

enhanced to accommodate a shrinking reservoir boundary,
caused by water influx. This is achieved by discretizing
the transient solution in time and using the PSS water
influx equations to predict the advancement of the aquifer
front. The solution still assumes single phase flow, but
can now more accurately estimate the time to water
breakthrough.

2. Two phase flow (after M. Abbaszadeh et al)

The previously discussed model can also be modified to

accommodate a region of two-phase flow (located
between the inner region - hydrocarbon phase and outer
region - water phase). Thus, geometrically, the overall
model is three zone composite. The pressure transient
solution for the two-phase zone is calculated by
superimposing the single phase pressure solution on a
saturation profile determined using the Buckley-Leverett
equations.
 Water Drive (Aquifer) Models: Example
Example F Data Chart
14000 Example F Blasingame Typecurve Analysis
Legend 101
22 Pressure
Actual Gas Data 13000 8

6
20 12000 5
4
18 11000
3

10000 2
16

9000
14 1.0
8000 8

Pressure, psi
Gas, MMscfd

12 6

Normalized Rate
7000 5
4
10 6000
3

5000 2
8

4000
6

-Gulf coast gas 3000

10-1
8
-Boundary dominated
-Pressure support evident
4

condensate reservoir
6
2000 5
4
2
1000
3

0 0 2
Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct
2002 2003

10-2
2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78
10-3 10-2 10-1 1.0 101 102
Material Balance Pseudo Time
Example F Agarwal Gardner Rate vs Time Typecurve Analysis Example F Blasingame Typecurve Analysis
101
8

1.0
8
Transient Water Drive 6
5 PSS Water Drive Model
6
5
Model 4

3
4
2
3

2
1.0
8
Normalized Rate, Derivative

6
10-1
Normalized Rate

5
8
4
6 3
5
4

3 k = 8.5 md 2
k = 3.1 md
2
s=0 10-1 s = -4
8

10-2 OGIP = 12 bcf 6

5
OGIP = 13.5 bcf
IWIP = 47 MMbbl
4

M = 0.001
8

6 3

PI (aq) = 0.59 bbl/d/psi

5
2
4

10-2
2 3 4 5 6 7 8 9 -1 2 3 4 5 6 7 89 2 3 4 5 6 7 89 1 2 3 4 5 6 7 8 2 3 4
10 1.0 10 102 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78
10-3 10-2 10-1 1.0 101 102
Material Balance Pseudo Time
Material Balance Pseudo Time
Multiple Well Analysis
 Multi-well / Reservoir-based Analysis-
Available Methods

1. Empirical- Group production decline plots

2. Material Balance Analysis- Shut-in data only

3. Reservoir Simulation

4. Semi-analytic production data analysis methods

- Blasingame approach
 Multi-well Analysis- When is it required?

1. Situations where high efficiency is required

- Scoping studies / A & D
- Reserves auditing

2. Single well methods sometimes don’t apply

- Interference effects evident in production / pressure
data- Wells producing and shutting in at different times
- Predictive tool for entire reservoir is required
- Complex reservoir behavior in the presence of
multiple wells (multi-phase flow, reservoir
heterogeneities)
 Multi-well Analysis- When is it not required?

The vast majority of production data can be analyzed

effectively without using multi-well methods

1. Single well reservoirs

2. Low permeability reservoirs

- Pressure transients from different wells in reservoir
do not interfere over the production life of the well

3. Cases where “outer boundary conditions” do not change

too much over the production life of the well
- Wide range of reservoir types
 Identifying Interference

Well A Well B

Rate is adjusted at Well A Response at Well B

Q Q
 Correcting Interference Using
Blasingame et al Method

Define a “total material balance time” function

Q tot QA + QB
tce = ⇒ (for analyzing Well A)
q qA

tce is used in place of tc to plot the data in the typecurve match

 Multi-Well Analysis as a Typecurve Plot

MBT is corrected for

Analysis of Well A: interference caused
by production from
Well B

log(q/Δp)

log(tc) tcA tce

tce= (QB +QA)/qA

Also applies to Agarwal-Gardner, NPI and FMB

 Multi-Well Analysis- Example
Well 1 Data Chart
6.00 36000
2.80
Legend
Pressure 34000
5.50
2.60
-Three well system Actual Gas Data
Pool Production 32000
-“Staggered” on-stream dates Water Production

-High permeability reservoir

5.00 2.40 30000

28000
4.50 2.20
26000
2.00
4.00
Aggregate production of well group 24000

1.80
Oil / Water Rates, bbl/d

22000
3.50

Pressure, psi
Gas, MMscfd

1.60 20000

3.00 18000
1.40
16000
2.50 1.20
14000
1.00 12000
2.00

0.80 10000
1.50
8000
0.60
1.00 6000
0.40
4000
0.50 0.20 Production history of well to be analyzed
2000

0.00 0.00 0
1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
 Multi-Well Analysis- Example

Well 1 Blasingame Typecurve Analysis

Well 1 Blasingame Typecurve Analysis
101
101
7
7
5
5
4
4
3
3
2
2

1.0
1.0
7
7
Normalized Rate

Normalized Rate
5
5
4
4
3
3
2
2

10-1
10-1
7
“Leaky reservoir” diagnostic 7
5
4
5 Corrected using multi-well model
Total OGIP = 7 bcf
4
3
3
2
2

10-2
10-2
2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78
10-3 10-2 10-1 1.0 101 102 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78
10-3 10-2 10-1 1.0 101 102
Material Balance Pseudo Time
Material Balance Pseudo Time
 Multi-Well Analysis- Example

Well 1 Flowing Material Balance Well 1 Flowing Material Balance

Legend 1900 Legend 2000
P/Z Line P/Z Line
Flowing P/Z* 1800 Flowing P/Z*
1800
1700

1600
1600
1500
Total OGIP = 7.0 bcf
OGIP for subject well = 3.5 bcf
1400
1400
1300

1200
1200
1100

P/Z*, psi
P/Z*, psi
1000
1000
900

800
800
700

600 600
500

400 400
300

200 200
Original Gas In Place Original Gas In Place
100

0 0
0.00 0.40 0.80 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00 4.40 4.80 5.20 5.60 6.00 6.40 6.80 7.20 7.60 0.00 0.40 0.80 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00 4.40 4.80 5.20 5.60 6.00 6.40 6.80 7.20 7.60 8.00
Cumulative Production, Bscf Cumulative Production, Bscf
Overpressured Reservoirs
 Overpressured Reservoirs

1. Analysis methods are the same as normally pressured case

2. Additional parameters to be aware of

• Formation compressibility
• In-situ water compressibility
• Compaction effects (pressure dependent permeability)

3. Two models available, depending on required complexity

• p/z* model (accounts for constant cf, cw and co in
material balance equation
• Full geomechanical model (accounts for cf(p) and k(p))
 Compresibilities of gas and rock

Compressibility vs. Pressure (Typical Gas Reservoir)

3.00E-04
gas

Formation Formation
2.50E-04 energy is energy may Formation energy is critical in this region
negligible in be influencial
Compressibility (1/psi)

this region in this region

2.00E-04

1.50E-04

1.00E-04

5.00E-05 formation

0.00E+00
0 2000 4000 6000 8000 10000 12000
Reservoir Pressure (psi)
 p/z* Model – Corrects Material Balance

⎛ p⎞ ⎡
p 1 Gp ⎤⎥
= ⎜⎜ ⎟⎟ ⎢1−
⎜ ⎟ ⎢
⎥
z ⎢⎣1 − cf ( pi − p )⎥⎦ ⎝ z ⎠ i ⎢⎣ OGIP ⎥⎦
⎡ ⎤

*
Flowing MB
p ⎛⎜ p ⎞⎟ ⎡⎢ Gp ⎤⎥
= ⎜⎜ ⎟⎟ ⎢1− ⎥
z ⎝ z ⎠ i ⎢⎣ OGIP ⎥⎦

( μ ct ) i t q (t ) Typecurves
tca =
q ∫0 μ ct ⎡⎣⎢1 − cf ( pi − p)⎤⎦⎥ dt
Geomechanical Model – Corrects Well Productivity

In the standard pressure transient equations, permeability is usually considered to be

constant. There are several situations where this may not be a valid assumption:

1. Compaction in overpressured reservoirs

2. Very low permeability reservoirs in general
3. Unconsolidated and/or fractured formations

One way to account for a variable permeability over time is to modify the definition of
pseudo-pressure and pseudo-time.

2qpi 1.417e6 * Tq ⎛ re 3 ⎞
Δp p * = ta * + ⎜ ln − ⎟
( μctZ )iGi kih ⎝ rwa 4 ⎠
where

2 pi k ( p) pdp Pressure dependent

Δpp = ∫ *
permeability included in
ki pwf μz pseudo-pressure and pseudo-
time
(μct)i t k dt
= ∫
*
ta
ki 0 μct
 Overpressured Reservoirs - Example
Blasingame Typecurve Analysis
101
8

6
5 Gulf Coast, deep gas condensate reservoir
4

1.0
8

6
Normalized Rate

5
4

10-1
8

6 Boundary dominated flow

5
4 OGIP = 17 bcf
3

10-2
2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78
10-3 10-2 10-1 1.0 101 102
Material Balance Pseudo Time
 Overpressured Reservoirs - Example
Radial Model
218 Prod and Pressure Data History Match
80 18000

70 16000

60 14000

Good flowing pressure match,

12000
50 Poor shut-in pressure match
Rate, MMscfd

Pressure, psi
OGIP = 17 bcf
40 10000

30 8000

6000
20

4000
10

2000
0
June July August September October
2003
 Overpressured Reservoirs - Example
Radial Model
218 Prod and Pressure Data History Match
80 18000

70 16000

60 14000

Good flowing pressure match,

50 Good shut-in pressure match 12000
Rate, MMscfd

Pressure, psi
OGIP = 29 bcf
40 10000

30 8000

6000
20

4000
10

2000
0
June July August September October
2003
 Overpressured Reservoirs - Example
k (p) Permeability
218 Prod and Pressure Data k (p)
1.05
Legend
1.00 Default
Custom
0.95 Interpolation
0.90

0.85

0.80

0.75

0.70
Assumed permeability profile
0.65

0.60

0.55
k / ki

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00
0 500 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 13000 14000 15000 16000 17000 18000
Pressure, psi(a)
Horizontal Wells
 Horizontal Wells

Horizontal wells may be analyzed in any of three different

ways, depending on
completion and petrophysical details:

1. As a vertical well,
• if lateral length is small compared to drainage area

2. As a fractured well,
• if the formation is very thin
• if the vertical permeability is high
• if the lateral is cased hole with single or multiple stage
fractures
• to get an idea about the contributing lateral length

3. As a horizontal well (Blasingame model)

• all others
Horizontal Wells – Blasingame Typecurves
The horizontal well typecurve matching procedure is based on a square shaped reservoir with
uniform thickness (h). The well is assumed to penetrate the center of the pay zone.

The procedure for matching horizontal wells is similar to that of vertical wells. However, for
horizontal wells, there is more than one choice of model. Each model presents a suite of
typecurves representing a different penetration ratio (L/2xe) and dimensionless wellbore radius
(rwD). The definition of the penetration ratio is illustrated in the following diagram:

Plan

Cross Section

L
h rwa
L

2 rwa
rwD =
L
2xe

The characteristic dimensionless parameter for each suite of horizontal typecurves is defined as
follows:

L
LD =
2βh
Where is the square root of the anisotropic ratio:

kh
β=
kv
For an input value of “L”,
 Horizontal Wells – Example
Unnamed Well Blasingame Typecurve Analysis
102
8
6

4 L/2xe = 1
3

2
rwD = 2e-3
Ld = 5
101
8
Le = 1,968 ft
6

4 k (hz) = 0.18 md
3

2
k (v) = 0.011 md
Normalized Rate

OGIP = 1.1 bcf

1.0
8
6

4
3

10-1
8
6

4
3

10-2
2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78
10-3 10-2 10-1 1.0 101 102
Material Balance Pseudo Time
Oil Wells
 Oil Wells

¾ Analysis methods are no different from that

of gas reservoirs (in fact they are simpler)
provided that the reservoir is above the
bubble point

¾ If below bubble point, a multi-phase

capable model (Numerical) must be used
¾Include relative permeability effects
¾Include variable oil and gas properties
 Oil Wells – Example
example7 Data Chart
4000
190 Legend
0.11 Pressure 3800

- Pumping oil well

Actual Gas Data
180
Oil Production 3600
Water Production

- Assumed to be pumped off

170 0.10
3400
160
3200
0.09
150 Producing GOR ~ constant 3000
140 (indicates reservoir pressure is above
2800
0.08 bubble point
130
2600
120 0.07 2400
Liquid Rates, bbl/d

110

Pressure, psi
Gas, MMscfd

2200

100 0.06
2000

90 1800
0.05
80 1600

70 1400
0.04

60 1200

50 0.03 1000

40 800
0.02
30 600

20 400
0.01
10 200

0 0.00 0
Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct
2001 2002
Oil Wells – Example

Rs input from production data,

Pbp and co calculated using
Vasquez and Beggs
 Oil Wells – Example
example7 Blasingame Typecurve Analysis
101
8

6
5
4

3 k = 1.4 md
s = -3
2
OOIP = 2.4 million bbls

1.0
8

6
Normalized Rate

5
4

10-1
8

6
5
4

10-2
2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78 2 3 4 5 6 78
10-3 10-2 10-1 1.0 101 102
Material Balance Time
 Oil Wells – Example
example7 Numerical Radial Model - Production Forecast

300 4000

Legend 3800
280 History Oil Rate
240 month forecast Flow Press 3600
260 EUR = 265 Mbbls Syn Rate
History Reservoir Press 3400
Forecasted Press
240 Forecasted Reservoir Press 3200
Forecasted Rate
3000
220
2800
200
2600

180 2400
Oil Rate, bbl/d

Pressure, psi
2200
160
2000
140
1800

120 1600

1400
100
1200
80
1000

60 800

600
40
400
20
200

0 0
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