Production decline analysis is a traditional means of identifying well production problems and predicting well performance and life based on real production data. It uses empirical decline models that have little fundamental justifications. These models include
•
Exponential decline (constant fractional decline)
•
Harmonic decline, and
•
Hyperbolic decline.
prediction of original oil in place using material balance simulation. It's also useful for future reservoir performance and predict ultimate hydrocarbon recovery under various types of primary driving mechanisms.
prediction of original oil in place using material balance simulation. It's also useful for future reservoir performance and predict ultimate hydrocarbon recovery under various types of primary driving mechanisms.
Overview of Reservoir Simulation by Prem Dayal Saini
Reservoir simulation is the study of how fluids flow in a hydrocarbon reservoir when put under production conditions. The purpose is usually to predict the behavior of a reservoir to different production scenarios, or to increase the understanding of its geological properties by comparing known behavior to a simulation using different geological representations.
calculating reservoir pressure, knowing Depth of gas-oil. oil water interface, GOC AND WOC, numeric method to calculate interface. importance of isobaric maps in estimating reservoir pressure.
That is my presentation for my grad research about reservoir geomechanics, hope you find it useful, and my source book was reservoir geomechanics for prof Mark Zoback, soon the PDF copy will be available as well.
The problem of water and gas coning has plagued the petroleum industry for decades. Water or gas encroachment in oil zone and thus simultaneous production of oil & water or oil & gas is a major technical, environmental and economic problems associated with oil and gas production. This can limit the productive life of the oil and gas wells and can cause severe problems including corrosion of tubulars, fine migration, hydrostatic loading etc. The environmental impact of handling, treating and disposing of the produced water can seriously affect the economics of the production. Commonly, the reservoirs have an aquifer beneath the zone of hydrocarbon. While producing from oil zone, there develops a low pressure zone as a result of which the water zone starts coning upwards and gas zone cones down towards the production perforation in oil zone and thus reducing the oil production. Pressure enhanced capillary transition zone enlargement around the wellbore is responsible for the concurrent production. This also results in the loss of water drive and gas drive to a certain extent.
Numerous technologies have been developed to control unwanted water and gas coning. In order to design an effective strategy to control the coning of oil or gas, it is important to understand the mechanism of coning of oil and gas in reservoirs by developing a model of it. Non-Darcy flow effect (NDFE), vertical permeability, aquifer size, density of well perforation, and flow behind casing increase water coning/inflow to wells in homogeneous gas reservoirs with bottom water are important factors to consider. There are several methods to slow down coning of water and/or gas such as producing at a certain critical rate, polymer injection, Downhole Water Sink (DWS) technology etc.
Shubham Saxena
B.Tech. petroleum Engineering
IIT (ISM) Dhanbad
Reservoir engineers cannot capture full value from waterflood projects on their own. Cross-functional participation from earth sciences, production, drilling, completions, and facility engineering, and operational groups is required to get full value from waterfloods. Waterflood design and operational case histories of cross-functional collaboration are provided that have improved life cycle costs and increased recovery for onshore and offshore waterfloods. The role that water quality, surveillance, reservoir processing rates, and layered reservoir management has on waterflood oil recovery and life cycle costs will be clarified. Techniques to get better performance out of your waterflood will be shared.
Overview of Reservoir Simulation by Prem Dayal Saini
Reservoir simulation is the study of how fluids flow in a hydrocarbon reservoir when put under production conditions. The purpose is usually to predict the behavior of a reservoir to different production scenarios, or to increase the understanding of its geological properties by comparing known behavior to a simulation using different geological representations.
calculating reservoir pressure, knowing Depth of gas-oil. oil water interface, GOC AND WOC, numeric method to calculate interface. importance of isobaric maps in estimating reservoir pressure.
That is my presentation for my grad research about reservoir geomechanics, hope you find it useful, and my source book was reservoir geomechanics for prof Mark Zoback, soon the PDF copy will be available as well.
The problem of water and gas coning has plagued the petroleum industry for decades. Water or gas encroachment in oil zone and thus simultaneous production of oil & water or oil & gas is a major technical, environmental and economic problems associated with oil and gas production. This can limit the productive life of the oil and gas wells and can cause severe problems including corrosion of tubulars, fine migration, hydrostatic loading etc. The environmental impact of handling, treating and disposing of the produced water can seriously affect the economics of the production. Commonly, the reservoirs have an aquifer beneath the zone of hydrocarbon. While producing from oil zone, there develops a low pressure zone as a result of which the water zone starts coning upwards and gas zone cones down towards the production perforation in oil zone and thus reducing the oil production. Pressure enhanced capillary transition zone enlargement around the wellbore is responsible for the concurrent production. This also results in the loss of water drive and gas drive to a certain extent.
Numerous technologies have been developed to control unwanted water and gas coning. In order to design an effective strategy to control the coning of oil or gas, it is important to understand the mechanism of coning of oil and gas in reservoirs by developing a model of it. Non-Darcy flow effect (NDFE), vertical permeability, aquifer size, density of well perforation, and flow behind casing increase water coning/inflow to wells in homogeneous gas reservoirs with bottom water are important factors to consider. There are several methods to slow down coning of water and/or gas such as producing at a certain critical rate, polymer injection, Downhole Water Sink (DWS) technology etc.
Shubham Saxena
B.Tech. petroleum Engineering
IIT (ISM) Dhanbad
Reservoir engineers cannot capture full value from waterflood projects on their own. Cross-functional participation from earth sciences, production, drilling, completions, and facility engineering, and operational groups is required to get full value from waterfloods. Waterflood design and operational case histories of cross-functional collaboration are provided that have improved life cycle costs and increased recovery for onshore and offshore waterfloods. The role that water quality, surveillance, reservoir processing rates, and layered reservoir management has on waterflood oil recovery and life cycle costs will be clarified. Techniques to get better performance out of your waterflood will be shared.
Metal cutting tool position control using static output feedback and full sta...Mustefa Jibril
In this paper, a metal cutting machine position control have been designed and simulated using
Matlab/Simulink Toolbox successfully. The open loop response of the system analysis shows that the system needs
performance improvement. Static output feedback and full state feedback H 2 controllers have been used to increase
the performance of the system. Comparison of the metal cutting machine position using static output feedback and
full state feedback H 2 controllers have been done to track a set point position using step and sine wave input signals
and a promising results have been analyzed.
The Krylov-TPWL Method of Accelerating Reservoir Numerical Simulationinventionjournals
Because of the large number of system unknowns, reservoir simulation of realistic reservoir can be computationally demanding. Model order reduction (MOR) technique represents a promising approach for accelerating the simulations. In this work, we focus on the application of a MOR technique called Krylov trajectory piecewise-linear (Krylov-TPWL). First, the nonlinear system is represented as a weighted combined piecewise linear system using TPWL method, and then reducing order of each linear model using Krylov subspace. We apply Krylov-TPWL method for a two-phase (oil-water) reservoir model which is solved by full implicit. The example demonstrates that which can greatly reduce the dimension of reservoir model, so as to reduce the calculation time and improve the operation speed.
An Inventory Management System for Deteriorating Items with Ramp Type and Qua...ijsc
The present paper deals with an inventory management system with ramp type and quadratic demand rates. A constant deterioration rate is considered into the model. In the two types models, the optimum time and total cost are derived when demand is ramp type and quadratic. A structural comparative study is demonstrated here by illustrating the model with sensitivity analysis.
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
ACEP Magazine edition 4th launched on 05.06.2024Rahul
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
Water billing management system project report.pdfKamal Acharya
Our project entitled “Water Billing Management System” aims is to generate Water bill with all the charges and penalty. Manual system that is employed is extremely laborious and quite inadequate. It only makes the process more difficult and hard.
The aim of our project is to develop a system that is meant to partially computerize the work performed in the Water Board like generating monthly Water bill, record of consuming unit of water, store record of the customer and previous unpaid record.
We used HTML/PHP as front end and MYSQL as back end for developing our project. HTML is primarily a visual design environment. We can create a android application by designing the form and that make up the user interface. Adding android application code to the form and the objects such as buttons and text boxes on them and adding any required support code in additional modular.
MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software. It is a stable ,reliable and the powerful solution with the advanced features and advantages which are as follows: Data Security.MySQL is free open source database that facilitates the effective management of the databases by connecting them to the software.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
plant. If a power plant could recover and reuse a portion of this
moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
from flue gas is to use a condensing heat exchanger. The power
plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
1. Chapter 8: Production Decline Analysis
8.1 Introduction
Production decline analysis is a traditional means of identifying well production
problems and predicting well performance and life based on real production data. It uses
empirical decline models that have little fundamental justifications. These models include
• Exponential decline (constant fractional decline)
• Harmonic decline, and
• Hyperbolic decline.
While the hyperbolic decline model is more general, the other two models are
degenerations of the hyperbolic decline model. These three models are related through
the following relative decline rate equation (Arps, 1945):
d
bq
dt
dq
q
−=
1
(8.1)
where b and d are empirical constants to be determined based on production data. When d
= 0, Eq (8.1) degenerates to an exponential decline model, and when d = 1, Eq (8.1)
yields a harmonic decline model. When 0 < d < 1, Eq (8.1) derives a hyperbolic decline
model. The decline models are applicable to both oil and gas wells.
8.2 Exponential Decline
The relative decline rate and production rate decline equations for the exponential decline
model can be derived from volumetric reservoir model. Cumulative production
expression is obtained by integrating the production rate decline equation.
8.2.1 Relative Decline Rate
Consider an oil well drilled in a volumetric oil reservoir. Suppose the well’s production
rate starts to decline when a critical (lowest permissible) bottom hole pressure is reached.
Under the pseudo-steady state flow condition, the production rate at a given decline time
t can be expressed as:
⎥
⎦
⎤
⎢
⎣
⎡
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
=
s
r
r
B
ppkh
q
w
e
c
t wf
472.0
ln2.141
)(
0 μ
(8.2)
where tp = average reservoir pressure at decline time t,
c
wfp = the critical bottom hole pressure maintained during the production decline.
The cumulative oil production of the well after the production decline time t can be
expressed as:
8-1
2. dt
s
r
r
B
ppkh
N
t
w
e
o
c
t
p
wf
∫
⎥
⎦
⎤
⎢
⎣
⎡
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
=
0 472.0
ln2.141
)(
μ
(8.3)
The cumulative oil production after the production decline upon decline time t can also
be evaluated based on the total reservoir compressibility:
( t
o
it
p pp
B
Nc
N −= 0 ) (8.4)
where = total reservoir compressibility,tc
iN = initial oil in place in the well drainage area,
0p = average reservoir pressure at decline time zero.
Substituting Eq (8.3) into Eq (8.4) yields:
( t
o
it
t
w
e
o
c
t
pp
B
Nc
dt
s
r
r
B
ppkh wf
−=
⎥
⎦
⎤
⎢
⎣
⎡
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
∫ 0
0 472.0
ln2.141
)(
μ
) (8.5)
Taking derivative on both sides of this equation with respect to time t gives the
differential equation for reservoir pressure:
dt
pd
Nc
s
r
r
ppkh t
it
w
e
c
t wf
−=
⎥
⎦
⎤
⎢
⎣
⎡
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
472.0
ln2.141
)(
μ
(8.6)
Since the left-hand-side of this equation is q and Eq (8.2) gives
dt
pd
s
r
r
B
kh
dt
dq t
w
e
⎥
⎦
⎤
⎢
⎣
⎡
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
472.0
ln2.141 0μ
(8.7)
Eq (8.6) becomes
dt
dq
kh
s
r
r
Nc
q
w
e
it ⎥
⎦
⎤
⎢
⎣
⎡
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
=
472.0
ln2.141 μ
(8.8)
8-2
3. or the relative decline rate equation of
b
dt
dq
q
−=
1
(8.9)
where
⎥
⎦
⎤
⎢
⎣
⎡
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
s
r
r
Nc
kh
b
w
e
it
472.0
ln2.141 μ
. (8.10)
8.2.2 Production Rate Decline
Equation (8.6) can be expressed as:
dt
pd
ppb tc
t wf
=−− )( (8.11)
By separation of variables, Eq (8.11) can be integrated
∫∫ −
=−
t
wf
p
p
c
t
t
t
pp
pd
dtb
0
)(0
(8.12)
to yield an equation for reservoir pressure decline:
( ) btcc
t epppp wfwf
−
−+= 0 (8.13)
Substituting Eq (8.13) into Eq (8.2) gives well production rate decline equation:
bt
w
e
o
c
e
s
r
r
B
ppkh
q wf −
⎥
⎦
⎤
⎢
⎣
⎡
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
=
472.0
ln2.141
)( 0
μ
(8.14)
or
btc
o
it
epp
B
Nbc
q wf
−
−= )( 0 (8.15)
which is the exponential decline model commonly used for production decline analysis of
solution-gas-drive reservoirs. In practice, the following form of Eq (8.15) is used:
bt
ieqq −
= (8.16)
where qi is the production rate at t = 0.
It can be shown that b
n
n
e
q
q
q
q
q
q −
−
====
12
3
1
2
...... . That is, the fractional decline is constant
8-3
4. for exponential decline. As an exercise, this is left to the reader to prove.
8.2.3 Cumulative Production
Integration of Eq (8.16) over time gives an expression for the cumulative oil production
since decline of
∫∫
−
==
t
bt
i
t
p dteqqdtN
00
(8.17)
i.e.,
( )bti
p e
b
q
N −
−= 1 . (8.18)
Since , Eq (8.18) becomesbt
ieqq −
=
( qq
b
N ip −=
1
). (8.19)
8.2.4 Determination of Decline Rate
The constant b is called the continuous decline rate. Its value can be determined from
production history data. If production rate and time data are available, the b-value can be
obtained based on the slope of the straight line on a semi-log plot. In fact, taking
logarithm of Eq (8.16) gives:
( ) ( ) btqq i −= lnln (8.20)
which implies that the data should form a straight line with a slope of -b on the log(q)
versus t plot, if exponential decline is the right model. Picking up any two points, (t1, q1)
and (t2, q2), on the straight line will allow analytical determination of b-value because
( ) ( ) 11 lnln btqq i −= (8.21)
and
( ) ( ) 22 lnln btqq i −= (8.22)
give
( ) ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
=
2
1
12
ln
1
q
q
tt
b . (8.23)
If production rate and cumulative production data are available, the b-value can be
obtained based on the slope of the straight line on an Np versus q plot. In fact, rearranging
Eq (8.19) yields:
8-4
5. pi bNqq −= (8.24)
Picking up any two points, (Np1, q1) and (Np2, q2), on the straight line will allow analytical
determination of b-value because
11 pi bNqq −= (8.25)
and
22 pi bNqq −= (8.26)
give
12
21
pp NN
qq
b
−
−
= . (8.27)
Depending on the unit of time t, the b can have different units such as month-1
and year-1
.
The following relation can be derived:
dma bbb 36512 == . (8.28)
where ba, bm, and bd are annual, monthly, and daily decline rates.
8.2.5 Effective Decline Rate
Because the exponential function is not easy to use in hand calculations, traditionally the
effective decline rate has been used. Since for small x-values based on
Taylor’s expansion, holds true for small values of b. The b is substituted by
, the effective decline rate, in field applications. Thus Eq (8.16) becomes
xe x
−≈−
1
be b
−≈−
1
'b
( t
i bqq '1−= ) (8.29)
Again, it can be shown that '1......
12
3
1
2
b
q
q
q
q
q
q
n
n
−====
−
.
Depending on the unit of time t, the can have different units such as month-1
and year-
1
. The following relation can be derived:
'b
( ) ( ) ( )36512
'1'1'1 dma bbb −=−=− . (8.30)
where a, , and b d are annual, monthly, and daily effective decline rates.'b 'b m '
8-5
6. Example Problem 8-1:
Given that a well has declined from 100 stb/day to 96 stb/day during a one-month period,
use the exponential decline model to perform the following tasts:
a) Predict the production rate after 11 more months
b) Calculate the amount of oil produced during the first year
c) Project the yearly production for the well for the next 5 years.
Solution:
a) Production rate after 11 more months:
( ) ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
=
m
m
mm
m
q
q
tt
b
1
0
01
ln
1
/month04082.0
96
100
ln
1
1
=⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
=
Rate at end of one year
( )
stb/day27.61100 1204082.0
01 === −−
eeqq tb
mm
m
If the effective decline rate b’ is used,
/month04.0
100
96100
'
0
10
=
−
=
−
=
m
mm
m
q
qq
b .
From
( ) ( )
/year3875.0'
getsone
04.01'1'1
1212
=
−=−=−
y
my
b
bb
Rate at end of one year
( ) ( ) stb/day27.613875.01100'101 =−=−= ybqq
b) The amount of oil produced during the first year:
8-6
7. ( ) /year48986.01204082.0 ==yb
stb858,28365
48986.0
27.6110010
1, =⎟
⎠
⎞
⎜
⎝
⎛ −
=
−
=
y
p
b
qq
N
or
( )
( ) stb858,281
001342.0
100
day
1
001342.0
42.30
1
96
100
ln
365001342.0
1, =−=
=⎟
⎠
⎞
⎜
⎝
⎛
⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
=
−
eN
b
p
d
c) Yearly production for the next 5 years:
( )
( ) stb681,171
001342.0
27.61 365001342.0
2, =−= −
eN p
( )
stb/day54.37100 )2(1204082.0
2 === −−
eeqq bt
i
( )
( ) stb834,101
001342.0
54.37 365001342.0
3, =−= −
eN p
( )
stb/day00.23100 )3(1204082.0
3 === −−
eeqq bt
i
( )
( ) stb639,61
001342.0
00.23 365001342.0
4, =−= −
eNp
( )
stb/day09.14100 )4(1204082.0
4 === −−
eeqq bt
i
( )
( ) stb061,41
001342.0
09.14 365001342.0
5, =−= −
eNp
In summary,
8-7
8. Year
Rate at End of Year
(stb/day)
Yearly Production
(stb)
0
1
2
3
4
5
100.00
61.27
37.54
23.00
14.09
8.64
-
28,858
17,681
10,834
6,639
4,061
68,073
8.3 Harmonic Decline
When d = 1, Eq (8.1) yields differential equation for a harmonic decline model:
bq
dt
dq
q
−=
1
(8.31)
which can be integrated as
bt
q
q
+
=
1
0
(8.32)
where q0 is the production rate at t = 0.
Expression for the cumulative production is obtained by integration:
∫=
t
p qdtN
0
which gives:
( bt
b
q
N p += 1ln0
). (8.33)
Combining Eqs (8.32) and (8.33) gives
( ) ( )[ qq
b
q
N p lnln 0
0
−= ]. (8.34)
8-8
9. 8.4 Hyperbolic Decline
When 0 < d < 1, integration of Eq (8.1) gives:
∫∫ −=+
tq
q
d
bdt
q
dq
0
1
0
(8.35)
which results in
( ) d
dbt
q
q /1
0
1+
= (8.36)
or
a
t
a
b
q
q
⎟
⎠
⎞
⎜
⎝
⎛
+
=
1
0
(8.37)
where a = 1/d.
Expression for the cumulative production is obtained by integration:
∫=
t
p qdtN
0
which gives:
( ) ⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
+−
−
=
−a
p t
a
b
ab
aq
N
1
0
11
1
. (8.38)
Combining Eqs (8.37) and (8.38) gives
( ) ⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
+−
−
= t
a
b
qq
ab
a
N p 1
1
0 . (8.39)
8.5 Model Identification
Production data can be plotted in different ways to identify a representative decline
model. If the plot of log(q) versus t shows a straight line (Figure 8-1), according to Eq
(8.20), the decline data follow an exponential decline model. If the plot of q versus Np
shows a straight line (Figure 8-2), according to Eq (8.24), an exponential decline model
should be adopted. If the plot of log(q) versus log(t) shows a straight line (Figure 8-3),
according to Eq (8.32), the decline data follow a harmonic decline model. If the plot of
Np versus log(q) shows a straight line (Figure 8-4), according to Eq (8.34), the harmonic
decline model should be used. If no straight line is seen in these plots, the hyperbolic
8-9
10. decline model may be verified by plotting the relative decline rate defined by Eq (8.1).
Figure 8-5 shows such a plot. This work can be easily performed with computer program
UcomS.exe.
q
t
Figure 8-1: A Semilog plot of q versus t indicating an exponential decline
q
pN
Figure 8-2: A plot of Np versus q indicating an exponential decline
8-10
11. q
t
Figure 8-3: A plot of log(q) versus log(t) indicating a harmonic decline
q
pN
Figure 8-4: A plot of Np versus log(q) indicating a harmonic decline
8-11
12. q
tq
q
Δ
Δ
−
Exponential Decline
Harmonic Decline
Hyperbolic Decline
Figure 8-5: A plot of relative decline rate versus production rate
8.6 Determination of Model Parameters
Once a decline model is identified, the model parameters a and b can be determined by
fitting the data to the selected model. For the exponential decline model, the b-value can
be estimated on the basis of the slope of the straight line in the plot of log(q) versus t (Eq
8.23). The b-value can also be determined based on the slope of the straight line in the
plot of q versus Np (Eq 8.27).
For the harmonic decline model, the b-value can be estimated on the basis of the slope of
the straight line in the plot of log(q) versus log(t) shows a straight line, or Eq (8.32):
1
1
0
1
t
q
q
b
−
= (8.40)
The b-value can also be estimated based on the slope of the straight line in the plot of Np
versus log(q) (Eq 8.34).
For the hyperbolic decline model, determination of a- and b-values is of a little tedious.
The procedure is shown in Figure 8-6.
8-12
13. 1. Select points (t1, q1)
and (t2, q2)
2. Read t3 at
3. Calculate
4. Find q0 at t = 0
5. Pick up any point (t*, q*)
6. Use
7. Finally
q
t
1
2
213 qqq =
21
2
3
321 2
ttt
ttt
a
b
−
−+
=⎟
⎠
⎞
⎜
⎝
⎛
a
t
a
b
q
q
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+
=
*
0
*
1 ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
*
*
0
1log
log
t
a
b
q
q
a
a
a
b
b ⎟
⎠
⎞
⎜
⎝
⎛
=
q3
t3
(t*, q*)
Figure 8-6: Procedure for determining a- and b-values
Computer program UcomS.exe can be used for both model identification and model
parameter determination, as well as production rate prediction.
8.7 Illustrative Examples
Example Problem 8-2:
For the data given in Table 8-1, identify a suitable decline model, determine model
parameters, and project production rate until a marginal rate of 25 stb/day is reached.
Table 8-1: Production Data for Example Problem 8-2
t (Month) q (STB/D) t (Month) q (STB/D)
301.1912.00
332.8711.00
367.8810.00
406.579.00
449.338.00
496.597.00
548.816.00
606.535.00
670.324.00
740.823.00
818.732.00
904.841.00
301.1912.00
332.8711.00
367.8810.00
406.579.00
449.338.00
496.597.00
548.816.00
606.535.00
670.324.00
740.823.00
818.732.00
904.841.00
90.7224.00
100.2623.00
110.8022.00
122.4621.00
135.3420.00
149.5719.00
165.3018.00
182.6817.00
201.9016.00
223.1315.00
246.6014.00
272.5313.00
90.7224.00
100.2623.00
110.8022.00
122.4621.00
135.3420.00
149.5719.00
165.3018.00
182.6817.00
201.9016.00
223.1315.00
246.6014.00
272.5313.00
8-13
14. Solution:
A plot of log(q) versus t is presented in Figure 8-7 which shows a straight line.
According to Eq (8.20), the exponential decline model is applicable. This is further
evidenced by the relative decline rate shown in Figure 8-8.
Select points on the trend line:
t1= 5 months, q1 = 607 STB/D
t2= 20 months, q2 = 135 STB/D
Decline rate is calculated with Eq (8.23):
( )
1/month1.0
607
135
ln
205
1
=⎟
⎠
⎞
⎜
⎝
⎛
−
=b
Projected production rate profile is shown in Figure 8-9.
1
10
100
1000
10000
0 5 10 15 20 25 30
t (month)
q(STB/D)
8-14
15. Figure 8-7: A plot of log(q) versus t showing an exponential decline
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
3 203 403 603 803 1003
q (STB/D)
-Δq/Δt/q(Month-1
)
Figure 8-8: Relative decline rate plot showing exponential decline
0
100
200
300
400
500
600
700
800
900
1000
0 10 20 30
t (month)
q(STB/D
40
)
Figure 8-9: Projected production rate by an exponential decline model
Example Problem 8-3:
For the data given in Table 8-2, identify a suitable decline model, determine model
parameters, and project production rate till the end of the 5th
year.
8-15
16. Table 8-2: Production Data for Example Problem 8-3
t (year) q (1000 STB/D) t (year) q (1000 STB/D)
5.682.00
5.811.90
5.941.80
6.081.70
6.221.60
6.371.50
6.531.40
6.691.30
6.871.20
7.051.10
7.251.00
7.450.90
7.670.80
7.900.70
8.140.60
8.400.50
8.680.40
8.980.30
9.290.20
5.682.00
5.811.90
5.941.80
6.081.70
6.221.60
6.371.50
6.531.40
6.691.30
6.871.20
7.051.10
7.251.00
7.450.90
7.670.80
7.900.70
8.140.60
8.400.50
8.680.40
8.980.30
9.290.20
4.033.90
4.093.80
4.163.70
4.223.60
4.293.50
4.363.40
4.443.30
4.513.20
4.593.10
4.673.00
4.762.90
4.842.80
4.942.70
5.032.60
5.132.50
5.232.40
5.342.30
5.452.20
5.562.10
4.033.90
4.093.80
4.163.70
4.223.60
4.293.50
4.363.40
4.443.30
4.513.20
4.593.10
4.673.00
4.762.90
4.842.80
4.942.70
5.032.60
5.132.50
5.232.40
5.342.30
5.452.20
5.562.10
Solution:
A plot of relative decline rate is shown in Figure 8-10 which clearly indicates a
harmonic decline model.
On the trend line, select
q0 = 10,000 stb/day at t = 0
q1 = 5,680 stb/day at t = 2 years
Therefore, Eq (8.40) gives:
1/year38.0
2
1
680,5
000,10
=
−
=b .
Projected production rate profile is shown in Figure 8-11.
8-16
17. 0.1
0.15
0.2
0.25
0.3
0.35
0.4
3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
q (1000 STB/D)
-Δq/Δt/q(year-1
)
Figure 8-10: Relative decline rate plot showing harmonic declineplot showing harmonic decline
0
2
4
6
8
10
12
0.0 1.0 2.0 3.0 4.0 5.0 6.0
t (year)
q(1000STB/D)
Projected production rate by a harmonic decline modelProjected production rate by a harmonic decline modelFigure 8-11:Figure 8-11:
Example Problem 8-4:Example Problem 8-4:
For the data given in Table 8-3, identify a suitable decline model, determine model
rate till the end of the 5th
year.
Sol
ecline rate is shown in Figure 8-12 which clearly indicates a
line model.
Select poin
parameters, and project production
ution:
A plot of relative d
hyperbolic dec
ts:
8-17
18. t1 = 0.2 year , q1 = 9,280 stb/day
t2 = 3.8 years, q2 = 3,490 stb/day
Read from decline curve (Figure 8-13) t3 = 1.75 yaers at q3 = 5,670 stb/day.
Read from decline curve (Figure 8-13) q0 = 10,000 stb/day at t0 = 0.
Pick up point ( = 6,280 stb/day).
Projected production rate profile is shown in Figure 8-14.
Table 8-3: Production Dat for Example Problem 8-4
stb/day5,670)490,3)(280,9(3 ==q
217.0
)75.1(28.32.0
2
=
−+
=⎟
⎞
⎜
⎛ b
)8.3)(2.0()75.1( −⎠⎝ a
t*
= 1.4 yesrs, q*
a
5.322.00 5.322.00 3.344.00 3.344.00
5.461.90
5.611.80
5.771.70
5.931.60
6.101.50
6.281.40
6.471.30
6.671.20
6.871.10
7.091.00
7.320.90
7.550.80
7.810.70
8.070.60
8.350.50
8.640.40
8.950.30
9.280.20
9.630.10
5.461.90
5.611.80
5.771.70
5.931.60
6.101.50
6.281.40
6.471.30
6.671.20
6.871.10
7.091.00
7.320.90
7.550.80
7.810.70
8.070.60
8.350.50
8.640.40
8.950.30
9.280.20
9.630.10
3.413.90
3.493.80
3.563.70
3.643.60
3.713.50
3.803.40
3.883.30
3.973.20
4.063.10
4.153.00
4.252.90
4.352.80
4.462.70
4.572.60
4.682.50
4.802.40
4.922.30
5.052.20
5.182.10
3.413.90
3.493.80
3.563.70
3.643.60
3.713.50
3.803.40
3.883.30
3.973.20
4.063.10
4.153.00
4.252.90
4.352.80
4.462.70
4.572.60
4.682.50
4.802.40
4.922.30
5.052.20
5.182.10
t(year) q(1000STB/D) t(year) q(1000STB/D)
( )( )
75.1
)4.1(217.01log
280,6
=
+
⎠⎝=a
000,10
log ⎟
⎞
⎜
⎛
( ) 38.0)758.1(217.0 ==b
8-18
20. 0
2
4
6
8
10
12
0.0 1.0 2.0 3.0 4.0 5.0 6.0
t (year)
q(1000STB/D)
Figure 8-14: Projected production rate by a hyperbolic decline model
* * * * *
Summary
This chapter presented empirical models and procedure of using the models to perform
production decline data analyses. Computer program UcomS.exe can be used for model
identification, model parameter determination, and production rate prediction.
References
Arps, J.J.: “ Analysis of Decline Curves,” Trans. AIME, 160, 228-247, 1945.
Golan, M. and Whitson, C.M.: Well Performance, International Human Resource
Development Corp., 122-125, 1986.
Economides, M.J., Hill, A.D., and Ehlig-Economides, C.: Petroleum Production
Systems, Prentice Hall PTR, Upper Saddle River, 516-519, 1994.
Problems
8.1 For the data given in the following table, identify a suitable decline model, determine
model parameters, and project production rate till the end of the 10th
year. Predict
yearly oil productions:
8-20
21. Time (year) Production Rate (1,000 stb/day)
0.1 9.63
0.2 9.29
0.3 8.98
0.4 8.68
0.5 8.4
0.6 8.14
0.7 7.9
0.8 7.67
0.9 7.45
1 7.25
1.1 7.05
1.2 6.87
1.3 6.69
1.4 6.53
1.5 6.37
1.6 6.22
1.7 6.08
1.8 5.94
1.9 5.81
2 5.68
2.1 5.56
2.2 5.45
2.3 5.34
2.4 5.23
2.5 5.13
2.6 5.03
2.7 4.94
2.8 4.84
2.9 4.76
3 4.67
3.1 4.59
3.2 4.51
3.3 4.44
3.4 4.36
8.2 For the data given in the following table, identify a suitable decline model, determine
model parameters, predict the time when the production rate will decline to a
marginal value of 500 stb/day, and the reverses to be recovered before the marginal
production rate is reached:
8-21
22. Time (year) Production Rate (stb/day)
0.1 9.63
0.2 9.28
0.3 8.95
0.4 8.64
0.5 8.35
0.6 8.07
0.7 7.81
0.8 7.55
0.9 7.32
1 7.09
1.1 6.87
1.2 6.67
1.3 6.47
1.4 6.28
1.5 6.1
1.6 5.93
1.7 5.77
1.8 5.61
1.9 5.46
2 5.32
2.1 5.18
2.2 5.05
2.3 4.92
2.4 4.8
2.5 4.68
2.6 4.57
2.7 4.46
2.8 4.35
2.9 4.25
3 4.15
3.1 4.06
3.2 3.97
3.3 3.88
3.4 3.8
8.3 For the data given in the following table, identify a suitable decline model, determine
model parameters, predict the time when the production rate will decline to a
marginal value of 50 Mscf/day, and the reverses to be recovered before the marginal
production rate is reached:
8-22
23. Time
(Month)
Production Rate
(Mscf/day)
1 904.84
2 818.73
3 740.82
4 670.32
5 606.53
6 548.81
7 496.59
8 449.33
9 406.57
10 367.88
11 332.87
12 301.19
13 272.53
14 246.6
15 223.13
16 201.9
17 182.68
18 165.3
19 149.57
20 135.34
21 122.46
22 110.8
23 100.26
24 90.72
8.4 For the data given in the following table, identify a suitable decline model, determine
model parameters, predict the time when the production rate will decline to a
marginal value of 50 stb/day, and yearly oil productions:
Time
(Month)
Production Rate
(stb/day)
1 1810
2 1637
3 1482
4 1341
5 1213
6 1098
7 993
8-23