Annual Report – 2007
November, 2007
OUTLINE• CPARM Overview• Production Optimization and Flood Efficiency
Evaluation Using Capacitance Resistive Model and Injection Rates
• Using Semicontinuous Gains in Capacitance-Resistive Modeling of Large Scale Reservoir Systems
• The Cost of Errors in Estimates Used in Concept Selection
• A Discrete-Time Approach for Modeling Two-Factor Mean-Reverting Stochastic Processes
• Value of Information in E&P• Calibration of likelihood of HC recovery from
reservoir datasets• Multi-period Models of E&P Project Portfolios
CPARM Overview
CPARM Objectives
• To perform research into ways that decisions regarding hydrocarbon exploration and production can be improved
• Improve profitability of E&P operations by research in– Methods– Processes– Culture– Tools
DRA Through the Life of a Field
• We repeat the Bayesian update every time new information is acquired and plot the NPV distributions together.
• The figure represents a new type of DRA display that can easily communicate how uncertainty evolves through the life of a field.
CPARM Industrial SponsorsChevronStatoilHydroDevon EnergyLandmarkBP
The CPARM Team
* Graduated
CPARM FacultyLarry W. Lake, PGEBob Gilbert, CELeon Lasdon, IROMSanjay Srinivasan, PGEJim Dyer, IROMChris Jablonowski, EER/PGETom Edgar, ChEEmilio Núñez, CPARM
CPARM StudentsMin Chen, IROM*Wei Chen, IROMNamhong Min, CERobert MacAskie, PGE*Morteza Sayarpour, PGEAviral Sharma, PGEYonghoon Lee, CE*Dan Weber,ChEHariharan Ramachandran, PGE
CPARM Publications - 2007• Theses
– Robert MacAskie – “The Value of Oil Price Forecasts”– Azeez Lawal – “Applications of Sensitivity Analysis in Petroleum Engineering”
• Dissertation– Min Chen – “Inevitable Disappointment and Decision Making Based on Forecasts”
• Publications– Jablonowski and MacAskie – The Value of Oil Price Forecasts (SPE 107570)– Hultzsch, Lake and Gilbert – Decision and Risk Analysis through the Life of the
Field (SPE 107704)– Chen and Dyer – Inevitable Disappointment in Projects Selected Based on
Forecasts (SPE 107710)– Lasdon, Faya, Lake, Dyer and Chen – constructing Oil Exploration and
Development Project Portfolios Using Several Risk Measures – A Realistic Example (SPE 107708)
– Faya, Lake and Lasdon – Beyond Portfolio Optimization (SPE 107709)– Liang, Weber, Edgar, Lake, Sayarpour and Al-Yousef – Optimization of Oil
Production Based on a Capacitance Model of production and Injection Rates (SPE 107713)
– Hahn, Dyer and Brandao - Using Decision Analysis to Solve Real Option Valuation Problems: Building a Generalized Approach (SPE 108066)
– Gilbert, Lake, Jablonowski, Jennings and Nunez – A Procedure for Assessing the Value of Oilfield Sensors (SPE 109628)
Appraisal and Conceptual
AnalysisGATE GATEEvaluate
Alternatives GATE
Define Selected
AlternativeGATEExecute Operate
Inevitable Dis-
appointment
Portfolio Optimization
Uncertainty Updating
Concept Selection & Development Optimization
Real Options
Dry Gas Model;
Compare MC & Decision
Trees
Portfolio Management and Project Selection
UT Tank Model
Enhance & Sens.
Analysis
Addressing Risks Throughout the E&P Asset Lifecycle
VOI; Impact of Estimates & Methods
Cost and Schedule Estimating; Execution Risk Management
HSE Risk Management
Real-Time Optimization and Risk Management
Tool Development:
Life Cycle Assessments
Cost Modeling of Wells and Facilities
Contracting Strategies
(lump sum v cost plus?)
MPD & Blowouts;
Drlg Safety; Offshore
Spills Likelihood Functions from Data Analysis
Real Time Optimization
Under Uncertainty
Production Optimization and Flood Efficiency Evaluation Using
Capacitance Resistive Model and Injection Rates
Morteza SayarpourLarry W. Lake (Supervisor)
OutlineBackgroundCapacitance–Resistive Model (CRM)Rate Measurement ErrorBHP Variation EffectField-Case Examples
Reinicke / West Texas MESL Field - (4 Inj., 6 Pro.)Big Field (17 Inj., 30 Pro.)Other Fields (Alaska, Angola, GOM, Seminole, SWCF, …)*
Summary, Recommendations and Future Work
Background
– Albertoni-Lake (2003)• Interwell Connectivities (SPE 75225 & 83381)
– Yousef-Gentil-Lake (2006): • Capacitance Model (SPE 95322 & 99998)
– Ximing et al. (2006): • Oil Production Optimization (SPE 107713)
– Sayarpour et al. (2007): • Capacitance-Resistive Model (SPE 110081)
CRM: Injection-Production Signals
P1
P20
250
500
750
1000
0 30 60 90Time, days
RB/D
I2
I3
I4
0
250
500
750
1000
0 30 60 90Time, days
RB/D
I1
0
250
500
750
1000
0 30 60 90Time, days
RB/D
P3
P4
P5
P6
0
250
500
750
1000
0 30 60 90Time, days
RB/D
Injectors Producers
1τ
f12 =40%
f11 =60%f11
2τf12
CRM: Total Production
⎟⎟⎠
⎞⎜⎜⎝
⎛=
JVc ptτ[ ]*1)()(
)()(
0 Ieetqtqtt
⎟⎟⎠
⎞⎜⎜⎝
⎛−+=
Δ−
Δ−
ττ
)( tq Field)( tI Field
Fτ
i6
i1i2
i3
i4i5
qj(t)
f1jf11 f12 f13
f2j
f6j
f4j
f3j
f5j
jτ
Volume Change = ( Injection Rate – Production Rate)
ii
qj
fijijτ
iij ifI =*∑=
=injN
iiijifI
1
*
fieldII =*CRMT
CRMIPCRMP
CRM: Oil Fractional-Flow Equations
)(11)(
tWORqqq
tfwo
oo +
=+
=
q0(t) = fo(t)q(t)
( )βα )(11)(
tCWItfo +=
( ))(loglog1)(
1log tCWItfo
βα +=⎟⎟⎠
⎞⎜⎜⎝
⎛−
Injection & ProductionData
CRM Total Production Match
CRM Fitting Parameters
Oil Fractional-Flow Match
Empirical Power-LawModel
Injection Rate
CRM Total ProductionPrediction
Oil Fractional-Flow Prediction
Oil Production Optimization
Workflow
Oil Production Match
0
1,000
2,000
3,000
4,000
0 1,000 2,000Time, Days
Rate
, RB/
D
I1 I2 P1CRMP
Injection Signal – Production Response(2 Inj, 2 Pro)
Injection/Production Rate Errors
0
1,000
2,000
3,000
4,000
0 1,000 2,000Time, Days
Rate
, RB/
D
I1 I2 P1CRMP
Synfield: Streak Case
Streak Case: Varying BHP
2,400
2,500
2,600
2,700
2,800
0 200 400 600 800 1,000 1,200Time, Days
Rat
e, R
B/D
150
200
250
300
Prod
ucer
s B
HP,
psi
QinjPwf ,psi
Streak Case: Varying BHP
2,400
2,500
2,600
2,700
2,800
0 200 400 600 800 1,000 1,200Time, Days
Rat
e, R
B/D
150
200
250
300
Prod
ucer
s B
HP,
psi
Qinj
Qpro Eclipse
Pwf ,psi
Streak Case: Varying BHP
2,400
2,500
2,600
2,700
0 200 400 600 800 1,000 1,200
Rat
e, R
B/D
150
200
250
Prod
ucer
s B
HP,
psi
2,800
Time, Days
300QinjQpro EclipseQpro est CRMTPwf ,psi
τField , Days 17q0 Field, RB/D 0J field, (RB/D)/psi 2.703Error, RB/D 5.57
Field Example 1: Reinecke
• West Texas, reefal reservoir
• 1950 discovered• Depth 6700 ft• Water flooding 1972-
1995• Major production from
south dome (1 mi2) • 46 MMBO
(Saller et al. 2004)
CRMTCRMT Match for Reinecke fieldREINECKE FIELD
0
2000
4000
6000
8000
10000
12000
1970 1975 1980 1985 1990 1995
Time, year
Oil
Prod
uctio
n (S
TB/d
ay)
0
10
20
30
40
50
60
70
80
90
Wat
er In
ject
ion
& P
rodu
ctio
n, 1
000
STB
/day
Oil Productionqw inj Water injectionWater Production
J/ctPv=0.40 and qoi= 4735 STB/DR =0.964
0
10000
20000
30000
40000
50000
60000
70000
1970 1975 1980 1985 1990 1995Time, year
Tota
l Pro
duct
ion,
STB
/day
q pro M BEq pro fie ld
0
2000
4000
6000
8000
10000
12000
14000
16000
1970 1975 1980 1985 1990 1995Time, year
qo, S
TB /
day
q oil fieldqo, Kovalqo, Koval&Gravityqo, Coreyqo, Corey&Gravity
Field Example 2: MESL
Offshore Field
• 4 Injectors
• 6 Producers P2, P6 Horizontal
• kh ~ 1400 md
• kv ~ 166 md
• Porosity ~ 15%
•Oil Gravity ~ 24 API
•Oil Viscosity ~ 3 cp
MESL Field: Injection/Production Rates
30,000
40,000
50,000
60,000
0 1,000 2,000 3,000Time, Days
Tota
l rat
e, R
B/D
ProductionInjection
MESL Field: Total Production Match
40,000
46,000
52,000
58,000
0 1,000 2,000Time, Days
Tota
l rat
e, R
B/D
Field CRMTCRMP CRMIP
P5 Total Production Match
5,000
8,000
11,000
0 1,000 2,000Time, Days
Tota
l rat
e, R
B/D
FieldCRMPCRMIP
MESL Field: Oil Production Match
0
10,000
20,000
30,000
40,000
50,000
0 1,000 2,000Time, Days
Oil
rate
, STB
/D
FieldCRMP
MESL CRMP Parameters
P1 (j=1) P2 (j=2) P3 (j=3) P4 (j=4) P5 (j=5) P6 (j=6)f 1j (i=1) 0.293 0.473 0.035 0.088 0.079 0.015f 2j (i=2) 0.038 0.090 0.440 0.195 0.315 0.003f 3j (i=3) 0.010 0.080 0.374 0.246 0.185 0.101f 4j (i=4) 0.065 0.020 0.328 0.293 0.070 0.182
τ j , Days 47 67 89 220 47 300
q j (t 0 ), RB/D 3000 3000 3000 3437 3000 6649
P1 (j=1) P2 (j=2) P3 (j=3) P4 (j=4) P5 (j=5) P6 (j=6)α j 6.5E-14 3.2E-15 2.0E-12 5.6E-13 8.9E-13 2.3E-14β j 1.9276 2.0288 1.6650 1.7742 1.7630 2.0275
• Total Production
• Oil Fractional Flow
Field Example 2: MESL
47%
24%
44%
29%
19%
19%
37%
31%
18%
29%
33%
Field Example 2: MESL Constraints,
RB/D
100 < I1 < 21000
100 < I2 < 24000
100 < I3 < 24000
100 < I4 < 7500
IField < 56000
I1 I2 I3 I421000 24000 10900 100
Production Response of Optimized Injection
6,000
8,000
10,000
12,000
2,000 2,500 3,000Time, Days
Oil
Rat
e, S
TB/D
BaseOptimized
6% increase
●P1▼I2
▼I3
▼I4
▼I6
▼I7
▼I9
▼I10
___I5
▼I12
▼I13
▼I14
▼I15▼I16
▼I17
▼I1
●P2●P15
●P3
●P4
●P13
___P19
●P5
●P12
●P20
●P26
●P21
●P6
●P7
●P8●P9
●P23
●P10
●P25●P##
●P22●P##
___P##●P27
___P29
●P17
●P18
●P11
●P14
___P16
●P##
●P28
___P30
●P##
____P##
____P##
____P##
____P##
____P##
____P##
____P##
___I8
▼I11
Field Example 3: “Big Field”
17 Injectors
30 Producers
Field Example 2: CRMP Match
Field Injection
Field Production
CRMP Match
0
50,000
100,000
150,000
200,000
250,000
300,000
0 2,000 4,000 6,000 8,000 10,000 12,000 14,000
Time, Days
Rat
e, B
/D
CRM Individual Well Match
0
1,500
3,000
4,500
7,000 8,000 9,000 10,000 11,000 12,000Time, Days
Prod
uctio
n B
/D
P14CRMP P14
0
10,000
20,000
30,000
0 2,000 4,000 6,000 8,000 10,000 12,000 14,000
Time, Days
Prod
uctio
n B
/D
P24CRMP P24
0
3,000
6,000
9,000
12,000
7,000 8,000 9,000 10,000 11,000 12,000
Time, Days
Prod
uctio
n B
/D
P10CRMP P10
0
2,000
4,000
6,000
8,000
10,000
6,000 8,000 10,000 12,000Time, Days
Prod
uctio
n B
/D
P12CRMP P12
●P1▼I2
▼I3
▼I4
▼I6
▼I7
▼I9
▼I10
___I5
▼I12
▼I13
▼I14
▼I15▼I16
▼I17
▼I1
●P2●P15
●P3
●P4
●P13
___P19
●P5
●P12
●P20
●P26
●P21
●P6
●P7
●P8●P9
●P23
●P10
●P25●P##
●P22●P##
___P##●P27
___P29
●P17
●P18
●P11
●P14
___P16
●P##
●P28
___P30
●P##
____P##
____P##
____P##
____P##
____P##
____P##
____P##
___I8
▼I11
CRMT Waterflood Efficiency
Area 2
~25 % of Injection
~105 Days Response Time
~28 % of Injection
~300 Days Response Time
Area 1
●P1▼I2
▼I3
▼I4
▼I6
▼I7
▼I9
▼I10
___I5
▼I12
▼I13
▼I14
▼I15▼I16
▼I17
▼I1
●P2●P15
●P3
●P4
●P13
___P19
●P5
●P12
●P20
●P26
●P21
●P6
●P7
●P8●P9
●P23
●P10
●P25●P##
●P22●P##
___P##●P27
___P29
●P17
●P18
●P11
●P14
___P16
●P##
●P28
___P30
●P##
____P##
____P##
____P##
____P##
____P##
____P##
____P##
___I8
▼I11
11%
27%
22%18%
12%
14%
19%
CRM Waterflood Efficiency & Interwells connectivities
53 % of Injection
200 Days Response Time
Summary
• CRM Analytical Solutions are Developed• BHP Variation for CRM is Validated • CRM Handles Rate Measurement Errors • CRM History Matching Ability
Demonstrated by Field Studies• CRM Quantifies Connectivities and Flood
Efficiency
Recommendation & Future Work
• Field Real Time Optimization• Regional Flood Efficiency Evaluation• Optimization of large fields (1000s of wells)• Enhanced displacement models…
–Improved waterflood–Carbon dioxide flood
Using Semicontinuous Gains in Capacitance-Resistive Modeling of
Large Scale Reservoir SystemsDaniel Weber
Supervisors:Thomas F. Edgar
Larry W. LakeLeon S. Lasdon
Overview• Objective• History Matching and Optimization Review• Homogeneous Reservoir
– Description– Continuous Gains– Semicontinuous Gains
• Heterogeneous Reservoir– Description– Continuous Gains– Semicontinuous Gains
• Oil Model Match• Optimal Injections
Objective
• Calculating many parameters by nonlinear regression may lead to statistically insignificant parameters
• Removal of these parameters leads to– Models that have fewer parameters– Matrices that are more sparse
• Can we find a parameter set that is simple without creating a large model error?
History Matching and Optimization
• Fit capacitance-resistive model (CRM) to total production data using nonlinear regression– Gains are non-negative and must sum to one for each
injector– Using single objective function – all producers are fit
at the same time• Fit oil fractional flow model to oil production data
using nonlinear regression• Use CRM and oil fractional flow model to
optimize future injection to maximize net present value (NPV) of the reservoir
Capacitance-Resistive Model
Where = total production rate of producer j= injection rate of injector i = weight (gain) between injector I and
producer j= time constant for producer j
ijλ
dtdq
Iq jji
n
iijj
i
τλ −=∑=1
jτ
jqiI
Oil Fractional Flow Modeljojoj qfq =
jbj
oj CWIaf
+=
11
Where = oil production rate of producer j= oil fraction of producer j = cumulative water injected in all injectors
, = model parameters for producer jja
ojqojf
CWIjb
Optimization formulationMaximize net present value
Subject to• CRM• Fractional flow model
• Upper limit on total injection rate
• Upper limits on rate of eachinjector
( ) ( )∑∑∑∑= == =
Δ+
−Δ+
=i tp t n
i
n
kkik
wn
j
n
kkojk
o ttIir
pttqir
pNPV1 11 1
)(1
)(1
ikii utIl ≤≤ )(
TOT
n
iki ItI
i
≤∑=1
)(
Reservoir Map• 25 injectors• 20 producers• Homogeneous
properties
• 25x20=500 gains• 20 time constants• 20 initial rates• 540 total
parameters
InjectorsProducers
Characteristics of the Data Set• Injection rates, total production rates and oil
production rates every 5 days for 100 months (600 time steps)
• Injection rates sampled from a normal distribution about a mean (different mean for each injector)
• Mean injection rate increases by 100 stb/day at some point during the first 25 months
• Total production and oil production rates modeled with ECLIPSE
Injection Historyst
b/da
y
Time (days)
0 1000 2000 3000400
600
8001
0 1000 2000 3000600
800
10002
0 1000 2000 3000500
1000
15003
0 1000 2000 3000200
400
6004
0 1000 2000 30000
5005
0 1000 2000 30000
200
4006
0 1000 2000 3000
600
800
10007
0 1000 2000 3000500
1000
15008
0 1000 2000 30000
200
4009
0 1000 2000 3000200
400
60010
0 1000 2000 30000
50011
0 1000 2000 3000
600
800
100012
0 1000 2000 30000
500
100013
0 1000 2000 3000600
800
100014
0 1000 2000 3000500
1000
150015
0 1000 2000 30000
500
100016
0 1000 2000 30000
500
100017
0 1000 2000 3000
600
800
100018
0 1000 2000 3000500
1000
150019
0 1000 2000 3000500
1000
150020
0 1000 2000 30000
50021
0 1000 2000 3000600
800
100022
0 1000 2000 3000500
1000
150023
0 1000 2000 3000200
400
60024
0 1000 2000 3000
600
800
100025
Total Injection History
0 500 1000 1500 2000 2500 30001.35
1.4
1.45
1.5
1.55
1.6
1.65
1.7x 104
Time (days)
stb/
day
Saturation Map (t=1000 days)
Saturation Map (t=2000 days)
Saturation Map (t=3000 days)
GAMS
• General Algebraic Modeling System• Modeling language that allows use of
many different optimization algorithms (linear, nonlinear, mixed integer, etc.)
• Can interface with MS Excel (input and output)
Continuous Gains
• Finds parameters using nonlinear regression, finding the minimum sum of squared errors using CONOPT
• CONOPT is a nonlinear programming (NLP) code that uses a generalized reduced gradient (GRG) algorithm to find a local optimum (it is one of the solvers in GAMS)
• Resulting gains are “messy” – many gains close to but not equal to zero
Total Production Match
0 1000 2000 3000500
1000
15001
0 1000 2000 3000500
1000
15002
0 1000 2000 3000500
1000
15003
0 1000 2000 3000500
1000
15004
0 1000 2000 3000500
1000
15005
0 1000 2000 3000500
1000
15006
0 1000 2000 3000500
1000
15007
0 1000 2000 3000500
1000
15008
0 1000 2000 3000500
1000
15009
0 1000 2000 3000500
1000
150010
0 1000 2000 3000500
1000
150011
0 1000 2000 3000500
1000
150012
0 1000 2000 3000500
1000
150013
0 1000 2000 3000500
1000
150014
0 1000 2000 3000500
1000
150015
0 1000 2000 3000500
1000
150016
0 1000 2000 3000500
1000
150017
0 1000 2000 3000500
1000
150018
0 1000 2000 3000500
1000
150019
0 1000 2000 3000500
1000
150020
stb/
day
Time (days)
Total Production Match
0 500 1000 1500 2000 2500 3000700
720
740
760
780
800
820
840
860
880
900Producer 1
Time (days)
Tota
l Pro
duct
ion
(stb
/day
)
DataModel
Gains and Time Constants
InjectorProducerGain > 0.300.30 > Gain > 0.0
Semicontinuous Gains
1. Fit parameters using continuous gains with CONOPT
2. If fij < tol, then fij=03. If fij < tol, then constrain fij > tol4. Refit with fewer variables (some gains
are fixed) using CONOPT
Gains and Time Constants
InjectorProducerGain > 0.300.30 > Gain > 0.0
Model Fit Comparison
0 2 4 6 8 10 12 14 16 18 200
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
Producer
Ave
rage
Abs
olut
e E
rror
ContinuousSemicontinuous
∑=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−
=tn
kobsk
estk
obsk
t qqqabs
nerr
111
Should We Use SemicontinuousGains?
• Fewer nonzero gains• Runs relatively
quickly• Easily implemented
• Solution is still dependent on starting point (using NLP)
• Solution has slightly greater average error
Advantages Disadvantages
Heterogeneous Case
• Region of 1000 md between upper nine spot patterns
• Region of 500 md between leftmost nine spot patterns
• Permeability of 2 md everywhere else
Saturation Map (t=1000 days)
Saturation Map (t=2000 days)
Saturation Map (t=3000 days)
Continuous Gains and Time Constants
InjectorProducerGain > 0.300.30 > Gain > 0.0
Semicontinuous vs. Continuous
Model Fit Comparison
0 2 4 6 8 10 12 14 16 18 200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Producer
Ave
rage
Abs
olut
e E
rror
ContinuousSemicontinuous
Summary
• Semicontinuous gains provide a simple model without compromising model error
• Can be effective regardless of level of heterogeneity in the reservoir
Future Work
• Weighted history matching• Benchmark for optimal injections• Testing with field cases
The Cost of Errors in Estimates Used in Concept Selection
Chris Jablonowski
Chai Wiboonkij-Arphakul
Mark Neuhold
Department of Petroleum and Geosystems Engineeringand Energy and Earth Resources Graduate Program
Agenda
• Overview and motivation• Framework, model, and procedure• Results• Conclusions• Wrap up
Overview and Motivation
Concept Selection Drives Value Creation
• The ultimate value derived from an asset is largely determined in concept selection, driven by:– The degree of project team integration– The quality of scenario analysis, quantitative
analysis, and optimization• Failures in concept selection also
contribute to differences between estimated and actual (value) for the concept selected
Concept Decisions Are Based on Uncertain Estimates
ReservoirReservoir
•Reserves and Rates
•Rock Properties
•Fluid Properties
FacilitiesFacilities
•Cost & Schedule
•Future Optionality
•Operability
WellsWells
•Cost & Schedule
•Completion Performance
•Reliability
Research Objective
• This research assesses the relative value of accuracy in the estimates used in concept selection – We examine the loss in project value caused
by making concept selection decisions with inaccurate estimates
– A value of information (VOI) framework demonstrates which estimates “matter most”in terms of increasing accuracy
Research Objective:The Value of Information
• Historically, VOI studies have been on valuing the impact of reservoir uncertainty– For example, valuing new seismic data or
additional wells to reduce uncertainty in the reserves estimate
• This work examines other estimates that influence initial facility design and expansion:– Facility costs– Schedule (expansion timing)
Research Objective:The Value of Information
• We ask: how much to uncertainties in schedule estimates matter?
• What impact do over/under estimates in the cost or schedule have on concept selection and project value?
• How do these impacts compare to the impact of reservoir uncertainty?
Framework, Model, and Procedure
Competing Hypotheses: Reservoir Properties
#1#2
Discovery well is a success and results confirm pre-drilling expectations
Appraisal well is a success, but results are contrary to pre-drilling expectations
• For a cost function of the following form:
• What is the value of reducing the uncertainty in estimates of the parameters?
Competing Hypotheses: Facility Costs
Facility CAPEX = b0 + b1cap0eplat
Estimating the Value of Information
Optimize w/ Current
Hypothesis
Optimize w/ Alternate
Hypothesis
Obtain New Information?
Optimize w/ Current
Hypothesis
New Information Supports Current Hypothesis
New Information Supports Alternate Hypothesis
No
Yes
Current Hypothesis Realized
Alternate Hypothesis Realized
Alternate Hypothesis Realized
Current Hypothesis Realized
B
A
C
The Asset Team’s Decision
• The Decision: Must select an initial facility capacity
• Capacity may expand in the future (an option)
• Uncertain variables are as follows:– Reservoir size– Facility cost parameters (initial and
expansion cost)– Timing of expansion
Development Optimization Model
• A Development Optimization Model models the scenarios and to compute project values
• Risk-neutral profit maximizer• One oil reservoir; reservoir variables:
– Ultimate recoverable reserves– Ultimate recovery per well
• Steel piled jacket
Development Optimization Model
• Costs– OPEX– CAPEX (costs: initial facility, facility
expansion, drilling)• Expansion project allowed• Choice variables
– Production rates– Initial capacity and expansion decision– Drilling schedule (well count and timing)
• Optimization model to simulate project outcomes given realizations of the uncertain variables (Monte Carlo)– Uncertain variables are revealed in period 1
• Step 1: Model the decision using the current hypotheses (A)
Estimating the Value of Information
Selecting the Initial Capacity
Cumulative Distribution Function of NPV @ Initial Facility Capacity = 10,000 mbopy
0.0
0.2
0.4
0.6
0.8
1.0
300 500 700 900 1100NPV, $mm
P10 = 635.88P50 = 797.60P90 = 945.01Expected Value = 794.55
P50
P10
P90
Expected Value
Optimal Initial Platform Capacity
700
750
800
850
4000 6000 8000 10000 12000 14000 16000Initial Platform Capacity, mbopy
NPV
, $M
M
FAC 0 = 11000 mbopy
recommend an initial capacity of 11000 mbopy
Selecting the Initial Capacity
• Step 2: Initialize the project with the initial facility capacity determined in Step 1, but reveal the alternate hypothesis (B)
• Step 3: Model the decision using the alternate hypothesis (C)
• If the alternate hypothesis is true, the value of knowing this is equal to C-B
Estimating the Value of Information
Results
Results: Reserves Case I
Std. Dev. (mmbbls) 80 160 240
8 -3.24 -0.32 -3.6424 -5.31 0.00 -3.6440 -5.31 -0.25 -2.79
Mean (mmbbls)
If the team’s current hypothesis for reserves is (80,8)…
…and the alternate hypothesis of (160,24) is true… then the loss in project value is -3.24%
Results: Reserves Case II and III
Std. Dev. (mmbbls) 40 80 120
4 -4.46 -0.13 -0.8512 -4.46 0.00 -0.8520 -1.92 -0.20 0.00
Mean (mmbbls)
Std. Dev. (mmbbls) 10 20 30
1 -4.27 0.00 -1.033 -4.27 0.00 -1.035 -6.68 0.00 -1.03
Mean (mmbbls)
Case II
Case III
Results: Facility Cost (b0)
Std. Dev. (mmbbls) -10 20 30
2.5 -1.37 -0.19 -0.205.0 -2.86 0.00 -1.36
10.0 -0.36 -0.58 -1.40
Mean (mmbbls)
Facility CAPEX = b0 + b1cap0eplat
Results: Facility Cost (b1)
Std. Dev. (mmbbls) 0.00075 0.001 0.002
0.000175 -1.80 -0.24 -0.360.000200 -0.76 0.00 -0.400.000250 -1.23 -2.36 -0.52
Mean (mmbbls)
Facility CAPEX = b0 + b1cap0eplat
Results: Facility Cost (eplat)
Std. Dev. (mmbbls) 2.0 2.5 2.7
0.05 -11.48 -2.21 -0.760.10 -4.04 0.00 -0.390.20 -6.59 -0.35 -1.11
Mean (mmbbls)
Facility CAPEX = b0 + b1cap0eplat
Results: Expansion Timing
Alternate Hypothesis 2 3 5 7
Year 2 0.00 -4.17 -4.29 -4.53Year 3 -1.31 0.00 -1.08 -1.30Year 5 -2.36 -2.00 0.00 -2.48Year 7 -2.30 -2.71 -2.48 0.00
Expansion Year
Conclusions
Conclusions (1)
• The model and general approach used in this study demonstrate a transparent and consistent means to assess the cost of inaccurate estimates across different project disciplines
• The cost of erroneous estimates for initial costs, expansion costs, and expansion timing are comparable to the cost of erroneous reserve estimates– Inaccurate cost and schedule estimates do
more than irritate—they destroy project value
Conclusions (2)
• The cost of underestimating expected reserve volume tends to be larger than the cost of overestimating
• Aggressive cost estimates are more destructive to project value than conservative estimates
• Erroneous cost estimates, whether aggressive or conservative, create an incentive to over-invest in initial facility capacity
Conclusions (3)
• Conservative schedule estimates for the timing of expansion are generally more destructive to project value than aggressive estimates
• Generally, the cost of erroneous estimates is more sensitive to errors in estimates of means than errors in estimates of variances
Wrap Up
Future Work
• Integrate a reservoir model• Expand investigation to include uncertainty in:
– Recovery per well– Wells CAPEX – Facility and wells reliability– Functional form of cost functions– Expansion constraints– Schedule assumptions (initial install and expansion)– Commodity price– Extreme events (hurricanes)
Acknowledgements
• The Cockrell School of Engineering
• The Jackson School of Geosciences
• Tim Taylor (UT-Austin) and Bill Lamport (OPE, Inc.) for comments and suggestions
• Simon Richards (Editor, SPE Projects, Facilities and Construction), and two anonymous reviewers
A Discrete-Time Approach for Modeling Two-Factor Mean-
Reverting Stochastic Processes
W.J. Hahn and Jim Dyer
Modeling UncertaintyThe most common stochastic process model is a Geometric Brownian Motion (GBM)
Assumed in the Black-Scholes-Merton formula and in the equivalent discrete-time binomial lattice of Cox, Ross, & Rubinstein (CRR)
Ten Sample Paths for a GBM Process
15
20
25
30
35
40
0 1 2 3 4 5 6 7 8 9Time Period
Valu
e ($
)
90% C.I.
10% C.I.
Lognormal D
ist'n
Select probability (p) and up and down moves (u,d) so that expected value and variance equal a lognormal distribution
Lattice can also be represented as a binomial tree
For risk-neutral process: ,
Lognormal D
istribution ⇒
Discrete Models: Binomial Lattice
udeu t 1 and == Δσ
dudrp
−−+
= 1
pV 0
Time 0 1 2 3
p
p
pp
p1-p
1-p
1-p
1-p
1-p
1-p
V u
V d
V ddV ddd
V d
V u
V uuuV uu
⇒
V ddd
V d
V d
V d
V uuu
p
p
p
p
p
p
p
1-p
1-p
1-p
1-p
1-p
1-p
1-p
V u
V u
V u
V 0
V u
V uu
V 0
V 0
V d
V dd
Binomial Tree ImplementationImplementation of a tree in DPL: Up
2.86804 .462 [2.86804]
Down 1.28869 .538
[1.28869]
T3 Up .462
[2.01863]
Up 1.28869 .462
[1.28869]
Down 0.579048 .538
[0.579048]
T3 Down .538
[0.907029]
T2 Up
.462 [1.42079]
Up 1.28869 .462
[1.28869]
Down 0.579048 .538
[0.579048]
T3 Up .462
[0.907029]
Up 0.579048 .462
[0.579048]
Down 0.260183 .538
[0.260183]
T3 Down .538
[0.407555]
T2 Down .538
[0.6384]
T1 [1]
Up
Down
Up
Down
T3
Up
Down
T2T1
Abandon -Invest
Abandon -Invest+T1 Payoff
Abandon -Invest+T1,T2 Payoffs
Up -Invest+T1,T2,T3 Payoffs
Down -Invest+T1,T2,T3 Payoffs
Continue
T3 PriceMove
Up
Down
T3 Option
Continue
T2 PriceMove
Up
Down
T2 Option
Continue
T1 PriceMove
T1 Option
Add decision nodes to model flexibility
LimitationsGBM’s are convenient for modeling, but what if the value of the underlying asset does not follow a GBM?
eg,. Mean-Reverting commodity price uncertainty
Can result in significant errors in valuation
GBM and M-R Processes with σ=20%
15
20
25
30
35
40
1 2 3 4 5 6 7 8 9 10Time Period
Oil
Pric
e ($
)
GBM - MeanGBM - 10% CIGBM - 90% CIM-R MeanM-R - 10% CIM-R - 90% CI
Modeling Mean-Reverting ProcessesTrinomial Trees
Computationally intensive and often difficult to implement
Monte Carlo SimulationCan model any process, but can be difficult to model options
Proposed alternative: a General Binomial Approximation
Similar to conventional binomial approachNodes still reconnect (can still represent with lattice or tree), but probabilities can be different for each node
Allows drift rate to depend on current value of variable
Invalid probabilities (>1 or <0) of up and down moves censored to [0,1]
Convergence obtained by reducing time increments
General Binomial ApproximationY is a variable (e.g., log of price) that follows a general process with drift function and variance function
Y is to be modeled in discrete time increments of length Δt
Binomial approximation:
where is censored between [0,1]
),( tYtYYt σΔ+≡+
),( tYtYYt σΔ−≡−
),(2),(
21
tYtYtqt σ
μΔ+≡
),( tYσ),( tYμ
Y
tΔ
tq−1
tq
Implementation for a simple M-R processExample implementation for a simple one factor mean-reverting process:
= , =
Proof of convergence demonstrated by Nelson & Ramaswamy (1990)
Dealing with Large TreesFor long-term problems, or high degree of accuracy, DPL or recursive tree algorithm limited to ~ 30 periods
Address by switching to a recombining lattice algorithm
Advantages compared to the other methodsAbility to handle multiple/complex options
More straightforward computationally relative to Hull’s trinomial tree method
Can also extend to a two factor stochastic process with multivariate binomial approximation
)( tYY −κ σ),( tYμ ),( tYσ
Implementation for two-factor processTwo-factor process (Schwartz and Smith, 2000):
Variation in long-term mean follows an ABM process
Short-term deviation from long-term mean follows a simple mean-reverting process
Represent Up and Down moves with a four-branch node:
Cannot directly censor invalid probabilities as before
ξ+Δξ,χ+Δχ
ξ−Δξ,χ−Δχ
ξ−Δξ,χ+Δχ
ξ+Δξ,χ−Δχpdu
pdd
puupud
ξχ
tttY χξ +=
puu/p χ+Δχ
Δχ
p ξ+Δξ pud/p χ−Δχ
ξ
1-p ξ−Δξ pdu/(1-p) χ+Δχ
Δχ
pdd/(1-p) χ−Δχ
Implementation for two-factor processUse Bayes’ Rule break four-branch node for joint process into a marginal-conditional sequence (sequence of two chance nodes)
Marginal for : Conventional binomial approximation
Conditional for : General Binomial approximation
Censor as necessary
Diverge
Revert
Up
Down
DeviationL-T Mean
Expanded View
Deviation
L-T Mean
ξχ
χ
ξ
Implementation for two-factor processImplement binomial approximation in decision tree:
To obtain convergence, we again reduce the time increment (and increase number of nodes)
Can use a 2-D lattice when number of periods becomes largeWe now have a discrete-time binomial model of a two-factor mean-reverting process
Allows us to use a more robust commodity price model in ROV problemsFacilitates relatively straightforward evaluation of options
Yes -Invest
Yes Payoff_1
Yes Payoff_2
DivergePayoff
Revert Payoff
Up
Down
X3
No
Mean 3Diverge
Revert
Abandon 3
Up
Down
X2
No
Mean 2 Diverge
Revert
Abandon 2
Up
Down
X1
No
Mean 1
Abandon 1
ApplicationIllustrative Real Options Applications
Section 7 of Schwartz & Smith
Long-term investmentOption to develop property for $800,000
3-year construction lag before oil production begins
Production starts at 5,000 barrels/year, with exponential decline of 5% per year
Short-term investmentOption to develop property for $40,000
Oil production begins immediately at 1,000 barrels/year, with 40% exponential decline per year
Value using risk-neutral techniques, given two-factor parameter estimates in the paper
5% risk-free discount rate
Value Functions for Optimal SolutionsSchwartz and Smith solve for the optimal exercise strategy and value over a range of possible prices
Results are depicted using 3-D value functions
Planar surfaces with coordinates for project value, equilibrium price, and deviation
Upper surface is value function with optimal exercise of option
Lower surface is value function with immediate exercise
Dark area is where two functions coincide
Solution by Dynamic ProgrammingSchwartz and Smith obtain these results assuming infinite horizon (i.e., perpetual option), using a dynamic programming approach formulated as a linear program and solved using CPLEXWe investigate whether our discrete two-factor model can be implemented in a lattice algorithm to obtain similar results when we use a long (i.e., 25 year) horizon.
If so, this model can then be used to solve finite-horizon problems, as indicated by Schwartz & Smith
We use an algorithm similar to the two-dimensional lattice model described by Clewlow and Strickland for modeling two GBM processes
Model a GBM for the equilibrium levelModel a one-factor mean-reverting Ornstein-Uhlenbeck process for the deviation, utilizing the general binomial approach and Bayes Rule as described earlierResults shown on following slides
Solution by Multivariate Lattice ModelShort-term Investment, Immediate Exercise
Approximately matches lower value function in Figure 5a of S&S
$8 $10 $12 $14 $17 $21 $25 $31 $37 $45
-50%
-35%
-14%12%
47% -$30,000
-$10,000
$10,000
$30,000
$50,000
$70,000
$90,000
Equilibrium Price ($/bbl)
Deviation
Short-Term Investment
Solution by Multivariate Lattice ModelShort-term Investment, Optimal Exercise
Approximately matches upper value function in Figure 5a of S&S
$8 $10 $12 $14 $17 $21 $25 $31 $37 $45
-50%
-35%
-14%12%
47% -$30,000
-$10,000
$10,000
$30,000
$50,000
$70,000
$90,000
Equilibrium Price ($/bbl)
Deviation
Short-Term Investment
Solution by Multivariate Lattice ModelIn addition to the general position of the two value functions, the following table shows general agreement for the regions over which the two value functions coincide
Corresponds to dark shaded area in Figure 5a of S&S
Difference (Option Value)8.00$ 8.81$ 9.70$ 10.68$ 11.77$ 12.96$ 14.27$ 15.71$ 17.31$ 19.06$ 20.99$ 23.11$ 25.45$ 28.03$ 30.87$ 34.00$ 37.44$ 41.23$ 45.40$ 50.00$
80% 19747 17968 16046 14077 11962 9786 7532 5182 2869 441 0 0 0 0 0 0 0 0 0 068% 20138 18406 16527 14614 12554 10436 8261 5946 3752 1412 0 0 0 0 0 0 0 0 0 057% 20493 18801 16970 15081 13084 11000 8884 6672 4448 2303 81 0 0 0 0 0 0 0 0 047% 20856 19202 17414 15565 13625 11592 9542 7404 5260 3189 1095 0 0 0 0 0 0 0 0 037% 21213 19577 17837 16027 14116 12160 10125 8082 6013 3887 2024 48 0 0 0 0 0 0 0 028% 21559 19967 18244 16499 14635 12728 10775 8752 6781 4763 2906 1190 0 0 0 0 0 0 0 020% 21884 20332 18650 16945 15128 13254 11380 9382 7510 5591 3689 2169 558 0 0 0 0 0 0 012% 22189 20670 19028 17349 15580 13753 11905 10019 8139 6329 4559 2961 1752 514 0 0 0 0 0 05% 22503 21011 19415 17759 16048 14264 12464 10663 8818 7111 5440 3849 2790 1695 966 262 0 0 0 0
-2% 22809 21336 19766 18161 16459 14748 12981 11196 9482 7697 6194 4702 3417 2678 1902 1496 1124 761 427 62-8% 23104 21668 20122 18568 16910 15243 13531 11784 10144 8456 6993 5615 4425 3425 2771 2345 2013 1919 1819 1727
-14% 23370 21969 20455 18926 17312 15653 14024 12301 10690 9131 7553 6417 5295 4438 3834 3265 2869 2612 2460 2352-20% 23636 22264 20781 19282 17708 16099 14497 12862 11275 9758 8377 7074 6060 5235 4531 4221 3906 3735 3590 3439-25% 23908 22550 21112 19629 18099 16533 14947 13409 11854 10398 9122 7826 6955 6096 5417 5024 4632 4494 4426 4401-30% 24170 22837 21411 19983 18466 16952 15421 13867 12432 10956 9683 8583 7516 6878 6272 5845 5564 5288 5086 4890-35% 24421 23123 21722 20335 18853 17378 15902 14365 13020 11609 10399 9319 8293 7635 6937 6618 6336 6179 6135 6099-39% 24643 23372 22002 20624 19189 17721 16289 14823 13414 12164 10917 9909 9045 8253 7779 7291 6985 6738 6528 6458-43% 24875 23629 22288 20934 19537 18105 16715 15297 13957 12711 11536 10539 9602 8970 8419 8048 7814 7616 7537 7455-47% 25107 23866 22561 21227 19847 18468 17073 15736 14436 13169 12140 11093 10327 9638 9079 8665 8369 8173 8058 8022-50% 25333 24122 22821 21540 20185 18839 17506 16159 14922 13708 12608 11716 10882 10246 9761 9329 9103 8875 8712 8566
Solution by Multivariate Lattice ModelLong-term Investment, Immediate Exercise
Approximately matches lower value function in Figure 5b of S&S
$8 $10 $12 $14 $17 $21 $25 $31 $37 $45
-50%
-35%-14%
12%47% -$500,000
$0
$500,000
$1,000,000
$1,500,000
$2,000,000
Equilibrium Price ($/bbl)
Deviation
Long-Term Investment
Solution by Multivariate Lattice ModelLong-term Investment, Optimal Exercise
Approximately matches upper value function in Figure 5b of S&S
$8 $10 $12 $14 $17 $21 $25 $31 $37 $45
-50%
-35%-14%
12%47% -$500,000
$0
$500,000
$1,000,000
$1,500,000
$2,000,000
Equilibrium Price ($/bbl)
Deviation
Long-Term Investment
Solution by Multivariate Lattice ModelAgain, in addition to the general position of the two value functions, the following table shows general agreement for the regions over which the two value functions coincide
Corresponds to dark shaded area in Figure 5b of S&SDifference (Option Value)
8.00$ 8.81$ 9.70$ 10.68$ 11.77$ 12.96$ 14.27$ 15.71$ 17.31$ 19.06$ 20.99$ 23.11$ 25.45$ 28.03$ 30.87$ 34.00$ 37.44$ 41.23$ 45.40$ 50.00$ 80% 389079 355917 320699 284166 247357 208468 171159 133022 97053 63956 34791 12769 0 0 0 0 0 0 0 068% 389257 356110 320921 284414 247623 208764 171457 133403 97456 64295 35132 12916 0 0 0 0 0 0 0 057% 389423 356274 321117 284628 247836 209045 171684 133689 97848 64573 35470 13053 0 0 0 0 0 0 0 047% 389598 356472 321329 284864 248095 209343 172024 134021 98256 65012 35796 13283 0 0 0 0 0 0 0 037% 389766 356670 321530 285082 248354 209587 172368 134344 98545 65464 36024 13524 0 0 0 0 0 0 0 028% 389941 356867 321748 285322 248631 209867 172697 134731 98915 65942 36486 13772 0 0 0 0 0 0 0 020% 390113 357046 321957 285554 248870 210150 172954 135070 99311 66276 36995 13935 0 0 0 0 0 0 0 012% 390280 357217 322155 285772 249093 210439 173213 135366 99715 66599 37530 14340 0 0 0 0 0 0 0 05% 390453 357418 322366 286007 249357 210728 173570 135704 100100 67064 37968 14899 0 0 0 0 0 0 0 0
-2% 390618 357611 322565 286219 249613 210964 173899 136028 100379 67517 38228 15449 150 0 0 0 0 0 0 0-8% 390793 357804 322782 286462 249885 211248 174212 136415 100765 67967 38708 15987 807 0 0 0 0 0 0 0
-14% 390956 357968 322977 286677 250098 211520 174431 136709 101143 68234 39228 16259 1293 0 0 0 0 0 0 0-20% 391125 358150 323180 286902 250337 211813 174729 137023 101555 68616 39726 16452 1494 0 0 0 0 0 0 0-25% 391294 358348 323383 287127 250597 212078 175083 137350 101890 69084 40109 16700 1628 0 0 0 0 0 0 0-30% 391460 358539 323587 287345 250858 212326 175406 137696 102196 69530 40336 16928 1720 0 0 0 0 0 0 0-35% 391632 358725 323801 287586 251118 212611 175697 138069 102590 69949 40670 17138 1994 0 0 0 0 0 0 0-39% 391792 358883 323990 287791 251321 212880 175910 138343 102963 70182 40987 17240 2191 0 0 0 0 0 0 0-43% 391960 359072 324193 288018 251568 213169 176233 138659 103359 70514 41301 17385 2339 0 0 0 0 0 0 0-47% 392121 359263 324386 288228 251818 213405 176569 138970 103644 70862 41504 17542 2428 0 0 0 0 0 0 0-50% 392290 359454 324596 288458 252085 213671 176886 139339 103955 71219 41719 17695 2521 0 0 0 0 0 0 0
Use of Multivariate Lattice ModelWe implemented the multivariate lattice model in a relatively simple Microsoft Excel/Visual Basic framework
Relatively simple and efficient algorithm:Programmed in Excel with VBA (<100 lines of code)
To obtain the value functions on the previous slides, the equilibrium price and deviation were discretized into 20 steps each for a total state space of 400 different states
Algorithm solved the entire problem (400 solutions for value of option with 25-year life) in less than 5 minutes
This methodology can be applied to many different types of options with different types of underlying assets and processes
Value switching option with two underlying assets that each follow a one-factor mean-reverting price process
Value of Information in E&PValue of Information in E&P
Namhong Min, Bob Gilbert, and Larry W. LakeNamhong Min, Bob Gilbert, and Larry W. Lake
Background
Economic Decision Making
Technical Uncertainty
Development Options
Exploration DataSpatial VariabilityUpscalingReservoir Simulation
Net Present ValueVolatilityReal Option Valuation
Value of Information
Goals
1. Develop practical methods for value of information analyses
2. Integrate technical models with economic decision making
3. Identify where best to expend research resources in this area
Outline
1. Introduction 2. Theoretical Development3. Practical Implementation4. Illustrative Examples
1. Introduction
Decision Tree
Start Phase 2 (Infill Wells)
10th Year
in First Year
Never
Phase I Development
100 acre Well Spacing
Reservoir Scenario Reservoir Performance → Expected Economic Performance
Discounted Cash Flow Analysis
h, φ, Soi, k, s
Expected Production Rate, qosciDecay Constant, λLimiting Production Rate, qLIM
Development CostOperating CostOil ValueDiscount RateEconomic Limit
0
10
20
30
40
50
60
70
80
90
0 5 10 15 20
Time (year)
Pro
duct
ion
Rat
e (S
TB/d
ay)
-2
-10
1
23
4
5
6
0 5 10 15 20
BenefitTotal CostNet Profit
Pre
sent
Val
ue ($
MM
)Time (year)
Decision Criteria
Decision Criteria
1016
2228
34
12
46
7
0
0.05
0.1
0.15
LN(k)Porosity
(%)
PMF
Modeling UncertaintyAverage Reservoir Properties
Time
Production Rate
Average Well
Individual Wells
Modeling UncertaintyWell Variability
Start Phase 2 (Infill Wells)
10th Year
in First Year
Never
$ - 5,561
$ 152,138
$ - 892,133
Expected Net Profit (NPV/100 acres)
Initial Decision
Value of Information
Likelihood Function
Time
Production Rate
0 110
1622
2834
12
46
7
0
0.05
0.1
0.15
0.2
LN(k)Porosity
(%)
PMF
1016
2228
34
12
46
7
0
0.05
0.1
0.15
0.2
LN(k)Porosity
(%)
PMF
-2.0
-1.0
0.0
1.0
2.0
-2.0
-1.0
0.0
1.0
2.0 Likelihood
NormalizedPeak Production Rate
NormalizedDecay Constant
Value of Information
Oil Value ($/STB)
VI($ NPV/100 acres)
0
100,000
200,000
300,000
400,000
500,000
600,000
0 10 20 30 40 50 60 70 80 90 100 110
Prior Decision No Go 2nd Well at Year 2
Options somewhere between No Go & Year 2
2. Theoretical Development
Bayes’ TheoremThe Basis for VOI Analyses
VI($ NPV/100 acres)
P State of Nature i Information( )=P Information State of Nature i( )P State of Nature i( )
P Information State of Nature i( )P State of Nature i( )⎡⎣
⎤⎦
all i∑
“Updated Probability”(what we want)
“Likelihood Function” (what does the available information say)
“Prior Probability” (what do we know with no information)
Prior Probability Is Foundation
Objective: Prior probabilities should be unbiased (that is, include “no”information). If they are unintentionally biased, then everything else that goes into the decision is also biased.
Conventional ApproachBernoulli’s Principle of Insufficient Reason
If a decision maker is completely ignorant as to which state of nature will occur, then the decision maker should behave as if the states are random (equally likely).
Conventional Approach Is Pervasive
1. Used implicitly in all conventional statistical methods (such as maximum likelihood)
2. Used implicitly or explicitly in assigning probabilities in all formal decision analyses
3. Used explicitly in the Principle of Maximum Entropy for Information
ExampleConventional Approach for Priors
Option A
Option B
State 1
State 2
State 3
State 1
State 2
State 3
Utility
0
1
2
3
2
0
ExampleConventional Approach for Priors
Utility
0
1
2
3
2
0
0
0.2
0.4
0.6
0.8
1
1 2 3
State
Pro
babi
lity
1/3 1/3 1/3
Option A
Option B
State 1
State 2
State 3
State 1
State 2
State 3
ExampleConventional Approach for Priors
We Unintentionally Insert Bias in Decision.
0
0.2
0.4
0.6
0.8
1
A B
Preferred Option
Pro
babi
lity
1/3
2/3Option B
Option A
State 2
State 3
State 1
Utility
Option B
Option A
Option B
Option A
3
0
2
1
0
2
CPARM Conclusion
Conventional approach for non-informative priors is wrong.
Intuitive Support for CPARM Conclusion
1. It is not possible to apply the conventional approach consistently.
Inconsistency in Conventional Approach
Prob
abili
ty
Den
sity
Prob
abili
ty
Den
sity
Permeability
Log of Permeability
Which distribution is non-informative?
OR
Prob
abili
ty
Den
sity
Permeability
Inconsistency in Conventional ApproachTo which input parameters do we apply it?
k1 k2
Flow
keff = 1/(1/k1 + 1/k2)
Inconsistency in Conventional ApproachTo which input parameters do we apply it?
12
1
20
0.1
0.2
0.3
0.4
0.5
k1
k2
Prob
abili
ty
0
0.1
0.2
0.3
0.4
0.5
0.5 0.67 1
keff
Prob
abili
ty
OR
0
0.1
0.2
0.3
0.4
0.5
0.5 0.67 1
keff
Prob
abili
ty
Practical Support for CPARM Conclusion
Applying Principle of Insufficient Reason to input does necessarily not produce “insufficient reason”in the output.
Journel and Deutsch Example
Input Permeability Field
Cumulative Production
High Entropy Low Entropy
Low Entropy High Entropy
Theoretical Support for CPARM Conclusion
Decision theorists (such as Luce and Raiffa) show that Bernoulli’s Principle is not consistent with the fundamental axioms of decision theory.
Luce and Raiffa Example
S1 S2
A1 11 0A2 0 10
States of Nature
DecisionAlternatives
Consequences (Utilities)
Luce and Raiffa Example
S1 S2
0101/2
A2 05.55.0
A1 11
1/2
States of Nature
DecisionAlternatives
Probabilities
ExpectedUtility
Non-Informative Probabilities
Luce and Raiffa Example
S1 S2
0101/2
A2 05.55.0
A1 11
1/2Probabilities
ExpectedUtility
S1 S2a S2b S2c S2d S2e
0101/6
00010 101/6 1/6
101/6
0101/6
1.88.3A2 0
A1 11
1/6Probabilities
ExpectedUtility
Luce and Raiffa ExampleAdding Duplicate States of Nature Should
Not Change Decision
S1 S2
0101/2
A2 05.55.0
A1 11
1/2Probabilities
ExpectedUtility
S1 S2a S2b S2c S2d S2e
0101/6
00010 101/6 1/6
101/6
0101/6
1.88.3A2 0
A1 11
1/6Probabilities
ExpectedUtility
Luce and Raiffa ExampleS1 S2
0101/2
A2 05.55.0
A1 11
1/2Probabilities
ExpectedUtility
S1 S2a S2b
0A2 0 10 10 6.7
91/3
3.70
01/3
6.0A3 9
A1 11
1/3Probabilities
ExpectedUtility
Luce and Raiffa ExampleAdding Alternative Should Not Change
Non-Optimal Decision to OptimalS1 S2
0101/2
A2 05.55.0
A1 11
1/2Probabilities
ExpectedUtility
S1 S2a S2b
0A2 0 10 10 6.7
91/3
3.70
01/3
6.0A3 9
A1 11
1/3Probabilities
ExpectedUtility
CPARM Approach
Convention: If a decision maker is completely ignorant as to which state of nature will occur, then the decision maker should behave as if thestates are random (equally likely).
CPARM: If a decision maker is completely ignorant as to which state of nature will occur, then the decision maker should behave as if thepreferred decision alternative is random.
An Unbiased Starting Point
0
0.2
0.4
0.6
0.8
1
A B
Preferred Option
Pro
babi
lity
1/2 1/2
0
0.2
0.4
0.6
0.8
1
1 2 3
State
Pro
babi
lity
1/4 1/4
1/2
Option B
Option A
State 2
State 3
State 1
Utility
Option B
Option A
Option B
Option A
3
0
2
1
0
2
CPARM Approach Is SoundAdding Duplicate States of Nature Should
Not Change Decision
S1 S2
0101/2
A2 05.55.0
A1 11
1/2
ExpectedUtility
S1 S2a S2b S2c S2d S2e
010
1/10
00010 10
1/10 1/1010
1/10
010
1/10
5.55.0A2 0
A1 11
1/2Probabilities
ExpectedUtility
Probabilities
CPARM Approach Is SoundAdding Alternative Should Not Change
Non-Optimal Decision to Optimal
S1 S2
0101/2
A2 05.55.0
A1 11
1/2Probabilities
ExpectedUtility
S1 S2a S2b
0A2 0 10 10 5.0
91/4
5.50
01/4
6.8A3 9
A1 11
1/2Probabilities
ExpectedUtility
3. Practical Implementation
S1 S2a S2b S2c S2d S2e
0A2 0 10 10 10 10 10
9
0000
0 0 0 0A3 9
A1 11
A1>A3>A2 A2>A3>A1 A2>A1=A3
Implementation of CPARM Approach
Implementation of CPARM Approach
A3A1> A2>
A1A3>
A1A3=
A2>
1/2
1/2 1/4
1/4
1/4
1/4
1/2 1/2
Implementation of CPARM Approach
A3A1> A2>
A1A3>
A1A3=A2>
21
41
41
Probability
P(S2a)
P(S2b)+P(S2c)+P(S2d)+P(S2e)
P(S1)
Apply Principle of insufficient reason to lumped states
P(S2b)=P(S2c)=P(S2d)=P(S2e)=1/16
Implementation of CPARM Approach
S1 S2a S2b S2c S2d S2e
0 0100
1/16
A2 0 10 10 10 10 5.0A3 9 9 0 0 0 6.8
1/2 1/4 1/16 1/16 1/16
0 5.500A1 11
Probabilities
ExpectedUtility
S1 S2a S2b
0A2 0 10 10 5.0
91/4
5.50
01/4
6.8A3 9
A1 11
1/2Probabilities
ExpectedUtility
4. Illustrative Examples
• Decision criteria: Maximizing the Expected Value of the Net Profit ($ Net Present Value)
• Decision: The Number of Wells/Unit Production Time Schedule(12241 Alternatives)
Unit 1 Unit 2Crossflow
Export Line
Unit 1 Unit 2
Reservoir Heterogeneity Example
Unit 1 Unit 2Communication between units
Export Line
Unit 1 Unit 2
Unit 1 Unit 2Drainage area (acre) 5000 2500Permeability (md) 100 50Oil value ($/STB) 40 25Development cost ($ MM/well) 20 20Facility cost ($ MM) 10 10
qLim = 20000 STB/day
Net pay = 300 ftPorosity = 20 %
Reservoir Heterogeneity Example
Transmissibility(STB/psi/day)
0
10
20
Uncertainty Modeling
• Uncertain Variable- Transmissibility between two units
• Transmissibility- Degree of the connectivity- Unit: STB/psi/day
• 11 scenarios
90
100
Reservoir Heterogeneity Example
Decision Tree Transmissibility(STB/psi/day)
Option 1: 1 well for each unit and 1 year production
Option 12241: No go
Year 1 2 3 4 5 6 7 8 9 10Unit 1Unit 2
4 wells3 wells
Option6819
0
10
20
90
100
Reservoir Heterogeneity Example
Non-Informative Probability DistributionsTransmissibility(STB/psi/day)
Best Option
0102030405060708090100
6870
6874
6877
6822
12200 Unit 1 4 wellsUnit 2
1 2 3 4 5 6 7 8 9 10
Unit 1 4 wellsUnit 2 4 wells
Unit 1 4 wellsUnit 2 3 wells
Unit 1 4 wellsUnit 2 4 wells
Unit 1 4 wellsUnit 2 4 wells
Reservoir Heterogeneity Example
Non-Informative Probability Distribution
0
0.1
0.2
0.3
0.4
0.5
0 10 20 30 40 50 60 70 80 90 100
Transmissibility (STB/psi/day)
Probability
Reservoir Heterogeneity Example
VPI
$ 7.8 MM
$ 12.9 MM
Convention
Decision
CPARM
1 2 3 4 5 6 7 8 9 10
Unit 1: 4 wells
Unit 2: 4 wells
Decision Analysis Results - Low Well Cost
1 2 3 4 5 6 7 8 9 10
Unit 1: 4 wells
Unit 2: 4 wells
Reservoir Heterogeneity Example
VPI
$ 5.8 MM
$ 29.4 MM
Convention
Decision
CPARM
Decision Analysis Results - Medium Well Cost
1 2 3 4 5 6 7 8 9 10
Unit 1: 4 wells
Reservoir Heterogeneity Example
1 2 3 4 5 6 7 8 9 10
Unit 1: 4 wells
Unit 2: 1 well
VPI
$ 0.4 MM
$ 2.2 MM
Convention
Decision
CPARM
1 2 3 4 5 6 7 8 9 10
Unit 1: 4 wells
Decision Analysis Results - High Well Cost
1 2 3 4 5 6 7 8 9 10
Unit 1: 4 wells
Reservoir Heterogeneity Example
Decision Analysis Results - Well Cost of Unit 2
0
10
20
30
40
0 50 100 150
ConventionalCPARM
Well Cost for Unit 2 ($MM/Well)
VPI($ MM)
Reservoir Heterogeneity Example
Value of Information Analysis
• Information- Inferred from communication between units- T*: estimated value of T- Imperfect information
P(T* 20|T 20) = 0.8, P(T*>20|T 20) = 0.2P(T* 20|T>20) = 0.3, P(T*>20|T>20) = 0.7
Reservoir Heterogeneity Example
Value of Information Analysis
T* 20
T* > 20
Purchase Info.
No Info.
Option 1Option 2
Option 12241
Option 1Option 2
Option 12241
Option 1Option 2
Option 12241
T=0
T=10
T=100
Transmissibility(STB/psi/day)
Reservoir Heterogeneity Example
Value ofInformation
$ 1.8 MM
$ 4.2 MM
Convention
Decision
CPARM
1 2 3 4 5 6 7 8 9 10
Unit 1: 4 wells
Unit 2: 3 wells
Value of Information AnalysisReservoir Heterogeneity Example
1 2 3 4 5 6 7 8 9 10
Unit 1: 4 wells
Unit 2: 4 wells
• Objective: Maximize Net Profit ($ NPV)
• Decision: The Number of Wells/Unit Production Time Schedule
(48,000 Alternatives)
Unit 2Unit 1
Gathering Center
Export Line
Landmark Example
Porosity
20 %
30 %
40 %
50 %
60 %
74 (md)
330 (md)
1500 (md)
6600 (md)
16 (md) 50 ($/STB)
30 ($/STB)
70 ($/STB)
Medium
Low
High
Permeability Oil Value Cost
$ MM Dev. Cost Opr. Cost
Low 6 0.3
Medium 18 0.8
High 24 1.3
Uncertainty Modeling50625 branches considering both units
Landmark Example
Landmark ExampleNon-Informative Priors
Unit 1 Unit 2
16743301480
663420
3040
50600
0.2
0.4
0.6
φ (%)
k (md)
412
3390245
1525
3545
550
0.2
0.4
0.6
k (md)φ (%)
0.1820.280
0.538
00.10.20.30.40.50.60.7
Low Medium High
PMF
Cost
0.2620.350 0.389
00.10.20.30.40.50.60.7
30 50 70
PMF
Oil Price
0.493
0.259 0.247
00.10.20.30.40.50.60.7
30 50 70
PMF
Oil Price
0.433
0.300 0.267
00.10.20.30.40.50.60.7
Low Medium High
PMF
Cost
Implications of CPARM Approach for Non-Informative Priors
• The proposed approach provides truly non-informative prior probability distributions for the purposes of decision making.
• The exercise of establishing non-informative prior probabilities is relatively straightforward and insightful.
• This approach provides a rational, objective and consistent starting point for assessing probabilities without or before soliciting subjective information.
• Any and all additional information about input parameters can be included through Bayes’ Theorem.
• For the same variable, the non-informative probability distribution that applies to one decision may be different than the non-informative probability distribution that applies to another decision.
Summary1. Method is Defensible, Consistent and Objective
• Provides Rational Basis as Starting Point for Establishing Probability Distributions
• Captures Significance of Extremes in and Complex Relationships between Input Variables
2. Method is PracticalReadily Implemented on Problems with 10,000’s of Decision Alternatives and Possible Outcomes
3. Future Work• Apply to Real Examples• Automate Method• Develop/Calibrate Realistic Likelihood Functions based
on Exploration Information and Well Performance
Calibration of likelihood of HC recovery from reservoir datasets
Aviral SharmaSanjay Srinivasan
Larry Lake
Research Objectives• Explore if data analysis could yield:
a fast proxy for assessing reservoir performance
• Explore if analysis could yield:
{ }.....wells,geology,|factorrecovery lhd
⎭⎬⎫
⎩⎨⎧
..... char., prod. ty,permeabili porosity model, aldeposition|analog geological
P
Precursor to mp statistics based reservoir modeling
Toris Data set
• A fairly complete set of reservoir, fluid and operational characteristics of over 1300 reservoirs
• Data includes:– Information on wells– Reservoir properties– Geological descriptors– Reservoir fluid properties– Production characteristics
TORIS DATA
Inference of the likelihoodThe following simplification is assumed:Data groups:
g – variables that relate to the geologic model
f – variables that relate to the fluid properties
w – variables that relate to the wells
The groups are constructed such that g, f and ware orthogonal to each other
• By Bayes’ Rule:
Inference of the likelihood
)(
)()()()(
)|()|()|(),,(
)()|,,(),,|(
rfP
rfPfPwPgP
rffPrfwPrfgPfwgP
rfPrffwgPfwgrflhd
fwg ⋅⋅⋅=
⋅⋅⋅⋅⋅=
⋅=
πππ
πg is the relative updating of the prior geological model due to aparticular observation of rf and similarly for πw and πf
Due to the prior assumption of independence between g, w and f
Calculating likelihoodsTasks• Constructing orthogonal projections of data in
order to arrive at groups g, w and f– A physics based classification scheme– Statistical classification schemes (PCA, k-mean
declustering etc.)• Calibration of likelihood functions• Checking the validity of the calibrated functions
(jack-knife analysis)
Principal Component AnalysisPrincipal Component Analysis
• Lumps the correlated features• Steps involved are:
Calculate the covariance matrixCalculate the eigenvectors and eigenvalues Sort the eigenvalueObtain reduced set of eigenvectors forms
Smith (2002)
PCA• Before doing PCA on the data, box plot has
been shown for 19 variables
• Box plot tells us about the possible outliers in the data set.
• This is one of the way to identify the outliers and range of variables.
+ shows outliers in the features.
Box Plot of 19 VariablesBox Plot of 19 Variables
0 1000 2000 3000 4000 5000 6000 7000H
Phi Soi Sw i
D K P
D ip VD P D S SC B o API
viscocity WellAcres N o.Prod
N o.InjectorQ
W ellD ensity
Values
Col
umn
Num
ber
Standardizing VariableStandardizing Variable
• Since all the variables have different units, we standardized all variables before doing PCA.
• Dividing each column by its std.dev.• We have 24 reservoirs for which all the
nineteen variables are available.
• First 10 PC’s take care of 90% variability in data.
• From the first 10 components, the variables receiving more weights in each PC are highlighted.
Results of the PCA analysis
Data reductionData reduction
•We have19 PCs and the associated variance contribution of each component.
•From the figure, its clear that the first 10 PC takes care of 90% variability in the data set without loss of any significant information.
Principal Components
PCAPCAPC 1 2 3 4 5 6 7 8 9 10
H 0.13798 0.19191 -0.31104 0.32321 -0.1248 0.070608 -0.10145 0.63245 -0.34675 0.20358
Phi 0.32716 0.072613 -0.08188 -0.02506 0.094791 -0.14376 0.024017 -0.23122 0.30747 0.23914
Soi -0.14795 0.51771 0.037901 -0.2563 0.10088 -0.03499 0.069493 0.10372 0.12894 0.19313
Swi 0.14836 -0.51664 -0.03912 0.25751 -0.10641 0.038007 -0.07627 -0.10247 -0.12514 -0.19541
D -0.34852 -0.11762 -0.11813 0.042604 -0.07654 0.008154 0.029467 -0.03892 -0.23621 0.18231
K 0.25145 0.050712 0.09303 -0.46239 -0.09664 -0.00371 -0.11495 0.17266 -0.12997 -0.22988
P -0.35412 -0.11571 -0.10613 -0.03611 0.067951 -0.03453 -0.11569 -0.03584 -0.04518 -0.0045
Dip 0.13684 0.24203 -0.23124 0.40741 0.11731 0.061465 0.026238 -0.53411 -0.21281 0.22058
VDP -0.22104 0.31346 -0.03674 -0.08185 -0.04653 0.30289 0.40279 -0.20477 -0.19955 -0.23815
DS -0.27495 0.22054 -0.11213 0.11323 0.012424 -0.26771 -0.33718 0.038322 0.086362 0.10187
SC 0.071868 -0.14388 -0.55863 -0.18957 -0.10882 -0.21814 0.079757 0.094915 0.17375 0.1683
Bo -0.30763 0.032501 -0.01602 -0.04586 0.024808 -0.01195 -0.55327 -0.19962 -0.08178 -0.07179
API -0.34914 -0.10434 0.11182 0.084681 0.0207 0.080941 -0.16527 0.12135 0.23942 -0.05442
viscocity 0.19552 -0.044 0.32906 -0.29404 -0.18206 -0.09453 -0.29458 -0.19236 -0.36445 0.48006
WellAcres -0.23634 -0.23923 0.16835 0.04335 -0.24357 -0.02355 0.32145 0.030775 0.31123 0.48391
No.Prod 0.085035 -0.14131 0.041864 -0.04042 0.58225 0.63345 -0.15772 0.14925 0.11298 0.28903
No.Injector -0.01919 -0.0898 0.15873 0.074195 0.66288 -0.57896 0.18298 0.10912 -0.1865 -0.04675
Q 0.009558 -0.12637 -0.53964 -0.36382 0.15892 0.037408 -0.10246 -0.16788 0.09202 -0.07238
WellDensity 0.23083 0.22775 0.11269 0.29568 -0.09491 -0.03987 -0.27505 -0.01019 0.45359 -0.17888
Principal Component ScoresPrincipal Component Scores
-6 -4 -2 0 2 4-4
-3
-2
-1
0
1
2
3
8
23 24
4
22
16
2118 15
1910
2
716
20
3
14
12 5
9
1713
11
PC1
PC2
•After Performing PCA on 24 observations and 19 variables, the principal component scores for each of the twenty reservoirs were obtained.
•Figure alongside shows the 1st versus 2nd principal scores for the twenty four reservoirs.
•As seen in the PC space, we can see some grouping of reservoirs and some outliers.
Principal ScoresPrincipal Scores
-6 -4 -2 0 2 4-4
-3
-2
-1
0
1
2
3
23
24
422
8
1
6
10
11
19
18
153 12
21
16
20 14
13172
5
9
First score
Third
sco
re
Variables after PCAVariables after PCAH ============ Geology
Phi ============ Geology
Soi ============ Geology
K ============ Geology
Dip ============ Geology
VDP ============ Geology
Viscosity ============ Fluid
Well Acres ============ Well
No. Prod ============ Well
No. Injec ============ Well
Well density ============ Well
Cluster AnalysisCluster Analysis
• Some clustering of reservoirs is observed in the previous plot in the principal score space.
• To determine the grouping in reservoirs,cluster analysis has been done.
• Cluster analysis or segmentation analysis is a way to create groups of objects such that the profiles of objects in the same cluster are very similar and the profiles of objects in different clusters are quite distinct.
Cluster AnalysisCluster Analysis• Types:
1. Hierarchical Clustering2. K-Means Clustering
• K-Means clustering has been used because it can handle large volumes of data.
• K-means treats each observation as an object having a location in space. It finds a partition in which objects within each cluster are as close to each other as possible, and as far from objects in other clusters as possible. Five different distance measures can be tried, depending on the data being analyzed.
Silhouette plotSilhouette plot• To get an idea of how well-separated the
resulting clusters are, we made a silhouette plot using the cluster indices output from k-means.
• The silhouette plot is a measure of the distance from each point in one cluster to points in the neighboring clusters.
• The higher that measure, the more reliable is the discrimination between the two clusters.
• Mean of silhouette values give the average distance between the clusters.
Cluster AnalysisBased on 24 Observations
Cluster AnalysisBased on 24 Observations
0 0.2 0.4 0.6 0.8 1
1
2
3
1
2
3
S ilhouette Value
Clu
ster
0 0.2 0.4 0.6 0.8 1
1
2
3
4
5
Silhouette Value
Clu
ster
0 0.2 0.4 0.6 0.8 1
1
2
3
4
5
6
Silhouette Value
Clu
ster
0 0.2 0.4 0.6 0.8 1
1
2
3
4
Silhouette Value
Clu
ster
3 clusters 4 clusters
5 clusters 6 clusters
Means of SilhouetteMeans of SilhouetteNumber of Clusters Mean (Silh.)
3 0.3684
4 0.3972
5 0.3263
6 0.3642Max. distance
between clusters
Based on the above result – the reservoir data set can be optimally split into four clusters
• Cluster analysis was done using the principal component scores that were in turn derived using standardized data.
• From the cluster analysis and the quantitative analysis of silhouette plot it appears that the reservoir dataset can optimally be split into 4 clusters.
Cluster Analysis - Results
Linear Regression on New Data in PC Coordinates
Linear Regression on New Data in PC Coordinates
• Regression was done on the new data in the PC coordinate system.
• R2 = 0.365, shows that the linear regression explains 36% variability in the data.
• From the residual map, we can see that there are outliers in the data at row 7 and 23.
5 10 15 20
-0.2
-0.1
0
0.1
0.2
Residual Case Order Plot
Res
idua
ls
Case Number
Residual Plot for Linear Regression on the New Data in PC Space
Residual Plot for Linear Regression on the New Data in PC Space
Linear Regression (LR)Linear Regression (LR)• After removing the outliers from the data, we
came up with new regression coefficients. • R2 = 0.78, which is higher than previous case.• After removing the outliers, we get a better
regression line that covers 78% of variability in the data.
• However, the predictive power of the LR is poor when used in a validation mode. The correlation between actual recovery values and the predicted values for a validation set is -0.08
Perform LR on ClustersPerform LR on Clusters
• One way to improve the predictive ability of the linear regression is to segregate the dataset into clusters and then fit a LR in the principal component space.
• To perform LR on clusters, we have retained all reservoirs that have the 11 variables obtained from the first step PCA.
LR in Cluster 1 (with 11 variables)LR in Cluster 1 (with 11 variables)• Correlation coefficient between actual and
predicted recovery factor values = 0.68
0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9outlier
PR Calculated
PR A
ctua
l
This result is to be compared with the -0.08 that we obtained prior to cluster analysis
Cluster No.2 (with 11 variables)Cluster No.2 (with 11 variables)
0.05 0.1 0.15 0.2 0.250.1
0.2
0.3
0.4
0.5
0.6
Calculated PR
Act
ual P
R
Correlation coefficient = 0.64
Some conclusions• A viable scheme for computing the likelihood of recovery
factors given the reservoir related variables is by orthogonalizing the data and employing a hypothesis of conditional independence
• Principal component analysis is a preferred method for data orthogonalization
• Cluster analysis coupled with principal component analysis significantly improves the predictive power of linear regression.
• Cluster analysis results suggest that the TORIS data set might have to be further segregated into 4 clusters prior to calibration of the likelihood functions.
Multi-period Models of E&P Project Portfolios
Wei Chen, Leon Lasdon, and Jim Dyer
Motivation• Traditional single period project selection methods
may undervalue the projects• Multiperiod models capture
– Project uncertainties as they occur, and their reduction as new information becomes available
– flexibility and optionality: make later decisions based on the most recent information
– Multiperiod budget limits and common uncertainties which relate the projects
Methodology• Model each project as a multistage decision tree,
coupled by budget limits and revenue sharing• Represent each tree as a mixed integer linear program
(MILP): binary decision variables for the choices at each decision node, and constraints to ensure that you can select a decision at a node only if you arrive at that node.
• Additional continuous variables for costs and revenues• Reservoir behavior based on a tank model with uncertain
parameters. Code this in Excel or GAMS, generate many scenarios.
• Code the MILP in the General Algebraic Modeling System, solve it with CPLEX
• Create some moderate and large size model instances and investigate their behavior
Decision tree for a single oilfield
Results to Date
• GAMS/CPLEX quickly and accurately solves problems with about 220,000 constraints, 200,000 variables, and 6000 binary variables in times of between 10 and 90 seconds.
• These problems have 3 projects, 10 years, and 4 E&P stages