Annual Report – 2007cpge.utexas.edu/sites/default/files/research/cparm/cparm_ann_rpt_2007.pdfI2 I3...

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

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

0 30 60 90Time, days

Injectors Producers

f12 =40%

f11 =60%f11

2τf12

CRM: Total Production

⎟⎟⎠

⎞⎜⎜⎝

JVc ptτ[ ]*1)()(

0 Ieetqtqtt

⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

)( tq Field)( tI Field

f1jf11 f12 f13

Volume Change = ( Injection Rate – Production Rate)

fijijτ

iij ifI =*∑=

iiijifI

fieldII =*CRMT

CRMIPCRMP

CRM: Oil Fractional-Flow Equations

)(11)(

tWORqqq

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 2,000Time, Days

I1 I2 P1CRMP

Injection Signal – Production Response(2 Inj, 2 Pro)

Injection/Production Rate Errors

0 1,000 2,000Time, Days

I1 I2 P1CRMP

Synfield: Streak Case

Streak Case: Varying BHP

0 200 400 600 800 1,000 1,200Time, Days

QinjPwf ,psi

0 200 400 600 800 1,000 1,200Time, Days

Qpro Eclipse

Pwf ,psi

0 200 400 600 800 1,000 1,200

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

1970 1975 1980 1985 1990 1995

Time, year

Oil Productionqw inj Water injectionWater Production

J/ctPv=0.40 and qoi= 4735 STB/DR =0.964

1970 1975 1980 1985 1990 1995Time, year

q pro M BEq pro fie ld

1970 1975 1980 1985 1990 1995Time, year

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

ProductionInjection

MESL Field: Total Production Match

40,000

46,000

52,000

58,000

0 1,000 2,000Time, Days

Field CRMTCRMP CRMIP

P5 Total Production Match

11,000

0 1,000 2,000Time, Days

FieldCRMPCRMIP

MESL Field: Oil Production Match

10,000

20,000

30,000

40,000

50,000

0 1,000 2,000Time, Days

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

Field Example 2: MESL Constraints,

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

10,000

12,000

2,000 2,500 3,000Time, Days

BaseOptimized

6% increase

●P1▼I2

▼I10

▼I12

▼I13

▼I14

▼I15▼I16

▼I17

●P2●P15

●P13

___P19

●P12

●P20

●P26

●P21

●P8●P9

●P23

●P10

●P25●P##

●P22●P##

___P##●P27

___P29

●P17

●P18

●P11

●P14

___P16

●P##

●P28

___P30

●P##

____P##

▼I11

Field Example 3: “Big Field”

17 Injectors

30 Producers

Field Example 2: CRMP Match

Field Injection

Field Production

CRMP Match

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

CRM Individual Well Match

7,000 8,000 9,000 10,000 11,000 12,000Time, Days

P14CRMP P14

10,000

20,000

30,000

0 2,000 4,000 6,000 8,000 10,000 12,000 14,000

Time, Days

P24CRMP P24

12,000

7,000 8,000 9,000 10,000 11,000 12,000

Time, Days

P10CRMP P10

10,000

6,000 8,000 10,000 12,000Time, Days

P12CRMP P12

●P1▼I2

▼I10

▼I12

▼I13

▼I14

▼I15▼I16

▼I17

●P2●P15

●P13

___P19

●P12

●P20

●P26

●P21

●P8●P9

●P23

●P10

●P25●P##

●P22●P##

___P##●P27

___P29

●P17

●P18

●P11

●P14

___P16

●P##

●P28

___P30

●P##

____P##

▼I11

CRMT Waterflood Efficiency

Area 2

~25 % of Injection

~105 Days Response Time

~28 % of Injection

~300 Days Response Time

Area 1

●P1▼I2

▼I10

▼I12

▼I13

▼I14

▼I15▼I16

▼I17

●P2●P15

●P13

___P19

●P12

●P20

●P26

●P21

●P8●P9

●P23

●P10

●P25●P##

●P22●P##

___P##●P27

___P29

●P17

●P18

●P11

●P14

___P16

●P##

●P28

___P30

●P##

____P##

▼I11

22%18%

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

Iq jji

τλ −=∑=1

Oil Fractional Flow Modeljojoj qfq =

oj CWIaf

Where = oil production rate of producer j= oil fraction of producer j = cumulative water injected in all injectors

, = model parameters for producer jja

ojqojf

Optimization formulationMaximize net present value

Subject to• CRM• Fractional flow model

• Upper limit on total injection rate

• Upper limits on rate of eachinjector

( ) ( )∑∑∑∑= == =

−Δ+

=i tp t n

o ttIir

pttqir

pNPV1 11 1

ikii utIl ≤≤ )(

iki ItI

≤∑=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

Time (days)

0 1000 2000 3000400

0 1000 2000 3000600

0 1000 2000 3000500

0 1000 2000 3000200

0 1000 2000 30000

0 1000 2000 3000

0 1000 2000 3000500

0 1000 2000 30000

0 1000 2000 3000200

0 1000 2000 30000

0 1000 2000 3000

100012

0 1000 2000 30000

100013

0 1000 2000 3000600

100014

0 1000 2000 3000500

150015

0 1000 2000 30000

100016

0 1000 2000 30000

100017

0 1000 2000 3000

100018

0 1000 2000 3000500

150019

0 1000 2000 3000500

150020

0 1000 2000 30000

0 1000 2000 3000600

100022

0 1000 2000 3000500

150023

0 1000 2000 3000200

0 1000 2000 3000

100025

Total Injection History

0 500 1000 1500 2000 2500 30001.35

1.7x 104

Time (days)

Saturation Map (t=1000 days)

• 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

150010

0 1000 2000 3000500

150011

0 1000 2000 3000500

150012

0 1000 2000 3000500

150013

0 1000 2000 3000500

150014

0 1000 2000 3000500

150015

0 1000 2000 3000500

150016

0 1000 2000 3000500

150017

0 1000 2000 3000500

150018

0 1000 2000 3000500

150019

0 1000 2000 3000500

150020

Time (days)

Total Production Match

0 500 1000 1500 2000 2500 3000700

900Producer 1

Time (days)

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

Model Fit Comparison

0 2 4 6 8 10 12 14 16 18 200

Producer

ContinuousSemicontinuous

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

t qqqabs

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

Continuous Gains and Time Constants

Semicontinuous vs. Continuous

Model Fit Comparison

0 2 4 6 8 10 12 14 16 18 200

Producer

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

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

Current Hypothesis Realized

Alternate Hypothesis Realized

Current Hypothesis Realized

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)

Selecting the Initial Capacity

Cumulative Distribution Function of NPV @ Initial Facility Capacity = 10,000 mbopy

300 500 700 900 1100NPV, $mm

P10 = 635.88P50 = 797.60P90 = 945.01Expected Value = 794.55

Expected Value

Optimal Initial Platform Capacity

4000 6000 8000 10000 12000 14000 16000Initial Platform Capacity, mbopy

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

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

4 -4.46 -0.13 -0.8512 -4.46 0.00 -0.8520 -1.92 -0.20 0.00

Mean (mmbbls)

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)

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)

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)

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

0 1 2 3 4 5 6 7 8 9Time Period

90% C.I.

10% C.I.

Lognormal D

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

−−+

Time 0 1 2 3

V ddV ddd

V uuuV uu

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]

.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]

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

T3 Option

Continue

T2 PriceMove

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%

1 2 3 4 5 6 7 8 9 10Time Period

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),(

tYtYtqt σ

μΔ+≡

),( tYσ),( tYμ

tq−1

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

puupud

tttY χξ +=

puu/p χ+Δχ

p ξ+Δξ pud/p χ−Δχ

1-p ξ−Δξ pdu/(1-p) χ+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

DeviationL-T Mean

Expanded View

Deviation

L-T Mean

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

Mean 3Diverge

Revert

Abandon 3

Mean 2 Diverge

Revert

Abandon 2

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

-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

-14%12%

47% -$30,000

-$10,000

$10,000

$30,000

$50,000

$70,000

$90,000

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

-35%-14%

12%47% -$500,000

$500,000

$1,000,000

$1,500,000

$2,000,000

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

-35%-14%

12%47% -$500,000

$500,000

$1,000,000

$1,500,000

$2,000,000

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

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

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 5 10 15 20

Time (year)

0 5 10 15 20

BenefitTotal CostNet Profit

)Time (year)

Decision Criteria

LN(k)Porosity

Modeling UncertaintyAverage Reservoir Properties

Production Rate

Average Well

Individual Wells

Modeling UncertaintyWell Variability

Start Phase 2 (Infill Wells)

10th Year

in First Year

$ - 5,561

$ 152,138

$ - 892,133

Expected Net Profit (NPV/100 acres)

Initial Decision

Likelihood Function

Production Rate

LN(k)Porosity

2.0 Likelihood

NormalizedPeak Production Rate

NormalizedDecay Constant

Oil Value ($/STB)

VI($ NPV/100 acres)

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

1/3 1/3 1/3

Option A

Option B

State 1

State 2

State 3

State 1

State 2

State 3

We Unintentionally Insert Bias in Decision.

Preferred Option

2/3Option B

Option A

State 2

State 3

State 1

Utility

Option B

Option A

Option B

Option A

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

Permeability

Log of Permeability

Which distribution is non-informative?

Permeability

Inconsistency in Conventional ApproachTo which input parameters do we apply it?

keff = 1/(1/k1 + 1/k2)

Inconsistency in Conventional ApproachTo which input parameters do we apply it?

0.5 0.67 1

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

A1 11 0A2 0 10

States of Nature

DecisionAlternatives

Consequences (Utilities)

0101/2

A2 05.55.0

States of Nature

DecisionAlternatives

Probabilities

ExpectedUtility

Non-Informative Probabilities

0101/2

A2 05.55.0

1/2Probabilities

ExpectedUtility

S1 S2a S2b S2c S2d S2e

0101/6

00010 101/6 1/6

0101/6

1.88.3A2 0

1/6Probabilities

ExpectedUtility

Luce and Raiffa ExampleAdding Duplicate States of Nature Should

Not Change Decision

0101/2

A2 05.55.0

1/2Probabilities

ExpectedUtility

0101/6

00010 101/6 1/6

0101/6

1.88.3A2 0

1/6Probabilities

ExpectedUtility

Luce and Raiffa ExampleS1 S2

0101/2

A2 05.55.0

1/2Probabilities

ExpectedUtility

S1 S2a S2b

0A2 0 10 10 6.7

6.0A3 9

1/3Probabilities

ExpectedUtility

Luce and Raiffa ExampleAdding Alternative Should Not Change

Non-Optimal Decision to OptimalS1 S2

0101/2

A2 05.55.0

1/2Probabilities

ExpectedUtility

S1 S2a S2b

0A2 0 10 10 6.7

6.0A3 9

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

Preferred Option

1/2 1/2

1/4 1/4

Option B

Option A

State 2

State 3

State 1

Utility

Option B

Option A

Option B

Option A

CPARM Approach Is SoundAdding Duplicate States of Nature Should

Not Change Decision

0101/2

A2 05.55.0

ExpectedUtility

00010 10

1/10 1/1010

5.55.0A2 0

1/2Probabilities

ExpectedUtility

Probabilities

CPARM Approach Is SoundAdding Alternative Should Not Change

Non-Optimal Decision to Optimal

0101/2

A2 05.55.0

1/2Probabilities

ExpectedUtility

S1 S2a S2b

0A2 0 10 10 5.0

6.8A3 9

1/2Probabilities

ExpectedUtility

3. Practical Implementation

0A2 0 10 10 10 10 10

0 0 0 0A3 9

A1>A3>A2 A2>A3>A1 A2>A1=A3

Implementation of CPARM Approach

A3A1> A2>

1/2 1/4

1/2 1/2

A3A1> A2>

A1A3=A2>

Probability

P(S2a)

P(S2b)+P(S2c)+P(S2d)+P(S2e)

Apply Principle of insufficient reason to lumped states

P(S2b)=P(S2c)=P(S2d)=P(S2e)=1/16

0 0100

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

6.8A3 9

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 %

Transmissibility(STB/psi/day)

Uncertainty Modeling

• Uncertain Variable- Transmissibility between two units

• Transmissibility- Degree of the connectivity- Unit: STB/psi/day

• 11 scenarios

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

Non-Informative Probability DistributionsTransmissibility(STB/psi/day)

Best Option

0102030405060708090100

12200 Unit 1 4 wellsUnit 2

1 2 3 4 5 6 7 8 9 10

Unit 1 4 wellsUnit 2 4 wells

Non-Informative Probability Distribution

0 10 20 30 40 50 60 70 80 90 100

Transmissibility (STB/psi/day)

Probability

$ 7.8 MM

$ 12.9 MM

Convention

Decision

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

$ 5.8 MM

$ 29.4 MM

Convention

Decision

Decision Analysis Results - Medium Well Cost

1 2 3 4 5 6 7 8 9 10

Unit 1: 4 wells

1 2 3 4 5 6 7 8 9 10

Unit 1: 4 wells

Unit 2: 1 well

$ 0.4 MM

$ 2.2 MM

Convention

Decision

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

Decision Analysis Results - Well Cost of Unit 2

0 50 100 150

ConventionalCPARM

Well Cost for Unit 2 ($MM/Well)

VPI($ MM)

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

Value of Information Analysis

T* > 20

Purchase Info.

No Info.

Option 1Option 2

Option 12241

Option 1Option 2

Option 12241

Option 1Option 2

Option 12241

Transmissibility(STB/psi/day)

Value ofInformation

$ 1.8 MM

$ 4.2 MM

Convention

Decision

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

74 (md)

330 (md)

1500 (md)

6600 (md)

16 (md) 50 ($/STB)

30 ($/STB)

70 ($/STB)

Medium

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

φ (%)

k (md)

3390245

k (md)φ (%)

0.1820.280

00.10.20.30.40.50.60.7

Low Medium High

0.2620.350 0.389

00.10.20.30.40.50.60.7

30 50 70

Oil Price

0.259 0.247

00.10.20.30.40.50.60.7

30 50 70

Oil Price

0.300 0.267

00.10.20.30.40.50.60.7

Low Medium High

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

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

)()()()(

)|()|()|(),,(

)()|,,(),,|(

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 ip VD P D S SC B o API

viscocity WellAcres N o.Prod

N o.InjectorQ

W ellD ensity

Values

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

2118 15

•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

153 12

First score

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

0 0.2 0.4 0.6 0.8 1

S ilhouette Value

0 0.2 0.4 0.6 0.8 1

Silhouette Value

0 0.2 0.4 0.6 0.8 1

Silhouette Value

0 0.2 0.4 0.6 0.8 1

Silhouette Value

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

• 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

Residual Case Order Plot

Case Number

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

0.9outlier

PR Calculated

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

Calculated PR

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

Annual Report – 2007cpge.utexas.edu/sites/default/files/research/cparm/cparm_ann_rpt_2007.pdfI2 I3...

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