History Matching and Uncertainty Quantification with Produced Water Chemistry

8/10/2019 History Matching and Uncertainty Quantification with Produced Water Chemistry

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MSc (Petroleum Engineering)Project Report 2012/2013

Obinna NwaforH00137989

History Matching and Uncertainty

Quantification with Produced Water

Chemistry

Heriot-Watt UniversityInstitute of Petroleum Engineering

Supervisor-


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Declaration:

IObinna Nwaforonfim tht thi wok bmittd fo mnt i

my own and is expressed in my own words. Any uses made within it of theworks of other authors in any form (e.g. ideas, equations, figures, text,tables, programs) are properly acknowledged at the point of their use. Alist of the references employed is included.

Signd..

Dt20August, 2013


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Acknowledgements

I would like expressly my profound gratitude to my Supervisor Oscar Vasquez for his tireless effortin guiding me through this work. He has taken me from reservoir studies, and mixed in a little bit ofchemistry and lots mathematics. Without his kind reassurance, I would not have come this far.

My gratitude also go to th whol Untinty research group at IPE, Heriot-Watt University, ledby Mike Christie. I am very grateful to Mike, especially for those solutions he throws my way at themoments my wits were fully spent. Vasily Demyanova has been to me a great teacher and guide,especially with the woolly concepts of uncertainty quantification. He never once got deterred by mysilly questions, which popped out every morning. I am truly thankful. Dan Arnolds has beeninvlbl to m in thing ndtnding tht n wy. He takes time to lead them back-in,ensuring they were securely tucked into their beds.

I would not have had the courag to tk on mthmtil optimition withot thencouragement from Eric Mackay. I am truly thankful to his insight, especially those words aboutft job pnttion. My gtitd lo go to th Chplin of Hiot -Watt University,Alastair Donald, every Sunday I found my way back to him for the spiritual fodder that kept megoing.

I appreciate Epistemy Ltd for the Raven software used for this study, and the Computer Support

Team who kept the systems running in spite of the mounting pressure.

I wish to acknowledge all my lectures and tutors at the IPE over the last one year, they gave me a

little bit of themselves, which I have put together in this work.

My parents, especially mother, my sisters and brother in-law have been a rock to me throughoutmy one year stay in Edinburgh which culminated in this work. Thank you for being there, caring

and offering your words and resources. I thank the Almighty God for his grace every step of the

way.


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Abstract

History matching is used in reservoir calibration. Conventional history matching could be improvedby addition of more constraints to be matched. The injected sea water, produced as part of associated

water could have the potential of serving as an additional constraint. Such data can be obtainedcheaply by using ion in sea water as natural tracers.

This study aims to determine the extent of improvement to history matching and reduction ofuncertainty in forecasts brought about by the addition of injected sea water production data to thehistory matching process. The study was carried out using the PUNQS3 reservoir model. It is asynthetic reservoir model that has been used for similar studies and severally to test out methods inhistory matching and uncertainty quantification. The uncertain parameters in the PUNQS3 model arethe porosity and permeability.

Two cases of automatic history matching were carried out. One involved the use of injected sea

water tracer production data as additional constraint. The other involved using only the conventionalproduction data. The automatic history matching was based on multi-objective particle swarmoptimization (PSO). Which is a nature inspired stochastic optimization technique. The uncertaintyquantification was done using Neighbourhood Approximation method (NA-Bayes) based on aBayesian framework. The result for the two cases was compared on the basis of advance of theirpareto front of models ensemble towards lower misfit values. The quality of history matching andsize of uncertainty were also considered. They all show indications that addition of injected sea waterproduction data improves the history matching process, reduced uncertainty in forecast, as well ascreates a more robust uncertainty quantification. However the improvements were not large, butcould be more significant for more complex reservoir history matching problems.


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Table of ContentsAcknowledgements ......................................................................................................................... ii

Abstract ........................................................................................................................................... iii

1 Aims......................................................................................................................................... 12 Introduction ............................................................................................................................. 2

2.1 Reservoir History Hatching .............................................................................................. 2

2.2 Automatic History Matching and Mathematical Optimization ........................................ 3

2.2.1 Construction of Inverse Problems and Mathematical Optimization Problems ......... 4

2.3 Value of improved history matching ................................................................................ 4

2.4 Tracing Injected Sea Water in Hydrocarbon .................................................................... 5

2.4.1 Origin of Oilfield Water ............................................................................................ 5

2.4.2

Composition of Oil Field Water ................................................................................ 6

2.4.3 Compatibility of Formation Water and Sea Water .................................................... 6

3 Review of Concepts and Studies ............................................................................................. 9

3.1 Stochastic Optimization Techniques for History Matching.............................................. 9

3.1.1 Genetic Algorithm ................................................................................................... 10

3.1.2 Differential Evolution .............................................................................................. 10

3.1.3 Ant Colony Optimisation......................................................................................... 11

3.1.4 Particle Swarm Optimisation ................................................................................... 12

3.1.5 Use of Particle Swarm Optimization for Reservoir History Matching OptimizationProblem 14

3.1.6 Multi-Objective Particle Swarm Optimization. ....................................................... 14

3.1.7 Optimal Solutions in Multi-Objective Optimization: Pareto Front ......................... 15

3.2 Uncertainty Quantification for Forecast ......................................................................... 16

3.2.1 Definition of Uncertainty ......................................................................................... 16

3.3 Uncertainty Quantification Method: Neighbourhood Approximation Using BayesTheorem (NA-Bayes) ................................................................................................................ 17

3.3.1 Bayes Theorem ........................................................................................................ 18

3.3.2 Likelihood of Observation ....................................................................................... 193.3.3 Bayes Integral: Resampling by NA-Bayes .............................................................. 20

3.3.4 Neighbourhood Approximation and MCMC Walk ................................................. 20

3.3.5 Bayesian Credible Intervals ..................................................................................... 22

3.4 Review of Empirical Studies on Improving History Matching by Adding Data onInjected Sea Water Production .................................................................................................. 22

3.5 Problem Statement .......................................................................................................... 24


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4 Data Summary: PUNQS3 Reservoir Model .......................................................................... 27

4.1 Origin .............................................................................................................................. 27

4.2 Available Reservoir Description ..................................................................................... 27

5 Methods ................................................................................................................................. 295.1 Work Flow Diagram ....................................................................................................... 29

5.2 PUNQS3 Problem: Uncertain Parameters ...................................................................... 30

5.3 Modifications to the PUNQS3 and Historical Data ........................................................ 30

5.4 Parameterisation .............................................................................................................. 30

5.4.1 Regions and Justification/ Generation of Simulation Files ..................................... 31

5.4.2 Correlation of Porosity and Permeability ................................................................ 33

5.4.3 Parameter Distribution ............................................................................................. 34

5.5 Selection of Histories to Match and Objective Functions for Optimisation ................... 34

5.5.1 Case 1: ..................................................................................................................... 35

5.5.2 Case 2: ..................................................................................................................... 37

5.6 Generation of Truth Case Histories ................................................................................ 37

5.7 Variance for History Data ............................................................................................... 38

5.8 Optimisation Algorithm and Setup ................................................................................. 39

5.8.1 Setup of PSO run ..................................................................................................... 40

5.8.2 Ensemble of Models: convergence of ensemble; number of Ensembles ................ 41

5.9 Uncertainty Quantification: ............................................................................................ 41

5.9.1 Prior Probabilities .......................................................................................... 425.9.2 Likelihood of Models ................................................................................ 425.9.3 Monte Carlos Integration of the Bayesian Integral ................................................. 43

5.9.4 Bayesian Credible intervals. .................................................................................... 44

5.10 Analysis of Results ...................................................................................................... 44

5.10.1 Research Question 1 ................................................................................................ 44



6 Results ................................................................................................................................... 46

6.1 Research Question 1 ....................................................................................................... 46

6.1.1 Identification of Pareto Solutions Models for Case 1 and Case 2 ........................... 46



7 Discussion and Conclusion .................................................................................................... 50


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7.1 Improvement to History Matching.................................................................................. 50

7.2 Economic Value .............................................................................................................. 52

7.3 Reduction of Uncertainty ................................................................................................ 52

7.4 Conclusion ...................................................................................................................... 538 References ............................................................................................................................. 55

9 Appendix ............................................................................................................................... 58

9.1 Detailed Comparison of Front Advance by Region ........................................................ 58

9.1.1 Case 1: Excluding Sea Water Tracer Production Data ............................................ 59

9.1.2 Case 2: Excluding Sea Water Tracer Production Data ............................................ 60

9.2 3 Dimensional Pareto Plot for Case 1 ............................................................................. 60

9.3 Field Oil Production Total (FOPT) Intervals for Case1, C-Case and Case 2 ................. 61

9.4 Comparison of History Match for Best Five Models of Case 1 and Case 2 ................... 64

Well Water Production Rate ...................................................................................................... 64

9.5 Well Bottom Hole Pressure ............................................................................................ 67

9.6 Comparison of History Match for Most Pareto Models for Case 1(Iteration 493 Run2)Case 2(Iteration 262 Run3)-Well Bottom Hole Pressure .......................................................... 70

9.7 Comparison of History Match for Most Pareto Models for Case 1(Iteration 493 Run2)Case 2(Iteration 262 Run3) - Well Water Production Rate ....................................................... 73

9.8 Field Oil Production Total .............................................................................................. 76


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Table of FigureFIGURE 3-1VORONOI CELLS FOR TEN RANDOM POINTS (MODELS) IS A SOLUTION SPACE. B. THE UPDATED

VORONOI CELLS AFTER 100 POINTS ARE SAMPLED AND INTERPOLATED USING GIBBS SAMPLER. .................. 21

FIGURE 4-1 EXPECTED FACIES WITH ESTIMATES FOR WIDTH AND SPACING OF MAJOR FLOW UNITS ...................... 28

FIGURE 5-1WORK FLOW DIAGRAM ............................................................................................................................. 29

FIGURE 5-2 INJECTORS AND PRODUCER WELLS ON THE PUNQS3 RESERVOIR MODEL .............................................. 30

FIGURE 5-3SAMPLE OPTIMIZATION RESULT-QUALITATIVE PERMEABILITY MAP-LIGHT BLUE SHOWS HIGHER PERM.

........................................................................................................................................................................... 32

FIGURE 5-4 3 LAYER 3 PARAMETERISATION ................................................................................................................ 32

FIGURE 5-5 LAYER 2 PARAMETERISATION .................................................................................................................. 32

FIGURE 5-6 LAYER 1 PARAMETERISATION ................................................................................................................... 32

FIGURE 5-7 LAYER 5 PARAMETERISATION ................................................................................................................... 32

FIGURE 5-8 LAYER4 PARAMETERISATION.................................................................................................................... 32

FIGURE 6-1 CASE 1: SWTP PARETO PLOT .................................................................................................................... 46

FIGURE 6-2 CASE 2: NSWTP PARETO PLOT .................................................................................................................. 46

FIGURE 6-3 COMPARISON OF PARETO BY TRADE-OFF REGIONS ................................................................................ 47

FIGURE 6-4 CASE2: NSWTP FOPT ................................................................................................................................ 48FIGURE 6-5 CASE1: SWTP FOPT ................................................................................................................................... 48

FIGURE 6-6 CASE 2: NSWTP-FWPT .............................................................................................................................. 49

FIGURE 6-7 CASE 1: SWTP-FWPT ................................................................................................................................. 49

FIGURE 6-8 TERMINAL FWPT UNCERTAINTY LESS FOR CASE 2 ................................................................................... 50

FIGURE 6-9 TERMINAL FOPT UNCERTAINTY LESS FOR CASE 1 .................................................................................... 50

FIGURE 9-1 CASE 1: SWTP PARETO PLOT IN REGIONS ................................................................................................ 59

FIGURE 9-2 CASE 2: NSWTP PARETO PLOT IN REGIONS .............................................................................................. 60

FIGURE 9-3 3D PARETO PLOT CASE 1: INCLUDE SEA WATER TRACER PRODUCTION DATA ........................................ 60

FIGURE 9-4 CONTROL CASE -C FOPT-SWTP SHOWS SAME TREND AS CASE 1: SWTP ................................................. 61

FIGURE 9-5 CASE 1: SWTP FOPT FULL PLOT ................................................................................................................ 62

FIGURE 9-6 CASE 1: SWTP FOPT FULL PLOT ................................................................................................................ 62

FIGURE 9-7 WWPR PRO-1-BEST FIVE MODELS OF EACH CASE .................................................................................... 64

9-8 WWPR PRO-4 BEST FIVE MODELS OF EACH CASE ................................................................................................. 64

9-9 WWPR PRO-5-BEST FIVE MODELS OF EACH CASE................................................................................................. 65

9-10 WWPR PRO-11-BEST FIVE MODELS OF EACH CASE............................................................................................. 65



9-13 WBHP PRO-1-BEST FIVE MODELS OF EACH CASE ................................................................................................ 67

9-14WBHP PRO-4-BEST FIVE MODELS OF EACH CASE ................................................................................................ 67

9-15 WBHP PRO-5-BEST FIVE MODELS OF EACH CASE ................................................................................................ 68

9-16 WBHP PRO-11-BEST FIVE MODELS OF EACH CASE .............................................................................................. 68



9-19 WBHP PRO-1 CASE1 VS. CASE2 MOST PARETO OPTIMAL MODELS .................................................................... 70

9-20 WBHP PRO-5 CASE1 VS. CASE2 MOST PARETO OPTIMAL MODELS ................................................................... 709-21 WBHP PRO-4 CASE1 VS. CASE2 MOST PARETO OPTIMAL MODELS ................................................................... 71

9-22 WBHP PRO-11 CASE1 VS. CASE2 MOST PARETO OPTIMAL MODELS ................................................................. 71



9-25 WWPR PRO-1 CASE1 VS. CASE2 MOST PARETO OPTIMAL MODELS .................................................................. 73

9-26 WWPR PRO-5 CASE1 VS. CASE2 MOST PARETO OPTIMAL MODELS ................................................................... 73

9-27WWPR PRO-5 CASE1 VS. CASE2 MOST PARETO OPTIMAL MODELS .................................................................... 74

9-28 WWPR PRO-11 CASE1 VS. CASE2 MOST PARETO OPTIMAL MODELS ................................................................. 74


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9-31 FIELD OIL PRODUCTION TOTAL ........................................................................................................................... 76

TablesTABLE 5-1 LIST OF PARAMETER FOR PUNQS3 RESERVOIR .......................................................................................... 33

TABLE 5-2 DISTRIBUTION OF PARAMETER .................................................................................................................. 34

TABLE 6-1 MISFIT OF FIELD OIL PRODUCTION TOTAL FOPT FROM TRUTH CASE ........................................................ 47

TABLE 6-2 MISFIT OF FIELD WATER PRODUCTION TOTAL FWPT FROM TRUTH CASE ................................................ 47

TABLE 6-3 MISFIT FOPT FROM P(50) AS A MEASURE OF SPAN OF UNCERTAINTY ENVELOPE .................................... 48

TABLE 6-4 MISFIT FWPT FROM P(50) AS A MEASURE OF SPAN OF UNCERTAINTY ENVELOPE ................................... 48

TABLE 9-1 COMPARISON OF FRONT ADVANCE BY REGION ........................................................................................ 58


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1 AimsAutomatic history matching involves the use of mathematic optimization algorithms to determine

perform reservoir model calibration. However useful, it never yields unique single answers nor

eliminate totally the uncertainties associated with the reservoir model. The value of a well

calibrated reservoir model is the reduction in uncertainty or increase in reliability of the

performance data it provides to the decision process, involving the huge financial resources

invested to exploit hydrocarbon reservoirs. Hence, this study is a quest for improvement to the

history matching process.

It has been proposed by several previous studies that since injected sea water carried

complementary information on flow paths within the offshore reservoirs, its use as an additional

constraint in history matching could greatly improve the process, yielding highly reliable models

and reducing uncertainty in forecasts. In this view, the specific objectives of this study are:

To carry out a comparative history match of a synthetic reservoir model in two cases, one

constrained additionally by production data of injected sea water.

To effect the generation of sea water production data using the equivalent of natural water

tracers in the reservoir model.

To establish using an appropriate measure, if the addition of injected sea water production

data in one case has impacted in it, a better performance of the history matching process.

To establish using an appropriate measure, if the addition of injected sea water production

data in one case has impacted in it, a reduction in the size of uncertainty associated with

the forecasted performance of the reservoir.

To make conclusions on the potential of injected sea water production data in improving

reservoir history match results


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2 Introduction

2.1 Reservoir History HatchingIt is common to use numerical simulators to predict the performance of a reservoir. The

reliability of a reservoir simulation result is dependent on the inputs to the reservoir model

description. This input can be classified as relating to static (geological) properties description, or

dynamic (fluid flow) properties. Such information is gathered in the course of exploration and

appraisal. Static data will include time independent information derived from cores, wire line

logs, seismic surveys, etc. Dynamic data are time dependent data derived from flow relations,

they relate to reservoir properties such as relative permeability, fluid saturations, viscosity, flow

rate, fractional flows, etc. (Cheng et al., 2004, p.1). It is impossible to eliminate all uncertainties

in a reservoir.

History Matching is a major technique for calibrating the reservoir model in order to

maximize reliability of simulation results. It is the fine tuning of estimated reservoir description

parameters to match known past performance of the reservoir such as fluid rates, well bottom-

hole pressures, field average pressures, etc. History matching is an inverse problem, we attempt

to use the observed data about a reservoir to predict its properties. As is typical of inverse

mathematical problem, the solutions are never unique (Cunha, Prais, & Rodrigues, 2002, p.1).

Conventional history matching involves manual variation of field description parameters.

Simulation runs a made for each variation of model parameters, and the simulation results are

compared with the historical values. This is expensive in terms of human labour and computing

time. It is also highly subjective as the iteration direction depends on experience and insight.


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2.2 Automatic History Matching and Mathematical Optimization

Several History matching techniques have been studied and applied in the quest to

automate the process of finding solutions to history matching problems. They take the approach

of treating the inverse problem as a mathematical optimisation problem, in which a defined

objective function is either maximized or minimized. This objective function takes the form of a

function of the difference between observed history data and simulated result data (Cunha et al.,

2002).

Cunha et al. (2002, p.2) also indicates that automatic history matching can be broadly

classified into two groups, gradient based techniques and stochastic techniques. (Sarma,

Durlofsky, Aziz, & Chen, 2007, p.1) identified the streamline based history matching technique

as a class of its own. Each method has its own limitations and strengths.

The deterministic or gradient based techniques uses gradients of the mathematical model,

related to the parameterised properties of the model, to minimize the objective function which is

based on misfits between historical data and simulated results (Cunha et al., 2002, p.2). They are

known to converge very fast. However, they are poorly adapted to the multi-modal and non-

unique nature of solutions to history matching problems. Sarma et al.(2007) and Cunha et

al.(2002) agree that gradient based minimization is easily trapped into local minima point.

Stochastic history matching techniques have the exact opposite properties to gradient

techniques. They require a large number of simulation, hence, convergence and computing time

is quite significant. However, they are not easily trapped in local minima point, rather they effect

a more efficient search of the solution space. Sarma et al. (2007, p.1) noted that stochastic

techniques more easily honour complex geological models as they treat the simulator as a black


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box. The Streamlined based history matching techniques are limited by their inability to model

complex physics.

2.2.1

Construction of Inverse Problems and Mathematical Optimization Problems

As has been discussed earlier, history matching an inversion process where historical

production data of a reservoir is used to improve the estimates of parameters which characterize

the reservoir. Sarma et al.(2007, p.3) expressed the general construction of history matching

problem as mathematical optimization as follows:

( )

EQ. 2-1Where

Y refers to model parameter to be estimated; Yprior refers to the initial parameter estimates

C refer to the covariance which with Yprioris determined from the initial geological model

X refers to the states of the reservoir at various time N in Simulations. Such that f

n(X

n+1,

Y) simply refers to production data.

gn (Xn+1, Xn, Yn) represents the equations to which the simulator constrains the

reservoir model, linking the Parameters Y and reservoir states or results X.

The Lagrangian Ln

(Xn, Y

n) is the estimate of error between the observed data Dobs and the

simulated result fn(X

n+1, Y) also referred to as Misfit M.

is the variance of the data2.3 Value of improved history matchingHistory matching is applied to calibrate a reservoir model. Calibration has the singular

purpose of ensuring that simulations results are very reliable. Simulations results are applied to

Subject to the conditions:

Initial Conditions expressed as values X0


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field development planning, field optimization, economic evaluation of fields, testing of solution

ideas for field exploitation. These activities all have an economic value in the chain relevant to

production of the hydrocarbon.

It is impossible to accurately model every aspect of a real reservoir, hence every

simulation results has an attached uncertainty. Where history matching has been effectively

applied to a reservoir model, the uncertainty in the simulation results can be significantly

reduced. Statistical techniques can be used to quantify uncertainty.

The economic value of reservoir history matching lies in the reduction of uncertainty.

Uncertainty reduction facilitates the decision making process and risk management. The value of

improved history matching will stem from having results with fewer solution models and a direct

reduction in the uncertainty. The economic value of history matching will vary for each reservoir.

It is possible to quantify this economic value if we can quantify the reduction in uncertainty of

economic parameters resulting from the application.

2.4 Tracing Injected Sea Water in Hydrocarbon

2.4.1 Origin of Oilfield Water

The US Geological survey indicates that ground water constitute only about 1.7% of the

all the water on earth (U.S. Geological Survey, n.d.). Ground water in oil fields come from

various sources (Collins, 1975, p.194) classified as follows:

Meteoric Water: - Water tht h ntly bn in tmophi iltion

Sea Water: - This refers to water from modern sea, used for water flooding.

Interstitial Water:- Water occupying the pore spaces in formation rock (aquifer water).

Connate Water: - Refers to interstitial water of syngeneic origin with the formation rock..


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Digenetic Water: -Water with chemical or physical change from rock sedimentation.

The injected sea water applies for offshore field, where sea water being the most available water

source, is used for water flooding. Produced water from an oil reservoir, for the purpose of this

study will be grouped into two, namely: Injected Sea Water and Formation Water. The formation

water is made up of connate water, aquifer or interstitial water and digenetic water.

2.4.2 Composition of Oil Field Water

The water in modern sea is generally saline. The salt or ion composition is mainly of

hloid (Cl), odim (N+), sulphate (SO24), mgnim (Mg2+), calcium (Ca2+), and

potassium (K+). They constitute about 90 percent of all the salt in sea water. While inorganic

carbon, bromide, boron, strontium, and fluoride constitute the other major dissolved content of

seawater. MacKenzie (2013) gives a full list sea water composition.

Formation water cannot be ascribed a single composition as they come from various

sources and pass through various physical and chemical processes. In a study of water from

various geological aged rocks, Collins, (1975, p.216) concluded that the water were not of the

same chemical composition, and have evolved considerably compared to the modern sea water.

Any water in the reservoir can be modified by four major processes, dilution by meteoric

water or fresh water, reaction with minerals in the rock formation, clay membrane filtration and

ion exchange, mixing of sea water and aquifer water resulting in precipitations.

2.4.3 Compatibility of Formation Water and Sea Water

Vazquez, McCartney, et al.(2013,p.1) observed that injected sea water and formation

water

can be quite incompatible for mixing. Mixing of both waters could result in several possible

geochemical reactions which may lead to scale precipitation. Precipitation of insoluble


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compounds result in formation damage through reduction of permeability and porosity of the

reservoir. Collins (1975, p.367) identified the ions responsible for formation of scale from water

mixing as Ca

+2

,Sr

+2

, Ba

+2

, Fe

+2

, SO4-2

, HCO3-

. The time and actual precipitation of the scale may

be subject to other environmental changes such as pressure and temperature changes or factors

that affect concentration of the brines. The most notorious of the scales is barium sulphate

BaSO4which is highly insoluble and often impossible to remove once formed.

2.4.3.1 Conservation of Natural Tracers within Reservoir

Valestrand et al.(2008, p.2) defined tracers as inert chemical or radioactive compounds

used

to label fluids or track fluid movements. Artificial water tracers are used for inter-well tracer

tests. The interest of this study lies on natural water tracers. Even though ions sea water are

affected by chemical activities, Huseby et al., (2009, p.2) indicated that in most cases ions in sea

water only react moderately with the formation water. Such ions can be used as natural tracers of

sea water. Ions which may be used for such application include SO42-

, Mg

2+

, K

+

, Ba

2+

, Sr

2+

, Ca

2+

,

Cl-(Huseby et al., 2009, p.2).

The second option for natural tracers of water are isotopes. Hydrogen isotopes are the

best being abundant in water. Another isotope is Strontium 87Sr, a radiogenic isotope found in

high concentration in potassium rich rocks (Huseby et al., 2009, p.2). The high concentration is

transferred to formation waters with which such rocks have equilibrated. The ratio of 87Sr to the

more abundant 86Sr isotope can be used as tracers for formation water.

The choice of natural water tracers might be an economic decision rather than a choice

based on quality. Ion content data of produced water are routinely analysed as part of the flow

assurance, hence has little extra acquisition cost compared to isotopes. For this study, the


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assumptions is that there are scale risks in our synthetic reservoir which exclude the use of SO42-

as a tracer. The alternative choice is the use of Cl- ions as tracers. These ions do not move in

between reservoir phases and are not subject to portioning effects (Valestrand et al., 2008, p.2).


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3 Review of Concepts and Studies

3.1 Stochastic Optimization Techniques for History Matching

A major problem of history matching techniques is the possibility of existence of local

minima in the solution space (Hajizadeh, Demyanov, Mohamed, & Christie, 2011 p.211). It has

earlier been mentioned that gradient based history matching techniques lack the ability to

navigate through such a parameter space without being trapped in a local minima. Secondly,

accurate forecast of future reservoir performance is very important for decision making.

Hajizadeh et al.(2011, p.210) reported that simple optimization techniques have been

found to be inadequate for history matching problems. He further noted that even the Monte

Carlo Approaches were not intelligent enough for the optimization task. These problems speak to

a need for powerful optimization techniques which is able to navigate through multi-modal

parameter space to identify fitting solution models, as well as execute such task in feasible time.

Stochastic optimization algorithm have these characteristics and several have been developed

since the 1990s. Schulze-riegert & Ghedan (2007, p.1) lists some of them as follows:

Evolutionary Algorithms; Gradient techniques; Response surface modelling and optimisation on

the response surface; Hybrid schemes which couple different optimisation techniques; Ensemble

Kalman Filter Techniques. More recent algorithms were described by (Hajizadeh et al., 2011,

p.212) as follows:

Evolution Algorithms

Evolutionary Strategies (ES); Genetic Algorithm[6,7]; Differential Evolution (DE) [14]

Swarm Intelligence Algorithms

Ant Colony Optimisation (ACO); Particle Swarm Optimisation (PSO) ;Neighbourhood

Algorithm


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3.1.1 Genetic Algorithm

The genetic optimisation algorithm is a form of evolutionary algorithm based on

techniques or concepts of natural evolution and the genetic system, such

asinheritance,mutation,selection, andcrossover. An optimisation problem is defined, its

objective function is defined as the fitness function and the solution models sought are encoded

with chromosome or a string of bits such as binary numbers (Obitko, 1998). Each bit is related to

a parameter of the solution model by a function (Tatiana Tambouratzis, 2013, p.163).

It starts with the initiation of a generation of randomly generated solutions to the problem.

The process of selection follows, in which the solution are ranked based on specified fitness

criteria such as the value of the objective function. The best solutions of the population are

selected to breed a new generation. Breeding of a new set of solutions is achieved through

genetic operations. Two or more parent solution models are randomly selected from the set of

bests for the generation. There chromosomes or encoding are treated as genetic identities on

which the genetic operations are performed. Popular operations are mutation and crossover,

others include regrouping, migration, extinction, roulette wheel selection, elitism, etc. (Tatiana

Tambouratzis, 2013, p.163).A generation ends when its population size is reached. The objective

function is observed to approach closer to the target solution or fitness value with each new

generation.

3.1.2 Differential Evolution

Differential Evolution is a form of Evolution Algorithm, hence it shares the same

description and procedures with Genetic Algorithm, but differs in method of evolution. Creation

of the new generation occurs before selection is done. It used uses a specified mutation and

recombination operation to create the new generation. Rather than chromosomes, differential
http://en.wikipedia.org/wiki/Heredityhttp://en.wikipedia.org/wiki/Mutation_(genetic_algorithm)http://en.wikipedia.org/wiki/Selection_(genetic_algorithm)http://en.wikipedia.org/wiki/Crossover_(genetic_algorithm)http://en.wikipedia.org/wiki/Crossover_(genetic_algorithm)http://en.wikipedia.org/wiki/Selection_(genetic_algorithm)http://en.wikipedia.org/wiki/Mutation_(genetic_algorithm)http://en.wikipedia.org/wiki/Heredity


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algorithm uses vectors of solution model parameters to represent solutions in the parameter space

(Hajizadeh, 2011, p.71). For mutation, three solutions from the population are randomly selected

(x1g, x2g, x3g where g denotes the generation). A mutation or mutant vector is calculated as

follows.

(Hajizadeh, 2011, p.72) EQ. 3-1The factor F controls the rate of evolution, ranging from 0 to 1. In recombination or cross over

operation, each member of the population is crossed with the mutated vector to produceand offspring

. The selection process is then implemented in which the objective function

for the parent and offspring are compared, the one with lower objective function isretained as for a minimization problem (Hajizadeh et al., 2011, p.218). EQ. 3-2

3.1.3 Ant Colony Optimisation

Swarm Intelligence algorithms are optimisation techniques based on the collective

behaviour organism such as insects, birds, etc. (Leonor Melo Francisco Pereira, 2013, p.179).

Swarm intelligence algorithms are the most recent of optimisation algorithms, they include the

Ant Colony Optimisation, Bee Colony Optimisation, and Particle Swarm Optimisation.

The Ant Colony Optimization algorithm is based on the behaviour of ants as they search

for food. Ants leave pheromones to mark their paths so other ants can follow them as they search

for food. The strength of the pheromone markers decay with time, the more frequently a path has

been taken, the stronger the accumulation of pheromones from various ants, and the more likely

this path will be taken by other ants. The longer the path taken is, the more time for the

pheromones to dissipate. Hence, the less likely other Ants will follow the path. The information


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shared through pheromone distribution allows ants to translate from random exploration to the

shortest path to the nearest food source. This behaviours of ants are modelled in optimization as

solution searches by a number of ants in a colony. When one ant finds a better solution based on

defined criteria, a pheromone like property of its search influences the other ants to search more

in the vicinity of the solution found (Hajizadeh et al., 2011, p.214). This however risks trapping

the process in a local minima, hence some parameters or weighting are applied to the decision

criteria so that the ants at times deviate from the pheromone informed decisions in order to create

a better exploration of the solution space.

3.1.4 Particle Swarm Optimisation

Particle Swarm Optimization (PSO) was originated by Kennedy J, Eberhart R in 1995

Hajizadeh et al., 2011, p.219). It is an optimization techniques based on the movement of flock of

birds and school of fishes. A set of particles, each representing a solution to the defined problem,

is randomly initialized. Subsequently two factors influence the next movement of each particle as

they search for better solution:

1. pbest: - a particles best known position in terms of the optimization target.

2. gbest: - a global best known position from among the swarm of particles.

They are updated in each iteration and used to calculate the particles next movement as follows:

Updating:

EQ. 3-3 where EQ. 3-4Velocity:

( ) ( ) EQ. 3-5


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EQ. 3-6(Hajizadeh et al., 2011, pp. 219-220)

Where

and

are as defined earlier

denotes a particular particle in the swarm is the velocity of the particle of index in a specified dimension for the next

iteration in a multi-dimensional solution space.

and are randomly generated real numbers between 0 and 1 and are weighting used to control the focus of exploitation of a local area versus

exploration of the solution space.

is a weighting factor which controls the rate of convergence of the algorithm represents the iteration count

The particle swarm optimization has various variants from this basic definition. They may differ

on the method of updating the velocity, choice of gbest, updating of particle position, etc. Below

is a flow diagram of the process.

Start

Initialize Z particles in

Solution space i=1

Evaluate Particles

Position for fitness

Compare current

Particles position with

pBest

Update the Value of

PBest

Do for Particle i, for

i=1 to Z

Update gBest

Calculate Particle is

new position using

EQ.3-5

Update Particle is

position

Yes i=Z

Is stopping

criterion met?

Yes

End

No

Next Iterations

Set i=1

Figure 3-1 Work Flow for Particle Swarm Optimization


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3.1.5 Use of Particle Swarm Optimization for Reservoir History Matching OptimizationProblem

(Hajizadeh et al., 2011) conducted a comparative study of various stochastic optimization

algorithms. This study was conducted on two relatively well known reservoir models, the Teal

South reservoir and the PUNQS3 reservoir model with 45 parameters. The authors considered the

following algorithms: Differential Evolution- Best Variant; Differential Evolution; Particle

Swarm Optimization; Ant Colony Optimization; Neighbourhood Algorithm. The study reached

the following conclusions (Hajizadeh et al., 2011, p.238):

1. All the stochastic algorithms performed well compared to gradient based algorithms.

2. That for all the algorithms studied, Differential Evolution-Best and Particle Swarm

Optimization had the fastest convergence, as well as achieved the lowest misfit solutions.

3. That all the algorithms had uncertainty bounds which included the truth case

Ant Colony Optimization had the smallest span of uncertainty, followed by Particle Swarm

Optimization, these differences are very small. Several other studies have been conducted using

particle swarm optimisation (Lina Mohamed, Christie, & Demyanov, 2011), (Linah Mohamed,

Christie, & Demyanov, 2009), (Hajizadeh, 2011b), (Arnold, Vazquez, Demyanov, & Christie,

2012), (Vazquez, MacMillan, et al., 2013), (Vazquez, McCartney, et al., 2013).

3.1.6 Multi-Objective Particle Swarm Optimization.

History matching problems will often have to consider different kinds of data. For

example history matching of a reservoir may involve data from the well bottom-hole pressure,

gas oil ratio, production volumes or rates, water cuts, etc. These data come in different numerical

ranges, a summation of all into a misfit definition will result in those with high numerical range

overshadowing those with low numerical range. An inefficient optimization will result.


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Where F is the objective function. EQ.

3-7

The solution to this problem for single objective optimization is to apply weights c1, c2

and c3 (see EQ. 3-7) in order to control the contribution of misfit components. Determination of

values of these weight is debateable. An alternative approach to this problem is the use of multi-

objective optimization. Data of different numerical range and type are separated different

objectives for concurrent optimization. It eliminates arbitrary combination of dissimilar data into

misfit functions. In a recent study, Lina Mohamed et al.(2011) conducted a comparative study of

Single Objective Particle Swarm Optimization (SOPSO) and Multi-Objective Particle Swarm

Optimization (MOPSO). The study was conducted as a history matching task on IC Fault

Reservoir model from Imperial University, UK. The authors concluded as follows:

That the MOPSO was twice faster than SOPSO in convergence and obtained good fitting

models to the history matching problem.

That MOPSO obtained a more diverse set of solution models compared with SOPSO.

That while SOPSO gave a narrower uncertainty range, MOPSO resulted in a more robust

and more accurate uncertainty definition which included the truth case.

This study will use Multi-objective Particle Swarm Optimization for the mentioned benefits.

3.1.7 Optimal Solutions in Multi-Objective Optimization: Pareto Front

In single objective optimization problems, an optimal solution is selected based on the

value of the misfit function. In multi-objective optimization the task of selecting optimal

solutions becomes a bit more complex as the solutions represent different trade-off between

objectives in terms of dominance (Lina Mohamed et al., 2011,p.2). Veldhuizen & Lamont (1997,


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p.2) indicated that one of the key ideas for dealing with this problem of selecting optimal

solutions is the concept of Pareto Optimality. Pareto optimal solutions stems from the definition

of Pareto Dominance.

Let us view two solutions of a multi objective optimization problem as vectors and, wherethere are p objective functions.

and is said to dominate if and only if is partially less than (for minimization problems).i.e

{ } { } EQ. 3-8Based on this definition we now define the pareto optimal solution set as solutions that are not

dominated by any other solution. This means that for N number of evaluated solutions, solution

is pareto optimal if { } { }

{ }

EQ. 3-9

The pareto set when plotted graphically is called the pareto front. It physically represents the

range of optimal trade-off between the objectives.

3.2 Uncertainty Quantification for Forecast

3.2.1 Definition of Uncertainty

Uncertainty is the lack of assurance about the truth of a statement or the exact magnitude

of an unknown measurement or number (Schulze-Riegert & Ghedan, 2007, p.2). Uncertainty

may result from an actual lack of knowledge, a difficulty in measurement or errors in

measurement.


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For a reservoir, the major driver of uncertainty is heterogeneity of the reservoir. Most

reservoir data are only reflective of measurements at the well location. Inferences have to be

made about locations in between wells. Heterogeneities occur at all scales of the reservoir, from

microscopic pore scale to the megascopic properties.

Heterogeneity in reservoir characterisation translates to uncertainty in simulation outputs

and forecasts. Uncertainties in reservoir characterisation were grouped by Schulze-riegert &

Ghedan (2007, p.7) as follows.

Uncertainty in Geological Data: Uncertainties due to measurement errors, selection or

interpretation of geological data of the reservoirs

Uncertainty of Geological Data: Uncertainties inherent from the complexity of reservoir

geology or lithology. Issues and interpretation from of sedimentation, lithology, and mapping.

Uncertainty in Dynamic Reservoir Data: Uncertainty in properties that affect the flow of

fluids.

Uncertainty in Reservoir Fluids Data: Composition of reservoir fluids retains some

uncertainties as to the extent to which obtained samples are representative of the whole field.

The many sources of uncertainties means that it is impossible to totally eliminate uncertainty

from reservoir model. It is imperative that we quantify the uncertainty in the results of reservoir

models and simulation to better inform of the limits of their applicability.

3.3 Uncertainty Quantification Method: Neighbourhood Approximation Using Bayes

Theorem (NA-Bayes)

Bayesian framework was used for uncertainty quantification in this study. It has been

applied severally in in similar studies by several scholars (Christie, Demyanov, & Erbas, 2006),

(Christie, Subbey, & Sambridge, 2002), (Sambridge, 1999), (Hajizadeh et al., 2011).


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It is important to note that uncertainty quantification is also necessitated by the

inadequacy of misfit values as a measure of accuracy of a solution model. By definition misfit is

a function of the difference between observed values and simulated values of reservoir

performance.

EQ. 3-10However it is not just simulated values that are subject to inaccuracies, observed values are

subject to measurement errors. Accounting for this error, we refine the misfit definition as

] (Christie et al., 2006, p.145)EQ. 3-11

Hence in actual fact, misfit values are the difference in errors associated with observed data and

simulated data. Misfit may not reflect direct relations to the truth value of a reservoirs

performance.

3.3.1 Bayes Theorem

Bayes theorem is a method of inference which allows us to update the probability

estimate for a hypothesis as added evidence is acquired about the hypothesis (Christie et al.,

2006, p.4).

EQ. 3-12Bayes rule stated above can be summarised as stating that P(H) - Priorprobability of the hypothesis before any Evidence; P(H|E) - Posterior probability of

hypothesis given the evidence; P(E|H)-Likelihood of the evidence given the hypothesis or

Probability of observing the evidence in the event of the hypothesis being true; P(E) -

Marginal Probability of the evidence independent of any particular hypothesis.


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Relating Bayes rule to reservoir history matching problems, the models M under investigation are

the hypothesis H, the historic production data or Observation O is the Evidence. We can rewrite

the Bayes Rule using the new notation as follows: EQ. 3-13(Christie et al., 2006, p.3)

is the posterior probability, while is the prior probability. The marginalprobability of the evidence which is the denominator is now expressed as the Bayesianintegral. The Evaluation of the posterior probabilities entails the evaluation of three elements

1. Prior Probabilities 2. Likelihood of

Observations

3. Bayesian integral expressing

Probability of the observation

The prior probabilities are evaluated from the prior information obtained from the reservoir on

the variability of the parameters describing the uncertain properties of the reservoir.

3.3.2 Likelihood of Observation

The evaluation of the likelihood is based on Gaussian error statistics. The likelihood is defined as

the negative exponent of the misfit between observations and simulation values. This is

expressed in mean squares form below:

(Christie et al., 2006, p.4) or (Sambridge, 1999, p.3) EQ. 3-14

EQ.

3-15

(Sambridge, 1999, p.3). Where C is the covariance matrix of the observation data. For a single

parameter the definition of misfit simplifies as EQ. 3-16


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Where do is the observed data, g (m) is a function of the model, or simulation result for the

model.

3.3.3

Bayes Integral: Resampling by NA-Bayes

A Bayesian integral is contained in the definition of the normalizing factor in theexpression for Bayes rule. Bayesian integrals are evaluated using Monte Carlos integration:

EQ. 3-17Using Monte Carlos Integration

EQ. 3-18

Where is the density distribution of the sampled models drawn from the solution space inthe forward solution earlier described? Difficulties arise in evaluating the relation as the density

distribution with which the models space is sampled in the optimization setups are usually

unknown. The solution to this problem is to re-sample the solution space in such a way that the

density distribution of the samples equals the probability distribution, hence they cancel out.

EQ. 3-19This can be executed using Neighbourhood Approximation Algorithm

3.3.4 Neighbourhood Approximation and MCMC Walk

Makov Chain Monte Carlos random walk have the unique property that when used to

sample a given probability distribution based on set rejection criteria, the Markov Chain will

have a distribution that is equal to the probability distribution of the sampled ensemble.This property is what we need to evaluate the Bayesian integrals using Monte Carlo integration.

However, one of the impediments to using this approach is that we do not have a full

detailed description of the probability distribution in the solution space. The ensemble of models


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are only representative. This is where neighbourhood approximation of the solution space comes

in. Using voronoi cells, the entire volume of the solution space is described by an approximation

of the actual probability. Voronoi cells are nearest neighbour regions in the solution space

defined around each model in the solution ensemble (see figure 3-1), they have the properties of

being space filling polyhedral, with their size, shape and volume automatically adapted to the

distribution of the models in the ensemble (Sambridge, 1999a, p.4). The spacing filling attribute

allows the points within a voronoi cell to be assigned a probability equal to the probability of the

model around which it is defined.

Figure 3-2Voronoi cells for ten random points (models) is a solution space. b. The updated

Voronoi cells after 100 points are sampled and interpolated using Gibbs sampler.

The sampling of the approximate probability distribution formed using voronoi cells is now

accomplished using the MCMC random walk. The MCMC variant used is the Gibbs Sampler

Algorithm. The Gibbs Sampler selects a model by taking random sized step in the direction of

each dimension of the solution space in turn (Sambridge, 1999). When it has stepped in all

dimensions, a parameter vector representing a model results. The Gibbs sampler implement


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typical MCMC rejection check on each step, this turns the sampling to select models of high

probability, hence following probability distribution in the solution space (Sambridge, 1999a,

p.5).

3.3.5 Bayesian Credible Intervals

A Bayesian credible interval is an interval in which the probability of find a truth caseor say the solution model is (Levy, 2007). It offers a convenient way of expressinguncertainty.

3.4

Review of Empirical Studies on Improving History Matching by Adding Data onInjected Sea Water Production

One of the more recent studies on using tracer information for improved history matching

of a reservoir was carried out by Valestrand et al.(2008). The study investigated the effect of the

use of partitioning gas tracer data in the history match of a reservoir for the determination of the

permeability and transmissibility Multipliers. This study was carried out on the synthetic

reservoir with various realistic features coupled. Ensemble Kalman filter (EnKF) was used in

updating the reservoir properties permeability and transmissibility. The estimation using the gas

tracer data was successful in estimating transmissibility and permeability that provide as good

match to history data, the control case without the gas tracer data was not successful in this

regard. Valestrand et al.(2008) concluded that partitioning gas tracer data was of crucial

importance in successfully estimating the reservoir properties.

Huseby et al.(2009) carried out a similar study using the Ensemble Kalman filter method.

Their study however was focused on water tracers and considers ordinary and natural water

tracers. The study was conducted using the model 2 of the 10 th SPE Comparative Solution

Project. The model is derived from a part of the North Sea Brent Sequence. The inversion


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problem was to estimate the reservoir permeability and porosity. Three cases of history matching

were considered: using oil rate and water cut production data only; using the natural tracer SO42-

with oil rate and water cut production data; and using ordinary inter-well tracer data. The three

cases were compared based on mean square error of their estimation the true porosity and

permeability of the reservoir. The result showed that for both porosity and permeability, there

were only slight differences between estimations done without tracers, and with ordinary tracers.

However, the estimations done using natural tracers showed a marked improvement in quality

with much lower error values.. The study also noted the lack of explanation for the better

performance of natural tracers than ordinary tracers, since the former do not carry

complementary information on water injection sources. The authors concluded that tracer data

were underexploited as a source of knowledge about reservoirs.

Arnold et al.(2012) carried out a study on the value of adding produced water chemistry

as tracer of injected sea water, to further constrain the history matching of the PUNQS3 reservoir

model. The study was carried out using single objective Particle Swarm Optimisation. It

considered well bottom-hole pressure, oil production rate, gas oil ration, water production rate

and well tracer data. The cases with tracer data and without tracer data were compared. Based on

misfit calculated as mean square error from the history date, it was found that out of five trials of

each case, only one case with tracer data achieved very low misfit. A second comparison was

made on the bases of clustering of the solution models in parameter space, and did not find any

considerable improvement due to tracer data. However the study found that tracers reduced the

number of acceptably matched minima points from the parameter space. The study concluded

that adding tracer data did not harm nor greatly improve the quality of the history match, but

made significant improvements to forecast.


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Vazquez et al.(2013) extended the methods of the study by Arnold et al.(2012) to an

actual real field, the Janice Field. The reservoir model provided for the study was history match

by the operator. The aim of the study was not strictly comparative, the study analysis found that

the conventionally matched model was mismatched to produced sea water fraction of the three

wells considered. Based on this, it identified new geological uncertainties in the reservoir model

and a new history match was carried out using Particle Swarm Optimisation and sea water

production tracer data. In terms of sea water fraction, it achieved a significantly better match for

one well, and slightly better match for the remaining two well. Using the Bayesian framework,

the study made uncertainty quantifications for the field forecasts, and developed scale risk

assessment for the field.

Vazquez et al.(2013) conducted the most recent study on effect of the composition of

produced water on history matching a one dimensional reactive reservoir model. The study was

not comparative of the non-tracer, rather it investigated connectivity and reactions between the

producer and injector in two cases which differed based on the amount of produced water

chemistry data available. It concluded that produced water chemistry had the potential to provide

the information sought on dispersivity of the connection flow paths between the wells, it also

considered it a successful application of Particle Swarm Optimisation to a reactive reservoir

model.

3.5 Problem Statement

The problem to be investigated by this study is the determination of the extent of improvements

to reservoir history match and uncertainty quantification induced by the used of produced water

chemistry data to specify historic production of injected sea water fractions. It has been proposed

by several studies reviewed in the preceding section that since injected sea water carried


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complementary information on flow paths within the reservoir, its addition as a constraint to

reservoir calibration should improve the quality of history match and also reduce the amount of

uncertainty in forecast. Specifically this study aims to answer the following questions.

Question 1: Does adding natural tracer data reduce the mean square error misfit achieved by

sampled models with reference to oil rate and bottom-hole historical data?

Question 2:Does adding natural tracer data generally reduce the range of uncertainties specified

by Bayesian credibility intervals over the history match and forecast period?

Question 3: Does adding natural tracer data reduce the range of uncertainties specified by

Bayesian credibility at the terminal point of the forecast period?

While none of the earlier works has sought to compare the effect of adding natural sea water

tracer data to history matching by measuring the range of uncertainties, Arnold et al.(2012) had

observed that it made improvements to forecast generally. The comparison by misfit of sampled

models has been a bit more complicated due to earlier studies use us single optimization

techniques, which requires addition or removal of tracer data points before misfit values could be

compared (Arnold et al., 2012, p.6). Also, the combination and weighting of the objective

function for oil rate, water rate and natural chemical tracers for sea water in earlier studies

prefixes the combinational relationship between these objectives. While this does not hamper the

optimisation schemes from finding good solution models, it does limit the extent of exploration

in the search for good fitting models. This study will be carried out using multi-objective particle

swarm optimisation to allow a free comparison of misfit values and also maximise the space

searched by the optimisation algorithm. The choice of optimisation algorithm has also been

informed by the earlier reported works on comparative study of optimisation techniques (Linah


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Mohamed et al., 2009) and multi-objective particle swarm optimisation (Lina Mohamed et al.,

2011). This value of improving history matching has been earlier discussed in the introduction.


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4 Data Summary: PUNQS3 Reservoir Model4.1 OriginThe PUNQS3 is a synthetic reservoir model based on an actual reservoir developed by Elf

petroleum. The synthetic case was initially developed for the PUNQ (production forecasting with

uncertainty quantification) project sponsored by the European Community. In the PUNQ project

ten partners from industry, research institutes and universities are collaborating on research on

uncertainty quantification methods for oil production forecasting (Soleng, 1999, p.1). It has

however become a benchmark for testing methods in history matching and uncertainty

quantification (Arnold et al., 2012, p.2).

The reservoir model consist of 19x28x5 grid blocks, of which 1761 blocks are active. It is

bounded to the east and south by a fault, the north and west are linked to a strong aquifer. It also

includes a gas cap, while six well are located around the gas oil contact. There were no injector

wells since the reservoir had a strong aquifer support.

The production scheduling is based on the real reservoir. Wells are under production constraint

based on flow. The scheduled flow periods are for a first year of extended well testing, followed

by a three year shut-in period, before field production commences. During field production, two

weeks shut-in period for each year is included for each well to collect shut-in pressure data. Total

production period is for 16.5years or 6025 days.

4.2 Available Reservoir DescriptionThe geological description of the reservoir as provided by imperial college is given below

(Gologil Diption fo PUNQS3 Rvoi Modl, n.d.).

The layer thickness is of the order of 5meters in thickness, it played a major role in geological

interpretation. The sediments were deposited in a deltaic, coastal plain environment. Layers 1, 3,


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and 5 consist of fluvial channel fills encased in floodplain mudstone. Layer 2 represents marine

or lagoonal clay with some distal mouthbar deposits; and layer 4 represents a mouthbar or

lagoonal delta encased in lagoonal clays.

Ly 1, 3, nd 5 hv lin tk of high -porous sands ( > 20 %), with an azimuth

somewhere between 110 and 170 degrees SE. These sand streaks of about 800 m wide are

embedded in a low porous shale matrix ( < 5 %). The width and the spacing of the streaks vary

somewhat between the layers. A summary is given in the table below.

In layer 2 is a marine or lagoonal shale in which distal mouthbar or distal lagoonal delta occur.

They translate into a low-porous ( < 5%), shaly sediment, with some irregular patches of

somewhat higher porosity ( > 5%).

Ly 4 ontin mothb o lgoonl dlt within lgoonl ly, o flow nit i expected

which consists of an intermediate porosity region ( ~ 15%)with an approximate lobate shape

embedded in a low-porosity matrix ( < 5%). The lobate shape is usually expressed as an ellipse

(ratio of the axes= 3:2) with the longest axis perpendicular to the paleocurrent (which is between

110 and 170 degrees SE).

Layer Facies Width Spacing

1 Channel Fill 800 m 2-5 km

2 Lagoonal Shale

3 Channel Fill 1000 m 2-5 km4 Mouthbar 500-5000 m 10 km

5 Channel Fill 2000 m 4-10 km

Figure 4-1 Expected facies with estimates for width and spacing of major flow units


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5 Methods

5.1 Work Flow Diagram

Start

Definition of Study

Objectives-PUNQS3

Reservoir Model

Parameterisation of

PUNQS3 Reservoir

Model

Define Uncertain

Parameter

Define History

Data to be

Matched

Generation of

Simulation Files

Determination of

Range of Uncertain

Parameters

Generation of

Distribution Files

for Optimisation

Generation of History

Data from Truth

Case: Blind to

Parameterisation

Determine Initial

Values of

Variance for

History Data

Modification of

PUNQS3 Model to

include Water

Injectors and Water

Tracers

Case1 PWC:

Selection of History

Data to Match

Include Sea

Water Production

Rate History

Determination of

Optimisation

Objective Functions

Setup Optimisation

Software: RAVEN

Execute Multi-

Objection

Optimisation : PSOfor 3000 Iteration

Determine

Convergence Point:

Number of Iterations

Repeat PSO

Optimisation for

Converged Number

of Iterations

Setup Ensemble for

Forecast and

Uncertainty

Quantification:

RAVEN

EXECUTE:

PPD Approx.

MCMC Walk

Is No. of Sampled

Models Adequate?

Review Variance

Values to Modify

Ensemble Density

Execute Two

additional Runs of

PSO Optimisation

Setup the 3

ensembles jointly for

forecast and

Uncertainty

Quantification:

RAVEN

EXECUTE

PPD Approx.;

MCMC Walk;

Simulations for

forecast;

EXECUTE: CalculateBayesian Credibility

Intervals P10, P50 &

P90

Case2 No PWC:

Selection of History

Data to match

Exclude Sea

Water Production

Rate History

Execute Three Runs

of PSO Optimisation

at Converging

Iteration Number

Setup 3 ensembles

jointly for forecast and

Uncertainty

Quantification:

RAVEN

EXECUTE

PPD Approx.;

MCMC Walk;

Simulations for

forecast;

EXECUTE: Calculate

Bayesian Credibility

Intervals P10, P50 &

P90

Result Analysis and

Conclusions

End

Figure 5-1Work Flow Diagram


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5.2 PUNQS3 Problem: Uncertain Parameters

The PUNQS3 reservoir model was generated synthetically. The reservoirs description (see

Data Summary) was developed to match the synthetic reservoir. The reservoir model without the

porosity and permeability data, was distributed to researchers who were asked to invert the

production data to estimate the permeability and porosity. This study will be working with the

same problem configuration to determine the porosity and permeability distributions for the 5

layers on the PUNQS3 model. However our focus is to assess impact of adding injected sea

water production data as an additional constraint to the history matching method.

5.3 Modifications to the PUNQS3 and Historical Data

The PUNQS3 was initially designed without any

injection wells due to the strong aquifer support

modelled. It is imperative that sea water is injected into

the reservoir for this study, hence four Injections wells

have been added to the reservoir as shown in the figure

above. With this modification, the original history data

distributed with the reservoir model can no longer be

used for this study. These modifications necessitate the

generation of a new history data using the truth case data

provided at the PUNQS3 website of Imperial College

(Gologil Diption fo PUNQS3 Rvoi

Modl, n.d.).

5.4 ParameterisationThe two uncertain reservoir properties were identified as porosity and permeability.

Figure 5-2 Injectors and Producer

Wells on the PUNQS3 Reservoir

odel


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5.4.1 Regions and Justification/ Generation of Simulation Files

The reservoir description for the PUNQS3 indicates the existence three layers (1, 3 and 5)

which have fluvial sand channels embedded in a flood plain. The description also highlights the

direction of these sand channels as between 110 to 170 degrees azimuth. The sand channels had a

minimum width of 800m and spacing of 2 to 5 km.

For these sand channels we assume that the channels direction is mid-way between the

specified range at 145 degrees azimuth. This is to reduce the number of required parameters,

variations of 35 degrees in azimuth is not expected to have a major impact on the reservoir

performance. The parameterisation will be based on regions to capture the heterogeneity across

layers of the reservoir. Since the position of the sand channels is unknown, as region scheme has

been adopted which allows flexibility in the location and width of the sand channels, while also

minimizing the resulting of geometrically unrealistic models.

As illustrated in the Figures 5-4 to 5-8 below, the parameter regions have been defined

diagonally in the approximate direction of 145 degrees azimuth. The Larger regions whose

width(448m) is about half the minimum sand channel width(800m) described is alternated with

two smaller regions of width (256m) in order to build in reasonable flexibility on the location of

the sand channels. This is at the expense of having some poor models with sand width less than

800m. The same parameter regions is used in layers 3 and 5, being mindful that the sand

channels in in layer 5 have widths of about 2km. This arrangement was adopted with the

expectation that the best models will identify several adjacent regions to be of similar high

permeability to form the required sand body. This effect was observed for one of the low misfit

models, its first layer is visualized in Figure 5-3. There are a total of 17 parameter regions in

layer 1, 14 parameter regions in layers 3 and 5 respectively, and 18 in layer 4.


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Note that similar colours are not related. They all mark different parameterisation regions.

Figure 5-6 Layer 1

ParameterisationFigure 5-5 Layer 2

ParameterisationFigure 5-4 3 Layer 3

arameterisation

Figure 5-8 Layer4

ParameterisationFigure 5-7 Layer 5

Parameterisation

Figure 5-3Sample

Optimization Result-Qualitative Permeability

ap-light blue shows highererm.


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Layers 2 and 4 from the reservoir description are low permeability shale or clay. Layer 2

has no prominent flow unit, while layer 4 has a lobate shaped flow unit embedded in the lagoon

clay. Layer 2 has been parameterised as a single parameter region (See Figure. 5-4). An

arrangement of several block region has been made, with small block interconnecting them. The

idea is to be able to form an approximation of a lobate shape in many ways with several

selections. On the total, the PUNQS3 reservoir model has been parameterised into 66 parameters.

The table below lists the parameters by layer and shows the naming convention.

Table 5-1 List of Parameter for PUNQS3 ReservoirLayer1 Layer2 Layer 3 Layer4 Layer5

$L1P $L2P $L3P2 $L4P1 $L5P2

$L1P1 $L3P3 $L4P2 $L5P3

$L1P2 $L3P4 $L4P3 $L5P4

$L1P3 $L3P5 $L4P4 $L5P5

$L1P4 $L3P6 $L4P5 $L5P6

$L1P5 $L3P7 $L4P6 $L5P7

$L1P6 $L3P8 $L4P7 $L5P8

$L1P7 $L3P9 $L4P8 $L5P9

$L1P8 $L3P10 $L4P9 $L5P10

$L1P9 $L3P11 $L4P10 $L5P11

$L1P10 $L3P12 $L4P11 $L5P12

$L1P11 $L3P13 $L4P12 $L5P13

$L1P12 $L3P14 $L4P13 $L5P14$L1P13 $L3P15 $L4P19 $L5P15

$L1P14 $L4PA

$L1P15 $L4PB

$L1P16 $L4PC

$L4PD

$L4PE

$L4PF

5.4.2 Correlation of Porosity and Permeability

Hajizadeh et al.(2011) in his study indicated the existence of a correlation between

porosity

and permeability for the PUNQS3 reservoir model. The report also included a correlation

between vertical and horizontal permeability. The relations are given below.


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These relations are used for this study to minimize the number of parameters required to fully

capture the uncertain permeability and porosity of the reservoir. The parameters will be used to

define porosity, while the porosity will be used to calculate the horizontal permeability according to above relations. Vertical permeability is calculated from the relations withhorizontal permeability. Permeability is assumed to be equal in all horizontal directions.

5.4.3 Parameter Distribution

The prior distribution of the parameters has been decided based on the PUNQS3 reservoir

description. The porosity variation are assumed to be uniform over the range. The range or

distribution were assigned based on the layers. Summary of distributions by layers is given

below.

Table 5-2 Distribution of ParameterLayer Distribution Porosity Range (%)

1 Uniform 15-30

2 Uniform 5-10

3 Uniform 15-30

4 Uniform 5-15

5 Uniform 15 -30

5.5 Selection of Histories to Match and Objective Functions for Optimisation

This study is comparative of two cases of history match. The first case use the additional

constraint of injected sea water produced data and is denotes Case 1:SWTP (Sea Water Tracer

Production). The second case excludes the water tracer data from the history match, and is

denoted Case2: NSWTP (No Sea Water Tracer Production). Case-C is a repeate of Case 1 made

as a control.


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5.5.1 Case 1:

5.5.1.1 Tracer Data

For this case tracers have been added to the injector wells. The same tracer is used for all

four injection wells to model the nature of natural water tracers which do not have

compli

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History Matching and Uncertainty Quantification with Produced Water Chemistry

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