efficient history matching in reduced reservoir models with pce-based particle filters

University of Stuttgart, Institute for Modelling Hydraulic and EnvironmentalSystems

Independent Junior Research Group: Stochastic Modelling ofHydrosystems

Jun.-Prof. Dr.-Ing. Wolfgang Nowak

diploma thesis

Ef icient History Matching forReduced Reservoir Models withPCE-based Bootstrap Filters

Roman HeimhuberMatriculation Number 2422563

Sonthofen, August 13, 2012

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Examiner: Jun.-Prof. Dr.-Ing. Wolfgang NowakProf. Dr.-Ing. Rainer Helmig

Supervisor:Ph.D. Inga BerreDr.-Ing. Sergey Oladyshkin

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Fur Faser.Und meine Mathelehrer.

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I hereby certify that I have prepared this thesis independently, and that only those sources,aids and advisors that are duly noted herein have been used and/or consulted.

Sonthofen, August 13, 2012

Roman Heimhuber

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Contents

Contents v

List of Figures vii

1 Introduction 31.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Summary of the State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 History Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 The Ensemble Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . 61.2.3 Particle Filters and Bootstrap Filters . . . . . . . . . . . . . . . . . . . 7

1.3 Approach and Bene its . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Multiphase Flow in Porous Media 92.1 Assumptions Relevant to this Thesis . . . . . . . . . . . . . . . . . . . . . . . 92.2 Basic De initions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Darcy's Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 The Multiphase Flow Differential Equation . . . . . . . . . . . . . . . . . . . 14

2.5.1 The Global Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5.2 The Pressure Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 152.5.3 The Pressure-Saturation Formulation . . . . . . . . . . . . . . . . . . 152.5.4 The Total Pressure-Velocity Formulation . . . . . . . . . . . . . . . . 16

2.6 The IMPE-Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6.1 A Remark on Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.6.2 The IMPE-Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.7 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 History Matching 203.1 History Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 History Matching as an Inverse Problem . . . . . . . . . . . . . . . . . . . . . 213.3 Bayesian Updating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Particle and Bootstrap Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4.2 Importance Sampling (IS) . . . . . . . . . . . . . . . . . . . . . . . . . 263.4.3 Sequential Importance Sampling (SIS) . . . . . . . . . . . . . . . . . . 283.4.4 Sequential Importance Resampling (SIR) . . . . . . . . . . . . . . . . 293.4.5 SIR: The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Contents vi

4 Polynomial Chaos Expansion 324.1 Response Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Polynomial Chaos Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3 Combining PCE and BF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Numerical Implementation 375.1 A Remark on the Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 375.2 Composition of the Reservoir Model . . . . . . . . . . . . . . . . . . . . . . . 375.3 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.4 Realization of the History Matching . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Results and Discussion 446.1 Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.2 Results: Time-Series of Different Length . . . . . . . . . . . . . . . . . . . . . 456.3 Results: Polynomial Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.4 Results: Fineness of Time-Series . . . . . . . . . . . . . . . . . . . . . . . . . . 526.5 Results: Number of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7 Summary and Conclusions 55

vii

List of Figures

1.1 Number of papers on history matching prepared by each year for SPE con-ferences and journals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Relative-permeability-saturation-relationships. . . . . . . . . . . . . . . . . . 14

3.1 Updating a particle cloud based on the weights. . . . . . . . . . . . . . . . . . 30

5.1 Model domain and wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2 Capillary pressure saturation relationship used in the example problem. . . 385.3 Parabolic relative permeabilities used in the example problem. . . . . . . . 395.4 Basic lowchart of the used code. . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.1 Results for ϕ and d = 2 (Simulation #8). . . . . . . . . . . . . . . . . . . . . . 486.2 Resulting pressure time-series for d = 1 (Simulation #1). . . . . . . . . . . . 486.3 Illustration of shrinking and shifting of posterior distributions. . . . . . . . 496.4 Comparison of response surfaces expanded for degrees d = 1, 2, 3. . . . . . 506.5 Resulting posterior pressure time-series for different polynomial degrees d. 516.6 Comparison of response surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . 516.7 Results for k1 and d = 3 (Simulation #19). . . . . . . . . . . . . . . . . . . . . 546.8 Results for ϕ and d = 3 (Simulation #19). . . . . . . . . . . . . . . . . . . . . 54

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List of Tables

6.1 History matching results for runs # 1 - 12. . . . . . . . . . . . . . . . . . . . . 476.2 History matching results for runs # 13 - 19. . . . . . . . . . . . . . . . . . . . 53

Nomenclature

Subscript De initionα Phasew Wetting phasen Non-wetting phaset Time-step1 : t All time-steps up to time step t

Symbol Unit De initiond − Polynomial degreeg m/s2 Gravitational constantG − Integration domainh m Hydraulic pressure headhi − Unknown coef icients in the polynomial basisk m2 Absolute permeabilityIN − Identitykr m2 Relative permeabilityN − Number of parametersNt − Number of offsprings related to a particleN (. . . , . . .) − Gaussian distributionp Pa Pressurepc Pa Capillary pressureP (A) − Probability of an event AP (A|B) − Conditional probability of A by given BP (A ∪ B) − Joint probability of A and Bp(x0) − Initial distributionp(x0:t|y1:t) − Posterior distributionp(xt|y1:t) − Marginal distributionS − Saturationq m3/s Darcy velocityV m3 Volumew − Model input parametersx − Hidden statey − ObservationΓ − Domain boundaryδ − Delta-dirac probabilityϵ − Measurement error

List of Tables x

Symbol Unit De initionE − Expectations for some functionκ − Chemical componentλ − Phase mobilityµ Pa · s Dynamic viscosityπ(x0:t|y1:t) − Importance functionϱ kg/m3 Densityϕ − Porosityω(x0:t) − Particle weightsωt − Normalized particle weightsΩ − ModelΩ − Reduced model

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Abstract

Abstract Underground low systems, such as oil or gas reservoirs and CO2 storage sites,are an important and challenging class of complex dynamic systems. Lacking informationabout distributed system properties (such as porosity, permeability, . . . ) leads to modeluncertainties up to a level where quanti ication of uncertainties may become the domi-nant question in application tasks. Calibrating models for underground reservoirs on pastproduction data (called historymatching) helps to reduce uncertainties and hence can im-prove the predictive power. However, history matching complex models is a very chal-lenging task. Usually, brute-force optimization approaches are not feasible, especially forlarge-scale computer-intensive simulations, or when using realistically large numbers ofuncertain parameters to be calibrated.We propose an advanced framework for model calibration and history matching based onthe polynomial chaos expansion (PCE). Our framework consists of twomain steps. In stepone, the original full complexmodel is projected onto a so-called response surface via PCE,which is a drastic and ef icient model reduction. Step two consists of Bayesian updating inorder to match the reduced model to available measurements of state variables or otherreal-time observations of system behavior. Because the reduced model is vastly more ef-icient than the original one, accurate updating methods (such as particle iltering) muchbeyond brute-force optimization or Ensemble Kalman Filtering become feasible.Individual work steps:

• Set up a synthetic reservoir model. Keep it reasonably cheap so that the followingcomputations are always swift. This phase uses the SinTef Open Source Matlab codefor reservoir simulation.

• Use thismodel, generate synthetic datausing a randomparameter set in a simulation-and data collection scenario from production history.

• Develop and test a PCE-based particle ilter for history matching, using the modeland data from the two previous steps.

• Asses the quality of model it to the data after history matching, and how good theinferred parameter values match the synthetic parameter values that were used togenerate the data.

• Discuss the advantages and disadvantages of PCE-based particle iltering.

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Abriss

UnterirdischeStromungssystemewie z.B. Ol- oderGasfelderundCO2 Speicherstattengehorenheutzutage zu denwohlwichtigsten undherausforderndsten dynamischen Systemen ihrerArt. Unvollstandige Kenntnis uber Systemparameter wie z.B. Porositat oder Permeabiliatkanndazu fuhren, dassdieUntersuchungundQuanti izierungderUnsicherheiten zur entschei-denden Frage im Anwendungsfall wird. Das Kalibrieren von Modellen fur unterirdischeSpeicher mit Hilfe von aufgezeichneten Daten des Reservoirs kann Unsicherheiten im Sys-tem reduzieren und somit zu einer erhoten Verlasslichkeit fuhren. Normalerweise sindauf brute-force oder Monte Carlo basierende Methoden, im Speziellen fur grossskalige,rechenintensive Modelle, oder wenn eine realistische Anzahl an unsicheren Parameternverwendet wird, nicht durchfurbar.Wir schlageneinenerweitertenAnsatz, basierendauf derpolynomial chaos expansion (PCE),vor. Unser Ansatz beinhaltet im Wesentlichen zwei Schritte. Im ersten Schritt wird dasursprungliche Modell mit Hilfe der PCE Technik auf eine sog. Antwort-Flache projiziert.Die PCE Technik ist eine wirksame und ef iziente Modelreduzierungstechnik. Im zweitenSchritt wird das reduzierte Modell mit Hilfe von Bayesian updating an vorhandene Mess-werte der Zustandsvariablen angepasst. Da das reduzierte Modell im rechnerischen Sinnewesentlich effektiver als das Urspungliche ist, werden genauere updating Techniken (z.B.Partikel-Filter), welche weit uber die Moglichkeiten von Ensemble Kalman Filtern hinaus-gehen, einsetzbar.Die einzelnen Arbeitsschritte gliedern sich wie folgt:

• Konstruieren eines synthetischen Reservoir-Modells. Das Modell soll einfach gehal-tenwerden, so dass die darauf folgendenBerechnungennicht zuviel Zeit in Anspruchnehmen. Fur diesen Schritt wird der SinTef Open Source Matlab Code fur Reservoir-Berechnungen verwendet.

• Generieren und Speichern von kustlichen Datenmit Hilfe von ZufallsparameternmitHilfe des synthetischen Modells aus dem vorherigen Schritt.

• Entwickeln und Testen eines PCE-basierten Partikel-Filters fur den history matchingSchritt unter Zuhilfenahme desModells und der Daten aus den vorherigen Schritten.

• Untersuchung der Qualitat des Anpassungschrittes des Modells mit den aufgezeich-neten Daten und wie sehr sich die Verteilungen der unsicheren Parameter vor undnach dem Anpassungsschritt angenahert haben.

• Diskussion der Vor- und Nachteile von PCE-basierten Partikel-Filtern.

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1 Introduction

1.1 Motivation

Underground low systems are and have always been a great focus of research. Since sev-eral years, miscellaneous topics and speci ic sites have increasingly been discussed, espe-cially in public. Prominent examples in Europe are CO2 sequestration sites, oil reservoirsand radioactivewaste disposal sites. Radioactivewaste disposal has been ahighly disputedtopic in german news formany years. The reasonwhy all these systems are in everybody'sattention, even outside of scienti ic milieus, is their increasing importance and constantthreat for nature and mankind.Apart from general risks, which are omnipresentwhenmaking these kind of enormous en-gineering changes in nature, the immanent risk is mainly related to the uncertainty of pre-dicting these systems and to the uncertainty of predicting the impact of human action onthese systems and the environment. Uncertainty often originates from the fact that mod-els are being used without detailed knowledge of all important parameters. When dealingwith reservoir simulation, rock parameters, such as porosity or conductivity are typicallyvery important, but not well known. In almost any case, natural systems are way too com-plex to describe properly on a conceptual model level. Beside that, a complete explorationof underground systems for their material properties is simply and obviously impossible.Even if all that information was known, the available computational power would be farfrom suf icient for solving detailed, highly resolving large-scale models. Ignoring theseproblems is no alternative. They are essential for preserving our living standard and, ofcourse, for environmental purposes. Since many years, scientists have been developingdifferent approaches which all try to get these issues under reasonable control.Basically, the common models can roughly be divided in two groups, deterministic modelsand stochastic models. Natural systems can bemodeled as deterministic systems, by claim-ing all processes, parameters, boundary and initial conditions to be known. As a result,deterministic models claim their results to be true, or assume that a single best predic-tion is suf icient. Stochastic models, in contrast, admit the inherent uncertainty, and henceprovide a range of possible predictions, with probabilities assigned to them. Uncertain pa-rameters are not assumed to be known, but probability functions are used to handle them.Often deterministic models are good enough for making basic predictions or to providefundamental system and process understanding. But, when there is need for reliable de-scription of complex, dynamic systems or for uncertainty quanti ication, to provide robustdecision support, they fail.

1 Introduction 4

The complexity, scale and uncertainty of natural systems are the primary reasons for mostof themodeling-dif iculties which have to be handled. Themain dif iculties are the scale ofreservoirs in general and scale issues from small to large scales in speci ic, the challengesof solving the coupled multiphase low differential equation systems and, of course, thelarge number and uncertainty of parameters backed up only by error-prone and scarcedata sets that characterize reservoirs only incompletely. All this leads to highly complexand computationally expensivemodels. Even in our days, where processing power becameeasily available and extremely cheap compared to past decades, there is still a great needfor novel approaches.A technique for handling the lack of prior information in reservoir simulation is historymatching. History matching uses existing recorded or real-time data on the state of thesystem (e.g. observation well pressure) to match it with model output, i.e., to calibrate themodel parameters. The matching is typically based on brute-force optimization. This is avery useful but expensive way for model calibration under uncertainty. Such approachesare usually not feasible for large-scale, complex and uncertain systems.This thesis is now focussing on an approach, which shall be helpful to make accurate his-torymatching techniques ef icient, even for large-scale, complex anduncertain systems.

1 Introduction 5

1.2 Summary of the State of the Art

1.2.1 History Matching

The relevance of and the interest in History Matching began to rise with developing geo-statistical science. In the ield of history matching, irst approaches have been made byMatheron [1963]. Later, Deutsch [1992] applied history matching to petroleum reservoirmodels.At this time, scientists began to implementmore andmore heterogeneity in reservoirmod-els [Oliver and Chen, 2011]. Evaluating the effect of more complex heterogeneity andthe large number of parameters let the demand for computational power increase sig-ni icantly. With the risen interest in heterogeneity, the interest in model calibration anduncertainty quanti ication went up as well. Figure 1.1 shows the annual number of publi-cations related to historymatching, whichwere published in the SPE (Society of PetroleumEngineers) Journal.

Figure 1.1: Number of papers on history matching prepared by each year for SPE conferencesand journals. [Corradi, 2010]

Theadvancementsmade in the ieldofHistoryMatching are versatile and sometimes closelyrelated to adaptions. Therefore, they will not be reviewed here in detail. Instead, the im-provements in History Matching shall be illustrated with a comparison of the state-of-the-art in 1993 and 2009, summarized from Oliver and Chen [2011].Makhlouf et al. [1993] published an example of an automatic History Matching techniquefor synthetic two- or three-phase reservoir data. The model domain is made up of 450grid-cells. 15 wells are implemented. For reducing the squared data mismatch function,they used 450 parameters, 1413 observed pressure data values and 785water cut and ratedata values. The simulation history was 10 years. In total, 11 cpu-hours were necessaryfor reducing the squared data mismatch function by a factor of 200.A state-of-the-art three-phasemodel of HistoryMatching in 2009 includes production dataat 30 wells, 20 production and 10 injection wells, and time-lapse seismic data for estimat-ing approximately 270.000 parameters. The used coarse grid which consists of 540.000

1 Introduction 6

cells was obtained by upscaling a high-resolution model with 20 million cells. The latterinclude horizontal and vertical permeabilities, porosities, net-to-gross ratio of each grid-block, luid contacts, fault transmissibility and relative permeability parameters. [Oliverand Chen, 2011, Peters et al., 2010].This comparison shows that there has been a great progress during the last two decades.It has become possible to history match quite large-scaled reservoir models with a greatamount of uncertain parameters. It is now possible to handle amultiplicity of different pa-rameters in a much higher number. However, until now there is no generally best method.Depending on the data and the problem, a suf icient method has to be chosen. Since thereare still many remaining challenges with complex geology, History Matching can not beconsidered as truly practical yet [Oliver and Chen, 2011]. The history matching techniquewhich probably gained the greatest interest during the last years is Ensemble Kalman Fil-tering.

1.2.2 The Ensemble Kalman Filter

The Ensemble Kalman Filter (EnKF) is a recursive ilter which is used in data assimila-tion for non-linear, inverse problems. It was irst generally introduced by Evensen [1994].Lorentzen et al. [2003] applied it later to the petroleum science. The EnKF is a advance-ment of the simple Kalman Filter (KF), introduced by Kalman [1960]. Due to its simpleconceptual formulation and easiness of implementation, the EnKF has gained great atten-tion. Evensen [2003] and Aanonsen et al. [2009] are describing practical adaptions of theEnKF for history matching problems. There are several examples where EnKFs are usedfor closed loop reservoirmanagement. Closed loopmeans that oil production is being con-trolled and optimized while accouting for data from real-time monitoring [Brouwer et al.,2004, Naevdal et al., 2006].The basic enhancements compared to the KF are the applicability to large-scale problems.For more details see Evensen [2003]. EnKFs are Monte-Carlo based mechanisms, using airst-order second-moment version of Bayesian updating. Thus they are closely related toparticle ilters (PF) because both ilters approximate Bayes' theorem. An important factand the biggest difference to PF is, that, in the EnKF, all the underlying distributions aretechnically supposed to be Multi-Gaussian for optimal performance [Evensen, 2003]. ThePF releases this restriction, but is typically more computer intensive.As a consequence of the previously described characteristics, an EnKF is basically not suit-able for stronglynon-linear andnon-Gaussianproblems,which is themajor shortcomingofthis technique. In fact, Nowak [2009] showed that the EnKF technique can be routed backto a certain type of implicit linearization. There are several approaches of Bengtsson et al.[2003] and Anderson and Anderson [1999]which try to build hybrid techniques, based onEnKFs, to handle the described issues. Bengtsson himself is describing their approach as''not yet overwhelming`` [Bengtsson et al., 2003].

1 Introduction 7

1.2.3 Particle Filters and Bootstrap Filters

Particle Filters are being used for data assimilation in several different scienti ic areas.Doucet et al. [2001] introduces particle iltering in detail. Among other application areassuch as object-tracking, statistics or econometrics [Djuric et al., 2003], PFs arewidely usedfor data assimilation in geophysical models. Van Leeuwen [2009] gives a general overviewand adaptions to marine and atmospheric applications.This thesis is concernedwith bootstrap ilters (BF) for historymatching. BFs are the limit-ing case of PFwhen only uncertain parameters arematched, but predicted states ofmodelsare left untouched. Predicted states will only be matched via changes in matched param-eters. The advantage of PFs and BFs is that they are precise implementations of Bayesianupdating for statistical inference and stochastic inversion at the limit of large ensemblesize.Within this context, particles are random sets of variables (here: parameter sets) withassociated weights. The weights are adjusted in such way, that ''good- itting'' sets get highweights and vice versa. In the next step, particles with low weights are being '' iltered''and only particles with high weights are taken into account for further steps. A detaileddescription of particle iltering is given in section 3.4.The biggest drawback concerning PFs and BFs are their extremely high computationalcosts. Nowadays, the available computational power is still not suf icient for a ef icientcombination of PFs with realistic large-scale models. Nevertheless, PFs receive more andmore attention.

1 Introduction 8

1.3 Approach and Bene its

The previous sections clearly describe the current shortcomings in the state-of-the-art.When it comes to stochastic, non-linear systems that involve non-Gaussian distributions,particle ilters and related techniques have, apparently, signi icant advantages over EnKFs.A two-phase reservoir model, as described later in this work, has strongly non-linear be-havior. The underlying uncertainty, e.g. permeability, is usually not considered to be Gaus-sian even before matching to data. The non-linear character of inversion and data assim-ilation will further contribute the non-multi-Gaussian character of the involved distribu-tions.The approach of this thesis is to use bootstrap iltering for history matching. The irst stepis to reduce the initial model to a so-called response surface. It can be seen as a black-box model built by analytic polynomials. The used technique is named polynomial chaosexpansion (PCE). PCE is a technique which is able to reproduce the non-linear behaviorof the multiphase low differential equation system properly. The polynomial responsesurface, once it is set up, can be evaluated vastly faster than the originalmodel. The secondstep is the historymatching step. Weuse a bootstrap ilter to implement Bayesian updatingon the reducedmodel. Because the response surface is so cheap to evaluate,we canoperatetheBootstrap ilter at the limit of large sample size, where it converges to the exact solutionof Bayes' theorem. The combination of model reduction with accurate Bayesian updatingin this approach seems to be promising. It allows for a new compromise between accuracyand computational ef iciency. Therefore, we expect better results for HM compared to theusage of EnKFs.As the goal of this thesis is fundamental method advancement, it does not require realdata, but works with numerically generated synthetic data. This allows to investigate themethodwithout the complications ofmodel error thatwould be unavailablewhenworkingwith real data.The used techniques are described in detail in chapters 2 to 4. Chapter 5 gives more infor-mation about the actual reservoir model used in this thesis, the numerics and the applica-tion to historymatching. In chapter 5, the individual steps of the historymatching processare described. Finally, chapter 6 is used for a discussion of this novel approach.

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2 Multiphase Flow in Porous Media

The purpose of this chapter is to explain the necessary theoretical background of the mul-tiphase low system considered in thiswork. For extensive derivations of equations, pleaseind suitable literature. Appropriate references will be given.Due to the extent and complexity of multiphase low in general, only the case relevant tothis thesis shall be discussed. The following section provides the basic de initions, whichare essential for global understanding.

2.1 Assumptions Relevant to this Thesis

A detailed and complete description of the model used for history matching and its imple-mentation is given in chapter 5. The explanations in this section are based on the followingassumptions:

• Oil and water are two immiscible luids, and there is no mass-transfer between thephases.

• The system is isothermal.• No hysteretic or dynamic effects in the multiphase constitutive relations are consid-ered.

These statements merely reduce the complexity of the physical reservoir model. Becausethey hold for both the synthetic reservoir data which is used for history matching and forthe forward reservoir model, they do not affect the history matching in any way. Throughsimplifying the reservoir model, we reduce the computational cost and therefore the com-putational time. Instead of investigating physical complexity, this allows for more detailedtests of the proposed tools for history matching.

2.2 Basic De initions

Porosity When dealingwith porousmedia, e.g. rock, porosity ϕ is de ined as the fractionof void space. The void space is the space where luids are able to reside or low. It isde ined by:

2 Multiphase Flow in Porous Media 10

ϕ = VV

VT

(2.1)

VV is the void volume in the medium and VT is the total volume of the medium.

Permeability The permeability k is a rock property which determines the ability of aporous media to let luids pass. The SI unit is m2. In petroleum engineering, the unitmilli-darcy (for short: mD) is common (1 D = 9.86923 · 10−13m2).

Saturation The saturation S is the volume fraction of a liquid or a phase α in the porespace of a medium.

Sα = Vα

VV

= Vα

VT · ϕ(2.2)

Vα is the volume of the phase, respectively luid. In a multiphase system, all saturationssum up to 1:

n∑α=1

Sn = 1 (2.3)

Component A component κ is a compound in the systemwith particular chemical char-acteristics. A component is chemically independent. The number of components deter-mine the minimum number of independent substances to de ine the composition of allphases [Helmig, 2008]. With nomass transfer considered, the concept of components willplay an insigni icant role in this thesis.

Phase A phase α is a region in a system with speci ic physical properties, which vary toother regions in the system. Regions with the same properties are called phases. Phasescan consist of several components.Multiphase systems often distinguish between wetting and nonwetting phases. The wet-ting phase is indicated by αw, the non-wetting phase by αn.Wetting is the ability of a phase to wet the surface of a porous medium. In the consid-ered two-phase system (oil and water), water is the wetting phase compared to oil. Inthree-phase systems, comprised of oil, water and gases, oil is a wetting phase compared togas.When taking non-isothermal effects into account, one would have to pay special attentionto thermo-dynamical luid properties. Changes in, e.g., temperature or pressuremight leadto phase changes or phase transformations, and could in luence luid properties such asviscosity or density.Capillary Pressure The discontinuity in luid pressure between any two immiscible lu-ids is called capillary pressure pc, e.g. Chen et al. [2006]. It is de ined as


pc = pn − pw (2.4)

wherein pn and pw are the particular pressures at the respective phase boundary. Whenworking on the macro-scale, capillary pressure is strongly dependent on the respectivesaturation. The capillary-pressure-saturation-relationship

pc = pc(Sw) (2.5)

is furthermore dependent on the surface tensionσ, porosityϕ, permeability k, and the con-tact angle Φ between the rock surface and the wetting phase. The latter is also dependedon the temperature and the chemical composition of the components.The pc(Sw) curve can vary in terms of the saturation history (drainage vs. imbibition), andmay depend on the speed of these processes. These phenomena are called hysteresis anddynamic effects, respectively. In this thesis, hysteresis and dynamic effects shall not be fur-ther considered for the sake of simplicity.

2.3 Continuity Equation

One of the irst steps towards understanding themultiphase low equation is conservationof mass. This is expressed through the continuity equation, which is derived from the gen-eral balance equation, e.g. Helmig [2008]. The balance equation for an Eulerian space isused in this form:

L(e) :=∫G

[∂e

∂t+ div(qe + w) − r

]dG = 0 (2.6)

L is the change of the respective ield quantity,e is the respective scalar ield quantity e = e(x, t), for which the balance equation is for-mulated,G is the integration domain, dG = dxdydz,qe describes the advective luxes, where q is a speci ic discharge, within the control vol-ume,w speci ies the diffusive part of the luxes andr is a source and sink term.The continuity equation results when balancing the luid-mass mg by using density ϱ asield quantity. Multiplyingwith the porosity ϕ accommodates to the fact that not the entirevolume is illed by the luid.


L(Φ, ϱ, v) :=∫G

[∂(ϕϱ)

∂t+ div(ϕϱva) − ϱq

]dG = 0. (2.7)

wherein

va = q

ϕ(2.8)

is the effective low velocity.Equation2.7, which is only valid for fully saturated single-phase low, has now tobewrittenfor each single phase:

L(Φα, ϱα, vα) :=∫G

[∂(ϕαϱα)

∂t+ div(ϕαϱαvaα) − ϱαqα

]dG = 0 (2.9)

Φα = SαΦ. (2.10)

The combination of 2.9 and 2.10 delivers the continuity equation for each phase α:

L(Sα, ϱα, vα) :=∫G

[∂(Sαϕαϱα)

∂t+ div(ϱαqα) − ϱαqα

]dG = 0. (2.11)

2.4 Darcy's Law

The Darcy-velocity for single-phase low in porous media can, under certain assumptions,be de ined as the following:

q = −k

µ· h. (2.12)

µ denotes the dynamic viscosity. q is the velocity vector

q =

qx

qy

qz

.

k represents the absolute permeability tensor

k =

kxx kxy kxz

kyx kyy kyz

kzx kzy kzz

.


If the Cartesian coordinate system is chosen in a way that its axes coincide with the princi-pal directionsof permeability, thepermeability tensork reduces to adiagonal tensor:

k =

kx 0 00 ky 00 0 kz

.

The gauge level h combines the elevation z and the hydraulic pressure head pρg:

h = z + p

ρg.

The gradient of the gauge level h is de ined as

h =

∂h∂x∂h∂y∂h∂z

.

According to Scheidegger [1974], Darcy's low-law can be extended to multiphase- lowproblems. Again, each single phase must be described separately. The extended Darcy'slaw is written as

vα = −krα

µα

k · (pα − ϱαg), (2.13)

wherein µα is the dynamic viscosity for the respective phase and krα names the relativepermeability for each phase. The relative permeability describes the dependency of thepermeability on the saturation.

0 ≤m∑

α=1krα(Sα) ≤ 1, with α = n, w. (2.14)

krα/µα = λα is often named the mobility of phase α.In addition, constraint 2.15 has to be ful illed:

0 ≤ krα ≤ 1, ∀ α. (2.15)

Beneathnumerousother approaches andmodels, popular relative-permeability-saturationfunctions are given by Brooks and Corey (1964) and Van Genuchten (1980). Basically, rel-ative permeability kr is describing the phenomenon that the permeability kα of a phase isincreasing with its saturation Sα. This behavior is strongly non-linear. Simpli ied, it can beexplained that a phase can only low in the fraction of the pore space that it is occupyingitself.


saturation S [-]w

relative permeability [-]

Figure 2.1: Relative-permeability-saturation-relationships according to Brooks and Corey andVan Genuchten. [Helmig [2008]; adapted]

2.5 The Multiphase Flow Differential Equation

2.5.1 The Global Formulation

The global form of themultiphase low differential equation follows from the combinationof equations 2.11 and 2.14. From this, one gets a coupled, dynamic system of partial dif-ferential equations consisting of

Lα(Sα, ϱα, pα) = ϕϱα∂Sα

∂t+ Sαϱα

∂ϕ

∂t+ ϕSα

∂ϱα

∂t− div

[ϱα

krα

µα

k (pα − ϱαg) − ϱαqα

],

(2.16)for each phase α and with the side conditions

m∑α=1

Sα = 1 (2.17)

and

pc = pn − pw = f(Sα). (2.18)

This equation is able to describe the simultaneous low for two or more immiscible luidsin a porous medium. The resulting coupled equation system is not ideal in the light ofnumerical implementation. There is a strong underlying non-linear dependency of the


saturations Sα on the respective capillary pressures pc and, furthermore, of the relativepermeability krα on the saturations Sα. These non-linearities get even stronger when, i.e.issures produce sudden changes in physical properties or low characteristics. In general,these non-linear behaviors are computationally hard to handle.To overcome this issue, there are several distinctive formulations of equation 2.16, whichuse different primary variables. According to the speci ic problem, one has to choose oneof these different formulations. It may, e.g., either depend on the primary variables or theparticular drawbacks coming with each formulations. Some important formulations willbe stated in subsections 2.5.2, 2.5.3 and 5.3.

2.5.2 The Pressure Formulation

Since the capillary pressure saturation relationship is strictlymonotonic decreasingwithina homogeneous medium, there is a unique inverse function

Sα = p[−1]c (pn − pw). (2.19)

pn and pw shall now be used as the primary unknowns. In that case, it follows for the wet-ting phase

L(pw) :=∂(ϕϱwp[−1]

c

)∂t

− div [λwϱwk (pw − ϱwg)] − ϱwqw (2.20)

and for the non-wetting phase

L(pn) :=∂(ϕϱn(1 − p[−1]

c ))

∂t− div [λnϱnk (pn − ϱng)] − ϱnqn. (2.21)

This system was irst introduced by Douglas et al., 1959, within the simultaneous solutionscheme in petroleum reservoirs, e.g. Chen et al. [2006].Thebiggest disadvantageof this formulation is, that it cannot be applied for heterogeneousmedia, e.g. Helmig [2008]. The necessary condition (eq. 2.19) that ∂pc/∂Sw = 0 is not fullilled in issures or next to transition zones.

2.5.3 The Pressure-Saturation Formulation

When pw and Sn are used as the primary variables, the following equation system can beobtained, e.g. Chen et al. [2006]:


Taking∇pc = ∇ (pw + pcn)

and

∂Sw

∂t= ∂

∂t(1 − Sn) = −∂Sn

∂t

into account, one gets for the wetting phase

L(pw, Sn) := −∂(ϕϱwSn)∂t

− div [λwϱwk(∇pw − ϱwg)] − ϱwqw, (2.22)

and for the non-wetting phase

L(pw, Sn) := ∂(ϕϱnSn)∂t

− div [λnϱnk(∇pw + ∇pc − ϱng)] − ϱnqn. (2.23)

This parabolic strongly-coupled equation systems comes alongwith several distinctive ad-vantages. Since the capillary pressure effects are directly implemented in equation 2.23,the equation is valid in regions with low ∂pc

∂Swgradients. Therefore, it can be used for is-

sured or fragmented porous media. Furthermore, it is valid for both homogeneous andheterogeneous media. Choosing pw and Sn as primary variables is not the only possiblechoice. Depending on the problem and its description, different combinations, i.e., pn/Sw

or pw/Sw are also possible.

2.5.4 The Total Pressure-Velocity Formulation

In the case of two incompressible luids, likewater andoil, this differential equation systemcan be solved successively, in a way that the pressure can be calculated in a seperate irststep. In the second step, the saturation can be calculated, and depends on the pressuresolution of the irst step.The Saturation equation:

ϕ∂Sw

∂t= qw − div

[Kfw(Sw)λn(Sw)

(dpc

dSw

∇Sw + (ϱn − ϱw)g∇Z

)+ fw(Sw)v

](2.24)

With the side-conditions


pc = pn − pw, and v = vn + vw. (2.25)

Furthermore, we are taking into account that

div(v) = q(p, S) ≡ qw(p, S) + qn(p, S) (2.26)

and

v = −K (λ(S)∇p − λw(S)∇pc − (λwϱw + λnϱn) g∇Z) . (2.27)

S is the saturation, qw/n the external volumetric low rate for the respective phase,K is theabsolute permeability tensor, fw is the fractional low function, λ is the mobility of the re-spective phase, ϱ represents the density, g the gravitational constant,Z the geodetic heightand v the absolute Darcy velocity.

Pressure equation:

div (Kλ∇p) = div(K(λw∇pc + (λwϱw + λnϱn)g∇Z) − q (2.28)

With p = pn, the pressure of the non-wetting phase.This formulation works best when solving the pressure equation implicitly and the satu-ration equation explicitly in time. This scheme is commonly known as the sequential im-plicit pressure-explicit saturation scheme (IMPES) and is widely used in petroleum reser-voir computation.

2.6 The IMPE-Scheme

2.6.1 A Remark on Stability

There is a basic difference between explicit and implicit schemes concerning stability. So-lutions of implicit schemes do not show oscillating behavior nor error propagation. Thus,they are absolutely stable. In contrast, explicit schemes are not necessarily stable. One hasto choose the time step ∆t suf iciently small in comparison to the the discretization steps.The implicit time step can be chosen independently of any stability-reasons. For detailedstability analysis see Helmig [2008] and Chen et al. [2004].


2.6.2 The IMPE-Scheme

The following description is proposed by Chen et al. [2004]. The basic spatial discretiza-tion method used for solving is a mixed inite-element method on a N0 = 13 · 13 structuredcartesian grid. We de ine now the time interval J = (0, T ); T > 0 and the time stepst = t0 < t1 < . . . < tN ; N ∈ N. The proposed scheme solves the pressure implicitly in aseparate irst step. The saturation S in 2.28 is known either from the initial condition orfrom the solution of equation 2.24 in later time steps.The difference between explicit and implicit is, that implicit solving schemes include thein luence of the unknown variables of the neighbor-cells or knots out of the following newtime step. Explicit schemes, in contrast, use only values of the previous old time step.Therefore, explicit schemes do not have to solve an equation.

Upon discretization of equations 2.24 and 2.28 for each time step ti, i ∈ 0, 1, . . . , N, pi

satis ies the implicit equation

− div [Kλ(Si)∇pi] = F (pi, Si). (2.29)

F (pi, Si) stands for the right-hand side of equation 2.28, and Si is given.Si+1 is obtained from the explicit relation

ϕ∂Si+1

∂t= G(pi, vi, Si) (2.30)

G(pi, vi, Si) stands for the right-hand side of equation 2.24.After all necessary boundary and initial conditions are ixed, equation 2.29 and Si are usedto compute pi; equation 2.27 is used to evaluatev. The next step usesSi, pi, vi and equation2.30 to compute Si+1.To avoid numerical problems and assure a stable solution, the time stepti = ti −ti has tobe suf iciently small. For stability analysis, detailed explanation and new approaches thereader should refer to Chen et al. [2004].

2.7 Boundary and Initial Conditions

The equations in the subsections of chapter 2.5 are formulating boundary and initial valueproblems. An essential part of this kind of problems consists of boundary and initial con-ditions. The entire model domain with its external boundary and the initial state has to bedescribed with suf icient conditions.The entire model domain is named G. Its boundary is termed Γ. Since the types of condi-tions on the boundary are often different on different parts of the boundary, the boundaryis often divided in parts Γ = Γ1 ∪ . . . ∪ Γn, with n ∈ N.


In general, there are three different kinds of boundary conditions (bc):Dirichletbc When thepressure is known in space and timeonsomesectionΓi, it reads

pα = uα,D on Γi. (2.31)

Neumann bc When the mass lux across some section Γi is know, it is

ϱαvα · n = uα,N on Γi. (2.32)

Here, n is the outward unit normal to Γi. For impervious boundaries, equation 2.32 be-comes uα,N = 0.Cauchy bc If the boundary is semipervious, it leads to amixed bc.

βpα + γϱαvα · n = uα,C on Γi, (2.33)

where β and γ are given functions.

20

3 History Matching

3.1 History Matching

A common task in reservoir engineering is to make predictions about the future and be-havior and performance of reservoirs and of installed wells, e.g. production rates. This re-quires to bring a (numerical) simulationmodel into correspondencewith the real reservoirand its properties as good as possible, e.g. by adjusting the model's parameters, boundaryand initial conditions such that the model matches observed data from the real reservoir.A widely-used approach to this challenge is history matching (HM). The procedure is any-thing but simple. ''History matching is as much art as science'' [Fanchi, 2005].Before going more into detail, the two basic terms forward problem and inverse problemshall be clari ied.Forward problem A forward problem is a deterministic model. Themodel parameters,e.g. physical properties, are known. It can be used to predict future outcomes of the systembehavior. Forward problems usually consist of differential equation systems with certainboundary and initial conditions.A forward problem of interest in petroleum engineering is, e.g., a simple one-phase steady-state low inporousmedia. According to section2.4 andequation2.12, it canbe formulatedas:

q = −k(x)µ

· h, (3.1)

with its boundary and initial conditions

dp

dx|x=L = − qµ

k(L)Aand p(0) = pa.

k(x) represents the permeability ield, A the cross section of the low, q the low-rate, pa

the pressure and µ the dynamic viscosity. The length of the system is described by L. Thevariables pa, A, q and µ are assumed to be constant.As all the variables aredetermined, the outcomeof the system is dependent on the constantout low q on the right side (x = L) and the constant pressurepon the left side (x = 0). Mostforward problems like this are or can be made well posed [Oliver et al., 2008]. Accordingto Hadamard [1902], problems are well posed, if they ful ill the following criteria:

3 History Matching 21

• The problem has a solution.• The solution is unique.• The solution is a continuous function of the problem data.

Inverse problem In the most general sense, an inverse problem, respectively inversemodeling, is the determination of feasible physical properties or information about physi-cal properties, given an observed response of a system [Oliver et al., 2008]. When restatingthe steady-state example of the previous paragraph, the inverse problemwould, e.g., be thedetermination of the permeability ield k(x) by using measured pressure data p(x) at cer-tain locations.A common nature of inverse problems is, that they are almost always ill posed. Ill posed,the opposite ofwell posed,means that different parameter values result in (almost) equallygood matches [Oliver et al., 2001, 2008]. When taking into account that measurements ingeneral, in our case pressure measurements, are noisy and thus not exact, and that thetheoretical model with its assumptions, e.g. constant viscosity µ, may not be correct, thereare errors which have to be regarded thoroughly. As a consequence, the de inition of in-verse problems can be reformulated in the sense that one has to determine feasible physi-cal properties, given inexact, respectively uncertain input data and an assumed theoreticalmodel, which relates the observed data to the model [Oliver et al., 2008].For inding an inverse problem solution which is as close as possible to reality, one usuallytries to include available prior information about the system. Examples might be prior in-formation based on core data fromwells or geological information from, e.g., seismic tests.Althoughoften implemented by simplemeans such as bounded optimization or Thikhonovregularization, this very important step inmodeling is related to Bayesian statistics. In theBayesian context, the prior uncertain information is usually included by using probabilitydensity functions. Detailed explanation will be given in chapter 3.3.Again, we can reformulate the de inition of inverse modeling to the following: Determina-tion of feasible physical properties, given prior information on model parameters, inexactmeasurements of observed parameters and an uncertain relation between model param-eters and the related data. From this de inition, one can formulate the essential question:''Howshould theprior probability density functionbemodi ied to include informationpro-vided by inexact measurements?'' [Oliver et al., 2008]. The answer to that question will begiven in this chapter.

3.2 History Matching as an Inverse Problem

Basically HM is a processwhere ''observed reservoir behavior is used to estimate reservoirmodel variables that caused the behavior'' [Oliver et al., 2001]. HM begins with choosingthe data which shall be compared. Depending on the area (e.g. petroleum engineering or


CO2-sequestration) there are commonly used data types, e.g. Fanchi [2005] or Oladyshkinet al. [2011a]:

• Gas-oil production rate.• Water-oil production rate.• CO2 or oil pressure at observation wells.• Oil production rates.• Dissolved CO2 concentration in brine.• Seismic tests.

In our case, we will use pressure time-series in observation wells for the matching. Themodel will be a reasonably simple arrangement of one production well, surrounded byfour observation wells. This is a typical schematic setup, often called the ive-spot problemin reservoir engineering. As we do not have realistic production history data at hand, thesame model is used to generate synthetic production history data.The selection of the data for history matching often has deeper consequences for the fol-lowing steps. If, e.g., recorded observation pressure is chosen to be compared with calcu-lated cell pressure, there might be corrections necessary. Depending, amongst others, onthe grid size, the cell-averaged pressure might differ from the actual observed well pres-sure. A widely used pressure correction is the Peaceman (1978, 1983) correction [Fanchi,2005]. Nevertheless, the pressures can be compared directly, if the goal is a pressure trendinstead of actual values. Corrections might also become necessary when using, e.g., satu-ration or dissolved concentration.The model can be assumed to be ''correct'' (in the light of the data only), if the data match.In case not, the model's parameters are adjusted. This process of adjusting is time con-suming for computationally expensive models and extremely dif icult [Chen et al., 2006].For manual HM outside the Bayesian context, several simple rules of thumb exist. Formatching the data, the parameter with the highest uncertainty and sensitivy should beadjusted at irst. Often, the porosity ϕ or the permeability k is adjusted. Also, HMmust notbe achieved through physically meaningless parameter adjustment. For example whenadjusting porosity, the property should lie within the common range of the given rocktype.

3.3 Bayesian Updating

The idea of Bayesian updating is going back to Thomas Bayes (1702 - 1761), an englishmathematician, who formulated Bayes' theorem. Basically, Bayes' theorem can be usedto update a prior probability estimation, when receiving new additional information. Thenew estimation is called posterior probability estimation.


In an arithmetic way, Bayes' theorem can be expressed through:P (A) is the prior probability of A,P (B) is the prior probability of B,P (A ∪ B) is the joint probability of A and B,P (B|A) is the conditional probability of B when A is already known,P (A|B) is the posterior probability for A when B is already known.

P (A|B) · P (B) = P (A ∪ B) = P (B|A) · P (A) (3.2)

P (A|B) = P (A ∪ B)P (B)

= P (B|A) · P (A)P (B)

(3.3)

A simple qualitative example:Bergen is supposed to be the city in Europewith the highest annual precipitation rate. Thefollowing numbers donot re lect the real situation, they are rather inspired by the personalimpression of the author. We are interested in the probability that it will rain on a day, forwhich the forecast predicted rain.We state that there is rain or snow on 250 days a year. In this example, we will only writerain.We have the following basic events:Event R: It is raining some time on any day of interest.Event F: The forecast predicts rain for that day.The probability P (R) can be calculated easily by 250/365 = 0.6849. Furthermore, we pre-tend to know P (F|R) = 0.8, the conditional probability of F given R. It means the prob-ability that the forecast correctly predicted rain for a day, on which it will actually rain.P (F|¬R) = 0.3 expresses the probability that the forecast wrongly predicted rain for aday, when you know that it did not rain that day.With this information, we can calculate the probability of the event P (F):

P (F) = P (R) · P (F|R) + P (¬R) · P (F|¬R) = 250365

· 0.8 + 115365

· 0.3 = 0.6425 (3.4)

Hence, P (F) is smaller than P (R). This implies that norwegian meteorologists are overlyoptimistic in the sense that they predict betterweather than it actuallymight be. The prob-ability we are looking for is the conditional probabilityP (R|F). According to equation 3.3,it can be calculated as the following:

P (R|F) = P (F|R) · P (R)P (F)

= 0.8528 (3.5)


Itmeans, the chance that there is rain on aday forwhich the forecast predicted rain is 85.28%. P (R|F) is apparently greater than P (R). This permits a relative obvious conclusion.When knowing the weather forecast, the chance to get surprised by the rain is lower thanwithout having any information about the weather.Given this information, we can also calculate the probability for the event that there is norain on a day for which the forecast did not predict rain:

P (¬R|¬F) = P (¬F|¬R) · P (¬R)P (¬F)

= 0.8 · 0.31510.3575

= 0.7051 (3.6)

Weassume theprobability for correct predictionof no rainP (¬F|¬R) tobe equal toP (F|R).If the used values would all be realistic, the fact that P (¬R|¬F) is around 15% lower thanP (R|F) would conclude, that the weather forecast in Norway is more accurate in predict-ing rain than sunshine.

Bayes' theorem can, for example, also be used for the determination of hydraulic conduc-tivity k of a certain soil. A practical way to do this is inding the grain-size distributioncurve. For this, the fractions of speci ic grain-size-classes are assessed.The logarithmic conductivity of sand is typically ln k = Y = −3, with a standard deviationof σ = 1. A measurement d results in a conductivity of ln kmeasured = Ymeasured = −3.5.The measuring error is assumed to be σ2

measure = 0.1.We can now formulate the a priori probability as

P (Y) = N (−3, 1). (3.7)

N (..., ...) means a Gaussian distribution with its parameters mean and the variance. Outof the measurement d and its error statistics we can express the possible truth Y behindthe observed value, regardless of the prior information, as:

P (Yd|Y) = N (−3.5, 0.1). (3.8)

Now, we can use Bayes' theorem to combine the prior information with the information inthe data:

P (Y|Yd) = P (Y) · P (Yd|Y)P (Yd)

= N (Y, σ2), (3.9)

with the analytical solution for the linear and Gaussian case:

Y = −3 + Cov(Y|Ym) · 10.1

· 3 − 3.5 (3.10)


and

σ2 = σ2 − Cov(Y|Ym) · 1σ2

measure + σ2 · Cov(Y|Ym) (3.11)

The correlationR2 betweenY andYmeasured can be calculated asR2 = 0.5. The covarianceis Cov(Y, Ymeasured) = σ · σmeasure · R2 = 0.05.Bayesian updating has become awidely used technique for historymatching, although theimplementation is not that trivial as it might seem in this section.

3.4 Particle and Bootstrap Filtering

Particle iltering (PF) is a strong tool for history matching. It is also known as sequentialMonte Carlo simulation or sequential Monte Carlo method. In general, Monte Carlo simu-lations are based on the law of large numbers. The basic theory behind the law of largenumbers can be summarized in the following way: An experiment with an uncertain out-come has a certain expected value E. When performing the same experiment many times,the averaged outcome should be close to the expected value E.The idea of PFs is to represent a posterior density or a hidden state of a system by a set ofrandomparticles. The particles consist of a point in the state space and associatedweights.The cluster of all particles approximates the probability density. The hidden state of asystem is a state which can not be determined due to distinctive reasons. In, e.g., objecttracking, the hidden state is the actual or future position of the object of interest. Sincethe object is moving and the measurements are defective, the position can not be locatedexactly. The hidden state in the area of reservoir modeling is usually given by the differentunseizable reservoir parameters, e.g. porosity ϕ.

In the following,weexplainparticle ilteringby stating the general problem in section3.4.1,followed by two techniques that derive PF from Monte Carlo Methods in sections 3.4.2 to3.4.5.

3.4.1 Problem Statement

We name the hidden states x at time step number t as xt (t ∈ N) and the correspondingobservations as yt. The initial distribution is denoted by p(x0). x0:t = x0, . . . , xt are thestates and y1:t = y1, . . . , yt the observations up to time t.Doucet et al. [2001] describes the threemain goals of PF as the following: The recursive in-time-estimation of the posterior distribution p(x0:t|y1:t), itsmarginal distribution p(xt|y1:t)and the expectations


I(ft) = Ep(x0:t|y1:t) [ft(x0:t)] =∫

ft(x0:t)p(x0:t|y1:t)dx0:t (3.12)

for some function of interest ft : ft → Rn. This includes, e.g., mean values, variances orexceedance probabilities for critical values.At any time t, the posterior distribution can be formulated using Bayes' theorem:

p(x0:t|y1:t) = p(y1:t|x0:t)p(x0:t)∫p(y1:t|x0:t)p(x0:t)dx0:t

(3.13)

The recursive formula for the joint distribution is

p(x0:t+1|y1:t+1) = p(x0:t|y1:t)p(yt+1|xt+1)p(xt+1|xt)

p(yt+1|y1:t)(3.14)

Themarginal distribution p(xt|y1:t) satis ies the following two recursive formulas forthe prediction

p(x0:t+1|y1:t+1) =∫

p(xt|xt−1)p(xt−1|y1:t−1)dxt−1 (3.15)

and the updatingp(xt|y1:t) = p(yt|xt)p(xt|y1:t−1)∫

p(yt|xt)p(xt|y1:t−1)dxt

. (3.16)

Equation3.16provides thehidden statesxt of the systemat time t, given all previousobser-vations y1:t. The following sections provide statistical sampling rules to evaluate equation3.16.

3.4.2 Importance Sampling (IS)

The underlying estimation technique of particle ilters is called sequential importance sam-pling (SIS) or sequential importance resampling, see sections 3.4.3 to 3.4.5. SIS and SIR arean advancement of importance sampling (IS). IS is a basic Monte Carlo mechanism whichcan be used for Bayesian updating or, more generally, for sampling from a distribution thatis not available in explicit form. When evaluating properties of distributions, one can usethe fact that certain random input parameters have a greater impact on the distributionthan others. For favoring the parameters with higher impact, IS uses random particleswith assigned weights. The weights are proportionally adjusted according to the impact,


respectively, importance of the certain input [Glynn and Iglehart, 1989]. In the following,the afore mentioned techniques are introduced according to Doucet et al. [2001].Firstly, we introduce an arbitrary proposal distribution, respectively, importance functionπ(x0:t|y1:t). We account for the possible dependency of π(·) on y1:t by writing π(·|y1:t). Weassume that π(·) includes the support of p(x0:t|y1:t).The identity I(ft), equation 3.12, can be written as

I(ft) =∫

ft(x0:t)ω(x0:t)π(x0:t|y1:t)dx0:t∫ω(x0:t)π(x0:t|y1:t)dx0:t

, (3.17)

wherein ω(x0:t) is de ined as the importance weight

ω(x0:t) = p(x0:t|y1:t)π(x0:t|y1:t)

.

A possible Monte Carlo estimation rule is obtained when sampling N identical and inde-pendently distributed particles

x[i]

0:t, i = 1, . . . , Naccording to π(x0:t|y1:t). A possible es-

timation of I(ft) is

IN(ft) ≈1N

∑Ni=1 ft(x[i]

0:t)ω(x[i]0:t)

1N

∑Nj=1 ω(x[i]

0:t)=

N∑i=1

ft(x[i]0:t)ω

[i]t , (3.18)

wherein ω[i]t stands for the normalized importance weights

ω[i]t = ω(x[i]

0:t)∑Nj=1 ω(x[i]

0:t).

Accommodating for the law of large numbers, one can state that

limN→+∞

IN(ft) = I(ft). (3.19)

This sampling method is a basic rule to accelerate the convergence of Monte Carlo esti-mations. However, in the context of Bayesian updating, it can be interpreted in a differentway. A possible approximation of the posterior distribution p(x0:t|y1:t) is


P (x0:t|y1:t) ≈ PN(dx0:t|y1:t) =N∑

i=1ω

[i]t δx[i]

0:t(dx0:t), (3.20)

where δx[i]0:t

denotes a delta-Dirac probability mass at the point x[i]0:t.

IN(ft) is obtained by integrating ft (x0:t) with respect to PN (dx0:t|y1:t):

IN(ft) =∫

ft(x0:t)PN (dx0:t|y1:t) . (3.21)

The biggest drawback of IS for Bayesian updating is that, in its simplest form, it is not feasi-ble for recursive-in-time-estimating. All used data have to be available before the estima-tion. Basically, when receiving new information about the system, the importance functionπ(·) changes. Therefore, all previous simulated trajectories

x[i]

0:t−1; i = 1, . . . , Nhave to

be computed anew. Hence, the whole sample including all weights would have to be com-puted again. This step gets computationally more and more expensive with proceedingtime. For handling this issue, sequential importance sampling techniques have been de-veloped.

3.4.3 Sequential Importance Sampling (SIS)

Basically, SISmodi ies IS in a way that an empirical estimate PN(dx0:t|y1:t) can be obtainedfor p(x0:t|y1:t) without recomputing the past trajectories

x[i]

0:t−1; i = 1, . . . , N. The key

idea, according to Doucet et al. [2001], is that ''The importance function π(x0:t|y1:t) at timet admits as marginal distribution at time t − 1 the importance function π(x0:t−1|y1:t−1).'' Itcan be expressed through

π(x0:t|y1:t) = π(x0:t−1|y1:t−1)π(xt|x0:t−1, y1:t). (3.22)

By iteration, one gets:

π(x0:t|y1:t) = π(x0)t∏

k=1π(xk|x0:k, y1:k). (3.23)

When adapting the prior distribution as the importance function, which is basically theidea of particle ilters, one obtains:


π(x0:t|y1:t) = p(x0:t) = p(x0)t∏

k=1p(xk|xk−1). (3.24)

Compared to IS, SISmethods aremore adequate for recursive Bayesian updating. However,the remaining problem of SIS methods is that the importance weight distribution ω

[i]t gets

more andmore skewedwith progressing time. The result is that, after a few time steps, allbut one single weight become 0 [Snyder et al., 2008]. To overcome this issue, an additionalstep is introduced, see section 3.4.4, which inally leads to the Bootstrap ilters.

3.4.4 Sequential Importance Resampling (SIR)

Themain goal of SIR is to eliminate lowweights and to boost higher ones. Basically, it is thesame approach like SIS. But, in contrast to SIS techniques, one replaces the empirical distri-bution by an according unweightedmeasurewhenever necessary. This is achieved by sub-stituting the normalized weights ω

[i]t by the use of N

[i]t , which is the number of offsprings

associated to a particle x[i]0:t. Offsprings are unweighted particles which are distributed ac-

cording to the weights received in the importance sampling step. The individual steps areexplained in detail in section 3.4.5.The empirical distribution

PN(dx0:t|y1:t) =N∑

i=1ω

[i]t δx[i]

0:t(dx0:t) (3.25)

is replaced by the unweighted measure

PN(dx0:t|y1:t) = 1N

N∑i=1

N[i]t δx[i]

0:t(dx0:t). (3.26)

For the integers N[i]t the constraint 3.27 has to be ful illed:

N∑i=1

N[i]t = N. (3.27)

Moreover, N [j]t = 0 means that a particle x[j]

0:t is not further considered in the selection stepand, consequently, dies. The key challenge of this technique is to chose N

[i]t in a way that

PN(dx0:t|y1:t) is close to PN(dx0:t|y1:t). This has to be ful illed in the sense that for anyfunction ft holds

∫ft(x0:t)PN(dx0:t|y1:t) ≈

∫ft(x0:t)PN(dx0:t|y1:t). (3.28)


One option to achieve this is to perform ''drawing with returning'' from the particles x[i]

with probabilities proportional to the weights ω[i]. After the particles x[i]0:t with weights

ω[i]0 > 0 have been chosen, their offspringsN

[i]t are redistributed according to theweighted

measurex[i]

0 , ω[i]0

. A possible algorithm is described in sectin 3.4.5.

3.4.5 SIR: The Algorithm

Figure 3.1: Updating a particle cloud based on the weights. (Doucet et al. [2001]; adapted)

The steps listed with bullet-points are the steps of the algorithm in general. The descrip-tions below the listings are referring to the example in igure 3.1.1)

• For i = 1, . . . , N sample x[i]0 ∼ p(x0) and set t = t + 1.

[Time step: t = 0] This step is called the initialisation step. The yellow dots are aset of random particles

x[i]

0 , N0drawn from the prior PDF, i.e., their density is not

yet proportional to the posterior PDF at time step t + 1. It is an approximation ofp(x0|y1:t−1). We have a total number of N = 10 particles x[i]

0 .2)

• For i = 1, . . . , N sample x[i]0 ∼ p(xt|x[i]

t−1) and set x[i]0:t =

(x[i]

0:t−1, x[i]t

).

• For i = 1, . . . , N evaluate the importance weights ω[i]t = p

(yt|x[i]

t

).


• Normalise the importance weights.

[Time step: t = 1] Within the importance sampling step the weights are adjusted bycomparing the particle stats individually with the given data from time step t = 0.The degree of matching data is measured by the likelihood that the respective statevalues of each particle could have produced the data. High accuracy in matching thedata results in high weights, low accuracy in low, respectively in zero weights. Theblue dots represent the weightedmeasure

x[i]

0 , ω[i]0

which results in an approxima-

tion p (x0, y1:t−1).

3)In the selection step the particles are, depending on the weights, again distributed.In areas with higher probability, the particle density will be increased. The ittestparticles result in the unweightedmeasure

x[i]

0 , N0, which is still an approximation

of p (x0, y1:t−1).4)

• Resample with replacement of N particles(x[i]

0:t; i = 1, . . . , N)from the set(

x[i]0:t; i = 1, . . . , N

)according to the weights.

• Set t = t + 1 and go to step 2).

The samplingorprediction step introduces variety. It results in themeasurex[i]

1 , N0,

which is an estimation of p (x1|y1:t−1) .

5)The blue dots in step 5 generate the weighted measure

x[i]

1 , ω[1]1

.

32

4 Polynomial Chaos Expansion

4.1 Response Surfaces

Within this thesis, polynomial chaos expansion (PCE) is used to generate a model responsesurface. This means that, the response surface will be constructed in a closed polyno-mial form, and orthogonal high-order polynomials are used. Response surface methodsare common, whenever the dependency of the model output on uncertain or otherwisevariable model input shall be characterized respectively approximated, e.g., Oladyshkinet al. [2011b]. A model response surface is an immense reduction of the model, whichcan be considered as a black box. In this case, we use the arbitrary PCE (aPC) technique.Common PCE techniques are based on Hermite polynomials, which form an ideal basis forGaussian distributed input variables. Before non-Gaussian input variables can be used, itis necessary to adequately transform them. Suitable transformations are called Gaussiananamorphosis or normal score transformation [Oladyshkin andNowak, 2012, Xiu andKar-niadakis, 2003]. However, the transformation-step leads to a poorer convergence of thePCE to approximate the model. The aPC is a generalization of PCE techniques to arbitraryinput distributions. Using the aPC technique, transformations of non-Gaussian input vari-ables become needless. Hence, the aPC technique leads to better convergence. The keydifference between achieving the aPCE and the common PCE techniques is that the poly-nomials of order 1 to d, that form the basis P [1], . . . , P [d], have to be found speci ically forthe distributions' input variables at hand [Oladyshkin and Nowak, 2012].One can de ine two different groups of input parameters. One class, consisting of designand control parameters, can be chosen by the modeler or engineer. The other class, inwhich uncertain parameters are grouped, can not be freely in luenced in the same way.Approaches which combine both, design and uncertain parameters, are called integrativeapproaches. They result in integrative multidimensional response surfaces, containing allintegral information about the system. This approach opens the path to robust designand decision making under uncertainty [Oladyshkin et al., 2011b, Oladyshkin and Nowak,2012]. However, in this work we have a non-integrative approach, since our parametersof interest are only uncertain reservoir parameters such as permeability and porosity, andwe are interested in history matching, not in robust design.

4 Polynomial Chaos Expansion 33

4.2 Polynomial Chaos Expansion

In the following, we use the recent aPC technique introduced by Oladyshkin and Nowak[2012]. Adetailedderivationof the expansion canbe found in thementioned citation.The model Ω(·) is a function of its input parameters w:

Ω(·) = f(w) (4.1)

The multivariable input w is de ined as:

w = w1, . . . , wNU,

wherein NU is the number of uncertain parameters. The polynomials P [d](w) of degree dare de ined for a single input parameter w as:

P [d](w) =d∑

i=0p

[d]i wi. (4.2)

p[d]i represent the unknown coef icients for the individual monomials wi of order i in thepolynomial basis P [d](w). In this work, the polynomials up to degree d are constructedby a moment-based technique. For the construction of the polynomials Pd, only statisticalmoments of the stochastic model input w and the orthogonal condition are used. In thecontext of polynomials, orthonormality is de ined as:

∫w

P [k](w)P [l](w)dΓ(w) = δkl, (4.3)

with the Kronecker-delta δkl = 1, if k = l, and zero otherwise.The following system of equations provides a general construction rule for constructingan orthogonal polynomial basis for arbitrary input distributions:

∫w

d∑i=0

p[d]i widΓ(w) = 0;

∫w

d∑i=0

p[d]i wi+1dΓ(w) = 0;

. . .∫w

d∑i=0

p[d]i wi+d−1dΓ(w) = 0;

p[d]d = 1. (4.4)


Γ denotes the probability measure. The above stated system of equations de ines the kth

orthogonal polynomial independent of all other polynomials from the basisP [d] (w)

.

Equation4.4 canbe reformulatedusing thede initionof thekth raw-momentµk =∫

w wddΓ(w)as the following:

d∑i=0

p[d]i µi = 0;

d∑i=0

p[d]i µi+1 = 0;

. . .d∑

i=0p

[d]i µi+d−1 = 0;

p[d]d = 1. (4.5)

This linear system of equations can be written in a more suitable matrix-form:

µ0 µ1 . . . µd

µ1 µ2 . . . µd+1... ... . . . ...

µd−1 µk . . . µ2d−10 0 . . . 1

p[d]0

p[d]1...

p[d]d−1p

[d]d

=

00...01

(4.6)

The only necessary condition to ensure that a orthogonal polynomial basis can be con-structed is that the square matrix of moments in equation 4.6 not singular.Finally, the expansion itself is achieved according toWiener [1938]. Themain difference is,that we do not use, as proposed by Wiener, Hermite polynomials. The above constructedpolynomials are used in equation 4.7:

Ω(w) = α0 +N∑

i=1αiP

[i]1 (wi)

+N∑

i=1

i∑j=1

αijP[i,j]2 (wi, wj)

+N∑

i=1

i∑j=1

j∑k=1

αijkP[i,j,k]3 (wi, wj, wk)

+ . . . (4.7)


The coef icients α quantify the dependence of the model output on its input. The totalnumber of expansion terms M is de ined as the following:

M = (NU + d)!NU !d!

. (4.8)

Within the last step of the PCE, the unknown coef icients αi in equation 4.7 have to beassigned. The technique used in this work is called the probabilistic collocation method(PCM). Basically, the coef icients are matched to model outputs. The model is evaluatedwith a certain number of parameter sets ωi, i = 1, . . . , M . These sets are called collocationpoints. The number of collocation points is chosen equal to M . Hence, M model evalua-tions are necessary. Then, we use the following linear system of equations to solve for theunknown coef icients αi:

MP (w)Vα(X) = VΩ(X) (4.9)

MP is the M × M matrix consisting of the polynomials P [j](w) evaluated at the M colloca-tion points wi,Vα is the M × 1 vector consisting of the unknown coef icients αi,VΩ is the M × 1 vector consisting of the model output Ω(wi) evaluated at the M collocationpoints wi.Vα andVΩ are space and time dependent, whileMP is independent of space and time.There are different ways for choosing the collocation points, see discussion in Oladyshkinet al. [2011b]. The solution of Vα is dependent on the choice of the collocation points. Ingeneral, the optimal choice corresponds to the roots of the polynomial one degree d higherthan the one used in the expansion itself [Villadsen and Michelsen, 1978]. For univariateexpansions, the number of these roots is exactly equal to M . For high dimensional prob-lems, thenumberof available roots isNd, which is larger thanM . Hence, a rule for choosinga good subset of the roots is necessary.In the PCM approach, the collocation points are chosen based on the distribution of theinput parameters. They are selected out of the most probable regions of the relevant pa-rameters [Li and Zhang, 2007]. Probable regions can be identi ied using veri ied infor-mation or empirical correlations, e.g., between porosity and permeability. Beneath that,many different techniques for constructing probability distributions of uncertain param-eters exist [Woodbury and Ulrych, 1993]. More details can be found in Oladyshkin et al.[2011b].

4.3 Combining PCE and BF

The new approach of this thesis is the combination of a PCE techniquewith a BF. This com-bination works basically analogue to sections 3.4.4 and 3.4.5. It is numerically a straight-


forward implementation of Bayes' theorem applied to a reducedmodel in form of the PCE.The BF algorithm is described in section 3.4.5.We assume a Gaussian error for the noisy measurements εy1:t . The probability densityfunction of εy1:t is proportional to:

p(εy1:t) ∝ exp

(−1

2εy1:t

T C−1εy1:t

), (4.10)

wherein C−1 represents the covariance matrix of the measurement errors, which is typi-cally a diagonal matrix to model independent measurement errors.According to chapter 4.1, the initial model Ω(w) is being reduced to a response surface bypolynomial chaos expansion. The reducedmodel is denoted here asΩ(w). In principle, weuse our reduced model Ω(w) as the function of interest, cf. section 3.4, and insert it intoequation 4.10. Together with

εy1:t = y1:t − Ω(w) (4.11)

equation 4.10 leads to the following likelihood function:

p(y1:t|w) ∝ exp

[−1

2(y1:t − Ω(w)

)TC−1

(y1:t − Ω(w)

)], (4.12)

which can be used in Bayes' theorem.The identity IN , see equation 3.28, can be reformulated using the reduced model.

IN(Ωt) ≈∫

Ωt(w)PN(dw|y1:t) = Ω(w) 1N

N∑i=1

N[i]t δw[i](dw) (4.13)

Since Ω is vastly cheaper to evaluate than Ω, we can easily afford to work with the limit oflarge sample size. The law of large numbers yields:

limN→+∞

IN(Ωt) = I(Ωt). (4.14)

The posterior distributions for the respective uncertain parameters at some updating stept can be obtained through:

p(k1/k2/ϕ)(w|y1:t) = p(y1:t|w)p(w)∫p(y1:t|w)p(w)dw

. (4.15)

37

5 Numerical Implementation

5.1 A Remark on the Problem Formulation

The core task of this thesis is not to perform a complex modeling exercise. Instead, thiswork features a rather simple model setup for evaluating the power of combining a PCE-based model reduction technique with a bootstrap ilter for history matching. The singlepurpose of the model described in this chapter is to produce synthetic reservoir data, e.g.pressure-time series, which are used for history matching, and then to be matched to thedata it generated using the newly proposed technique.

5.2 Composition of the Reservoir Model

The Model Domain The model is realized in a simple 2-dimensional model-domain.The domain itself is quadratic with a dimension of 130 · 130 meters. The grid is made of13 · 13 cells.Wells In the domain, there are 4 injection and 1 production wells placed. Nx and Nylocate the cells in which the wells are placed.ProductionWell (W1):Postition: Nx = Ny = 7Radius: 0.1 mValue: 1 · 105 [Pa] (bottom hole pressure)

InjectionWells (W 2, 3, 4, 5):Postition W2: Nx = Ny = 3Postition W3: Nx = 11; Ny = 3Postition W4: Nx = Ny = 11Postition W5: Nx = 3; Ny = 11Radii: 0.1 mValues: 2.0 [m3/day] (injection rates)

5 Numerical Implementation 38

0 2 4 6 8 10 12 140

2

4

6

8

10

12

14

inject ion2

inject ion3

inject ion2

inject ion3

productionproduction

inject ion1

inject ion4

inject ion1

inject ion4

Figure 5.1:Model domain and wells.

Fluid Properties We de ine the dynamic viscosity η and density ϱ for each luid:ηwater = 1 [mPa · s]; ηoil = 10 [mPa · s]ϱwater = 1000 [kg/m3]; ϱoil = 700 [kg/m3]

Capillary Pressure The capillary pressure pc is de ined through the wetting phase. Inthis case, we have chosen a linear capillary pressure saturation relationship pc(Sw) = −10·Sw + 10:

Figure 5.2: Capillary pressure saturation relationship used in the example problem.

Relative Permeability We have chosen parabolic relative permeability saturation rela-


tionships kr(Sw) = Sw2 and kr(Sn) = (1 − Sn)2:

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Saturation Sw

Rel

ativ

e pe

rmea

bilit

y k r

Relative permeability SwRelative permeability Sn

Figure 5.3: Parabolic relative permeabilities used in the example problem.

Boundary Conditions For the outer domain boundaries Γi, we have chosen Neumannboundary conditions. The mass lux of the respective phases across these boundaries issupposed to be zero:

ϱαvα · n = uα,N ≡ 0 , on Γi

Initial Conditions The necessary initial conditions to be set, are the initial wetting-phase and the initial non-wetting-phase saturations Swinitial

respectively Sninitial:

Swinitial= 0.2 and Sninitial

= 0.8.

5.3 Numerical Implementation

The MATLAB Reservoir Simulation Toolbox (MRST) is an open source MATLAB toolbox,which is developed by SINTEF. SINTEF is an independent scandinavian research organi-zation. An overview about MRST is given by Lie et al. [2011]. MRST's focus lies on rapidprototyping on complex grids and demonstration of the multi-scale simulation concepts.


However, since it is convenient for setting up and solving reservoir models, we use thispowerful toolbox for our simple simulation.This section focuses on the software, numerical schemeand solverwhich is used for solvingthe coupled two-phase differential-equation system. The chosen MRST solver is basicallyusing a IMPEs-scheme for solving the equation-system. We implemented a solver which isusing a IMPEs-scheme because it has several advantages compared to solvers which solvethe fully coupled system. First of all, the decoupling allows different discretization of theequations. This is, e.g., important, since the pressure has to be computed with local massconservation. Secondly, the method is computationally faster, because only n solutions ofthe equations are necessary. n is the number of unknowns, which is equal to the degree offreedom of the system. In contrast, the fully coupled system requires to solving the equa-tions m-times. m is usually twice as high as n [Freeden et al., 2010].

5.4 Realization of the History Matching

The codewhichwe use for realizing the polynomial chaos expansion and the particle ilter-ing has been originally coded by Sergey Oladyshkin[Oladyshkin et al., 2011a, OladyshkinandNowak, 2012]. Since its original application is a CO2 sequestration problemwhichwassimulated via the DuMuX simulation toolbox, several adaptions to make it work with ourtest case had to be done. Figure 5.4 shows the basic data low of the used code. Some ex-planations of the speci ic processes will be given in the following.


RunManager.m

IPCM_A_Snapshot.m PCE design parameters

deterministic model call reservoir_model.m

snapshotsIPCM_A.mat

IPCM_B_SnapshotReading.m

IPCM_B.mat

ParticleFilter_aIPCM_OnePointPlus.m

data fromreference snapshot

Post_Distibutions_PriorAndPost.m

Prior_MeanOutputPressure.m

Post_MeanOutputPressure.m

Stop

DB_Vector.mat

posteriorcalculations.m

Figure 5.4: Basic lowchart of the used code.

RunManager.m• Calls the different processes of the code.

IPCM A Snapshot.m• Initialization of uncertain anddesign parameters (numberN of parameters anddegree d of the polynomials) as speci ied in PCE design parameters.


• Load prior distributions from DB Vector.mat.• Computation of arbitrary polynomials according to equations 4.2 and 4.6.• Initialization of collocation points and storage in IPCM A.mat according to sec-tion 4.2.

• Computationof the snapshots at each collocationpoint via ''deterministicmodelcall'' to reservoir model.m.

IPCM B SnapshotReading.m• Reading the simulation output iles for all snapshots that result fromIPCM A Snapshot.m.

• Save variables in IPCM B.mat.ParticleFilter aIPCM OnePointPlus.m

• Loading synthetic reservoir production history from ''reference snapshot''.• Adding synthetic measurement error according to equation 4.10.• Realization of the polynomial chaos expansion for all quantities of interest ac-cording to equations 4.7, 4.8 and 4.9.

• Realization of the particle iltering according to section 3.4.5.

Post Distibutions PriorAndPost.m• Computation of posterior distributions for all quantities of interest using parti-cle weights.

Prior MeanOutputPressure.m• Computation of mean prior pressure by evaluating the response surface.

Post MeanOutputPressure.m• Computation of mean prior pressure by evaluating the response surface.

posteriorcalculations.m• Visualizing the resulting distributions using kernel smoothing density estima-tions with Gaussian kernels.

• Plotting reference, prior and posterior pressure time-series.• Calculating the effective sample size (ESS) according to equation 6.3.


• Evaluating the normalized root mean square error χ according to equation 6.1.

44

6 Results and Discussion

6.1 Test Procedure

The purpose of this chapter is to present the results and assess the quality of the historymatching. Firstly, the outcomes of matches which are realized with time-series of differ-ent lengths and degrees of expansion are presented. It should be expected that the resultsbecome more accurate with increasing expansion degree, and the posterior distributionsbecomemorenarrowwith longer data series used. Afterwards, based on the assessment ofthe irst set of matching results, parameters and data-aspects, e.g., ineness of time-seriesand degree of expansion are adjusted and their in luence shall be assessed.The quality of the matching and the particle iltering is quanti ied by different measures.On the one hand, we analyze the resulting posterior distributions of our parameters ofinterest. Therefore, we assess the statistical moments and the shape of the obtained dis-tributions, and compare them with the reference values of the parameters which wereused to generate the data. On the other hand, we have a look at the normalized root meansquare error (NRMSE) χ between the reference pressure pr,i and the pressure ppost,i. ppost,i

is obtained by evaluating the response surface using the posterior output distribution ofthe parameters of interest. Taking the mean in the squared errors is necessary since wemight comparematching resultswhich are obtained from time serieswith different length.For χ, normalizing becomes necessary if we compare results from time-series which areobtained with different levels of measurement error σϵ. The NRMSE χ is calculated in thefollowing way:

χ =

√√√√ 1Np

Np∑1

(pr,i − ppost,i

σϵ

)2

(6.1)

σϵ is the measurement error standard deviation, which is a constant value, de ined in thecode. Np represents thenumberof datapoints considered in thepressure time-series.The general reliability of the results is tested by the accuracy of the response surface.For analyzing the accuracy, we compute the root mean square error (RMSE) between a1-dimensional cut of the response surface for a respective parameter pRS

k1,k2,ϕ and the truepressurepF M

k1,k2,ϕ obtained fromtheoriginal forwardmodel. The response surface's 1-dimensionalcuts are obtained by ixing two of the three parameters and the time t. Here, we ix thevalues at a collocation point and time-step number 1. The third parameter serves as thevariable. The restriction to 1D cuts is necessary to avoid evaluating the original model ona high-dimensional grid of input parameters.

6 Results and Discussion 45

RMSE =

√√√√ 1Np

Np∑1

(pRS

k1,k2,ϕ − pF Mk1,k2,ϕ

)2 (6.2)

A prior assumption for a parameter of interest, which is un itting in the sense that thetrue value is in the low probability regions of the assumption, leads to a high number oflow, respectively, zero weights in the particle iltering procedure. As a consequence, manyparticles are dying. This phenomenon is called ilter degeneration. This problem can partlybe solved by the resampling step. But, with each dying particle, information gets lost. Theresampling step does not provide new nor recover old information. A measure for howwell the samples are distributed in the region of interest is the effective sample size (ESS).According to Bergman [1999] and Liu and Chen [1998], it can be estimated by

ESS ≈ 1∑i(w

[i]t )2

. (6.3)

Since our approachworks for arbitrary distributions, we do not choose Gaussian input dis-tributions. k1 and k2 are represented by log-normal distributions. The input distributionfor ϕ is based on beta-distributions.The following aspects are in the focus of this thesis:Polynomial degree The polynomial degree d of the polynomial expansion. Since the

response surface has to reproduce the dependency of the model input on the modeloutput, which is strongly non-linear, we expect a strong in luence of the expansiondegree on the quality of the results.

Pressure time-series The length of the interval which is used for the matching andthe number of time-steps in these intervals is assessed. More data values do notnecessarily imply more information, i.e., more data do not automatically trigger abetter history matching result. This may occur, when later parts of a time series donot conveynew insight, and is calleddata redundancy. For short time series, however,an increase in time series length or a iner time step size should always lead to animprovement.

Number of particles The number of particles used in the BF should strongly in luencethequality of the results. Themoreparticles areused, themoreweights arenon-zero.The number of particles which ''survive'', respectively contribute to thematching, asmeasured by the ESS, can be used as a measure for the quality of statistical samplingfrom the posterior pdf in the BF.

6.2 Results: Time-Series of Different Length

Table 6.1 provides an overview about the results of the different history matching runs.Length is the length of the time series in days, steps the number of time steps, d the polyno-


mial degree, particles the number of used particles and time the computational time for thewhole computation. k1, k2 and ϕ represent the mean of the posterior distributions.At irst sight, the results do not allow an obvious conclusion. The in luence of the length ofthe time-series does not seem to be signi icant in the analyzed range. The results obtainedfor different polynomial degrees show a differing behavior. When regarding table 6.1, wecan state that there is no proper improvement of χ for degree d = 2, 3. In contrast, degreed = 1 improves continuously with the extension of the time-series. Furthermore, we notethat the lowest χ is obtained by d = 1 and a length of 400 days: χ10 = 0.1626.

6.3 Results: Polynomial Degree

Wewant to have a closer look on the in luence of the polynomial degree. Figure 6.1 displaysexemplarily graphical results of the history matching for simulation # 8. As labeled in theplot, the irst graph shows the pdf of the input distribution for the respective parameter.The secondplot in igure 6.1 represents the posterior distribution. Additionally, the 1-d cutof the response surface for the respective parameter is plotted beneath. This compositionallows a quick and informative overview. The vertical red lines mark the true values ofthe parameters (k1true = 0.05 and k2true = 0.07 [darcy]; ϕtrue = 0.27 [-]). In igure 6.2,pressure time-series are plotted. Meanprior andmeanposterior pressure are the pressurevalues which are obtained by evaluating the response surface with the prior respectivelyposterior parameter distribution. With plotting ± 2 times the standard-deviation of therespective time-series as error-bars, we approximate the 95% con idence interval.


Table6.1:

Histo

rymatchingr

esultsforr

uns#

1-12

.po

sterio

rmean

#len

gth

steps

dpa

rticle

stim

ek

1k

2ϕ

RM

SE

k1

RM

SE

k2

RM

SE

ϕES

Sχ

[days]

[-][-]

[-][m

in]

[darcy]

[darcy]

[-][-]

[-][-]

[-][-]

115

015

110

0.000

6.35

0.069

70.0

877

0.255

24.0

484e

61.8

106e

71.0

754e

695

30.1

954

215

015

210

0.000

18.17

0.063

90.0

747

0.268

22.7

114e

61.7

886e

76.7

824e

513

580.2

209

315

015

310

0.000

31.73

0.059

30.0

681

0.273

91.5

874e

61.8

431e

71.1

441e

517

230.1

972

420

020

110

0.000

12.95

0.067

00.0

894

0.257

14.0

484e

61.8

106e

71.0

754e

695

30.1

644

520

020

210

0.000

23.03

0.062

40.0

754

0.272

42.7

114e

61.7

886e

76.7

824e

513

580.2

609

620

020

310

0.000

41.37

0.060

60.0

677

0.270

31.5

874e

61.8

431e

71.1

441e

517

230.1

994

730

030

110

0.000

19.08

0.068

30.0

891

0.258

44.0

484e

61.8

106e

71.0

754e

695

30.1

816

830

030

210

0.000

34.63

0.063

20.0

748

0.269

12.7

114e

61.7

886e

76.7

824e

513

580.2

552

930

030

310

0.000

60.80

0.058

80.0

682

0.275

81.5

874e

61.8

431e

71.1

441e

517

230.1

996

1040

040

110

0.000

25.76

0.066

30.0

894

0.254

54.0

484e

61.8

106e

71.0

754e

695

30.1

626

1140

040

210

0.000

47.15

0.062

90.0

752

0.268

42.7

114e

61.7

886e

76.7

824e

513

580.2

167

1240

040

310

0.000

81.20

0.059

80.0

679

0.269

61.5

874e

61.8

431e

71.1

441e

517

230.1

977


0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

1

2

3

4Prior Porosity

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

1

2

3

4

Posterior Porosity

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

5

10

x 106

Porosity phi [−]

Pres

sure

in [P

a]

Pressure Response Surface vs. Forward Model

Pressure RSPressue FM

Figure 6.1: Results for ϕ and d = 2 (Simulation #8).

−20 0 20 40 60 80 100 120 140 1600

2

4

6

8

10

12

14x 106

Time in [d]

Pres

sure

in [P

a]

Mean posterior pressureMean prior pressureReference Pressure

Figure 6.2: Resulting pressure time-series for d = 1 (Simulation #1).


Since the results donot really differwith regard to the lengthof the time-series, weevaluatethe in luenceof thepolynomial degree exemplarilywith the results of the time series of 150days.Effective Sample Size Our ESSs values are, without exception, greater than the neces-

sary value of about 100 particles.Posterior Mean Basically, we can note that the statistical means of the posterior dis-

tributions approach the true values of the parameters of interest. In case of a single-modal distribution, this may be a good indicator. But, as long as a distribution is uni-modal with distinctive extrema, this parameter is not necessarily signi icant. Thiscan be seen in igure 6.1, where the mean is close to the true value (ϕ83 = 0.2691).This could mislead to the conclusion that the posterior distribution merges well tothe true value. But, when regarding the posterior distribution for ϕ83, one can see alocal minimum around ϕ = 0.27, which is obviously a bad estimation.In igure 6.3 we can observe that the estimation of k1 is ill-posed. This phenomenonis typical for inversemodeling, and therefore not surprising. Nevertheless, the poste-rior distributions for the parameters of interest merge well and concentrate aroundtrue values. Figure 6.3 illustrates the shrinking and shifting of the posterior distri-butions for simulations #1,2 and 3.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

10

20

30

Distributions for k1

priord=1d=2d=3

0 0.015 0.03 0.045 0.06 0.075 0.09 0.105 0.12 0.135 0.150

10

20

30

40Distributions for k2

priord=1d=2d=3

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

2

4

6

Distributions for phi

priord=1d=2d=3

Figure 6.3: Illustration of shrinking and shifting of posterior distributions for simulations #1,2and 3.

Root Mean Square Error The RMSEs for k1 and ϕ show a signi icant trend. Theystrictly become smaller with increasing of the polynomial degree, up to d = 3. In


the case of ϕ, the RMSE for d = 3 is just around 10.64% of RMSE for d = 1, which isa great improvement. The RMSE for k2 does not show this continuous improvement.The different constructed response surfaces are shown in igure 6.4.

0 0.05 0.1 0.15 0.20

1

2

3x 107

Permeability k1 in [darcy]

Pres

sure

in [P

a]

Model Response Surfaces for k1

Pressure Forward Modeld=1d=2d=3

0 0.05 0.1 0.15 0.20

5

10

15x 106


Pres

sure

in [P

a]



0.1 0.2 0.3 0.4 0.50

5

10

15x 106

Porosity phi [−]

Pres

sure

in [P

a]

Model Response Surfaces for phi


Figure 6.4: Comparison of response surfaces expanded for degrees d = 1, 2, 3.

Normalized Root Mean Square Error As described above, we use the NRMSE χ as ameasurement for the matching accuracy. Comparing the values of the different de-grees, we can state that uneven degrees lead to better matching results than evendegrees. This is due to the non-linear and monotonic behavior of most of the modeldependencies. Even polynomial degrees are not monotonic. Hence, even degreesmay lead to worsematching results. Figure 6.5 shows a section of the three differentresulting pressure time-series. Degree d = 2 showswith the highest χ (χ2 = 0.2209)the worst matching result. To keep the plot clear, we do not plot the error bars.

Based on the observed results, we decided to run additional simulations for polynomialdegrees d = 4 and d = 5. The results in table 6.2 show the expected behavior. For theeven degree d = 4, we obtain worse results compared to d = 3 and d = 5. The improve-ment from simulation #13 to #18 is only 2.6%. In consideration of the more than doubledcomputational-time, the following simulations are performed with degree d = 3. Figure6.6 shows the response surfaces for simulations # 13,17 and 18.


40 50 60 70 80 90 100 110 120 130 1401.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5x 106

Time in [d]

Pres

sure

in [P

a]

Mean posterior pressure d=1Mean posterior pressure d=2Mean posterior pressure d=3Reference Pressure

Figure 6.5: Resulting posterior pressure time-series for different polynomial degrees d for simu-lations #1,2 and 3.

0 0.05 0.1 0.15 0.20

1

2

3x 107


Pres

sure

in [P

a]



0 0.05 0.1 0.15 0.20

5

10

15x 106


Pres

sure

in [P

a]



0.1 0.2 0.3 0.4 0.50

5

10

15x 106

Porosity phi [−]

Pres

sure

in [P

a]

Model Response Surfaces for phi


Figure 6.6: Comparison of response surfaces expanded with degrees d = 3, 4, 5 for simulations# 13, 17 and 18.


6.4 Results: Fineness of Time-Series

Based on the previous analyses, we decide to perform the assessment of the ineness of thetime-series with the 400-days time-series. Table 6.2 shows the results. Note that row 12and 13 are interchanged, in order to get a continuous increasing of the time-steps in thetable.Within this data, we can detect the afore mentioned phenomenon data redundancy. In ourcase, more time-steps in the time-series do not imply a better matching result. Additionaltime-steps do not provide more information to the history matching process. Hence, it isnot necessary to run the simulations with a higher number of time steps. The best match-ing result is obtained for only 20 time-steps.The values in 6.2, particularly χ13 and RMSE13, leaded us to the decision to use time-seriesof simulation #13 for a further assessment of the number of particles.

6.5 Results: Number of Particles

All previous simulations have beenmadewith 100.000 particles. Asmentioned before, weexpect a in luence of the number of particles on the matching result. After the previousanalyses, we decided to rerun simulation #13 under the same conditions, except that weincrease the number of particles tenfold to 1.000.000. The results are presented also intable 6.2.The results do not improve with this step. The reason might be that a sample size of100.000particles converges alreadywell towards the truth. For the theoretical background,we want to refer to equation 3.19. For a complete evaluation of the sample size, we per-form a simulationwith a reduced number of particles. In the irst step, we choose a samplesize of 10.000 particles. Still, the ESS should lie above 100 particles.The results of simulation #19 are presented in 6.2. The outcome con irms our guess. Both,the RMSE and χ deteriorate. It furthermore shows that the initial choice of 100.000 parti-cles was reasonable. Furthermore, with an ESS of 118, simulation #19 reaches the lowerboundary of reasonability. If possible, this should be avoided.Yet another interesting aspect of the reduced sample size is that the posterior distributionsfor k1 and ϕ show two, respectively, more than two distinctive extrema (see igures 6.7 and6.8). ϕ showed this behavior already before, but not that clearly. This phenomenon can beexplainedwith the reduced number of particles. Since the overall number of surviving par-ticles decreases, the single weight gains more absolute relevance. Thereby the chance formore extrema is raised. We can also note that by reducing the sample size by the factor of10, the ESS reduces by the factor of 15.


Table6.2:

Histo

rymatchingr

esultsforr

uns#

13-1

9.po

sterio

rmean

#len

gth

steps

dpa

rticle

stim

ek

1k

2ϕ

RM

SE

k1

RM

SE

k2

RM

SE

ϕES

Sχ

[days]

[-][-]

[-][m

in]

[darcy]

[darcy]

[-][-]

[-][-]

[-][-]

1340

020

310

0.000

40.72

0.061

30.0

688

0.295

91.5

874e

61.8

431e

71.1

441e

517

680.1

515

1240

040

310

0.000

81.20

0.059

80.0

679

0.269

61.5

874e

61.8

431e

71.1

441e

517

230.1

977

1440

050

310

0.000

98.77

0.059

50.0

677

0.262

21.5

874e

61.8

431e

71.1

441e

516

380.2

317

1540

080

310

0.000

155.2

50.0

579

0.068

00.2

652

1.587

4e6

1.843

1e7

1.144

1e5

1529

0.222

4

1640

020

31.0

00.00

038

9.43

0.060

80.0

683

0.292

21.5

874e

61.8

431e

71.1

441e

517

886

0.170

5

1740

020

410

0.000

65.70

0.056

90.0

678

0.300

24.2

407e

61.6

541e

73.1

603e

522

340.1

592

1840

020

510

0.000

102.7

20.0

561

0.066

70.3

018

3.047

6e6

2.556

9e7

5.608

3e5

2804

0.151

1

1940

020

310

.000

9.02

0.050

80.0

637

0.270

61.5

874e

61.8

431e

71.1

441e

511

80.1

641


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

10

20

30

Prior Permeability 1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

20

40

Posterior Permeability 1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

1

2

x 107


Pres

sure

in [P

a]



Figure 6.7: Results for k1 and d = 3 (Simulation #19).

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

1

2

3

Prior Porosity

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

2

4

6

Posterior Porosity

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

5

10

x 106

Porosity phi [−]

Pres

sure

in [P

a]



Figure 6.8: Results for ϕ and d = 3 (Simulation #19).

55

7 Summary and Conclusions

Summary In this work we presented a new history matching approach which uses aPCE-based particle ilter. The key idea behind this choice is to combine an accurate statis-tical method for history matching with an extremely fast reduced version of the model tobe matched. We pointed out that many frequently used parameter estimation techniquesare based on the assumption of Gaussian input distributions and on linear (or mildly non-linear) system behavior. This does not re lect natural behavior, especially for not for dy-namic reservoir models and multiphase low in porous media, strong assumptions andother techniques for handling this lack become necessary. Realistic, respectively, large-scale reservoirmodels areusually computationally tooexpensive forbrute-forceoptimidation-based or even statistical calibration methods.Our approach is to ind a compromise between accuracy and computational ef iciency. Weuse polynomial chaos expansion to reduce the model to a response surface that simulatesthe dependence of model responses on changes in the parameters to be itted. The recentarbitrary polynomial chaos technique is working for arbitrary input distributions and isable to represent the strong non-linear behavior of the dynamic multiphase low-system.Since the reduced reservoir model is computationally more ef icient, the used particle il-ter can be used with a great number of particles. The large sample size ensures a goodparameter estimation and accurate results. Up to presence, this combination of methodshas not been used before. The proposedmethodswere implemented and tested on a smallsynthetic example.Conclusions Based on the results of the performed test cases, the following conclusionscanbedrawn. (The conclusions refer to ourwork and arenot necessarily valid for a generalevaluation of the used techniques.)

• The proposed methods performed well. Synthetic parameter values that were usedto generate synthetic data setswere recoveredwell by themethod, and the computeduncertainty bounds are plausible.

• The aspect of computational feasibility has not been assessed in detail. But, even his-tory matching runs with a sample size of 1.000.000 particles were computed in fewhours. Despite the fact that we used a cheap reservoir model, our approach seems tobe feasible for large scale applications as well.

• One of the most important aspects of this work could also be proved. The arbitrarypolynomial chaos expansion method works well for arbitrary input distributions.Our uncertain input parameterswere constructedwith different statisticalmomentsand distributions. All uncertain parameters of interest converged well to the truevalues for all inspected input distributions.

7 Summary and Conclusions 56

• We found that uneven polynomial degrees work best for non-monotonic scenarios.In this work, degree d = 3 turned out to be a good compromise between accuracyand computational effort.

• Our assessment showed that a sample size of approximately 1700 particles after re-jection sampling showed good results. Under the investigated scenarios, this corre-sponds to N = 100.000 particles drawn from the prior distribution. When increasingthe number of particles, we did not obtain more accurate matching results. In con-trast, a reduction of the sample size led to worse results.

• As a side aspect, we analyzedwhat is a good sampling frequency and length of a pres-sure data time series, used as input data to history matching. We found that, quitein initively, only the periods of fast pressure change are information. High temporalresolution of the pressure data did not seem to be necessary.

Outlook Resulting from the drawn conclusions, work-steps for the basis of further re-search can be formulated:

• A more thorough test of the method could include a direct comparison to the brute-force approach that uses the original model.

• Instead of the used rejection sampling technique, one could try a weighting-basedscheme in the ilter and compare the performance.

• The new technique should be evaluated under more realistic conditions with, e.g.,large-scale reservoir models or with an increased number of uncertain parameters.

General evaluation Thiswork shows that a PCE-basedbootstrap ilter is a powerful toolfor reservoir history matching. Especially the fact that it works for arbitrary input param-eters and strong non-linear system behavior, provides a promising technique for many ac-tual applications. As soon as the technique is assessedwith large-scale applications, it maybecome a serious alternative to EnKFs and other parameter estimation techniques.

57

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efficient history matching in reduced reservoir models with pce-based particle filters

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