Inverse Modeling of Tight Gas Reservoirs

Transcription

1 Inverse Modelng of Tght Gas Reservors Der Fakultät für Geowssenschaften, Geotechnk und Bergbau der Technschen Unverstät Bergakademe Freberg engerechte Dssertaton Zur Erlangung des akademschen Grades Doktor-Ingeneur Dr.-Ing. vorgelegt von: geboren am Dpl.-Ing. Mtchedlshvl George n Tbls, Georgen Freberg, den:

2 Abstract The present thess focuses on the followng ssues: () nverse modelng technques for characterzaton of tght-gas reservors, () the numercal nvestgatons of advanced well stmulaton technques, such as hydraulc fracturng as well as underbalanced drllng, and () the statstcal analyses of results for dentfcaton of the optmal level of parameterzaton for calbrated model as qualty and quantty of the measured data justfes. In terms of a consderable ncrease the qualty of characterzaton of tght-gas reservors, the am of ths work was () an accurate representaton of technologcal aspects and specfc condtons n a reservor smulaton model, nduced after the hydraulc fracturng or as a result of the underbalanced drllng procedure and () performng the hstory match on a bass of real feld data to calbrate the generated model by dentfyng the man model parameters and to nvestgate the dfferent physcal mechansms, e.g. multphase flow phenomena, affectng the well producton performance. Due to the complexty of hydrocarbon reservors and the smplfed nature of the numercal model, the study of the nverse problems n the stochastc framework provdes capabltes usng dagnostc statstcs to quantfy a qualty of calbraton and the nferental statstcs that quantfy relablty of parameter estmates. As shown n the present thess the statstcal crtera for model selecton may help the modelers to determne an approprate level of parameterzaton and one would lke to have as good an approxmaton of the structure of the system as the nformaton permts. 2

3 Acknowledgment Frst of all, I would lke to thank my supervsor, Prof. Freder Häfner, for hs very knd encouragement, support and gudance durng ths work. I am very ndebted to Dr. Aron Behr for very useful dscussons and for the postve nfluence on my professonal career. I wsh to express my sncere thanks and grattude to Dr. Hans-Deter Vogt and Dr. Torsten Fredel, who gave useful contrbutons at varous tmes durng the development of ths thess. I am grateful to the staff of Insttute of Drllng Engneerng and Flud Mnng, TU Bergakademe Freberg. Fnally, my warmest thanks go to my famly for ther effort, moral support and love. 3

4 Content ABSTRACT 2 ACKNOWLEDGMENT 3 CONTENT 4 NOMENCLATURA 7 LIST OF FIGURES 11 LIST OF TABLES 13 1 STATEMENT OF THE PROBLEM 14 2 AN INTRODUCTION TO INVERSE PROBLEMS Reservor Modelng The Inverse Modelng The Statstcal Methods for Model Calbraton Statstcal Crtera of Parameter Estmaton wth Homogeneous Pror Dstrbuton Statstcal Crtera of Parameter Estmaton wth Gaussan Pror Dstrbuton Numercal Optmzaton Algorthms Nonlnear Last Squares Gradent Based Methods Global-Optmzaton Technques Drect Search Methods Neghbourhood Algorthm Analyses of Parameter Estmaton Uncertantes Optmalty and Model Identfcaton Crtera Kullback-Lebler Informaton Alternatve Model Selecton Crtera Applcaton to the Synthetc Feld Example Reservor Descrpton and Parameterzaton Smulaton Results 40 4

5 3 SIMULATION OF THE PRODUCTION BEHAVIOR OF HYDRAULICALLY FRACTURED WELLS IN TIGHT GAS RESERVOIRS Introducton to Fractured Well Smulaton Automatc Generaton of the Smulaton Model for Fractured Wells Local Grd Refnement Integraton of the Fracture Parameters Estmaton of the Flud Dstrbuton n the Invaded Zone Evaluaton of Exposure Tme Hstory-Matchng of a Case Study of Hydraulcally Fractured Well Feld Example Smulaton Model Hstory Match Cleanup wth hydraulc damage Combned effect of hydraulc and mechancal damage Dscusson of hstory match results Effect of cleanup on postfracture performance Identfcaton of the Leakoff Coeffcent by Hstory Matchng Hypothetcal Study Case Study Automatc Methods of Optmzng of Fractured Wells 79 4 SIMULATION OF INFLOW WHILEST UNDERBALANCED DRILLING (UBD) WITH AUTOMATIC IDENTIFICATION OF FORMATION PARAMETERS AND ASSESSMENT OF UNCERTAINTY Lterature revew Smulaton of Reservor Flow durng UBD Parameter Estmaton durng UBD Identfcaton Procedure Uncertanty Analyses Example Outlook: Optmzaton Approach for UBD 93 5

6 5 SUMMARY 94 BIBLIOGRAPHY 97 6

7 Nomenclatura Symbols Symbol Meanng Unt A area, m² senstvty matrx a,b,c coeffcents - b set of subsdary condtons b f fracture wdth m c proppant concentraton kg/m² C l leakoff coeffcent m/s 1/2 C l * Cov 0 Cov z d specfc leakoff coeffcent m/s 1/2 md γ pror parameter-covarance matrx covarance matrx of measured values dstance, m step sze F fractonal flow, dmensonless - F C fracture conductvty md*m F CD dmensonless fracture conductvty - g Gradent vector H Hessan matrx J Jacoban matrx I Unt matrx k permeablty, ndex m², Darcy - L lnear sze, m M number of grdblocks model - - N number of pumpng perods - N m N p number of measured data number of model parameters n coordnate normal to fracture plane m n s,n r number of Vorono cells p pressure, Pa set of physcal parameters R number 7

8 r ndex rght-hand-sde term m³ q mass source/snk set of control varables kg/(m³*s) - S saturaton - s 0 estmated error varance t tme sec u set of state varables V volume x coordnate, m set of spatal and tme varables x f fracture half length m Greek Symbols Symbol Meanng Unt α power factor, - fracture expanson factor φ porosty -, δ dfference, ncrement - ε error γ power factor - η net-to-gross thckness rato - µ dynamc vscosty Pa*s µ 0 vscosty rato - λ dsplacement drecton, - regularzaton factor, materal parameter Θ model parameter vector ρ densty kg/m³ σ effectve stress, MPa standard devaton ν velocty m/s Ω doman ω weght factor ξ self-smlar varable 8

9 Indces Index Meanng 0 ntal a absolute ad admssble cap capllary calc calculated const constant f fracture, fctve g gas ndex phase ndex, ndex of tme perod j ndex ndex of tme perod k ndex meas measured obs observed p proppant pr prmary pred predctve r relatve ref reference s slurry t total w water 9

10 Functons and Operators Functon AIC BIC d m J H I L p p 0 p *. Meanng Akake Informaton Crteron Bayesan Informaton Crteron Kashyap Index Leverett J-Functon statstcal entropy nformaton content partal dfferental operator, lkelhood probablty densty functon pror dstrbuton posteror dstrbuton dvergence operator Abbrevatons Abbrevaton BHP CSS DF FOPT GOR GPR LGR NPV OF THP TOL UBD WGR Meanng Bottom Hole Pressure Composte Scaled Senstvtes Dscount rate Feld Ol Producton Total Gas-Ol Rato Gas Producton Rate Local Grd Refnement Net Present Value Objectve Functon Tubng Head Pressure Tolerance Crtera Underbalanced Drllng Water Gas Rato 10

11 Lst of Fgures FIGURE 1: RANDOM POINTS IN A 2D SPACE FIGURE 2 A UNIFORM RANDOM WALK RESTRICTED TO A CHOSEN VORONOI CELL FIGURE 3: TOP STRUCTURE MAP OF THE PUNQ CASE WITH WELL LOCATIONS (FROM FLORIS ET AL. ) FIGURE 4: TRUTH CASE: HORIZONTAL PERMEABILITY FIELDS FOR LAYERS 1, 3, FIGURE 5: THE PRODUCTION DATA FOR THE WELL PRO FIGURE 6: THE PREDICTION RESULTS FIGURE 7: PRIMARY REFINEMENT OF FRACTURE FIGURE 8: SECONDARY REFINEMENT OF WELL AND FRACTURE FIGURE 9: (A) VARIATION OF THE FRACTURE WIDTH (FROM THE FRACTURE PACKAGE PROTOCOL). THE DASH LINE SHOWS THE FICTIVE WIDTH (0.1 M) OF THE FRACTURE IN THE SIMULATION MODEL (B) DISTRIBUTION OF THE PROPPANT CONCENTRATION AND FRACTURE CONDUCTIVITY BY THE ELLIPTICAL ZONES OVER THE FRACTURE (CORRESPONDS TO THE FRACTURE PACKAGE PROTOCOL) FIGURE 10: INTEGRATION OF THE FRACTURE INTO THE RESERVOIR MODEL (PERMEABILITY AND POROSITY DISTRIBUTION) FIGURE 11: WATER DISTRIBUTION AROUND THE FRACTURE IN THE SIMULATION MODEL FIGURE 12: PRODUCTION HISTORY OF CASE-STUDY WELL FIGURE 13: PERMEABILITY DISTRIBUTION WITHIN FRACTURE PLANE FIGURE 14: INITIAL WATER DISTRIBUTION AROUND THE FRACTURE AFTER THE LEAKOFF PROCESS FIGURE 15: PRODUCTION DATA AND HISTORY MATCH RESULTS OF CLEANUP PERIOD FIGURE 16: HISTORY MATCH OF POSTPRODUCTION PERIOD FIGURE 17: FUNCTIONS OF RELATIVE PERMEABILITY GAS-WATER AND CAPILLARY PRESSURE FIGURE 18: LONG TIME PRODUCTIVITY: COMPARISON OD PRODUCTIVITY OF UNDAMAGED WELL (IGNORING CLEANUP) AND DAMAGED WELL WITH 500 BAR DRAWDOWN FIGURE 19: SIMULATED GAS PRODUCTION RATE AND CUMULATIVE WATER RECOVERY FIGURE 20: DEPENDENCE ON VARIOUS PARAMETERS ON THE COEFFICIENT (A) NORMALISED MAXIMUM GAS RATE (B) NORMALISED GAS RATE SLOPE (C) NORMALISED GAS BREAKTHROUGH FIGURE 21: DEVELOPMENT OF WATER BACKFLOW FROM THE INVADED ZONE FIGURE 22: DEVELOPMENT OF GAS FLOW RATE AND WATER RECOVERY AT THE BOTTOM-HOLE DURING THE CLEANUP PERIOD FIGURE 23: PERMEABILITY CURVES MATCHED IN THE CASE STUDY AT FIXED γ =0 AND FIGURE 24: COMPARISON OF PRODUCTION DATA SIMULATED BY THE FIXED RELATIVE PERMABILITIES FOR DIFFERENT γ -EXPONENTS FIGURE 25: NORMALIZED COSTS OF THE FRACTURE TREATMENT VERSUS FRACTURE CONDUCTIVITY AT THE FIXED SLURRY VOLUME FIGURE 26: VARIATION THE ESTIMATED PARAMETERS DURING THE OPTIMIZATION PROCEDURE AND THE DEVELOPMENT OF THE NET PRESENT VALUE FIGURE 27: 3-D SURFACE OF THE OBJECTIVE FUNCTION FIGURE 28: OBJECTIVE FUNCTION CONTOUR GRAPHS

12 FIGURE 29: COMPONENTS FOR UBD FIGURE 30: WELLBORE GRID WITH SCHEMATIC OF BOUNDARY CONDITIONS DURING UBD FIGURE 31: UBD COUPLING FIGURE 32: MEASURED AND SIMULATED GAS RATES FIGURE 33 ESTIMATED AND TRUE PERMEABILITY VERSUS DRILLED DEPTH FIGURE 34: COMPOSITE SCALED SENSITIVITIES (CSS) FIGURE 35: CONTROL AND OPTIMIZATION LOOP FOR UBD PROCESSES

13 Lst of Tables TABLE 1: THE SET OF DIFFERENT PARAMETERIZED MODELS TABLE 2: RANKING THE MODELS WITH THE K-L DISTANCE TABLE 3: RANKING THE MODELS WITH DIFFERENT MODEL SELECTION CRITERIA TABLE 4: RESERVOIR AND WELL PARAMETERS

14 1 STATEMENT OF THE PROBLEM The economc vablty of an ol/gas feld development project s greatly nfluenced by the reservor producton performance under the current and future operatng condtons. In order to analyze the reservor performance and estmate reserves, engneers make use of numercal flow smulators that requre a parameterzaton of reservor propertes,.e. a reservor model, as nput. After the smulaton model s bult and the forecast problem can be solved,.e. we can forecast the response of the system for dfferent exctaton, as a result, dfferent management decsons can be compared and optmal decson can be selected based on certan crtera. In practce, however, t s very dffcult to construct an accurate smulaton model for a geologcal system. Assumng, that the governng mathematcal equatons of the constructed model satsfactorly descrbe the orgnal system the all physcal parameters are dffcult to measure accurately n the feld. Usually lmted Informaton on the geologcal and geophyscal background of the reservor s avalable from well tests, sesmc surveys, logs etc. The man source of nformaton from the system may come n the form of producton data, such as producton rates and pressure behavor, all of them ndrect measurements of the physcal parameters that descrbe the flud flow through the reservor. The goal s to estmate reservor parameters to be used as an nput n the flow smulator n order to descrbe satsfactorly the producton performance. Snce the nput-output relaton of a correct smulaton reservor model must ft the observed exctatonresponse relaton of the orgnal system, t s possble to ndrectly estmate the reservor model parameters by usng these observaton data. Model calbraton, nverse modelng, hstory matchng, parameter estmaton are terms descrbng essentally the same technque of adjustng the model parameters untl a close match between smulated and measured data s obtaned. Automatc model calbraton can be formulated as an optmzaton problem, whch has to be solved n the presence of uncertanty, because the avalable observatons are ncomplete and exhbt random measurement errors. Due to the complexty of many real systems under study the number of reservor parameters s usually larger than the avalable data set, therefore the soluton s non-unque and the nverse problem s ll-posed. Whle addng features to a model s often desrable to mnmze the msft functon between smulated and observed values, the ncreased complexty comes wth a cost. In general, the more parameters contaned n a model, the more uncertan are parameter estmates. Often t s advsable to smplfy some representaton of realty n order to acheve an understandng of the domnant aspects of the system under study. Inverse modelng provdes capabltes usng dagnostc statstcs to quantfy a qualty of calbraton and the nferental statstcs that quantfes relablty of parameter estmates and predctons. The statstcal crtera for model selecton may help the modelers to determne an approprate level of complexty of unknown parameters and one would lke to have as good an approxmaton of the structure of the system as the nformaton permts. 14

15 A key dffculty n choosng the most approprate reservor model s that several models may apparently satsfy the avalable nformaton and seem to provde more or less equvalent matches of the measured system responses. The quantfcaton of the model selecton uncertanty has an mportant bearng on the valdty of the model as a predctve tool and helps the engneers to take decson n a rsk prone mprovement. In ths work a methodology for the model selecton and nference s presented based on the Kullback-Lebler concept from nformaton theory, appled for the predctve model. In ths work nverse modelng s appled especally for characterzaton of tght-gas reservors. These are felds wth very low permeabltes (less than 0.1 md) n very challengng envronments. Durng the last decades great endeavor has been made to facltate new gas resources to contrbute to the energy supply guarantee, and the current trend n the German E&P ndustry s to nvolve those types of reservors nto the development and explotaton. The tght-gas felds are sedments of the North German Rotlegend as well as the upper Carbonferous form the prmary German tght-gas regons. Prospectve German tght-gas reserves are assumed to be as large as 300 bllon m³, wth a potental recovery factor of 30-50%. Ths could extend the strategc range of the local gas reserves another 7-8 years, provded an economcal explotaton (Lermann and Jentsch, 2003). The development of tght gas reservors mples the applcaton of advanced well stmulaton technques, manly concernng Hydraulc Fracturng and Underbalanced Drllng. Durng the hydraulc fracturng process specally engneered fluds are pumped at hgh pressure and rate nto the reservor nterval to be treated, causng a vertcal fracture to open. The wngs of the fracture extend away from the wellbore n opposng drectons accordng to the natural stresses wthn the formaton. Proppant, such as grans of sand of a partcular sze, s mxed wth the treatment flud keep the fracture open when the treatment s complete. Hydraulc fracturng creates hgh-conductvty communcaton wth a large area of formaton. Smultaneously, some of the flud leaks off nto the formaton and creates an nvaded zone around the fracture whch sometmes may mpar the producton as a consequence of formaton damage. UBD s the drllng process when the pressure of crculatng drllng flud s lower than the pore pressure of the target formaton of nterest. The most wdely recognzed beneft of UBD s the reducton of formaton damage by mnmzng the drllng flud leakoff and fnes mgraton nto the formaton. Especally n tght formaton the nvaded drllng flud can causes the sgnfcant reducton of gas producton do to the water blockng because of two-phase flow and capllary end effects. At the same tme UBD facltates the possblty for reservor characterzaton durng drllng. The nflow producton rate depends on the formaton propertes. Ths nformaton s analogue to the transent test data and applyng nverse modellng technques t s possble to estmate reservor model parameters (such as porostes, permeabltes, pore pressures and etc.). 15

16 In terms of a consderable ncrease the qualty of characterzaton of tght-gas reservors, the problem can be addressed by 1) an accurate representaton of technologcal aspects and specfc condtons n a reservor smulaton model, nduced after the hydraulc fracturng or as a result of the Underbalanced drllng procedure n the mmedate wellbore envronment and 2) Performng the hstory match on a bass of real feld data to calbrate the generated model by dentfyng the man model parameters and to nvestgate the dfferent physcal mechansms, e.g. multphase flow phenomena, affectng the well producton performance n tght gas formatons. To develop the concept for tght-gas model calbraton as well as for producton optmzaton an nterface module s developed to update automatcally a smulaton model at each varyng the parameters and to ncorporate the reservor smulator nto optmzaton algorthm. To accelerate hstory matchng or optmzaton procedures, automatc methods for solvng the nverse problems are presented. Therefore, a commercal reservor smulator s complemented wth dfferent local (gradent-based) and global (drect search algorthm) optmzaton technques. 16

17 2 An Introducton to Inverse Problems 2.1 Reservor Modelng Inverse modelng starts wth the formulaton of the so-called forward or drect problem. A model must be developed that s capable of smulatng the general features of the system behavor under measurement condtons. Ths step nvolves the mathematcal and numercal descrpton of the relevant physcal processes. The mult-phase flow of fluds n porous meda s governed by physcal laws and emprcal relatonshps, whch are vald n a broad varety of engneerng dscplnes. These laws are based on the conversaton of mass, momentum and energy. From a practcal standpont t s hopeless at ths tme to try to apply these basc laws drectly to the problems of flow n porous meda. Instead, a sememprcal approach s used where Darcy s law s employed nstead of the momentum equaton. Addtonally, several emprcal relatons, e.g., PVT-relatons, rock and flud propertes and mult-phase flow behavor, are necessary to formulate a flow equaton such that the representaton of the physcal problem s a realstc as possble. The startng pont for the mathematcal formulaton of the flow equaton s the applcaton of the mass conversaton law, whch can be stated n vectoral notaton:. ( ρ ν ) = ( ρφ ) + q ~ (2.1) t q ~ s negatve for a source and postve for a snk. When the entre pore space s occuped by several phases, the Equaton 2.1 s extended as follows. ( ρν ) = ( ρφs ) + q (2.2) t The left hand sde of Equaton 2.2 denotes the mass flow rate of phase by convecton wth a velocty ν, whereas rght hand sde descrbes the temporal accumulaton of mass plus sources/snk. Here, S represents the saturaton, ρ s the densty of the flud and φ the porosty of the matrx. In the cases of mult-phase flow a dstrbuted parameter model conssts of a set of partal dfferental equatons wth approprate ntal and boundary condtons. Its general form can be represented by ( u; q; p; b; x) = 0 L (2.3) Where L s a set of partal dfferental operators, u s a set of state varables, q s a set of control varables, p s a set of system parameters that characterze the geometry and/or physcal nature of the 17

18 system such as porosty, permeablty and etc, b represents a set of subsdary condtons that defne the ntal state of the system and the relaton to exchangng mass wth ts neghborng systems, x represents a set of spatal and tme varables. When ( q p, b), are gven, the problem of solvng the state varables u from Equaton 2.3 s called the forward problem. Varous numercal methods, such as the fnte dfference method, the fnte element method, and relevant software packages have been developed for solvng the forward problem to smulate the mult phase flow n geologcal envronment. The general form of a forward problem soluton can be represented by ( q; p; b x) u = M ; (2.4) Here M can be thought as a subroutne for solvng the forward problem ether by an analytcal method or by a numercal method wth nput ( q p, b), and output u. To represent not only physcal parameters but also the parameters defnng control varables and subsdary condtons,.e. ( q ; p; b) Θ =, (2.5) where Θ s called the vector of model parameters, or smply, the model parameter. Thus the Equaton 2.4 can be rewrtten n a compact form ( Θ) u = M (2.6) The conventonal process of constructng an envronmental model conssts of the followng fve steps: Defne the problem. To defne the objectves of model development and relevant state and control varables. Collect data. To collect hstorcal records and measurements on state varables, control varables and reservor parameters, to desgn and conduct feld experments Construct a conceptual model. To select a model structure based on approprate physcal rules, the data avalable, and some smplfyng assumptons Calbrate the model. To adjust the model structure and dentfy the model parameters such that the model outputs can ft the observaton data qute well. Assess the relablty. To use a determnstc or a statstc method to estmate the relablty of model predctons and model applcatons. 18

19 2.2 The Inverse Modelng In the above steps of model constructon, the key and also the most dffcult step s model calbraton. Wth a model gven by Equaton 2.4 the observed values of state varables obs u, at observaton locatons and tmes obs x can be expressed by the followng observaton equaton, obs ( q p; b x ) + ε obs u = M ; ; or obs ( Θ x ) + ε obs u = M ; (2.7) Where ε contans both model and observaton errors. The man objectve of model calbraton s the adjustment of the model structure and model parameters (n the most cases the unknown physcal, system parameters) of a smulaton model smultaneously or sequentally so as to make the nput-output relaton of the model ft any observed exctatonresponse relaton of the real system. If the model structure s determned (Equaton 2.3), the problem of only determnng some model parameters from the observed system states and other avalable nformaton s called Parameter dentfcaton. In certan sense, parameter dentfcaton s an nverse of the forward problem. The nverse problem seeks the physcal parameters p when the values of state varables obs u are measured and the snk/source term q as well as the ntal or boundary condtons b are known, whle the forward problem predcts the state varables u when the other parameters q ; p; b and values are gven. ( ) The forward problem of reservor modellng s well-posed,.e, ts soluton s always n exstence, unque and contnuously dependent on data. The nverse problem, however, may be ll-posed,.e., ts soluton (the dentfed parameters) may be non-unque and may be sgnfcantly changed when the observaton data only change slghtly. Examples of ll-posed Inverse problems can be found n Sun (1994). Some suffcent condtons have been derved n mathematcs for makng the nverse modellng well-posed. Unfortunately, these theoretcal results cannot be used drectly to real case studes n reservor modellng because of the followng dffcultes: Hgh degree of freedom. The unknown parameters are usually dependent on locaton and on tme. Very lmted data. The data for model calbraton are usually very lmted and ncomplete n both spatal and tme domans because the measurng of state varables s expensve. Large model errors. 19

20 The mathematcal models used to descrbe the complex flow mechansms n geologcal envronment are always based on some dealzed assumptons. As a result, sometmes t s do not excepted that the real values of the unknown parameters can be found through model calbraton. A model that fts the observatons best may not be the best one for predcton and management. Inverse reservor modellng s not smply a curve fttng problem. To fnd a satsfactory nverse soluton, we must systematcally consder the suffcent of data, the complexty of model structure, the dentfablty of model parameters, and the relablty of model applcatons The Statstcal Methods for Model Calbraton obs Snce the observed data vector u cannot be dentcal to the system responses calculated wth a numercal model because of measurement errors and the smplfed nature of the model (model structure error) t s absolutely necessary to study nverse problems n the stochastc framework. Accordng to the Equaton 2.7 and due to the random nature of ε the dentfed parameters are always assocated wth uncertantes and thus can be regarded as random varables. obs Thus, the nverse soluton can be smply defned as follows: by the ad of a model u M ( Θ; x ) The dea behnd the defnton can be seen from the followng two examples: 20 = to transfer the measurement nformaton (wth measurement errors) from the measurement space to the parameter space to decrease the uncertanty of the estmated parameters. The best method of parameter estmaton should extract nformaton from the observatons as much as possble and decrease the uncertanty of the unknown parameters as much as possble. For ths concept of parameter dentfcaton the followng tems are to be consdered How to measure nformaton and uncertanty How to transfer nformaton from the measurement space to the parameter space through a model How to estmate the uncertanty of the estmated parameters. If p ( Θ) s the jont probablty densty functon (pdf) of a parameter vector Θ. The uncertanty assocated wth p ( Θ) s measured by ts entropy (Bard, 1974): H ( Θ) ( ) Θ ( p) E(log p) = p log p Θ d (2.8) ( Ω) = E log denotes the mathematcal expectaton of ( ) Where ( p) space. log p and ( ) Ω s the whole dstrbuton

21 For the one dmensonal homogeneous dstrbuton n an nterval: H ( p) = log d. Where d s the length of the nterval. Ths means that the uncertanty of a homogeneous dstrbuton ncreases along wth d. 2 For the one dmensonal normal dstrbuton wth varance σ, 1 H ( p) = logσ + ( 1+ log 2π ). 2 Ths means that the uncertanty of a normal dstrbuton ncreases along wth ts varance. The negatve value of H ( p),.e. H ( p) = E (log p), s defned as the Informaton content of the dstrbuton p ( Θ). The Pror nformaton on parameters Θ can be descrbed by a pdf p ( Θ) 0, whch s called the pror dstrbuton of Θ. After transferrng the nformaton from the observaton data to the estmated parameters, we wll have a new pdf ( Θ) p, whch s called the posteror dstrbuton of Θ. The nformaton contents contaned n the pror and posteror dstrbutons are H p ) and H( p * ), respectvely. The dfference between them,.e. I ( p p ) H ( p ) H ( ), p 0 * 0 * * = (2.9) measures the nformaton content transferred from the observaton data. Therefore n the statstcal framework the problem s how to fnd the pror dstrbuton 0 ( Θ) p ( Θ). * ( 0 p and the posteror dstrbuton Pror Dstrbuton The followng two man types of pror dstrbuton of mult-varables are often used for parameter estmaton: Homogeneous Dstrbuton Ths type of dstrbuton s gven n range = ( Θ Θ ) of the estmated parameters, Θ U and pror dstrbuton can be expressed by p 1 Θ, when Θ ad V ( ) 0 = Θ ; ( Θ) 0 0 = Θ,. In ths case, the upper and lower bounds ad L U Θ L, are determned by pror nformaton, and the correspondng p otherwse; where V s the volume of the box Θ ad. 21

22 Normal Dstrbuton Ths type of dstrbuton s gven wth mean 0 Θ and covarance matrx ( ) Θ 0 Cov (an N p xn p matrx, where N p s the dmenson of Θ ). In ths case, Θ 0 are the guessed values of the estmated parameters based on avalable pror nformaton and Cov ( Θ 0 ) measures the relablty of these guessed values. Both types of Informaton can be usually derved from geostatstcal nformaton. Thus, the correspondng pror dstrbuton can be expressed by the followng N p -dmensonal Gaussan dstrbuton p 0 1 π (2.10) 2 N / 2 T 1 ( Θ) = ( 2 ) ( det Cov ( Θ )) exp [ Θ Θ ] Cov ( Θ )[ Θ Θ ] m The observaton Equaton 2.7 defnes the relatonshp between Θ and error vector ε. The posteror dstrbuton s the pdf of Θ for gven producton data obs merely the condtonal pdf ( u ) obs u through a model and an obs p Θ. On the other hand, the condtonal pdf ( u Θ) obs obs u for gven estmated parameters Θ. If there s no model error, ( u Θ) obs u. Thus s p s the pdf of p s equal to ( ε Θ) obs.e. the pdf of observaton errors when the estmated parameters Θ are gven. ( u Θ) p, p s usually called the lkelhood functon of observatons and s denoted by L ( Θ). Accordng to the Bayes s theorem: p obs ( Θ u ) = ( Ω ) p p obs ( u Θ) p0 ( Θ) obs ( u Θ) p ( Θ) 0 dθ (2.11) or ( Θ) = c L ( Θ) p ( Θ) p (2.12) 0 ( Θ) 1 obs Where constant c p( u Θ) p0 ( Θ) = d. The above equaton accomplshes the transfer of ( Ω) nformaton from observaton data to the estmated parameters. Snce the pror dstrbuton s gven, the posteror dstrbuton p ( Θ) s completely determned by the lkelhood functon ( Θ) constant factor. * 22 L and a Accordng the central lmt theorem whch states that the sum of a large number of ndependent, dentcally dstrbuted random varables, all wth fnte means and varances, s approxmately normally dstrbuted, t s assumed that the total observaton error vector ε s also normally (Gaussan) dstrbuted wth zero mean and constant covarance matrx of observaton data), the lkelhood functon can be specfed as Cov ε (a N m xn m matrx, where N m s the number

23 L N ( Θ) = ( ) m / 2 1/ 2 1 obs calc T 1 obs calc ( det Cov ) exp ( u u ( Θ M )) Cov ( u u ( Θ M )) π (2.13) ε 2 2 ε Substtutng (Equaton 2.10) and (Equaton 2.13) nto (Equaton 2.12), we may have an expresson of the posteror dstrbuton that contans both the pror nformaton and the nformaton transferred from the observaton data. On ths bass the crtera for parameter estmaton can be derved { [ ]} obs { p ( Θ) } = arg mn L ( Θ u ) Θ ˆ = arg mn Θ ln (2.14) Θ Substtutng (Equaton 2.12) nto (Equaton 2.14) and usng ln ( Θ) to replace ( Θ) { ln ( Θ) ln p ( Θ) } p p, follows: Θˆ = arg mn L Θ 0 (2.15) Statstcal Crtera of Parameter Estmaton wth Homogeneous Pror When ( Θ) 0 Dstrbuton p s gven by (8) we have the followng maxmum-lkelhood estmator (MLE): { ln[ L( Θ )]} subject to Θ Θ ad Θ ˆ = arg mn, (2.16) Θ If L ( Θ) can be expressed by (Equaton 2.13) and the covarance matrx Cov ε s ndependent of Θ the above MLE reduces to the followng generalzed least square estmator of the unknown parameters under the model M: Θ ˆ = arg Θ mn obs calc T 1 obs calc [( u u ( Θ M )) Cov ( u u ( Θ M ))] ε (2.17) 2 Furthermore, f the measured errors are equal (wth the varance σ ) and ndependent of each other, the above relatonshp reduces to the objectve functon (OF), expressed wth the sum of squared resduals: Θˆ = arg Θ mn { OF } = arg Θ mn N m = 1 u obs u calc σ ( Θ M ) 2 (2.18) 23