8/9/2019 Choudhary 2012

1/61

GENERATION OF MULTIPLE

HISTORY MATCH MODELS USING

MULTISTART OPTIMIZATION

A REPORT SUBMITTED TO THE DEPARTMENT OF ENERGY

RESOURCES ENGINEERING

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THEDEGREE OF MASTER OF SCIENCE

By

Manish K Choudhary

June 2012

8/9/2019 Choudhary 2012

2/61

8/9/2019 Choudhary 2012

3/61

 iii

I certify that I have read this report and that in my opinion it is fully adequate, in scopeand in quality, as partial fulfillment of the degree of Master of Science in PetroleumEngineering.

 __________________________________

Dr. Tapan Mukerji

8/9/2019 Choudhary 2012

4/61

8/9/2019 Choudhary 2012

5/61

 v

Abstract

Uncertainty in the geological model presents a key challenge in development decisions.

Production data from the field are acquired only at limited locations and are sparse.Time-lapse seismic data is available field-wide but has limited resolution. In addition,increasingly production logging data is being recorded in wells, which provideinformation regarding vertical heterogeneity between wells. The available data set is stillsparse for accurately modeling spatial distribution of reservoir properties. Hence,multiple geological realizations can exists which match the given production history andgenerate varying forecast, all of which should be analyzed for decision-making.

In my research thesis, two optimization algorithms have been tested for generatingmultiple history- matched geological models. The reservoir inversion problem has beenformulated using optimization technique, with an objective of minimizing the variance

 between observations and output of numerical models using one, two and all threedatasets as described above. Optimization is carried out in reduced model space. Modelreduction is achieved by spatial principal component analysis (PCA), where optimizationsearch space is projected to a subspace of much smaller dimension.

Local optimizers often tend to find solutions faster than global methods, though they can be trapped in local minima. Randomly generated multiple initial points can be optimizedin parallel to locate multiple models matching history. Hook-Jeeves direct search (HJDS)algorithm, simultaneous perturbation stochastic approximation (SPSA) algorithm has been used for optimization and results are compared with rejection sampler. The minima points identified through optimization represent geological models that are consistent

with the production history of the field. The methodology has been tested on threedifferent synthetic case studies with both categorical variable and continuous variablesThe optimization process locates geological models that are consistent with productionhistory but present a varying forecast which can help in decision analysis.

8/9/2019 Choudhary 2012

6/61

8/9/2019 Choudhary 2012

7/61

 vii

Acknowledgments

I wish to express my sincere gratitude to my advisor Dr. Tapan Mukerji for his guidance,

and support during my Masters research. His deep and insightful suggestions were the building blocks of this research. His continuous patience, motivation and enthusiasmhelped me at all times in my two years at Stanford. I deeply appreciate his tireless readingof my thesis.

I am greatly indebted to the faculty, staff, and students of Department of PetroleumEngineering for their support and contribution to my academic achievements.

I would also like to express my thanks to David Echeveria for his computer code, whichwas the starting point of this research. I would also like to thank Mehrdad Shirangi forexplaining SPSA algorithm to me.

I would like to thank the companies supporting the Stanford Center for ReservoirForecasting (SCRF) affiliate program at Stanford University for their financialcontribution that made this research possible.

Finally, I owe my greatest gratitude to my family for their love and devotion. Withouttheir support, I would have never made it this far. I am grateful to my friends for theirsupport and constant motivation to achieve my objective.

8/9/2019 Choudhary 2012

8/61

8/9/2019 Choudhary 2012

9/61

 ix

Contents

Abstract ............................................................................................................................... v 

Acknowledgments............................................................................................................. vii 

Contents ............................................................................................................................. ix 

List of Tables ..................................................................................................................... xi 

List of Figures .................................................................................................................. xiii 

1  Introduction ................................................................................................................. 1 

1.1  Methodology ........................................................................................................ 2 1.2  Principal Component Analysis ............................................................................. 3 1.3  K-Medoid Clustering............................................................................................ 3 

1.4  Optimization Algorithm ....................................................................................... 3 1.4.1  Hook Jeeves Algorithm................................................................................. 4 1.4.2  SPSA Algorithm ........................................................................................... 4

2  Testing the Idea - Case Study I & II ........................................................................... 5 

2.1  Model Description ................................................................................................ 5 2.2  Dimension Reduction ........................................................................................... 6 2.3  Observables and Objective Function.................................................................... 6 2.4  K - Medoid Clustering.......................................................................................... 7 2.5  Model Set - 1: Continuous Property based Model ............................................... 8 

2.5.1 

Model Generation ......................................................................................... 8 

2.5.2  Dimension reduction and Clustering ............................................................ 9 2.5.3  Optimization Results ..................................................................................... 9 2.5.4  Optimization Algorithm Efficiency ............................................................ 11 2.5.5  Forecast Uncertainty ................................................................................... 12 

2.6  Model Set - 2: Categorical Property based Model ............................................. 12 2.6.1  Model Generation ....................................................................................... 12 2.6.2  Dimension reduction and Clustering .......................................................... 13 2.6.3  Inclusion of Seismic Tomography Data ..................................................... 15 2.6.4  Optimization Results ................................................................................... 16 2.6.5  Optimization Efficiency .............................................................................. 18 

2.7  Conclusions ........................................................................................................ 19

3  Expanding the Idea – Case Study III ........................................................................ 21 

3.1  Model Description .............................................................................................. 21 3.1.1  Geology ....................................................................................................... 21 3.1.2  Petrophysics ................................................................................................ 22 3.1.3  Relative Permeability & Capillary Pressure ............................................... 23 3.1.4  Fluid Properties ........................................................................................... 23 

8/9/2019 Choudhary 2012

10/61

 x

3.1.5  Rock Physics Model ................................................................................... 24 3.1.6  Production Strategy ..................................................................................... 24 

3.2  Model Decomposition & Reconstruction ........................................................... 25 3.2.1  Principal Component Analysis ................................................................... 25 3.2.2  Dimension Reduction.................................................................................. 26 

3.2.3 

Model Generation ....................................................................................... 27 

3.3  History ................................................................................................................ 28 3.4  Optimization ....................................................................................................... 30 

3.4.1  Workflow .................................................................................................... 30 3.4.2  Observables & Response Function ............................................................. 30 

3.5  Results ................................................................................................................ 31 3.5.1  Flow Only ................................................................................................... 31 3.5.2  Flow & PLT Data........................................................................................ 32 3.5.3  Flow, PLT and Seismic Data Optimization ................................................ 34 

3.6  Rejection sampling algorithm ............................................................................ 36 3.7  Optimization Efficiency ..................................................................................... 37 

3.8 

Forecast Uncertainty .......................................................................................... 38 

3.9  Conclusions ........................................................................................................ 39

4  Summary ................................................................................................................... 41 

 Nomenclature .................................................................................................................... 43 

References ......................................................................................................................... 45 

8/9/2019 Choudhary 2012

11/61

 xi

List of Tables

Table 2-1: Dynamic properties of the model ...................................................................... 6 

Table 2-2: Porosity Variogram: Model - 1 ......................................................................... 8 

Table 3-1: Fluid Properties ............................................................................................... 24 

Table 3-2: Rock Physics Model ........................................................................................ 24 

Table 3-3: Weight for Optimization ................................................................................. 31 

8/9/2019 Choudhary 2012

12/61

8/9/2019 Choudhary 2012

13/61

8/9/2019 Choudhary 2012

14/61

 xiv

Figure 3-4: Relative permeability curveand Capillary pressure curve ............................. 23 

Figure 3-5: “Zamba” Production Strategy ........................................................................ 25 

Figure 3-6: Variance of first 3000 principal components ................................................. 26 

Figure 3-7: Cumulative variance of first 3000 principal components .............................. 26 

Figure 3-8: Reconstruction of model using limited set of principal components ............. 27 

Figure 3-9: Bounds for PCA coefficients ......................................................................... 27 

Figure 3-10: Randomly generated models using 3000 major principal components ....... 28 

Figure 3-11: Production history of the Zamba reservoir .................................................. 28 

Figure 3-12: Fluid saturations over time .......................................................................... 29 

Figure 3-13: Spinner survey data for well - 4 and well - 6 ............................................... 29 

Figure 3-14: Optimization workflow ................................................................................ 30 

Figure 3-15: Water Cut - flow only Optimization ............................................................ 32 

Figure 3-16: Layer 5 of history matched models: Flow only optimization ...................... 32 

Figure 3-17: Water cut - flow & PLT data optimization .................................................. 33 

Figure 3-18: PLT data match - End of 3 rd year for Well - 6 (Injector) ............................. 33 

Figure 3-19: PLT data match - End of 3 rd year for Well - 4 (Producer) ........................... 34 

Figure 3-20: History matched models: Flow & PLT data optimization ........................... 34 

Figure 3-21: Water cut - flow, PLT and seismic data optimization.................................. 35 

Figure 3-22: P-wave velocity match - flow, PLT and seismic data optimization ............. 35 

Figure 3-23: History matched models: Flow, PLT and seismic data optimization .......... 36 

Figure 3-24: Comparison with rejection sampler ............................................................. 36 

Figure 3-25: Optimization Code Efficiency for Hook – Jeeves algorithm ....................... 37 

Figure 3-26 : Optimization Code Efficiency for SPSA algorithm .................................... 38 

Figure 3-27: Forecast of history matched models ............................................................. 38 

8/9/2019 Choudhary 2012

15/61

  1

Chapter 1

1  Introduction

Selection of the best development plan for an oil field is one of the significant decisionsundertaken by oil-field operators. The decision is based on analysis of oil & gas forecastfrom multiple reservoir models generated using reservoir simulation. Numericalsimulation models require reliable estimates for various reservoir parameters that cannot be measured directly (Slater & Durrer, 1971; Friedmann, Chawathe, & Larue, 2003). Ofmany reservoir parameters, sub-surface heterogeneity i.e. spatial distribution of porosityand permeability is one of important sources of uncertainty (Stephen & MacBeth, 2008;Srinivasan & Deutsch, 2004). It is why operators rely on forecasts from multiplereservoir models to quantify the uncertainty and risk associated with development plan(Cruz, Horne, & Deutsch, 2004; Rivera, et al., 2007).

Limited data is available to classify the spatial distribution of reservoir propertiesaccurately, resulting in multiple geological models that match historical observations.Production data from the field is acquired only at limited locations and is sparse. Time-lapse seismic data is available field-wide but has limited resolution. Increasingly spinnersurveys (Kading, 1976)) are being recorded in wells. The data though being sparse, provides information about connectivity between wells and can be used for historymatching (Yoelin & Howald, 1970; Vogelij, Leach, & Kapteyn, 1993; Panda & Nottingham, 2011). These independently recorded observations from the field provideinformation about spatial distribution of rock parameters at different resolution, all ofwhich should be integrated for history matching and uncertainty quantification (Stephen,Soldo, MacBeth, & Christie, 2006; Litvak, Christie, Johnson, Colbert, & Sambridge,2005). Decisions pertaining to selection of best plausible development option should be based on performance analysis of all history matched subsurface realizations.

The process of history matching involves adjustment of model parameters to matchmodel response with the observed dataset. The traditional history matching approach orcascaded approach, involved independent adjustment of model parameters and is usuallyad-hoc. As such, it can result in models that are quite different from prior models. Themodern history matching approach calls for a closed-loop workflow where faciesdistribution is perturbed in a geological consistent manner to obtain a good match

 between model response and observed dataset. This ensures that posterior models havesimilar features as prior models (Caers, 2011). Different methodologies have been proposed to minimize the mismatch between model prediction and observed data -simulated annealing, pattern search, gradual deformation, Markov chain, iterative re-sampling, gradient based optimization, etc (Caers, 2011; Himmelblau, 1972).

The mismatch or error minimization problem can also be set up as an optimization problem. Many authors have proposed formulation of simultaneous matching of

8/9/2019 Choudhary 2012

16/61

 2

 production and seismic data as an optimization problem (Huang, Meister, & Workman,1998; Sarma, Durlofsky, Aziz, & Chen, 2006). The formulation can also be extended toother data set like spinner survey. A petroleum reservoir can have close to millionunknowns (porosity, permeability, facies etc.) and any optimization in the original spacewill be inefficient and would have extremely high computation costs. (Echeveria &

Mukerji, 2009) suggest using principal components to reduce the optimization searchspace while maintaining geological consistency. The process of optimization wouldlocate a model which has minimum variance across observed data points and estimatedvalues, in the local space (local minima) or across the entire solution space (globalminima). This process would only locate one solution model and hence does not provideany mean to capture production forecast uncertainty. This research thesis builds on earlier proposed optimization scheme of (Echeverria & Mukerji, 2009) to locate multiple historymatched models.

1.1  Methodology

The inversion process is formulated as an optimization problem, with the aim to reducingthe error between observations and calculated dataset, similar to the one proposed(Echeverria & Mukerji, 2009). Let   denotes the   dimensional space in which thereservoir models are defined,  is the set of admissible geological models which closelymatches observed data and g denotes the indicator property (facies) or reservoir property(porosity and/or permeability) which is used to define the geological model. The systemof models can be expressed as  ∈   ∪  . The optimization problem can be defined asin Equation (1-1)

∗ =    ⟦ℝ() −  ⟧  (1-1)where m ∪ Un  comprises all observed data in the inversion, and ℝ() ϵ Un representsthe numerically computed observables for the model . Varying weights/normalizationfor different components in the observables (production, seismic, flow survey etc.) can beincluded in the optimizer function (Euclidean norm in this case) to take into accountvariable uncertainty in the data acquisition & observations. The optimization problem isill conditioned due to much larger number of inversion parameters than observations(  >> ) and model reduction using principal component analysis have been proposedearlier. Optimization is carried out in subspace of lower dimension (principalcomponents:   

8/9/2019 Choudhary 2012

17/61

  3

1.2  Principal Component Analysis

Principal component analysis (PCA) is a mathematical procedure that linearly transformsan original set of variables into a substantially smaller set of linearly uncorrelatedvariables called principal components. The number of principal components is equal to

the dimensionality of the space that can be much smaller than the dimension of originalvariable set. The limited set of principal components represents all information of theoriginal data set (Jolliffe, 2002). The transformation generates principal components inthe decreasing order of variance i.e. the first principal component has the largest possiblevariance and accounts for maximum variability among the variable set, the second principal component has the second largest variance and is orthogonal to the first principal component. Each succeeding component add less and less variability to themodel and is orthogonal to all other principal components (Gass, 2007).

The principal components are independent of each other only if the variables arenormally distributed. PCA is a linear technique and any extension to non-linear variables

approximates the method. PCA is sensitive to the relative scaling of the originalvariables. The PCA technique is also called by other names such as discrete Karhunen– Loève transform (KLT), the Hotelling transform and proper orthogonal decomposition.

1.3  K-Medoid Clustering

Reservoir models are large dimensional models and it is often difficult to visuallycompare models with each other. Prior models generated using geostatistical techniquelook similar visually but can produce widely varying production response. likewise,models which look different may have similar oil & gas forecast. Since the localoptimizers are very sensitive to initial guess, it is important that the starting models be

spaced as far as possible to prevent convergence to same minima.K-Medoid clustering algorithm can be used in a multi-dimensional space (Caers, 2011) toselect initial models for optimization. A forward model computation may be tooexpensive so a proxy distance mapping can be used for clustering. The K-Medoidalgorithm attempts to cluster the dataset by minimizing the non-metric distance between points and selects models that are deemed center of the cluster. The selected prior modelsare likely to be spaced far apart and will generate different posterior models. (Caers,2011; Maulik, Bandyopadhyay, & Mukhopadhyay, 2011).

1.4  Optimization Algorithm

Local optimizers tend to be efficient than global optimizers as they quickly converge to alocal minima. A gradient-based optimization can further increase speed, but gradients arecostly to compute. Adjoint based simulator can help in accelerating the search but requireaccess to source code. In most cases, adjoints are unavailable in commercial simulators orthey are too complicated to work with. Two gradient free algorithms were used foroptimization during the study and are briefly described in the section below.

8/9/2019 Choudhary 2012

18/61

8/9/2019 Choudhary 2012

19/61

8/9/2019 Choudhary 2012

20/61

 6

The reservoir model represented an inverted five-spot pattern. All five wells weremodeled fully penetrating. Both sets of models were constrained using porosity/facies atthe well locations. PVT properties for the fluids in the reservoir were modeled as blackoil with no free gas in the system. The reservoir was considered saturated with oil andconnate water at the start of simulation. The reservoir model was simulated for water

flood scenarios. Oil-water relative permeability curves were generated using Coreyfunctions (Ahmed, 2010; Dake, 2001). Figure 2-2 shows the relative permeability curvesused in the model. The key dynamic parameters of the model are listed in Table 2-1. 

Table 2-1: Dynamic properties of the model

Property Value

Oil Viscosity 1.20 cpWater Viscosity 0.325 cpRock Compressibility 5 x 10-6 psi-1 Well Control Constant pressure injector and

Constant pressure producerProduction Strategy(Water Flood)

4 Water injectors in the Corner1 Oil Producer in the Center

2.2  Dimension Reduction

Large set of realizations are required for extracting principal component using PCAtechnique. An ensemble of 1000 realizations was generated for both model set usingSGSIM (Model – 1: Porosity based model) and SNESIM (Model – 2: Facies basedmodel). PCA was performed using MATLAB® software on both ensembles to generate principal components and their associated coefficients. The variance for each principalcomponent and cumulative variance was analyzed to select optimum number of principalcomponents for model reconstruction and thereby achieve dimension reduction.Additionally, original model realizations were compared with models reconstructed usinglimited set of principal components to verify the optimum number of principalcomponents.

Lower and upper bounds of the coefficients were also computed to determine the range ofcoefficient for each principal component. The range of coefficients was further extended by (+/-) twice the standard deviation of coefficients. New realizations for each modelwere constructed using limited set of principal components by selecting coefficientsrandomly sampled from the uniform distribution of the lower and upper bound.

2.3  Observables and Objective Function

The reservoir models were simulated for water flood for oil recovery. Four waterinjectors were located near the corners of the model that inject water at constant bottom-hole pressure. An oil producer was located at the center of the grid that is alsoconstrained by constant bottom-hole pressure. The five-spot well pattern described above

8/9/2019 Choudhary 2012

21/61

  7

is simulated using Stanford's GPRS3  simulator for a period of 90 days of reservoirexploitation. Flow data at wells is sampled at an interval of 10 days.

= �[

ℝ(

)

− 

(

)

]2

()2 + [

ℝ(

)

− 

(

)]2

()2   =

=0   = 0.5 = 0.5

(2-1)

The observable ℝ() for a model refers to the cumulative production and injection data ofthe wells over time, whereas () referred to the production and injection history dataset.The historical data set was generated by simulating the true reference model for waterflood. The response function () used for optimization was the normalized variance ofmodel response with respect to observed data set. Equal weighs for production andinjection data was used as a part of optimization algorithm. Equation (2-1) shows the

response function used for optimization algorithm. The optimization algorithm attemptedto minimize the response function by perturbing the coefficients for each principalcomponent.

2.4  K - Medoid Clustering

A large set of random realizations were generated for both types model. The large set ofrealization were clustered together to select 10 initial realizations for multi-startoptimization. Clustering was performed using multidimensional scaling (MDS) K-Medoid clustering (Scheidt & Caers, 2009) which helps in selecting models foroptimization, which are distinctively different from each other.

Figure 2-3: Proxy distance computation for clustering

A proxy distance was used for purpose of MDS clustering. The proxy distance that wasused for both model set was calculated using harmonic average of permeability in grid

3 GPRS – General Purpose Reservoir Simulator

http://pangea.stanford.edu/researchgroups/supri-b/research/research-areas/gprs

8/9/2019 Choudhary 2012

22/61

 8

cells between each injector-producer combination. Figure 2-3 shows a schematic of proxydistance computation. Permeability across the layers was arithmetically averaged invertical direction prior to computing harmonic averages along the injector – producer.This represented a four-dimensional vector for each realization that was mapped to a two-dimensional space for clustering

2.5  Model Set - 1: Continuous Property based Model

 2.5.1   Model Generation

The first model set used porosity as primary variable. Porosity was modeled as aGaussian variable using the histogram derived from well data and spatially correlated bya known variogram. The histogram of the porosity data modeled is shown in the Figure2-4. The parameters of the variogram model used for generating realizations are listed inTable 2-2. 

An ensemble of 1000 realizations was generated using SGSIM in SGeMS

4

 constrained tofive well locations for principal component analysis and dimension reduction.

Figure 2-4: PDF and CDF of porosity: Model Set - 1

Table 2-2: Porosity Variogram: Model - 1

Parameter Value

Nugget 10%Range Max – 10; Med – 5, Min – 2Model Exponential

4 SGeMS - Stanford Geostatistical Modeling Software: http:// sgems.sourceforge.net/

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.02

0.04

0.060.08

0.10

0.12

0.14

0.16

0.18

0.2

Porosit Fraction

       F     r     e     q     u     e     n     c     y

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.30.4

0.5

0.6

0.7

0.8

0.9

1

    C   u   m   u    l   a

   t    i   v   e    F   r   e   q   u   e   n   c   y

8/9/2019 Choudhary 2012

23/61

  9

 2.5.2   Dimension reduction and Clustering

PCA was carried out on model ensemble in MATLAB® software to generate principalcomponents and coefficients. The number of dimension was reduced to 50 afteranalyzing the variance associated with each principal component with intent to capture

approximately 50% of the total variance. A set of 100 new realizations was generated tousing limited set of principal components to select 10 starting models for optimization.Clustering was performed by using the proxy distance defined earlier (Section 2.4). TheMDS plot of the selected 10 models and all other models is shown in Figure 2-5. 

Figure 2-5: K-medoid clustering: Model Set - 1

 2.5.3 

Optimization Results

Figure 2-6: Cumulative oil production vs. Time – Initial and final models: Model Set - 1

8/9/2019 Choudhary 2012

24/61

 10

Figure 2-7: Cum. oil production vs. cum. water injection - Initial & final models: Model Set - 1

Hook-Jeeves algorithm was used as an optimization algorithm for perturbing models forhistory matching. Figure 2-6 plots the cumulative oil production vs. time and Figure 2-7 plots the cumulative oil production vs. cumulative water injection for the initial startingmodels and the final solution models for the period of history match. As it can beobserved, optimization results in reduction of mismatch with production history.

Figure 2-8: Layer 5 - History matched models: Model Set - 1

8/9/2019 Choudhary 2012

25/61

  11

The process of multistart optimization results in 10 different realization that match the production history. These realizations represent possible geological scenarios all of whichmatch production data within the chosen tolerance.

Figure 2-8 shows a cross section view (Layer 5) of the final models after optimization.As it can be observed, they represent widely different scenarios.

 2.5.4  Optimization Algorithm Efficiency

The variance of flow response with the historical data was computed (ResponseFunction) at end of every iteration to monitor efficiency of optimization. Figure 2-9 plotsthe response function for the realizations as it is perturbed through the iterations. At theend of Hook-Jeeves iteration, 80% of the initial models converged to a cost functionvalue of less than 10-2.

Figure 2-9: Optimization function efficiency: Model Set - 1

Multidimensional scaling can be also used to analyze the difference between the initial

guess models and the final optimized solution models. Figure 2-10 shows amultidimensional-scaled plot of initial models and final perturbed models along with thetrue solution mapped to two-dimension space. The response function value across tentime steps was used as distance between the models.

8/9/2019 Choudhary 2012

26/61

8/9/2019 Choudhary 2012

27/61

8/9/2019 Choudhary 2012

28/61

 14

Figure 2-14 compares the reconstructed models generated using 10, 30 and 100 principalcomponents. As it can be observed, model reconstructed with 100 principal componentsis very similar to model reconstructed with 30 principal components.

Figure 2-14: Model reconstruction: Model Set – 2 (Yellow- Sand; Black – Shale)

A set of 100 new random models were generated by using first 30 major principalcomponents. These models were clustered using proxy distance defined earlier (Section2.4)  to select ten starting models for optimization. The MDS plot of the selected ten

models and all other models is shown in Figure 15. The reconstructed values weremapped to porosity and permeability as inputs for flow simulation

Figure 2-15: K-medoid clustering: Model Set – 2 (Yellow- Sand; Green – Shale)

8/9/2019 Choudhary 2012

29/61

8/9/2019 Choudhary 2012

30/61

 16

 2.6.4  Optimization Results

Hook-Jeeves algorithm was used as an optimization algorithm for history matching. Theoptimization algorithm reduces the mismatch between model response and history

2.6.4.1 

Flow Only

Figure 2-17: Layer – 5: Flow only history matched models: Model Set – 2(Yellow- Sand; Green – Shale)

Figure 2-17 shows the X-Y cross section of the final solution models along with the truereference model. Here yellow color represents the sand and green represents the shalefacies. As it can be observed, final solution models vary with each other but at the sametime all models contain the trace of bottom channel facies in Layer – 5as in truereference model.

2.6.4.2  Flow and Seismic Data

Inclusion of cross-well tomography data provides additional information across the inter-well section. The data is not descriptive for the entire reservoir hence the impact ofseismic data does not drastically change the final solution models. Figure 2-17 shows theX-Y cross section of the final solution models along with the true reference model. Hereyellow represents the sand facies while green represents the shale facies. As it can beobserved, the solution models are different from true reference model and all modelscontains majority of sand facies along the lower edge of the model similar to true model.

True

8/9/2019 Choudhary 2012

31/61

  17

Figure 2-18: Layer – 5: Flow & seismic history matched models: Model Set – 2

(Yellow- Sand; Green – Shale)

2.6.4.3 

 Ensemble average and Variance

The ensemble of final solution was combined to compute E-type and variance.   Figure2-19 shows the E-type of first, fourth, seventh and ninth layer and Figure 2-20 displaysthe variance. The inclusion of cross-well seismic data slightly reduces the variance.

Figure 2-19: E- Type of final solution models: Model Set – 2 (Yellow – Sand; Green – Shale)

True

La er - 1 La er - 4 La er - 7 La er - 9

True Model

Production Data Only

Production & Seismic Data

8/9/2019 Choudhary 2012

32/61

 18

Figure 2-20: Variance of final solution models: Model Set – 2: (Yellow – Sand; Green – Shale)

 2.6.5 

Optimization Efficiency

Figure 2-21 plots the response function over the function evaluations. The optimizationalgorithm is able to reduce the response function value to less than 10-2. In addition, it can be observed that the inclusion of the cross-well seismic tomography data slows the

convergence algorithm and it takes longer to converge to values less than 10-2

.

Figure 2-21: Optimization efficiency : Model Set - 2

Layer - 1 Layer - 4 Layer - 7 Layer - 9

Production & Seismic Data

Production Data Only

True Model

8/9/2019 Choudhary 2012

33/61

  19

Forecast Uncertainty

The converged models were further simulated to quantify the forecasts and theirvariability. Figure 2-22 compares the mean and spread of prediction runs for both sets ofinversion with true model forecast. As expected, inclusion of seismic data in optimization

reduces the forecast uncertainty.

Figure 2-22: Forecast variability: Model Set – 2

2.7  Conclusions

Multi-start optimization in reduced model dimension works effectively with bothcontinuous property and categorical variable for generating multiple geological models.The resulting models are consistent with production history of the field but have differentforecast and hence represent possible subsurface realizations. Model dimension reductionusing PCA does cause of loss of geological information associated with higher order

Eigen vectors, but offers a reasonable model approximation. The technique also offersflexibility to include additional data such as seismic data or well survey data for reducingsolution space uncertainty.

8/9/2019 Choudhary 2012

34/61

8/9/2019 Choudhary 2012

35/61

8/9/2019 Choudhary 2012

36/61

 22

 3.1.2   Petrophysics

Both sand and shale have been considered porous and permeable in the reservoir model. The porosity in sand and shale facies were populated independently using SGSIM7 algorithm of SGeMS. Porosity variogram in sand was considered anisotropic with

continuity along the channel direction, while shale variogram was less anisotropic withmaximum, medium and minimum direction comparable with each other. Figure 3-2shows the porosity distribution in the reservoir. Figure 3-3 shows the porosity histogramof the sand and shale facies. The effective porosity in sand varies from 16% to 33% whileit varies from 3% to 16% in shale.

Figure 3-2: Porosity distribution in the reservoir

Figure 3-3: Histogram of porosity for Sand (Left) and Shale (Right)

Permeability in the grid cells was computed using a porosity permeability relationship. Amodified porosity permeability relationship from  Norne field (Suman & Mukerji, 2012)

was used for the model. Equation  (3-1) and Equation (3-2) represents the porosity permeability relationship used in the reservoir model for sand and shale respectively.

Sand log = 5.095 ∅ + 2.0580  (3-1)Shale log = 14.610 ∅ + 0.1745  (3-2)

7 SGSIM - Sequential Gaussian simulation

8/9/2019 Choudhary 2012

37/61

  23

 3.1.3   Relative Permeability & Capillary Pressure

The reservoir model is simulated for water flood scenario with both oil-water contact andgas-oil contact. A three-phase relative permeability model was used to model flow acrossgrid cells. Oil-water relative permeability for sand was generated using Corey functions

(Ahmed, 2010; Dake, 2001) with Corey exponent of 2.75 and 4 for oil and waterrespectively. The Corey exponent for oil and water in case of shale are 3.25 and 3.75respectively. Connate water saturation in sand and shale was assigned as 12% and 30%respectively.

Modified capillary pressure data from  Norne  field was used for sand and shale facies.The relative permeability model and capillary pressure model used in the model areshown in Figure 3-4. 

Figure 3-4: Oil – Water relative permeability curve (Left) and Capillary pressure curve (Right)

 3.1.4   Fluid Properties

The reservoir simulation model included a three phase fluid system - Oil, Water and Gas.An isothermal black oil formulation was used to describe fluid behavior. Oil in thereservoir was assigned an API gravity of 30o with a saturation pressure of 3600 psi. Thegravity of the gas in the reservoir was assigned as 0.85. The salinity of water wasassumed 50000 ppm. The reservoir temperature was set at 180  o  F. The model wasinitialized with an oil-water contact (OWC) within the structure at 6800 feet. No free gasexisted in the model at the time of initialization.

The PVT properties for oil and gas were generated for different reservoir pressures usingStanding correlation (Standing, 1947). PVT properties for water were calculated usingMcCain correlation (McCain, 1989). The key fluid properties are listed in Table 3-1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0

   R   e    l   a   t   i   v   e   P   e   r   m   e   a    b   i    l   i   t   y

   R   e    l   a   t   i   v   e   P   e   r   m   e   a    b   i    l   i   t   y

Water Saturation [fraction]

Kro - Shale

Kro - Sand

Krw - Shale

Krw - Sand

0

5

10

15

20

25

30

35

0.0 0.2 0.4 0.6 0.8 1.0

   C   a   p   i    l    l   a   r   y   P   r   e   s   s   u   r   e

    [   p   s

   i    ]

Water Saturation [fraction]

Shale

Sand

8/9/2019 Choudhary 2012

38/61

 24

Table 3-1: Fluid Properties

Fluid Property Value

Oil API 30o APIGas Gravity 0.85

Separator Gas Gravity 0.90Reservoir Temperature 180o FSaturation Pressure 3600 psiWater Salinity 50,000 ppm

 3.1.5   Rock Physics Model

Rock physics model defines how seismic attributes change due change in pressure andsaturation of the rock. An uncemented-sand (or “soft-sand”) is used for sand facies. Themodel assumes that the sand grains were deposited as a dense random pack of identicalspherical grains and average number of contacts per grain between five and nine (Mavko,

Mukerji, & Dborkin, 2009). Seismic attributes are defined for this type of setting usingHertz-Mindlin theory (Mindlin, 1949). Sand is assumed to be made of Quartz, Feldsparand fragments, while shale is made of clay and quartz.

Seismic attributes for shale facies are estimated using Gardner’s power law for () Pwave velocity (Gardner, Gardner, & Gregory, 1974) and “Mudrock line” for () S wavevelocity (Castagna, Batzle, & Eastwood, 1985). In-situ fluid properties are computedusing Batzle-Wang relations (Batzle & Wang, 1992) which are used to compute seismic properties for rock saturated with fluid mixture using Gassmann fluid substitution(Gassmann, 1951; Mavko, Mukerji, & Dborkin, 2009). The key parameters of rock physics model are listed in Table 3-2. 

Table 3-2: Rock Physics Model

Parameter Value

Sand Composition 80% Quartz, 15% Feldspar, 5 % Rock fragmentsShale Composition 85% Clay, 15% QuartzSand Vp & Vs Uncemented Soft Sand ModelShale Vp & Vs Gardner’s relation (Vp) & Castagna relation (Vs)

 3.1.6    Production Strategy

The “Zamba” reservoir model was simulated for water flood scenario. The top andmiddle rows of well were assigned as producers. The top row of the wells (Well – 1, 2and 3) were completed with preformation in all 25 layers, while the middle row of wells(Well – 4 and 5) were completed in top 20 layers. The middle group of wells wasconstrained at a rate of 13000 bbl. per day, while top group wells were constrained with amaximum liquid rate of 8000 bbl. per day each. A lower rate constraint was assigned ontop group of wells to minimize formation of secondary free gas due to pressure drop. All

8/9/2019 Choudhary 2012

39/61

  25

 producers had a minimum bottom-hole pressure (BHP) constraint 50 psi. Figure 5 showsthe location of wells in the reservoir

Figure 3-5: “Zamba” Production Strategy

Water was injected through the bottom row of wells (Well – 6, 7 and 8) to push the oil tothe producer and to maintain producers. The injectors were completed in all 25 layers andwere set to inject at a rate of 15000 bbl. per day with a maximum injection pressure of3500 psi. Injected water replaced the voidage created in the reservoir by the producersthereby maintaining pressure, at the same time pushing the water towards producers. Thereservoir model was simulated for 4 years of water flood.

Oil, gas and water production rate were recorded from the well at every month.Production rates from every layer were recorded once every year simulating a productionlogging survey – spinner survey in the well.

3.2  Model Decomposition & Reconstruction

 3.2.1   Principal Component Analysis

The multistart optimization methodology for history matching described in Section 1.1 iscarried out in a reduced model space. A large set of reservoir models – 10,000 modelswere generated with the same training image using SNESIM8  algorithm in SGeMSsoftware. The training image is indicative of the reservoir geology is a prior informationavailable with geoscientists

Principal component analysis was carried out the matrix of reservoir model to computethe major principal components. The principal components are set of values of linearlyuncorrelated variables, such that first principal component has the largest possiblevariance and each succeeding component in turn has the next highest variance. PCA of

8 SNESIM – Single Normal Equation Simulation

8/9/2019 Choudhary 2012

40/61

 26

the model set resulted in 10,000 principal components. Figure 3-6 and Figure 3-7 showsa plot of variance and cumulative variance of first 3000 principal components

Figure 3-6: Variance of first 3000 principal components

Figure 3-7: Cumulative variance of first 3000 principal components

 3.2.2   Dimension Reduction

Few of the models were reconstructed using limited set of principal components and werevisually compared with true model for consistency. Figure 3-8 shows a reconstruction oflayer 5 of the model number 1000 using 30, 70, 1000, 3000 and 5000 principalcomponents. As it can be observed, the reconstructed model closely resembles the truemodel when it is reconstructed using 3000 principal components. The first 3000 principalcomponents represent 90% of the total variance that can be inferred from Error!

Reference source not found. & Figure 3-7. This is in line with observation made by inother papers (Choudhary, Mukerji, & Echeverria, 2011; Echeverria & Mukerji, 2009)which suggests that models reconstructed with 25% of non-zero principal componentsclosely match the true model.

Additionally, it can be visualized that model reconstructed with 70 principal componentsreproduce the significant channels. The higher order principal components only add tofiner details to these channels.

8/9/2019 Choudhary 2012

41/61

  27

Figure 3-8: Reconstruction of model using limited set of principal components.(Yellow – Sand; Black – Shale)

 3.2.3 

 Model Generation

Figure 3-9: Bounds for PCA coefficients

The computational capacity limits the number of dimensions in which optimization can be performed, hence a smaller subset needs to be selected, which at the same time should be able to reproduce the key features. For the purpose of optimization for this case study,models were reconstructed using first 3000 principal components, while optimization wasonly performed in the first 70 principal components. Coefficients for each principalcomponent are selected sampled from a uniform distribution between minimum andmaximum coefficient value obtained as part of principal component analysis. The

8/9/2019 Choudhary 2012

42/61

 28

minimum and maximum bound for coefficients of the 3000 major principal vectors isshown in Figure 3-9 along with coefficients for a randomly generated model.

Figure 3-10: Randomly generated models using 3000 major principal components

(Yellow – Sand; Black – Shale)

Sand and shale facies is assigned in the model generated using with limited componentsusing a threshold equal to prior sand proportion. Figure 3-10 represents two-dimensionalX-Y slices (Layer 1) for three random realizations generated using 3000 principalcomponents.

3.3  History

Figure 3-11: Production history of the Zamba reservoir

The “Zamba” reservoir was simulated using Eclipse® for water injection for a period of 4years with wells on rate and pressure constraint. Injected water maintains the pressure ofthe reservoir and pushes the oil towards the producers. The simulated production for 4years is used as production history for purpose of optimization. Additionally, the layerwise flow rate data for the wells was recorded annually to simulate PLT survey. Figure11 shows the field production history of the reservoir.

8/9/2019 Choudhary 2012

43/61

8/9/2019 Choudhary 2012

44/61

8/9/2019 Choudhary 2012

45/61

  31

the model response with respect to observed data set is used as a response function foroptimization. The weight for each data type can be altered based on data quality.

 

=

�

ℝ() −  ()2

()2

=

=0  

 p = Field WC, Well WC and PLT data

(3-3)

Optimization was performed for three different scenarios - flow data only, flow data &PLT data and flow, PLT and seismic data. As the “ Zamba” reservoir is primarily underwater flood, only water cut at field level and well level was included for purpose of flowonly optimization. In case of flow and PLT data optimization, layer wise phase rates (oil,water and gas) for all eight wells combined with field level and well-level water cut wasused computing the response function. For the third scenario, Born-approximation(Mukerji, Mavko, & Rio, 1997; Lo & Inderwiesen, 1994) of P-wave velocity was for

every grid cell was also included in the response function computation for optimization.The seismic data is a field-wide data set can help in constraining facies away from wells.The weights used for all the three scenarios are listed in Table 3-3. 

Table 3-3: Weight for Optimization (Normalized by Total in Algorithm)

Flow Only Flow & PLT Data Flow , PLT and Seismic Data

Field Water Cut –10 Field Water Cut – 10 Field Water Cut – 10Well Water Cut – 20 Well Water Cut – 20 Well Water Cut – 20

Well PLT Data – 30 Well PLT Data – 30Seismic Data - 30

Total - 30 Total - 60 Total - 90

3.5  Results

A set of 10 randomly generated initial models was optimized using the optimizationalgorithms. The results for each of the scenario are discussed below.

 3.5.1   Flow Only

The response function for "Flow only optimization" was computed using water cutmeasurement at field and well level. The process of optimization resulted in sevenmodels with appreciable match. Figure 3-15 shows the water cut response of the reservoirand individual wells. At the end of 3rd  year, water breakthrough has already occurredthrough the middle row of producers (Well - 4 & Well - 5). No water breakthrough hasoccurred in top row of producers (Well - 1, Well - 2 & Well - 3) as such have not been plotted in  Figure 3-15.  The blue lines denotes models which match water-cut history,grey lines denotes model which did not match water-cut history and red points denotesthe history dataset.

8/9/2019 Choudhary 2012

46/61

8/9/2019 Choudhary 2012

47/61

  33

higher weight to PLT data should also be avoided due to inherent inaccuracies inmeasuring phase rates down-hole. The process of optimization resulted in 11 models withappreciable match. The water cut match of the model with respect to the history dataset isshown in Figure 3-17. 

Figure 3-17: Water cut - flow & PLT data optimizationRed – History, Blue – Models matching History, Grey – Models not matching History

The PLT data match for an injector (Well - 6) and a producer (Well - 4) is shown inFigure 3-18 and Figure 3-19 respectively. The optimization process results in models thatcapture all significant vertical flow rate variation in the well.

Figure 3-18: PLT data match - End of 3rd year for Well - 6 (Injector)

0 500 1000 15000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (Days)

    W   a   t   e   r    C   u   t    (    f   r   a   c   t    i   o   n    )

Well - 4

0 500 1000 15000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (Days)

    W   a   t   e   r    C   u   t    (    f   r   a   c   t    i   o   n    )

Well - 5

0 500 1000 15000

0.05

0.1

0.15

0.2

0.25

Time (Days)

    W   a   t   e   r    C   u   t    (    f   r   a   c   t    i   o   n    )

Field

8/9/2019 Choudhary 2012

48/61

 34

Figure 3-19: PLT data match - End of 3rd year for Well - 4 (Producer)

The XY slices of the fifteenth layer of solution models are shown in Figure 3-20 alongwith the “True Model”. The converged models do contain the major channel trends as inthe true reference models.

Figure 3-20: Layer 15 of history matched models: Flow & PLT data optimization

 3.5.3   Flow, PLT and Seismic Data Optimization

The third set of optimization was performed using all three data set. The P-wave velocitycube provides field-wide information at lower resolution. The change in P-wave velocity

8/9/2019 Choudhary 2012

49/61

  35

at each grid cell is a complex function of pressure and saturation. Changes in P-wavevelocity over time can only be resolved for a significant change in saturation and pressure. The water front movement in the reservoir represents a transition from oil-saturation to water-saturation and affects P-wave velocities in the grid cells. Sevenmodels converged to solution models. Figure 3-21 shoes the water-cut match of middle

row producers and field level water-cut with respect to history data set

Figure 3-21: Water cut - flow, PLT and seismic data optimization

Red – History, Blue – Models matching History, Grey – Models not matching History

The P-wave velocity match of selected models is shown in Figure 3-22. The convergedmodels capture the major channel trends in the reservoir that are seen in the true model.

Figure 3-22: P-wave velocity match - flow, PLT and seismic data optimization

0 500 1000 15000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (Days)

    W   a   t   e   r    C   u   t    (    f   r   a   c   t    i   o   n    )

Well - 4

0 500 1000 15000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (Days)

    W   a   t   e   r    C   u   t    (    f   r   a   c   t    i   o   n    )

Well - 5

0 500 1000 15000

0.05

0.1

0.15

0.2

0.25

Time (Days)

    W   a   t   e   r    C   u   t    (    f   r   a   c   t    i   o   n    )

Field

8/9/2019 Choudhary 2012

50/61

 36

The XY slices of the fifth layer of models that converged during iterations are shown inFigure 3-23 along with the “True Model”. As it can be observed, the converged solutionscapture major channel trends, but they still differ from each other due to different sandchannel placement in the reservoir.

Figure 3-23: Layer 15 of history matched models: Flow, PLT and seismic data optimization

3.6  Rejection sampling algorithm

Figure 3-24: Comparison with rejection sampler

8/9/2019 Choudhary 2012

51/61

  37

The results of optimization are compared with rejection sampler to estimate the efficiencyof optimization algorithms. The "rejection sampler" or acceptance-rejection method iscalled an "exact" sampler as it perfectly follows the Bayes’ rule. A rejection samplerrequires exhaustive evaluation of very large set of prior models. All models that do notmatch the historical data set are rejected, while retaining subset of models which history

(Caers, 2011).

For rejection sampling, 10,000 prior models generated for PCA were simulated for waterflood using Eclipse® and model response compared with the history. Rejection samplingwas carried out for all three scenarios - flow only and flow & PLT data and flow, PLTand seismic data optimization. Over 1000 models were selected during rejection samplingof flow only case scenario, 71 models were only selected for flow & PLT optimizationand 42 models matched the history data within tolerance for flow, PLT and seismic dataoptimization. Figure 3-24 compares the X-Y cross-section of the E-type of modelsobtained from rejection sampling with E-type obtained from multi-start optimization.

3.7  Optimization Efficiency

Figure 3-25: Optimization Code Efficiency for Hook – Jeeves algorithm

Convergence is slow with both the algorithm for a large model used in case study. While,Hook- Jeeves algorithm moves step-by-step towards the minima, SPSA algorithm usesstochastic gradient computed in randomly generated directions. The convergence forHook-Jeeves algorithm is shown in Figure 3-25 for first 40 iterations, while Figure 3-26shows convergence of SPSA algorithm for four convergence cycles. The lines in greendenote the flow & PLT optimization scenario while lines in red represent the flow, PLTand Seismic data optimization. The convergence rate for a large model like " Zamba" wasslow, due to complex spatial distribution of facies. Inclusion of additional dataset furtherslows the convergence.

8/9/2019 Choudhary 2012

52/61

 38

Figure 3-26 : Optimization Code Efficiency for SPSA algorithm

3.8 

Forecast Uncertainty

Figure 3-27: Forecast of history matched models

The history-matched models obtained from multi-start optimization were furthersimulated in forecast mode for three more years to ascertain the variability in the forecast.Figure 3-27 shows the water cut forecast obtained from history -matched models. As itcan be observed, water cut prediction varies within these models. The variation inforecast is less for flow only optimization due to tighter tolerance used for history -matching. The tolerance of flow & PLT data optimization and flow, PLT and seismicdata optimization was same. It can be observed that the uncertainty expressed by themodels from flow, PLT and seismic data is slightly less than the PLT data optimizationsimilar to the observation made in Section 2.6.4.  Inclusion of additional data set further

8/9/2019 Choudhary 2012

53/61

8/9/2019 Choudhary 2012

54/61

8/9/2019 Choudhary 2012

55/61

  41

Chapter 4

4  Summary

This thesis presents a workflow for generating multiple history-matched models usingoptimization in reduced dimensions. In this thesis, it has been demonstrated throughmultiple case studies that field observations i.e. production data, seismic data, spinnerdata etc. are too sparse for defining reservoir parameters accurately and multiple solutionmodels exists which closely match the historical observation.

In the thesis, it has been shown that principal component analysis (PCA) can be used toreduce the dimension of the reservoir model to less than ten percent of originaldimension. It is also shown that random model generated using PCA can be optimized inin a multistart approach to generate history-matched models that are different from eachother produce varying forecast for hydrocarbon production.

In the thesis, it has also been demonstrated that inclusion of spinner data representinginjector-producer connectivity and field–wide seismic can help in reducing theuncertainty of the posterior models. Any economic analysis of the field should be carriedout only after analyzing production forecast from all models that matched history.

It can also be concluded from Case III that PCA technique cannot accurately reproducefine scale features and other model reduction technique should be tested in cases wherefacies description is non-linear. The choice of model reduction technique and number ofreduced dimensions is a subjective discussion. The section should be based on analysis ofunconstrained prior models.

8/9/2019 Choudhary 2012

56/61

8/9/2019 Choudhary 2012

57/61

8/9/2019 Choudhary 2012

58/61

8/9/2019 Choudhary 2012

59/61

  45

References

Ahmed, T. (2010). Petroleum Engineering Handbook. Gulf Professional Publishing.Batzle, M., & Wang, Z. (1992). Seismic properties of pore fluids. Geophysics, 57 (11),

1396-1408.Caers, J. (2011). Modeling Uncertaininty in Earth Sciences. John Wiley and Sons.Castagna, J. P., Batzle, M. L., & Eastwood, R. L. (1985, April). Relationships between

compressional-wave and shear-wave velocities in clastic silicate rocks.Geophyiscs, 50(4), 571-281.

Castro, S., Caers, J., & Mukerji, T. (2005). The Stanford VI Reservoir. 18th Annual Affiliate Meeting. Stanford: SCRF.

Choudhary, M., Mukerji, T., & Echeverria, D. (2011). Generation of Multiple HistoryMatched Models using multistart optimisation. 24th Affiliate Meeting.  Stanford:

Stanford Center of Reservoir Forecasting.Cruz, P. S., Horne, R., & Deutsch, C. V. (2004). The Quality Map: A Tool for Reservoir

Uncertainty Quantification and Decision Making. SPE Reservoir Evaluation & Engineering, 7 (1), 6-14.

Dake, L. (2001). The practice of reservoir engineering. Elsevier.Echeveria, D., & Mukerji, T. (2009). Robust scheme for inversion of seismic and

 production data for reservoir facies modeling.  International Exposition and Annual Meeting. Houston: SEG.

Echeverria, D., & Mukerji, T. (2009). A Robust Scheme for Spatio-Temporal InverseModeling of Oil Reservoirs. 18th World IMACS/MODSIM Congress, (pp. 4206-4212). Cairns, Australia.

Friedmann, F., Chawathe, A., & Larue, D. (2003). Assessing Uncertainty in ChannelizedReservoirs Using Experimental Designs. SPE Reservoir Evaluation & Engineering, 6 (4), 264-274.

Gao, G., Li, G., & Reynolds, A. C. (2007). A Stochastic optimization algorithm forautomatic history matching. SPE Journal, 196 - 208.

Gardner, G. H., Gardner, L. W., & Gregory, A. (1974). Formation velocity and density -the diagnostic basic for stratigraphic traps. Geophysics, 39, 770-780.

Gass, P. C. (2007). Principal component analysis. Northridge: Calfiornia State university.Gassmann, F. (1951). Über die elastizität poröser medien. Viertel. Naturforsch. Ges.

 Zürich, 96 , 1-23.Gottfried, B. S., & Weisman, J. (1973).  Introduction to Optimization Theory. Englewood

Cliffs: Prentice Hall.Himmelblau, D. M. (1972). Applied Non Linear Programming. New York: McGraw-Hill.Hooke, R., & Jeeves, T. A. (1961, April). "Direct Search" Solution of numerical and

statistical problems. Journal of the ACM (JACM), 8 , 212-229.Huang, X., Meister, L., & Workman, R. (1998). Improving production history matching

using time-lapse seismic data. The Leading Edge, 1430-1433.Jolliffe, I. T. (2002). Principal Component Analysis  (Second ed.). New York: Springer-

Verlag.

8/9/2019 Choudhary 2012

60/61

8/9/2019 Choudhary 2012

61/61

Stephen, K. D., Soldo, J., MacBeth, C., & Christie, M. (2006, December). Multiple-Model Seismic and Production History Matching: A Case Study. SPE Journal,11(4), 418-430.

Stewart, R. R., & Domenico, S. N. (1991).  Exploration Seismic Tomography:Fundamentals. Tulsa: Society of Exploration Geophysicists.

Strebelle, S. (2002). Conditional simulation of complex geological structures using multi- point geostatistics. Mathematical Geology, 34, 1-21.Suman, A., & Mukerji, T. (2012). Sensitivity Study of Rock Physics Parameters for

Modeling the Time Lapse Signature of the Norne Field. 25th Annual Affiliate Meeting. Pacific Grove: SCRF.

Vogelij, H. N., Leach, M. J., & Kapteyn, P. K. (1993). Multilayer Flowmeter TestingCombined With 3D Field Modeling To Enhance the Ameland Field Development.SPE Formation Evaluation, 5-10.

Yoelin, S. D., & Howald, C. D. (1970). Production Logging as Used To Solve WaterInjection Problems in the Huntington Beach Offshore Field. Journal of PetroleumTechnology, 1083-1088.