Data Driven History Matching for Reservoir … · data driven history matching for reservoir production forecasting a thesis submitted to the department of energy resources engineering

DATA DRIVEN HISTORY MATCHING FOR RESERVOIR

PRODUCTION FORECASTING

A THESIS

SUBMITTED TO THE DEPARTMENT OF

ENERGY RESOURCES ENGINEERING

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

MASTER OF SCIENCE

Wenyue Sun

August 2014

I certify that I have read this thesis and that in my opinion it is fully

adequate, in scope and quality, as partial fulfillment of the degree of

Master of Science in Petroleum Engineering.

(Louis J. Durlofsky) Principal Adviser

ii

Abstract

Prior to performing reservoir forecasting, an inverse problem is often solved, in which

one or more models are inferred from historical data and prior geological information.

For large models, this procedure can be very expensive computationally, and it can

also be challenging to maintain geological realism in the resulting model. These

issues motivate the development of alternative approaches for reservoir forecasting.

In this thesis we develop one such procedure, which we refer to as data driven history

matching or DDHM.

In our DDHM method, multiple geostatistical reservoir models, consistent with

prior geological information, are first created and simulated to provide prior simu-

lation data. Reservoir forecasts are then obtained by linearly combining the prior

simulated data vectors with associated weights. The weights are computed through

linear regression on historical data. The maximum a posteriori (MAP) estimate

is computed with DDHM by performing singular value decomposition on an ap-

propriate data matrix. This procedure is applied for production forecasting for a

three-dimensional Gaussian model. The DDHM forecast shows clear improvement

compared with the forecast from the ‘best’ prior model. However, unphysical results

(e.g., negative rates) are observed in some cases, presumably due to the existence

of nonlinear features in the prior simulated data that are not modeled in the basic

DDHM procedure. To address this issue, we introduce a series of mapping operations

to transform the prior simulated data to data that are closer to Gaussian and more

nearly linearly correlated. This is shown to improve DDHM predictions and to lead

to MAP estimates that are physically reasonable and accurate, for both Gaussian and

channelized geological models.

We next incorporate the DDHM algorithm into a randomized maximum likeli-

hood (RML) procedure for uncertainty quantification. This RML-DDHM method is

iii

tested on Gaussian and channelized models. A computationally intensive rejection

sampling algorithm is used to provide reference P10-P50-P90 estimates of posterior

uncertainty. Reasonably close agreement between RML-DDHM and rejection sam-

pling is observed in both cases, which suggests that the RML-DDHM procedure is

able to provide useful uncertainty assessments. Finally, RML-DDHM is applied to

more complicated cases involving up to 16 wells. For the cases considered, the range

of forecast uncertainty is reduced significantly relative to the prior uncertainty, and

the true data consistently fall within or very near the RML-DDHM results. Taken in

total, our findings suggest that a data driven approach such as DDHM may eventually

represent a viable alternative to time-consuming model-based inversion procedures.

iv

Acknowledgements

First and foremost, I would like to express my sincere thanks to my advisor, Prof.

Louis Durlofsky, who has guided me over the past two years. I’ve learned a lot about

academic thinking during these two years through many discussions with him. I owe

him many thanks for all of our discussions and interactions, for his support of this

work, and for his careful reading of many drafts of this thesis.

I’d like to thank Dr. David Cameron, who was formerly a Ph.D. student in this

department. He offered many useful suggestions and helped me in my research when

I came to Stanford. I’m also grateful to Dr. Vladislav Bukshtynov and to Hai Xuan

Vo, who kindly provided assistance in this work. I’d like to acknowledge the Stanford

Smart Fields Consortium for providing financial support during my studies.

I’d also like to thank my colleagues and friends at Stanford. They helped me in

many ways and made my life here much more colorful. Special thanks to my parents

and brother, who supported me all the time over these years and motivated me to be

better and better!

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vi

Contents

Abstract iii

Acknowledgements v

1 Introduction 1

1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 History Matching Based on Model Inversion Theory . . . . . . 2

1.1.2 Ensemble Model Forecasting . . . . . . . . . . . . . . . . . . . 3

1.1.3 Uncertainty Quantification . . . . . . . . . . . . . . . . . . . . 4

1.2 Scope of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 DDHM Based on Prior Production Data 7

2.1 Basic DDHM Formulation . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Maximum a Posteriori Estimate . . . . . . . . . . . . . . . . . . . . . 9

2.3 Production Forecasts Using DDHM – Case 1 . . . . . . . . . . . . . . 13

2.4 Forward and Backward Mapping . . . . . . . . . . . . . . . . . . . . 20

2.4.1 Water Production Rate Mapping . . . . . . . . . . . . . . . . 20

2.4.2 Oil Production Rate Mapping . . . . . . . . . . . . . . . . . . 22

2.5 Production Forecasts using Mapped DDHM . . . . . . . . . . . . . . 28

2.5.1 Results for Production Forecasts – Case 1 . . . . . . . . . . . 30

2.5.2 Results for Production Forecasts – Case 2 . . . . . . . . . . . 31

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Uncertainty Quantification Based on DDHM 37

3.1 RML with DDHM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

vii

3.2 Rejection Sampling Procedure . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Uncertainty Quantification – Case 1 . . . . . . . . . . . . . . . . . . . 39

3.4 Uncertainty Quantification – Case 2 . . . . . . . . . . . . . . . . . . . 40

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Application to More Complicated Cases 49

4.1 Gaussian Model with 13 Wells . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Channelized Model with 16 Wells . . . . . . . . . . . . . . . . . . . . 58

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Summary and Future Work 69

Nomenclature 71

Bibliography 73

viii

List of Figures

2.1 Log permeability maps for four prior Gaussian models. Well locations

are also shown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Oil and water relative permeability curves . . . . . . . . . . . . . . . 14

2.3 Prior simulated production data for Case 1 . . . . . . . . . . . . . . . 16

2.4 Cumulative energy loss for principal components . . . . . . . . . . . . 17

2.5 DDHM results for Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6 Statistics for prior production data . . . . . . . . . . . . . . . . . . . 19

2.7 Prior water production rate mapping for two realizations . . . . . . . 21

2.8 Mapped water production rate after extrapolation . . . . . . . . . . . 22

2.9 Statistics for mapped prior WPR of Case 1 . . . . . . . . . . . . . . . 23

2.10 Correlation between peak oil rate time and water breakthrough time

for P2 (Case 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.11 Oil production rate mapping when breakthrough has occurred in the

observed data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.12 Oil production rate mapping when when breakthrough has not occurred 27

2.13 Backward mapping of OPR . . . . . . . . . . . . . . . . . . . . . . . 28

2.14 Statistics for mapped prior OPR of Case 1 . . . . . . . . . . . . . . . 29

2.15 DDHM results for Case 1 after mapping operations . . . . . . . . . . 32

2.16 Log permeability maps for four prior bimodal channelized models . . 33

2.17 Prior simulated production data for Case 2 . . . . . . . . . . . . . . . 34

2.18 DDHM results for Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1 Prediction results for Case 1 with RML-DDHM approach . . . . . . . 41

3.2 Uncertainty quantification (P10, P50 and P90) for Case 1 with RML-

DDHM and rejection sampling (RS) approaches . . . . . . . . . . . . 42

ix


DDHM and rejection sampling approaches (True Model 2) . . . . . . 43


DDHM and rejection sampling approaches (True Model 3) . . . . . . 44

3.5 Prediction results for Case 2 with RML-DDHM approach . . . . . . . 45


DDHM and rejection sampling approaches . . . . . . . . . . . . . . . 46

4.1 Log permeability maps for four prior Gaussian models . . . . . . . . . 50

4.2 RML-DDHM results for WIR (I1 – I4) . . . . . . . . . . . . . . . . . 52

4.3 RML-DDHM results for WPR and OPR (P1 and P2) . . . . . . . . . 53




4.7 RML-DDHM results for WPR and OPR (P9) . . . . . . . . . . . . . 57

4.8 Log permeability maps for four prior channelized models . . . . . . . 59

4.9 RML-DDHM results for WIR (I1 – I4) . . . . . . . . . . . . . . . . . 60





4.14 RML-DDHM results for WPR and OPR (P9 and P10) . . . . . . . . 65

4.15 RML-DDHM results for WPR and OPR (P11 and P12) . . . . . . . . 66

x

Chapter 1

Introduction

Reservoir forecasting represents an important but challenging problem. Traditional

approaches often require model inversion as a first step. We refer to model inversion

techniques as model driven history matching (MDHM). With these approaches, ob-

served reservoir behavior (such as production data) is used to determine parameters

in the reservoir model, and reservoir forecasts are based on the history-matched model

or models.

Though much research has been performed in the area of MDHM, it is still com-

plicated and time consuming to apply. Because real reservoir models may contain

a large number of grid blocks, and because inferring the model from observation

data usually entails a large number of forward reservoir simulations, significant com-

putational resources are typically required to generate history-matched models. In

addition, maintaining geological consistency during model calibration may be chal-

lenging. Such consistency is necessary if the model is to retain a degree of geological

realism.

It is also important to quantify prediction uncertainty. Due to our incomplete

knowledge of the reservoir and the limited amount of data available, history match-

ing problems are almost always ill-posed. This means that many reservoir models

may match historical observations equally well. However, sampling the posterior dis-

tribution of reservoir parameters correctly (i.e., generating history-matched models

that span the range of uncertainty) is very challenging.

Instead of directly addressing issues encountered in MDHM, our goal in this work

is to develop a novel “data driven history matching” (DDHM) procedure. The idea

1

2 CHAPTER 1. INTRODUCTION

with DDHM is to circumvent the complications associated with MDHM. DDHM will

be most useful when the final target of the history matching is reservoir forecasts

rather than history-matched models.

1.1 Literature Review

We begin our literature review with a brief discussion of the application of MDHM

for reservoir forecasting and the associated limitations. We then describe a data

driven approach developed for weather forecasting, which will be the starting point

for the approaches developed in this thesis. We also review some techniques developed

for uncertainty quantification based on both MDHM and DDHM. Because extensive

literature exists in many of these areas, we restrict our discussion to papers that are

most relevant to the work in this thesis.

1.1.1 History Matching Based on Model Inversion Theory

History matching based on model inversion theory has been studied for many years.

A detailed description of inverse modeling theory can be found in Tarantola [26].

Within the context of history matching in reservoir engineering, Oliver et al. [16]

provided a detailed description of the general theory and application of MDHM, and

Oliver and Chen [14] reviewed recent progress in this area.

The basic goal of MDHM is to find a reservoir model (or models) m that minimizes

the following equation:

J(m) = ‖g(m)− dobs‖2D +R, (1.1)

where J is the objective function, g(m) is the output of model m generated by

performing a flow simulation, dobs is the observed data, ‖ · ‖D designates the LD

norm, and R is a regularization term that is added to achieve a degree of geological

realism. The first contribution on the right hand side of (1.1) is generally referred to

as the data mismatch for m, and the second contribution as the model mismatch. For

the data mismatch, different weights can be included to account for different types

of data. A common choice is to use the inverse of the noise variance to weight the

squared data mismatch.

Oliver et al. [15] discussed aspects of the history matching problem that render this

1.1. LITERATURE REVIEW 3

inversion challenging. As noted earlier, these challenges relate to the computational

requirements associated with evaluating g(m) in (1.1) and the need to properly define

the regularization R. In addition, (1.1) represents a nonlinear optimization problem,

which may be difficult to solve and may display multiple local minima.

For production forecasting in which the desired result is not the reservoir model,

but rather actual forecasts, an approach that avoids model inversion is appealing.

We refer to such approaches as data driven history matching (DDHM). Next we will

review applications of this type of procedure in weather forecasting and subsurface

tracer problems. The application of DDHM for general reservoir simulation problems

represents a new and interesting research direction.

1.1.2 Ensemble Model Forecasting

In the context of weather forecasting, Krishnamurti et al. [9] proposed multimodel

ensemble forecast analysis, in which an ensemble of forecasts was first generated using

different physical models. These forecasts were not required to match the observed

data. A single “superior” forecast was then obtained by linearly combining all of the

forecasts along with associated weights. This can be expressed as:

d =Nm∑i=1

vidi, (1.2)

where d is an estimate of the behavior of the true model (the superior forecast), di

represents the simulation results from model i, vi is the weight for model i, and Nm

denotes the total number of prior simulation models used. Equation (1.2) was applied

for both the history-matching and prediction periods. The weights were estimated by

applying linear regression for the history-matching period, during which observation

data were available. Krishnamurti et al. concluded that multimodel ensemble fore-

casting provided better forecasts than any of the individual models used in (1.2). A

similar approach was subsequently applied for air quality forecasting in [12, 17].

Mallet et al. [12] extended the representation in (1.2) to sequential data assimi-

lation. With this approach, the weights are updated as new data are observed, and

thus the weights at each time depend on all past observations. Various methods (e.g.,

gradient descent, exponentially weighted average) were used for updating the weights,


and results were presented in terms of root mean square mismatch between the pre-

diction and observation data. All methods were shown to give better prediction than

the best model in the ensemble.

In [9, 12, 17], forecasts are accomplished without model inversion, so the proce-

dures are very efficient. This type of approach will provide the foundation for the

DDHM method developed in Chapter 2. However, [9, 12, 17] only provided results for

a single forecast; they did not consider how the algorithms sample the posterior prob-

ability of future behavior. Thus, modifications are required to quantify prediction

uncertainty, as will be discussed in detail in Chapter 3.

Another approach for data driven forecasting was proposed by Scheidt et al. [22].

Their framework was applied to a tracer flow problem with three upstream obser-

vation wells, and the objective was to predict the downstream tracer concentration

based on the upstream observed data. Prior geostatistical models were generated and

simulated to provide concentration data d (at observation well locations) and h (at

the downstream prediction location). Then nonlinear principal component analysis

was used to obtain low-dimensional representations of d and h, designated d∗ and

h∗, which provides a joint space (d∗,h∗). Then, given d∗obs (the low-dimensional rep-

resentation of observed data) from observation wells, they generated a predicted h∗

(called h∗pred) using a kernel smoothing approach employed in (d∗,h∗) space. This

approach was shown to be effective for the tracer flow problem considered, but it

relies heavily on the model reduction technique, since the joint space must be of very

low dimension (in their application, the dimension is 3) to allow effective estimation.

This may be problematic when the number of observed data becomes large.

1.1.3 Uncertainty Quantification

Multiple reservoir forecasts are required for uncertainty quantification in reservoir

management. Methods for quantifying uncertainty can be described in terms of the

way in which multiple samples are drawn from the conditional posterior distribu-

tion. Traditional rejection sampling [20] is a brute-force uncertainty quantification

approach that samples posterior models using a Bayesian treatment. Rejection sam-

pling is claimed to be capable of correctly sampling the posterior distribution, but the

number of simulations required is generally prohibitive, which makes it impractical

1.2. SCOPE OF THIS WORK 5

for real cases. Markov chain Monte Carlo (MCMC) is another approach that theoret-

ically samples the posterior distribution correctly. Though MCMC is more efficient

than rejection sampling, it is still very expensive in practice [11].

Due to the nonlinear correlation between observed data and reservoir parameters,

sampling the posterior distribution efficiently and correctly through a model driven

approach is challenging. Randomized maximum likelihood (RML) [8, 19] and ensem-

ble Kalman filter (EnKF) [1, 4] are two approaches claimed to be able to sample the

posterior distribution approximately. In RML [19], multiple model and data variables

are first jointly sampled from the prior distribution, and then the sampled models are

calibrated to the sampled data variables. Reynolds et al. proved theoretically that

RML samples the posterior distribution correctly when the observed data are linearly

correlated to the reservoir model variables, but no theoretical guarantee exists when

the correlation is nonlinear (as it is in reservoir simulation).

EnKF is a sequential data assimilation method that updates model variables and

state variables (e.g., phase saturation and pressure, production data) simultaneously.

Zafari and Reynolds [28] showed that EnKF and RML are equivalent for linear prob-

lems with Gaussian priors if the number of realizations within the ensemble goes to

infinity. For nonlinear problems (e.g., prior model variables are non-Gaussian, and/or

state variables are nonlinearly correlated to model variables), there are no general the-

oretical results indicating how well EnKF estimates the posterior distribution.

The approach developed by Scheidt et al. [22], discussed in Section 1.1.2, sam-

ples the posterior distribution directly without performing a model inversion. The

uncertainty quantification results they obtained matched the results from rejection

sampling (which required thousands of simulations). Thus, very substantial compu-

tational savings were achieved. However, as noted earlier, issues may arise with this

approach when many wells are included. Also, the way in which the measurement

error (noise) in the observed data propagates within this framework is not clear.

1.2 Scope of this Work

This thesis develops and applies a DDHM based approach for reservoir forecasting

and uncertainty quantification. The specific contributions are:


• We develop a basic DDHM method for reservoir forecasting. The initial imple-

mentation is analogous to the ensemble model forecasting approach developed

for weather forecasting applications [9]. Forecasting results for an oil-water

problem with a Gaussian geological model are presented.

• Based on the results observed in the initial implementation, we introduce a

series of data mappings. This reduces the non-Gaussianity and nonlinearity of

the prior simulated production data. Forecasting results for a Gaussian model

and a bimodal channelized model are presented.

• We extend the work in [9] to quantify prediction uncertainty. Principal com-

ponent analysis is used to obtain the maximum a posteriori estimate (MAP).

We then apply RML to sample the posterior production distribution within

the context of DDHM. Forecasting results for a Gaussian model and a bimodal

channelized model are presented and compared with the results from a rejection

sampling approach.

1.3 Thesis Outline

The thesis is organized as follows:

• Chapter 2 presents the basic DDHM formulation and describes appropriate

mappings for the prior simulated production data. Single forecast (MAP) results

are shown for a Gaussian model and a channelized model.

• Chapter 3 extends the DDHM algorithm to allow for uncertainty quantification.

Results using RML-DDHM are compared to those using a rejection sampling

approach.

• Chapter 4 presents RML-DDHM results for uncertainty quantification for mod-

els containing up to 16 wells.

• Chapter 5 provides a summary of this work and offers suggestions for future

research in this area.

Chapter 2

DDHM Based on Prior Production

Data

In this chapter, we formulate our data driven history matching (DDHM) procedure.

We present results using the basic implementation for a three-dimensional Gaussian

model. This motivates the need for mapping operations before performing DDHM,

which we develop and discuss. Using these procedures, we revisit the Gaussian model

and then consider a channelized example. In the next chapter, we will extend DDHM

to enable uncertainty quantification.

2.1 Basic DDHM Formulation

The basic idea of DDHM is that production data for an “unknown” reservoir can

be predicted by linearly combining the simulation data, assembled into data vectors,

from an ensemble of prior realizations. Given an ensemble of Nm prior reservoir

models m1,m2, · · · ,mNm , the simulated production data are generated via

di = g(mi), i = 1, 2, ..., Nm, (2.1)

where g designates solution of the reservoir flow equations, and di is a vector con-

taining the production data (water production rate, oil production rate, BHP, etc.)

at all time steps corresponding to both the history-matching and prediction periods

7

8 CHAPTER 2. DDHM BASED ON PRIOR PRODUCTION DATA

for model mi. Specifically, we write

di = [eT1,H , eT2,H , · · · , eTNw,H︸︷︷︸History-matching period

, eT1,F , eT2,F , · · · , eTNw,F︸︷︷︸Prediction period

]Ti , (2.2)

where Nw designates the number of wells, ej,H is a vector containing data values

corresponding to the history-matching period for well j (j = 1, 2, . . . , Nw), and ej,F

contains data values for the prediction period (the later part of the simulation) for

well j. We denote the amount of data during the history-matching and prediction

periods for well j as Nj,H and Nj,F , and the total amount of data during the history-

matching and prediction periods as Nh and Nf . Thus we have Nh =∑Nw

j=1Nj,H and

Nf =∑Nw

j=1Nj,F . The dimension of di is thus Nt = Nh +Nf .

In DDHM, dtrue (the vector of true data for both the history-matching and pre-

diction periods) is approximated as a linear combination of the prior data:

dtrue =Nm∑i=1

vidi + δ, (2.3)

where vi is the weight associated with model i, and δ designates the model error term.

Here we assume δ ≈ 0, which should be valid if enough prior models are simulated.

We have confirmed this numerically by constructing accurate representations of dtrue

using (2.3) with δ = 0. Then we can also write:

dtrue ≈Nm∑i=1

vidi. (2.4)

If there exists some measurement error ε, we have

dobs = dH,true + ε

≈Nm∑i=1

vidH,i + ε, (2.5)

where dobs is the observed data, dH,true is the true data during the history-matching

period, and dH,i is the data for model i during the history-matching period. We then

2.2. MAXIMUM A POSTERIORI ESTIMATE 9

approximate the future data, dF,est, as

dF,est =Nm∑i=1

vidF,i, (2.6)

where dF,i designates the data for model i during the prediction period. Note that,

with reference to (2.2),

dH,i = [eT1,H , eT2,H , · · · , eTNw,H ]Ti , dF,i = [eT1,F , eT2,F , · · · , eTNw,F ]Ti . (2.7)

The reservoir forecast is provided by (2.6) once the weights vi are determined. To

estimate the vector of weights v, (2.5) can be applied once we specify the form of the

error term ε. We then compute v via linear regression:

v = argminv‖Nm∑i=1

vidH,i − dobs‖22 +R, (2.8)

where R is a regularization term introduced to prevent overfitting. There are many

forms that can be used for the regularization term; e.g., R = λ‖v‖22 (Ridge Regres-

sion [7]), and R = λ‖v‖1 (Lasso Regression [27]), where λ is a tuning parameter that

serves to control the relative impact of the data mismatch and regularization terms.

A specific treatment for R will be introduced in Section 2.2.

Given v, production data dP can be predicted using

dP =Nm∑i=1

vidF,i. (2.9)

We next consider how to incorporate (2.9) within a Bayesian data assimilation frame-

work.

2.2 Maximum a Posteriori Estimate

Reservoir history matching problems are almost always ill-posed in the sense that

many possible combinations of reservoir parameters result in equally accurate matches

to the historical observations [14]. This indicates that there is uncertainty with respect


to the production forecast. Thus it is important to know how the algorithm samples

the posterior production distribution, especially for a history-matching algorithm

that provides a single forecast. We now consider DDHM within a Bayesian setting in

order to generate the most probable coefficients v. We first consider the maximum a

posteriori (MAP) estimate.

In standard (model-driven) history matching, Bayesian formulations are often

applied. This entails first defining a prior distribution for the model parameters.

With this approach, the most probable model is determined by minimizing a misfit

function S [26]:

S(m) = [g(m)− dobs]TC−1

D [g(m)− dobs] + [m− m]TC−1M [m− m], (2.10)

where CD is the data covariance matrix, which is usually a diagonal matrix with

each element representing the variance of the measurement error, CM is the model

covariance matrix defining the spatial correlation structure of the prior model, and m

is the prior model mean. The first term on the right hand side of (2.10) quantifies the

mismatch between observed and simulated data, and the second term can be viewed

as a regularization term that forces the history-matched model to be “close” to m.

If the equation d = g(m) for the forward problem is linear, we simply have

d = Gm. (2.11)

Then (2.10) becomes [26, pp. 65]:

S(m) = [Gm− dobs]TC−1

D [Gm− dobs] + [m− m]TC−1M [m− m]. (2.12)

In DDHM, we have d = Xv, where X = [d1,d2, . . . ,dNm ] is an Nt × Nm matrix

containing the prior simulated data. Thus, an equation of the form of (2.12) can be

applied to DDHM. Writing dobs,est = XHv, where dobs,est is estimated observed data,

and XH ∈ RNh×Nm contains the prior data over the history-matching period (i.e., the

first Nh rows of X), we have:

S(v) = (XHv − dobs)TC−1

D (XHv − dobs) + (v − v)TC−1V (v − v). (2.13)

Here CV ∈ RNm×Nm is the covariance matrix characterizing the correlation structure

2.2. MAXIMUM A POSTERIORI ESTIMATE 11

of v, and CD is as defined above. Because (2.13) involves only the multiplication of

relatively small matrices, DDHM is very efficient once we have performed the flow

simulations required to construct X.

Equation (2.10), used for MDHM, is applicable only for Gaussian models, which

are characterized by two-point spatial statistics. For non-Gaussian systems, (2.10)

can be used in conjunction with a reparameterization procedure [14, 21]. Similarly,

(2.13) is applicable only when v is multivariate Gaussian. Otherwise, some type of

reparameterization or mapping should be introduced.

It is not straightforward to assess the properties of v, since v is not given directly.

We do however know that d = Xv, which means d is a linear transformation of v.

This means that d will be multivariate Gaussian if v is multivariate Gaussian. Thus

we can check whether v can be approximated as multivariate Gaussian by assessing

d. In Section 2.4, we will investigate the properties of d for various cases.

To apply (2.13), we also need to calculate CV . Given d = Xv, we have:

Cd = E[(d− d)(d− d)T ]

= E[[X(v − v)][X(v − v)]T

]= XE[(v − v)(v − v)T ]XT

= XCV XT , (2.14)

where E[ · ] is the expectation operator, Cd (not to be confused with CD) is the

covariance matrix for d, and we have used the fact that X is fixed and thus can be

removed from the expectation operator. Equation (2.14) cannot be applied directly,

however, because X is not invertible (it is generally a nonsquare matrix).

Here we apply an approach that avoids calculating CV . This approach is based

on principal component analysis (PCA); refer to [10, 24] for further details on PCA.

We first redefine X as the following centered matrix:

X =[d1 − d, d2 − d, · · · , dNm − d

], (2.15)

where d =∑Nm

i=1 di is the mean of the prior simulated data vectors. We then perform

singular value decomposition (SVD) of the matrix Y = X/√Nm − 1, which gives

Y = UΣVT , (2.16)


where U is a unitary matrix containing the left singular vectors of Y, Σ is a diagonal

matrix containing the singular values of Y, and V is a unitary matrix containing the

right singular vectors of Y.

We then can express realizations of d as:

d = UΣξ + d = Φξ + d, (2.17)

where Φ = UΣ is the basis matrix, and ξ ∈ RNl×1 is a vector of independent

random Gaussian coefficients with zero mean and unit variance. The components

of Φ are ordered by the corresponding singular values. There are a maximum of

[min(Nm, Nt) − 1] nonzero singular values. We retain only Nl columns in Φ, where

Nl ≤ [min(Nm, Nt) − 1]. Thus Φ ∈ RNt×Nl . The determination of Nl is discussed

below.

Using (2.17), we now write dobs,est − dH = ΦHξ, where dH is the data vector

containing mean prior production data corresponding to the history-matching period,

and ΦH ∈ RNh×Nl contains the first Nh rows of Φ (which correspond to the history-

matching period). The MAP estimate can now be computed by minimizing:

S(ξ) = [ΦHξ− (dobs − dH)]TC−1D [ΦHξ− (dobs − dH)] + (ξ− ξ)TC−1

ξ (ξ− ξ), (2.18)

where ξ is the mean of ξ, and Cξ is the covariance matrix. Because ξ is a vector of

independent random Gaussian coefficients with zero mean and unit variance, we have

ξ = 0 and Cξ = I. Equation (2.18) can thus be written as:

S(ξ) = (ΦHξ + dH − dobs)TC−1

D (ΦHξ + dH − dobs) + ξTξ. (2.19)

Minimization of (2.19) is accomplished using the quasi-Newton line search algorithm

within Matlab.

DDHM based on (2.19) avoids the calculation of CV and involves ξ rather than v.

Because ξ is usually of lower dimension than v, this acts to simplify the minimization

problem. The value of Nl is often determined by applying an “energy” criterion [2].

The relative energy in the largest Nl eigenvalues is given by:

ENl=

∑Nl

i=1 λi∑min(Nm,Nt)−1i=1 λi

, (2.20)

2.3. PRODUCTION FORECASTS USING DDHM – CASE 1 13

where λi is the square of diagonal element i in matrix Σ. By specifying a value for

ENl(e.g., 0.99999), the value of Nl can be determined. The cumulative “energy loss”

from retaining only the Nl largest eigenvalues is given by 1− ENl.

2.3 Production Forecasts Using DDHM – Case 1

We now apply the DDHM procedure described above for a Gaussian model. The

realizations, four of which are shown in Fig. 2.1, are represented on a 30×30×5 grid,

with each grid block of size 25 m × 25 m × 5 m. The reservoir is heterogeneous with

permeability k generated using the sequential Gaussian simulation algorithm within

SGeMS [18]. All realizations are conditioned to honor permeability at well blocks.

The logarithm of permeability (md) is normally distributed with mean µln k of 3 and

standard deviation σln k of 1.5. We specify kx = ky and kz = 0.3kx. The porosity φ is

assumed to be constant at 0.2. Two producers and two injectors, all fully penetrating,

are introduced into the model, as shown in Fig. 2.1.

We consider oil-water systems. The initial saturation of oil and water are 0.9

and 0.1 respectively. The oil and water viscosities at standard conditions are 1.16 cp

and 0.31 cp. Relative permeability curves are shown in Fig. 2.2. We ignore capillary

pressure effects. Injectors operate at fixed bottom-hole pressures (BHPs) of 550 bar

(I1) and 500 bar (I2), and producers operate at fixed BHPs of 100 bar (P1) and

250 bar (P2). Initial reservoir pressure is 325 bar.

All simulations are performed using Stanford’s Automatic Differentiation-based

General Purpose Research Simulator, AD-GPRS [29]. The simulation period is 3000

days, with production data reported every ten days. The history-matching period

for Case 1 extends from 150 days to 400 days. We ignore data prior to this time

in order to avoid the early transient period. Measured data include water injection

rate (WIR) for injectors, and water production rate (WPR) and oil production rate

(OPR) for producers.

The observed data are generated synthetically by simulating one randomly selected

prior reservoir model. Data from this model are not included in the data matrix X.


Figure 2.1: Log permeability maps for four prior Gaussian models. Well locations arealso shown

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Sw

k r

krw

kro

Figure 2.2: Oil and water relative permeability curves


The observed data entail the simulation results for the true model over the history-

matching period, dH,true, plus measurement error, ε:

dobs = dH,true + ε. (2.21)

The measurement error ε is assumed to follow a multivariate independent Gaussian

distribution with zero mean and covariance CD [6]. The data covariance matrix CD

is diagonal, with elements representing the variance of the random variables in ε. In

this thesis, unless otherwise indicated, the standard deviation of measurement error

is set to be equal to 2% of the corresponding rate data, subject to minimum and

maximum values of 2 m3/d and 20 m3/d.

For Case 1, we use a total of 100 prior conditioned realizations. Simulation re-

sults for these realizations are shown in Fig. 2.3. We see that the ensemble of prior

production data (blue curves) displays considerable spread, especially for WIR and

WPR, even though all realizations are conditioned to well data. During the history-

matching period, we have 25 measured data points for each type of data (25 WIRs

for injectors, 25 WPRs and OPRs for producers) for each run. There are thus a total

of Nh = 150 observed data points per run. Our goal now is to predict the reservoir

production performance up to 3000 days.

The cumulative energy loss associated with principal values of X is shown in

Fig. 2.4. It is evident that most of the energy is carried by the first few components.

We seek to form a basis such that the fraction of the energy ignored is less than 10−5.

This requires us to retain the first 35 principal components. Then, by applying the

DDHM procedure described in Sections 2.1 and 2.2, the forecast results are obtained.

DDHM prediction results (blue dashed curves) are shown in Fig. 2.5. Results for

the “best” model in the ensemble of 100 prior models are also shown (solid black

curves). The best model mi is defined as the prior model with the minimum data

mismatch, with the data mismatch for any model calculated using:

Sp(mi) = (di − dobs)TC−1

D (di − dobs), (2.22)

where di is the simulated production data from model mi. We see that, for all wells,

DDHM results match the observed data better, and provide better predictions, than


0 500 1000 1500 2000 2500 30000

200

400

600

800

1000

1200

1400

1600

1800

Days

P1

wat

er r

ate

(m3 /d

ay)

Prior DataObserved Data"True" Future DataHM period

(a) Water rate (P1)

0 500 1000 1500 2000 2500 3000200

400

600

800

1000

1200

1400

1600

Days

P1

oil r

ate

(m3 /d

ay)


(b) Oil rate (P1)

0 500 1000 1500 2000 2500 30000

500

1000

1500

Days

P2

wat

er r

ate

(m3 /d

ay)


(c) Water rate (P2)

0 500 1000 1500 2000 2500 30000

200

400

600

800

1000

Days

P2

oil r

ate

(m3 /d

ay)


(d) Oil rate (P2)

0 500 1000 1500 2000 2500 3000200

400

600

800

1000

1200

1400

Days

I1 w

ater

rat

e (m

3 /day

)


(e) Injection rate (I1)

0 500 1000 1500 2000 2500 3000

500

1000

1500

2000

Days

I2 w

ater

rat

e (m

3 /day

)


(f) Injection rate (I2)

Figure 2.3: Prior simulated production data for Case 1


0 10 20 30 40 50 60 70 80 90 10010

−10

10−8

10−6

10−4

10−2

100

Index

Cum

ulat

ive

ener

gy lo

ss (

1−E

i)

Figure 2.4: Cumulative energy loss for principal components

the best prior model, most notably in WIR for I2 (Fig. 2.5f). These results show the

potential of DDHM for reservoir forecasting problems.

Because the majority of the computational effort for DDHM involves performing

forward simulations over prior models, which can be easily done in parallel, the elapsed

time for DDHM computations is actually equivalent to only one forward simulation

(assuming Nm processors are available). MDHM, by contrast, generally requires

O(10-1000) forward simulations depending on the size and complexity of the reservoir

model. In addition, MDHM computations cannot always be readily parallelized. We

expect that DDHM will become more efficient with respect to MDHM as the model

increases in size and complexity.

In Fig. 2.5c, we do observe a problem with this implementation of DDHM; namely,

that predicted WPR becomes negative. This is clearly unphysical and calls into

question some of the treatments in DDHM. To investigate this behavior, we now

assess the Gaussian character of the prior simulated production data. Fig. 2.6a shows

the prior WPR data for P2, and Fig. 2.6b shows the histogram of this data at 500

days. We can see that the histogram is clearly non-Gaussian. Also, in Fig. 2.6c

(and 2.6e), it is evident that the correlation between WPR (and OPR) at different

times is nonlinear, which indicates the existence of higher-order features. The current

DDHM implementation, however, only includes first-order features. Thus it may

not be appropriate to apply this framework directly to data with strong nonlinear


0 500 1000 1500 2000 2500 3000

0

500

1000

1500

Days

P1

wat

er r

ate

(m3 /d

ay)

DDHMBest Prior ModelObserved Data"True" Future DataHM period

(a) Water rate (P1)

0 500 1000 1500 2000 2500 3000200

400

600

800

1000

1200

1400

Days

P1

oil r

ate

(m3 /d

ay)


(b) Oil rate (P1)

0 500 1000 1500 2000 2500 3000−100

0

100

200

300

400

500

600

Days

P2

wat

er r

ate

(m3 /d

ay)


(c) Water rate (P2)

0 500 1000 1500 2000 2500 3000

100

200

300

400

500

Days

P2

oil r

ate

(m3 /d

ay)


(d) Oil rate (P2)

0 500 1000 1500 2000 2500 3000550

600

650

700

750

800

850

Days

I1 w

ater

rat

e (m

3 /day

)



0 500 1000 1500 2000 2500 30001000

1100

1200

1300

1400

1500

1600

1700

1800

Days

I2 w

ater

rat

e (m

3 /day

)



Figure 2.5: DDHM results for Case 1


0 500 1000 1500 2000 2500 30000

200

400

600

800

1000

1200

1400

Days

P2

wat

er r

ate

(m3 /d

ay)

Prior Data

(a) Prior WPR (P2)

0 200 400 600 8000

50

100

150

200

250

300

Rate (m3/day)

(b) Histogram of prior WPR at 500 days

0 100 200 300 400 5000

100

200

300

400

500

600

700

800

Rate (m3/day) at 500 days

Rat

e (m

3 /day

) at

800

day

s

(c) Correlation between prior WPR at 500days and 800 days

0 500 1000 1500 2000 2500 30000

200

400

600

800

1000

Days

P2

oil r

ate

(m3 /d

ay)

Prior Data

(d) Prior OPR (P2)

200 300 400 500 600 700 800200

300

400

500

600

700

800

Rate (m3/day) at 500 days

Rat

e (m

3 /day

) at

800

day

s

(e) Correlation between prior OPR at 500days and 800 days

Figure 2.6: Statistics for prior production data


correlation features. Note that this limitation is not caused by the use of PCA, but

by the use of the linear combination in (2.3). The approach we will apply to mitigate

these non-Gaussianity and nonlinearity issues is to map the high-dimensional prior

production data d to a new vector d. The projected data will be seen to be closer to

Gaussian and more nearly linear.

2.4 Forward and Backward Mapping

In this section, we introduce mapping operations to improve DDHM performance.

The basic idea is to map the original high-dimensional data vector d to another

high-dimensional data vector d that has better properties. We write the mapping as:

d = F (d), (2.23)

where d is of dimension Mt× 1 (Mt can differ from Nt). The backward mapping can

be expressed as:

d = F−1(d). (2.24)

Because our mapping operations are performed on a well-by-well basis, we con-

struct di (i = 1, 2, . . . , Nm) using:

di = [ eT1,H , eT2,H , · · · , eTNw,H , eT1,F , eT2,F , · · · , eTNw,F ]Ti , (2.25)

where ej is the mapped data vector corresponding to ej (ej = [eTj,H , eTj,F ]T , j =

1, 2, . . . , Nw). For the backward mapping, (2.2) is used to construct di once the ej

are backward-mapped from the predicted ej (j = 1, 2, . . . , Nw).

2.4.1 Water Production Rate Mapping

In Fig. 2.6b, we can see that the histogram is highly skewed, with many of the prior

WPRs at or close to zero at 500 days. This is because water breakthrough has yet to

occur or has just occurred at P2 in many of the runs. In Fig. 2.6a, we see that, post

breakthrough, the WPR curves tend to increase following a similar pattern. Based

on this observation, we represent prior WPR data in terms of breakthrough time tj,b

2.4. FORWARD AND BACKWARD MAPPING 21

and WPR after breakthrough time:

ej = [tj,b, eTj,t>tj,b ]T , j = 1, 2, . . . , Nprod, (2.26)

where Nprod is the number of production wells. Here we use ej to represent the WPR

data vector for well j. In the next section on OPR mapping, ej will instead represent

the OPR data vector for well j.

Fig. 2.7 shows prior WPR in the original data format and the mapped prior

WPR for two runs with different breakthrough times. The x-axis of Fig. 2.7b is

dimensionless time for well j, designated tj (here j = 2), and calculated as:

tj =t− tj,btt − tini

, (2.27)

where t is the actual time (in days), tt is the end of the prediction period (3000 days

in our case), and tini is the starting time for the history-matching period (150 days).

0 500 1000 1500 2000 2500 30000

200

400

600

800

1000

Days

P2

wat

er r

ate

(m3 /d

ay)

Prior Data

(a) Water rate (P2)

0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

t2

P2

wat

er r

ate

(m3 /d

ay)

Prior Data

(b) Mapped water rate (P2)

Figure 2.7: Prior water production rate mapping for two realizations

Different prior WPR curves may have different numbers of points after mapping.

This is due to the different breakthrough times. Since our procedure requires the

number of data points for each prior model to be the same, we extrapolate the WPR

out to tj = 1 for all wells. We use a polynomial extrapolation scheme, which results

in smooth curves, as shown in Fig. 2.8. Other extrapolation schemes could also be

used, but this method suffices for our purposes.


0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

t2

P2

wat

er r

ate

(m3 /d

ay)

Prior Data

(a) Mapped water rate (P2)

Figure 2.8: Mapped water production rate after extrapolation

In our cases, simulation data are reported every ten days. Thus breakthrough

time will be a multiple of ten. In addition, the mapped data from different prior

models will correspond to consistent time values. However, if the report times differ

between runs, simulated data from different models will have to be interpolated such

that all data vectors correspond to a consistent set of dimensionless times.

In Fig. 2.9, we present statistical data for the mapped WPR. This corresponds to

the original prior WPR shown in Fig. 2.6a. We see that the histogram of the mapped

prior WPR data at a certain time appears closer to Gaussian (though deviation from

Gaussianity is still evident), and the correlation is now closer to linear (compare

Fig. 2.9c to Fig. 2.6c). The histogram and correlations have also been checked at

other times, and improvements along the lines of those in Fig. 2.9 were generally

observed.

The backward mapping of WPR is straightforward. Given a predicted break-

through time tP,b and the corresponding mapped WPR after breakthrough (both

generated using DDHM) for well j, we can readily construct predicted ej by comput-

ing t from (2.27) and assigning WPR for t < tP,b to be zero.

2.4.2 Oil Production Rate Mapping

We can also improve DDHM predictions for oil rate by introducing analogous map-

pings. OPR, as shown in Fig. 2.6d, often increases slightly until a certain time (due


0 0.2 0.4 0.6 0.8 10

500

1000

1500

t2

P2

wat

er r

ate

(m3 /d

ay)

Prior Data

(a) Mapped prior WPR (P2)

0 200 400 600 800 10000

10

20

30

40

50

60

70

80

Rate (m3/day)

(b) Histogram of mapped WPR at t2 = 0.3

0 100 200 300 400 5000

200

400

600

800

1000

t2 = 0.1

t2=

0.3

(c) Correlation between mapped prior WPRat t2 = 0.1 and t2 = 0.3

Figure 2.9: Statistics for mapped prior WPR of Case 1


to mobility effects and the use of BHP control on all wells), and then starts to de-

crease. The time that OPR for a particular well starts to decrease is very near the

water breakthrough time for that well (as shown in Fig. 2.10). The details of this

behavior depend on the way the wells are controlled, but we do expect to commonly

observe plateau (or slightly increasing) oil production followed by a decline when

water breakthrough occurs. Thus we represent prior OPR data for well j as:

ej = [eTj,t≤tj,b , eTj,t>tj,b ]T , j = 1, 2, . . . , Nprod. (2.28)

Note that ej here represents OPR at well j. In the subsequent development we will

drop the subscript j. Equation (2.28) then becomes:

e = [eTt≤tb , eTt>tb ]T . (2.29)

In analogy to our approach for WPRs, OPRs are mapped such that pre-break-

through OPRs are only linearly combined with pre-breakthrough OPRs in the DDHM

procedure. Similarly, post-breakthrough OPRs are also treated separately.

0 400 800 1200 16000

400

800

1200

1600

Water breakthrough time (days)

Pea

k oi

l rat

e tim

e (d

ays)

Figure 2.10: Correlation between peak oil rate time and water breakthrough time forP2 (Case 1)

We consider two different scenarios in our OPR mapping: 1) water breakthrough

has occurred in the observed data, and 2) water breakthrough has not occurred in

the observed data. The treatments for these two scenarios are now described.


Scenario 1: water breakthrough has occurred in the observed data

Fig. 2.11a shows observed OPR (tB ≤ tH , where tB is the breakthrough time in the

true model and tH is the end of the history-matching period) together with two prior

simulated OPRs. One OPR displays earlier oil rate decline (Prior Data 1), and the

other later decline (Prior Data 2). For illustration purposes, the observed data (red

curve in Fig. 2.11a) are taken to be smooth (without measurement noise). Similar to

our approach for WPR mapping, we again map the prior OPR such that the decline

period starts at the observed breakthrough time. Because of the noise in the oil rate

data, the time that oil rate starts to decline cannot always be determined accurately

from observed data. Thus breakthrough time is used here as the starting time for oil

rate decline. This is an accurate approximation, as is evident in Fig. 2.10.

0 500 1000 1500 2000 2500 3000200

400

600

800

1000

1200

1400

Days

oil

rate

(m

3 /day

)

Prior Data 1Prior Data 2Observed DataHM Period

(a) Oil rate

0 0.2 0.4 0.6 0.8 1200

400

600

800

1000

1200

1400

t

oil

rate

(m

3 /day

)

Prior Data 1Prior Data 2Observed DataHM PeriodObserved BT

(b) Mapped oil rate

Figure 2.11: Oil production rate mapping when breakthrough has occurred in theobserved data

For prior data that display tb < tB (represented by Prior Data 1; here tb is the

breakthrough time of Prior Data 1), after mapping the decline period data to start at

tB (data during [tt− tB + tb, tt] are thus removed; recall tt is the end of the prediction

period), we assign the data between tb and tB as constant. This is expressed as:

e = [eTt<tb , eTtb︸︷︷︸0≤t≤tB

, eTt>tb︸︷︷︸tB<t≤1

]T (tB < tH , tB > tb), (2.30)

where etb is a vector with all elements equal to etb , which is the data value at time


tb in Prior Data 1, and the number of elements of etb is equal to the number of data

points required between tb and tB. This is determined by the number of observed

data points within this dimensionless time period. The dimensionless time here is

defined as:

t =t− tinitt − tini

. (2.31)

For prior data that display tb > tB (represented by Prior Data 2), after shifting

the decline period data (t > tb), the data originally between tB and tb are removed.

This is expressed as:

e = [ eTt≤tB︸︷︷︸0≤t≤tB

, eTt>tb︸︷︷︸tB<t≤1

]T (tB < tH , tB < tb). (2.32)

For this case, we also extrapolate the OPR data out to t = 1 using a polynomial

scheme. As in WPR mapping, interpolation would additionally be required when

the data points correspond to different values of t. Backward mapping for time is

accomplished using (2.31).

Scenario 2: water breakthrough has not occurred in the observed data

Fig. 2.12 shows observed OPR (tB > tH) together with two prior simulated OPRs.

One OPR declines before tH (Prior Data 1; tH is the end of the history-matching

period), and the other declines after tH (Prior Data 2). For both cases, we map the

prior OPR such that the decline period starts at tH (Fig. 2.12b).

For the portion of the curve before tH , there are two different situations, and

the treatments are very similar to those for Scenario 1. For prior data that display

tb < tH (represented by Prior Data 1; tb is the corresponding breakthrough time),

after mapping the decline period data (t > tb), we assign the data between tb and tH

as constant. This is expressed as:

e = [eTt<tb , eTtb︸︷︷︸0≤t≤tH

, eTt>tb︸︷︷︸tH<t≤1

]T (tB > tH , tH > tb), (2.33)

where the number of elements of etb is equal to the number of data points required

between tb and tH , as determined by the number of observed data points within this

time period.


0 500 1000 1500 2000 2500 3000100

150

200

250

300

350

400

450

500

550

Days

oil

rate

(m

3 /day

)


(a) Oil rate

0 0.2 0.4 0.6 0.8 1100

150

200

250

300

350

400

450

500

550

t

oil

rate

(m

3 /day

)


(b) Mapped oil rate

Figure 2.12: Oil production rate mapping when when breakthrough has not occurred

For prior data that display tb > tH (represented by Prior Data 2), after mapping

the decline period data (t > tb), the data originally between tH and tb are removed.


e = [ eTt≤tH︸︷︷︸0≤t≤tH

, eTt>tb︸︷︷︸tH<t≤1

]T (tB > tH , tH < tb). (2.34)

For this case, we also extrapolate the OPR data out to t = 1 using a polynomial

scheme. For both cases, interpolation may also be required, as noted earlier. Mapped

oil rate is shown in Fig. 2.12b. Backward mapping for rate data is different from

Scenario 1 due to the fact that we also need to predict the breakthrough time.

Fig. 2.13a shows DDHM results (without backward mapping) for Case 1 (Scenario

2). DDHM generates predicted breakthrough time (tP,b), and OPRs during the decline

period (tH < t ≤ 1). We then map the predicted OPR to start at tP,b and assign the

value of data between tH and tP,b (not generated by DDHM) using linear interpolation.


e = [ eT0≤t≤tH︸︷︷︸

tini≤t≤tH

, e∗T︸︷︷︸tH<t<tP,b

, eTtH<t≤tt︸︷︷︸tP,b≤t≤tt

]T , (2.35)

where e∗ is the interpolated OPR. After the shifting operation, the backward mapping

for time is calculated using (2.31). Backward mapped OPRs are shown in Fig. 2.13b.


0 0.2 0.4 0.6 0.8 1100

150

200

250

300

350

oil r

ate

(m3 /d

ay)

t

DDHM ResultsHM Period

(a) DDHM results before backward mapping

0 500 1000 1500 2000 2500 3000100

150

200

250

300

350

oil r

ate

(m3 /d

ay)

Days

DDHM ResultsHM PeriodPredicted BT

(b) DDHM results after backward mapping

Figure 2.13: Backward mapping of OPR

In Fig. 2.14, we present statistical data for the mapped OPR. This corresponds

to the original prior OPR data shown in Fig. 2.6d. We see that the histogram of

the mapped OPR at a certain time appears nearly Gaussian. The correlation is also

now closer to linear (compare to Fig. 2.6e). The histogram and correlation have been

checked for both scenarios at different times, and improvements of this type were

typically observed.

2.5 Production Forecasts using Mapped DDHM

We now apply DDHM using the mapping procedures described above. The equation

used is slightly different from that applied in Section 2.3. We now write (2.19) as

follows:

S(ξ) =

Nh∑i=1

wi[dP,i(ξ)− dobs,i]2 + ξTξ, (2.36)

where dP,i(ξ) = (ΦHξ+dH)i are the predicted data, and wi are the associated weights

(equal to 1/σ2i ). Because the mapping operation introduces new variables tb, which

are of a different form and involve different units than the rate data in d, the weight

for tb should be treated differently. The DDHM equation for the mapped data can

now be expressed as:

2.5. PRODUCTION FORECASTS USING MAPPED DDHM 29

0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

t

P2

oil r

ate

(m3 /d

ay)

Prior Data

(a) Mapped prior OPR (P2)

200 400 600 800 10000

10

20

30

40

50

60

70

Rate (m3/day)

(b) Histogram of mapped OPR at t = 0.12

200 400 600 800 1000200

300

400

500

600

700

t = 0.08

t=

0.22

(c) Correlation between mapped prior OPRat t = 0.08 and t = 0.22

Figure 2.14: Statistics for mapped prior OPR of Case 1


S(ξ) =

Mh,r∑i=1

wi[dP,i(ξ)− dobs,i]2 +

Mh,b∑i=1

ui[tP,i(ξ)− tobs,i]2 + ξTξ, (2.37)

where Mh,r is the number of observed rate data points (after mapping), Mh,b is the

number of water breakthrough times (which is equal to the number of production

wells), and wi and ui are the associated weights. When breakthrough has occurred in

the observed data, tobs is simply the observed breakthrough time. When breakthrough

has not occurred, we set tobs = tH . In this case the associated weight will only be

nonzero when the predicted breakthrough time is smaller than tH . Here we assign ui

by assuming the measurement of breakthrough time has normally distributed error

with zero mean and standard deviation σb. We then have ui = 1/σ2b (i = 1, . . . ,Mh,b).

Here we take σb = 5 days. A quasi-Newton line search algorithm within Matlab is

again used to minimize (2.37).

With the incorporation of the mapping operations, DDHM proceeds as follows:

1. Generate multiple prior reservoir models and simulate each one to obtain prior

simulation results.

2. Assemble prior simulated data into Nm vectors.

3. Define mapping rules for prior simulation data, and map all prior data vectors

d to d (also applies for observed data).

4. Perform SVD on mapped data matrix to construct basis matrix Φ, and obtain

MAP estimate for ξ by minimizing (2.37).

5. Generate predicted data using (2.17) and perform backward mapping.

2.5.1 Results for Production Forecasts – Case 1

The setup for Case 1 is as described in Section 2.3. The results using DDHM on

mapped data are shown in Fig. 2.15. A total of 30 principal components are retained

in Φ. We see that the predicted WPR for P2 is now positive (in contrast to the

negative values in Fig. 2.5c), and the predicted OPR at P2 does not have an unphysical

overshoot (as in Fig. 2.5d). The results using the mapping are more accurate overall,

though the prediction for I2 WIR does degrade slightly. We note that small negative


rate prediction is still possible (though this occurs rarely if at all) after mapping

operations.

In Fig. 2.15a, we see that water breakthrough has occurred for P1 during the

history-matching period. The DDHM results match the breakthrough time closely,

and show accurate predictions for both WPR and OPR. For P2, water breakthrough

has not occurred during the history-matching period (Fig. 2.15c). DDHM is still seen

to provide an accurate prediction for the breakthrough time and future WPR. Results

for WIR at I1 and I2 (Figs. 2.15e and 2.15f) also display reasonable accuracy.

2.5.2 Results for Production Forecasts – Case 2

We now apply DDHM for a bimodal channelized model. The simulation models,

shown in Fig. 2.16, are represented on a 60 × 60 grid, with each grid block of size

25 m × 25 m × 10 m. The reservoir is heterogeneous, with permeability constructed

using the following steps:

1. Generate a binary channelized system using Petrel [23]. All realizations are

conditioned to hard data at well locations, and all wells are in sand.

2. Generate sand permeability using sequential Gaussian simulation with mean

µln k of 5 and standard deviation σln k of 1. All realizations are conditioned to

hard data at well locations.

3. Generate mud permeability using sequential Gaussian simulation with mean

µln k of 1 and standard deviation σln k of 0.5.

4. Incorporate the Gaussian distributions into each facies using the cookie-cutter

approach [3].

The porosity is assumed to be constant at φ = 0.2. One producer and one injector

are located in opposite corners of the model, as shown in Fig. 2.16. The injector

is operated at a constant BHP of 550 bar, and the producer at a constant BHP of

100 bar. The history-matching period is from 150 days to 500 days, and the prediction

period is from 500 days to 3000 days.

The observed data are generated using the same approach as in Case 1. A total

of 30 prior realizations are used, and 20 principal components are retained in Φ. The


0 500 1000 1500 2000 2500 30000

200

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600

800

1000

1200

1400

1600

1800

Days

P1

wat

er r

ate

(m3 /d

ay)

DDHMBest Prior ModelObserved Data"True" Future DataHM Period

(a) Water rate (P1)

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400

600

800

1000

1200

1400

Days

P1

oil r

ate

(m3 /d

ay)


(b) Oil rate (P1)

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100

200

300

400

500

600

Days

P2

wat

er r

ate

(m3 /d

ay)


(c) Water rate (P2)

0 500 1000 1500 2000 2500 3000100

150

200

250

300

350

400

450

500

550

Days

P2

oil r

ate

(m3 /d

ay)


(d) Oil rate (P2)

0 500 1000 1500 2000 2500 3000550

600

650

700

750

800

850

Days

I1 w

ater

rat

e (m

3 /day

)



0 500 1000 1500 2000 2500 30001000

1100

1200

1300

1400

1500

1600

1700

1800

Days

I2 w

ater

rat

e (m

3 /day

)



Figure 2.15: DDHM results for Case 1 after mapping operations


prior simulation results are shown in Fig. 2.17. There is significant spread in these

results. For example, the breakthrough time for P1 ranges from less than 500 days

to over 1500 days.

The prediction results from DDHM (with mapping operations) and the best prior

model are shown in Fig. 2.18. We again observe a reasonable level of accuracy in the

DDHM results. These results again improve upon predictions based on the best prior

model.

Figure 2.16: Log permeability maps for four prior bimodal channelized models


0 500 1000 1500 2000 2500 30000

200

400

600

800

1000

1200

1400

1600

Days

P1

wat

er r

ate

(m3 /d

ay)

Prior DataObserved Data"True" Future DataHM Period

(a) Water rate (P1)

0 500 1000 1500 2000 2500 30000

200

400

600

800

1000

Days

P1

oil r

ate

(m3 /d

ay)


(b) Oil rate (P1)

0 500 1000 1500 2000 2500 3000200

400

600

800

1000

1200

1400

1600

Days

I1 w

ater

rat

e (m

3 /day

)


(c) Injection rate (I1)

Figure 2.17: Prior simulated production data for Case 2


0 500 1000 1500 2000 2500 30000

200

400

600

800

1000

1200

1400

Days

P1

wat

er r

ate

(m3 /d

ay)


(a) Water rate (P1)

0 500 1000 1500 2000 2500 30000

100

200

300

400

500

600

700

800

900

Days

P1

oil r

ate

(m3 /d

ay)


(b) Oil rate (P1)

0 500 1000 1500 2000 2500 3000700

800

900

1000

1100

1200

1300

1400

Days

I1 w

ater

rat

e (m

3 /day

)



Figure 2.18: DDHM results for Case 2


2.6 Summary

In this chapter, we developed the DDHM framework for maximum a posteriori esti-

mates. Results for a three-dimensional Gaussian model were presented. These results

suggested that mapping operations are needed before applying the DDHM procedure.

In Section 2.4, we introduced several specific mapping operations. Predictions using

DDHM with these mapping operations were presented in Section 2.5, and clear im-

provement was observed. In the next chapter, we will extend the DDHM framework

to include uncertainty quantification.

Chapter 3

Uncertainty Quantification Based

on DDHM

Our goal in this chapter is to generalize DDHM in order to perform uncertainty quan-

tification. We first introduce the randomized maximum likelihood (RML) method

developed for sampling the a posteriori probability density function (PDF). We then

formulate an RML procedure based on DDHM. Prediction results for Cases 1 and

2 (in Chapter 2) are presented, and comparisons with the results from a rejection

sampling approach are also shown. These results demonstrate that uncertainty quan-

tification for reservoir production forecasts can be obtained using our RML-DDHM

framework.

3.1 RML with DDHM

RML is a sampling approach for generating realizations conditioned to measured

data and prior constraints. RML was developed independently by Kitanidis [8] and

Oliver [13]. These authors showed that the RML method can sample the posterior

PDF correctly if the data are linearly related to the model. Gao et al. [6] applied RML

for quantifying uncertainty for the PUNQ-S3 reservoir model [5]. They concluded that

RML gave a reasonable characterization of the uncertainty in predicted performance,

even though the data in this case were nonlinearly related to the model. In this

section, we will implement RML within the context of DDHM, instead of within the

usual MDHM context.

37

38 CHAPTER 3. UNCERTAINTY QUANTIFICATION BASED ON DDHM

In [16, pp. 319-334], RML was applied to generate realizations for a linear problem

with Gaussian measurement errors and a Gaussian prior PDF for the model variables.

For such a case, RML is able to sample the posterior PDF correctly. Using the DDHM

representation, our problem is of this form.

In RML, multiple model and data variables are first sampled from the prior dis-

tributions. The prior model mean and observed data in (2.12) (MAP estimate) are

then replaced with the sampled model and data variables. By repeating this proce-

dure many times, multiple posterior realizations can be obtained. Within the DDHM

framework, the MAP estimate is obtained through minimizing:

S(ξ) = (ΦHξ + dH − dobs)TC−1

D (ΦHξ + dH − dobs) + (ξ − ξ)TC−1ξ (ξ − ξ). (3.1)

The RML-DDHM algorithm is then as follows:

1. Sample prior weights vector ξu,i from rows in matrix V. Here V appears in

the SVD Y = UΣVT (the reparameterization equation (2.16)). Row i in V

represents a lower-dimensional representation (ξi) of prior simulated data (di).

2. Perturb observed data by adding noise using (2.21). The perturbed data is

denoted du,i.

3. Replace dobs and ξ in (3.1) with du,i and ξu,i. Then compute conditioned weights

ξc,i by minimizing:

S(ξ) = (ΦHξ+ dH −du,i)TC−1

D (ΦHξ+ dH −du,i) + (ξ− ξu,i)T (ξ− ξu,i). (3.2)

4. Repeat 1–3 for i = 1, 2, . . . , Nc, where Nc is the number of RML predictions to

be generated.

3.2 Rejection Sampling Procedure

We use the rejection sampling approach to provide a reference estimate for the uncer-

tainty of production forecasts. The rejection sampling approach proceeds as follows

[16, 22]:

1. Sample m from its prior distribution.

3.3. UNCERTAINTY QUANTIFICATION – CASE 1 39

2. Sample a variable p from a uniform distribution over [0, 1].

3. Accept m as a posterior sample if p ≤ L(m)SL

, where L(m) is the likelihood

function, and SL is a value that is no less than the supremum of the likelihood

for any model.

The likelihood function L(m) is defined as:

L(m) = c exp

(−1

2(g(m)− dobs)

TC−1D (g(m)− dobs)

), (3.3)

where c is a normalization constant. It is evident that L(m) ≤ c,∀m, and thus we

set SL = c.

Because the rejection sampling approach requires a very large number of prior sim-

ulations as the amount of observed data increases, in Sections 3.3 and 3.4 we will only

use data at one time step (160 days) for each well. We also use a large standard de-

viation of measurement error. Then, a reasonable number of models can be accepted

by the rejection sampling algorithm, which enables uncertainty quantification.

3.3 Uncertainty Quantification – Case 1

We now apply RML-DDHM for uncertainty quantification for Case 1. Fig. 3.1

presents the RML-DDHM results for this case. The number of prior simulations

performed is Nm = 100. The standard deviation of measurement error is set to be

equal to 15% of the rate data, with the minimum and maximum error set to 5 m3/d

and 50 m3/d. The prior data for this case were shown in Fig. 2.3. We see that the

spread in the production forecasts decreases significantly after applying our RML-

DDHM procedure, particularly for WPR and WIR. This observation indicates the

impact of data assimilation even when the amount of observed data is very limited.

Fig. 3.2 presents the P10-P50-P90 quantiles of uncertainty estimated from RML-

DDHM and the rejection sampling approach. A total of 50 RML-DDHM results

were generated. In the rejection sampling procedure, we use the same error standard

deviations as in RML-DDHM; i.e., C−1D is the same in (3.2) and (3.3). A total of 236

out of 100,000 prior models are retained in the rejection sampling. We see in Fig. 3.2

that the general level of agreement between RML-DDHM and rejection sampling


is quite acceptable, which suggests RML-DDHM is able to provide a quantitative

indication of uncertainty.

Because all observed data are generated synthetically from a randomly selected

“true” model, we can repeat the uncertainty quantification process for several other

true models. We thus performed similar comparisons for many randomly chosen true

models. Figs. 3.3 and 3.4 show comparisons for two additional true models. True

Model 2 displays later breakthrough time for both production wells (Fig. 3.3), while

True Model 3 displays very late breakthrough time for P2 (Fig. 3.4). For True Model

2 and True Model 3, a total of 1528 and 1206 models (out of 100,000 prior models)

were accepted, respectively. We see that the quantiles estimated from RML-DDHM

are close to those from the rejection sampling procedure.

These comparison results demonstrate that, for this Gaussian model, our RML-

DDHM procedure can provide reasonable uncertainty estimates. This also suggests

that the multivariate Gaussian assumption is at least approximately valid for the

mapped production data, even though the flow equation d = g(m) is nonlinear.

3.4 Uncertainty Quantification – Case 2

We now apply RML-DDHM for uncertainty quantification for Case 2, considered

earlier in Section 2.5.2. Fig. 3.5 presents RML-DDHM results. We again only use

observed data at one time step at each well. The error settings are the same as

presented in Section 3.3. Prior data are shown in Fig. 2.17.

Fig. 3.6 presents the P10-P50-P90 quantiles of uncertainty estimated from RML-

DDHM and the rejection sampling approach. The number of prior simulations per-

formed for RML-DDHM is Nm = 100, and 50 predictions are generated. A total of

87 (out of 1000) models are accepted in the rejection sampling approach. We see

that the uncertainty quantiles estimated are generally close, which again suggests

RML-DDHM is providing a reasonable uncertainty assessment.


0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

Days

P1

wat

er r

ate

(m3 /d

ay)

DDHMObserved Data"True" Future DataHM Period

(a) Water rate (P1)

0 500 1000 1500 2000 2500 30000

500

1000

1500

Days

P1

oil r

ate

(m3 /d

ay)


(b) Oil rate (P1)

0 500 1000 1500 2000 2500 30000

200

400

600

800

1000

Days

P2

wat

er r

ate

(m3 /d

ay)


(c) Water rate (P2)

0 500 1000 1500 2000 2500 30000

100

200

300

400

500

600

700

Days

P2

oil r

ate

(m3 /d

ay)


(d) Oil rate (P2)

0 500 1000 1500 2000 2500 3000500

600

700

800

900

1000

1100

Days

I1 w

ater

rat

e (m

3 /day

)



0 500 1000 1500 2000 2500 30001000

1100

1200

1300

1400

1500

1600

1700

1800

1900

Days

I2 w

ater

rat

e (m

3 /day

)



Figure 3.1: Prediction results for Case 1 with RML-DDHM approach


0 500 1000 1500 2000 2500 30000

200

400

600

800

1000

1200

1400

1600

1800

Days

P1

wat

er r

ate

(m3 /d

ay)

DDHMRS"True" Future DataHM Period

(a) Water rate (P1)

0 500 1000 1500 2000 2500 3000200

400

600

800

1000

1200

1400

Days

P1

oil r

ate

(m3 /d

ay)


(b) Oil rate (P1)

0 500 1000 1500 2000 2500 30000

100

200

300

400

500

600

700

800

Days

P2

wat

er r

ate

(m3 /d

ay)


(c) Water rate (P2)

0 500 1000 1500 2000 2500 30000

100

200

300

400

500

600

Days

P2

oil r

ate

(m3 /d

ay)


(d) Oil rate (P2)

0 500 1000 1500 2000 2500 3000550

600

650

700

750

800

850

900

950

Days

I1 w

ater

rat

e (m

3 /day

)



0 500 1000 1500 2000 2500 30001000

1100

1200

1300

1400

1500

1600

1700

1800

1900

Days

I2 w

ater

rat

e (m

3 /day

)



Figure 3.2: Uncertainty quantification (P10, P50 and P90) for Case 1 with RML-DDHM and rejection sampling (RS) approaches


0 500 1000 1500 2000 2500 30000

200

400

600

800

1000

1200

1400

Days

P1

wat

er r

ate

(m3 /d

ay)


(a) Water rate (P1)

0 500 1000 1500 2000 2500 3000200

300

400

500

600

700

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900

1000

1100

Days

P1

oil r

ate

(m3 /d

ay)


(b) Oil rate (P1)

0 500 1000 1500 2000 2500 30000

100

200

300

400

500

600

700

Days

P2

wat

er r

ate

(m3 /d

ay)


(c) Water rate (P2)

0 500 1000 1500 2000 2500 300050

100

150

200

250

300

350

400

450

500

Days

P2

oil r

ate

(m3 /d

ay)


(d) Oil rate (P2)

0 500 1000 1500 2000 2500 3000550

600

650

700

750

800

850

900

Days

I1 w

ater

rat

e (m

3 /day

)



0 500 1000 1500 2000 2500 3000600

700

800

900

1000

1100

1200

1300

Days

I2 w

ater

rat

e (m

3 /day

)



Figure 3.3: Uncertainty quantification (P10, P50 and P90) for Case 1 with RML-DDHM and rejection sampling approaches (True Model 2)


0 500 1000 1500 2000 2500 30000

500

1000

1500

Days

P1

wat

er r

ate

(m3 /d

ay)


(a) Water rate (P1)

0 500 1000 1500 2000 2500 3000200

400

600

800

1000

1200

Days

P1

oil r

ate

(m3 /d

ay)


(b) Oil rate (P1)

0 500 1000 1500 2000 2500 30000

100

200

300

400

500

600

700

Days

P2

wat

er r

ate

(m3 /d

ay)


(c) Water rate (P2)

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100

150

200

250

300

350

400

450

500

Days

P2

oil r

ate

(m3 /d

ay)


(d) Oil rate (P2)

0 500 1000 1500 2000 2500 3000650

700

750

800

850

900

950

1000

1050

Days

I1 w

ater

rat

e (m

3 /day

)



0 500 1000 1500 2000 2500 3000600

700

800

900

1000

1100

1200

1300

Days

I2 w

ater

rat

e (m

3 /day

)



Figure 3.4: Uncertainty quantification (P10, P50 and P90) for Case 1 with RML-DDHM and rejection sampling approaches (True Model 3)


0 500 1000 1500 2000 2500 30000

200

400

600

800

1000

1200

Days

P1

wat

er r

ate

(m3 /d

ay)


(a) Water rate (P1)

0 500 1000 1500 2000 2500 30000

100

200

300

400

500

600

700

800

900

Days

P1

oil r

ate

(m3 /d

ay)


(b) Oil rate (P1)

0 500 1000 1500 2000 2500 3000500

600

700

800

900

1000

1100

1200

Days

I1 w

ater

rat

e (m

3 /day

)



Figure 3.5: Prediction results for Case 2 with RML-DDHM approach


0 500 1000 1500 2000 2500 30000

200

400

600

800

1000

1200

Days

P1

wat

er r

ate

(m3 /d

ay)


(a) Water rate (P1)

0 500 1000 1500 2000 2500 30000

100

200

300

400

500

600

700

800

Days

P1

oil r

ate

(m3 /d

ay)


(b) Oil rate (P1)

0 500 1000 1500 2000 2500 3000600

700

800

900

1000

1100

1200

Days

I1 w

ater

rat

e (m

3 /day

)



Figure 3.6: Uncertainty quantification (P10, P50 and P90) for Case 2 with RML-DDHM and rejection sampling approaches

3.5. SUMMARY 47

3.5 Summary

In this chapter, we introduced an RML procedure within the context of DDHM, and

applied it for Cases 1 and 2. The prediction uncertainty estimated from this approach

was seen to be close to that from a rejection sampling procedure, which demonstrates

the ability of RML-DDHM to perform uncertainty quantification. In the next chapter,

we will apply RML-DDHM for more challenging cases with more wells.


Chapter 4

Application to More Complicated

Cases

In Chapter 3, we applied DDHM for uncertainty quantification for a Gaussian model

with four wells and a bimodal channelized model with two wells. Here, we apply

DDHM for two more complex models with more wells. We demonstrate that reason-

able results can be achieved with an affordable number (500) of prior simulations. The

procedure used here (RML-DDHM) is the same as that applied in Chapter 3. This

chapter essentially entails a compilation of results. We do not discuss the well-by-well

results for the two cases in any detail.

4.1 Gaussian Model with 13 Wells

The three-dimensional Gaussian models, shown in Fig. 4.1, are represented on a

60 × 60 × 5 grid, with each grid block of size 25 m × 25 m × 15 m. Permeability

is generated using the sequential Gaussian simulation algorithm within SGeMS [18].

The parameter settings are the same as those for Case 1. The porosity is assumed

to be constant at φ = 0.25. Nine producers and four injectors, all fully penetrating,

are introduced into the models. All realizations are conditioned to hard data at well

locations.

The fluid properties are also the same as those for Case 1. Injectors operate

at fixed BHPs of 500 bar, and producers operate at fixed BHPs of 250 bar. The

simulation period is 3000 days. The history-matching period extends from 150 days

49

50 CHAPTER 4. APPLICATION TO MORE COMPLICATED CASES

Figure 4.1: Log permeability maps for four prior Gaussian models

4.1. GAUSSIAN MODEL WITH 13 WELLS 51

to 500 days. A total of 500 prior realizations are used, and 130 principal components

are retained. We generate 50 RML-DDHM results.

The prior simulation results are shown in the left columns in Figs. 4.2–4.7, and the

prediction results from RML-DDHM, along with the best prior model, are shown in

the right columns. We see that prediction uncertainty has been reduced considerably

after applying our RML-DDHM procedure. In addition, the true data fall within the

range of the RML-DDHM predictions in all but a few sets of results (Figs. 4.3b, 4.3d,

4.4b and 4.4d). The best prior model also provides relatively accurate results, but

discrepancies with the true data are evident in many cases (e.g., Figs. 4.2h, 4.3b, 4.3d,

4.4f, 4.5b, 4.5d). Thus the quantification of uncertainty provided by RML-DDHM is

very useful for this case.


0 500 1000 1500 2000 2500 30002000

4000

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14000

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I1 w

ater

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)


(a) Prior injection rate (I1)

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14000

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I1 w

ater

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)


(b) DDHM injection rate (I1)

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ater

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(c) Prior injection rate (I2)

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ater

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(d) DDHM injection rate (I2)

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6000

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I3 w

ater

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e (m

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)


(e) Prior injection rate (I3)

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6000

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I3 w

ater

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3 /day

)


(f) DDHM injection rate (I3)

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2000

2500

3000

3500

4000

4500

5000

Days

I4 w

ater

rat

e (m

3 /day

)


(g) Prior injection rate (I4)

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2000

2500

3000

3500

4000

4500

5000

Days

I4 w

ater

rat

e (m

3 /day

)


(h) DDHM injection rate (I4)

Figure 4.2: RML-DDHM results for WIR (I1 – I4)


0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

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P1

wat

er r

ate

(m3 /d

ay)


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1000

1500

2000

2500

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4000

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P1

wat

er r

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ay)


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1000

1500

2000

2500

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P1

oil r

ate

(m3 /d

ay)


(c) Prior oil rate (P1)

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500

1000

1500

2000

2500

Days

P1

oil r

ate

(m3 /d

ay)


(d) DDHM oil rate (P1)

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1000

2000

3000

4000

5000

6000

7000

Days

P2

wat

er r

ate

(m3 /d

ay)


(e) Prior water rate (P2)

0 500 1000 1500 2000 2500 30000

1000

2000

3000

4000

5000

6000

7000

Days

P2

wat

er r

ate

(m3 /d

ay)


(f) DDHM water rate (P2)

0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

Days

P2

oil r

ate

(m3 /d

ay)


(g) Prior oil rate (P2)

0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

Days

P2

oil r

ate

(m3 /d

ay)


(h) DDHM oil rate (P2)

Figure 4.3: RML-DDHM results for WPR and OPR (P1 and P2)


0 500 1000 1500 2000 2500 30000

100

200

300

400

500

600

700

800

Days

P3

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

100

200

300

400

500

600

700

800

Days

P3

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

100

200

300

400

500

600

700

Days

P3

oil r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

100

200

300

400

500

600

700

Days

P3

oil r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

3500

Days

P4

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

3500

Days

P4

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

Days

P4

oil r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

Days

P4

oil r

ate

(m3 /d

ay)





0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

Days

P5

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

Days

P5

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 3000200

400

600

800

1000

1200

1400

1600

1800

2000

Days

P5

oil r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 3000200

400

600

800

1000

1200

1400

1600

1800

2000

Days

P5

oil r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

3500

4000

4500

Days

P6

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

3500

4000

4500

Days

P6

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

3500

Days

P6

oil r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

3500

Days

P6

oil r

ate

(m3 /d

ay)





0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

3500

4000

4500

Days

P7

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

3500

4000

4500

Days

P7

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

Days

P7

oil r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

Days

P7

oil r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

1000

2000

3000

4000

5000

6000

Days

P8

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

1000

2000

3000

4000

5000

6000

Days

P8

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

3500

4000

Days

P8

oil r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

3500

4000

Days

P8

oil r

ate

(m3 /d

ay)





0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

Days

P9

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

Days

P9

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

200

400

600

800

1000

1200

1400

1600

1800

Days

P9

oil r

ate

(m3 /d

ay)



0 500 1000 1500 2000 2500 30000

200

400

600

800

1000

1200

1400

1600

1800

Days

P9

oil r

ate

(m3 /d

ay)



Figure 4.7: RML-DDHM results for WPR and OPR (P9)


4.2 Channelized Model with 16 Wells

The two-dimensional channelized model, shown in Fig. 4.8, is represented on a 60× 60

grid, with each grid block of size 25 m × 25 m × 10 m. The permeability field is

generated using the same general procedure described for Case 2 in Section 2.5.2.

The binary model now is generated using the ‘snesim’ geostatistical algorithm within

SGeMS [25]. The Gaussian realizations (ln k) with different means (-1 for mud and 4

for sand) and standard deviation (0.9 for mud and 0.6 for sand) are again simulated

independently using the sequential Gaussian simulation algorithm within SGeMS. All

realizations are conditioned to hard data at well locations. The porosity is assumed

to be constant φ = 0.3. A total of 12 producers and four injectors are drilled. In this

case, three production wells are drilled in shale (P6, P8, and P10), and all the other

wells are drilled in sand.

The fluid properties are the same as in previous cases. The simulation period is

2000 days and the history-matching period is from 150 days to 500 days. A total

of 500 prior realizations are used, and 150 principal components are retained. We

construct 50 RML-DDHM solutions.

The prior simulation results are shown in the left columns in Figs. 4.9–4.15, and

the prediction results from RML-DDHM and the best prior model are shown in the

right columns. For the injectors (Fig. 4.9), we see that the range of uncertainty for

WIR is largely reduced with the RML-DDHM procedure. The true data fall within

the RML-DDHM predictions for I1, I3, and I4. For I2, the true data are slightly

outside the RML-DDHM results. The best prior model is reasonably accurate for I1,

I2, and I3, though there is a large discrepancy for I4.

Results for the production wells also illustrate the ability of RML-DDHM to reduce

the range of uncertainty in the prior models. For all wells that display reasonable

flow rates, the true OPR and WPR data fall within, or very close to, the RML-

DDHM results. The best prior model is accurate for many quantities, but shows

large discrepancies in some cases (e.g., Figs. 4.11f, 4.12b, 4.15f). Note finally that

wells P6, P8 and P10 display very low flow rates. These are the wells that intersect

low-permeability shale.

4.2. CHANNELIZED MODEL WITH 16 WELLS 59

Figure 4.8: Log permeability maps for four prior channelized models


0 500 1000 1500 2000200

400

600

800

1000

1200

1400

1600

1800

Days

I1 w

ater

rat

e (m

3 /day

)


(a) Prior injection rate (I1)

0 500 1000 1500 2000200

400

600

800

1000

1200

1400

1600

1800

Days

I1 w

ater

rat

e (m

3 /day

)


(b) DDHM injection rate (I1)

0 500 1000 1500 2000

200

400

600

800

1000

Days

I2 w

ater

rat

e (m

3 /day

)


(c) Prior injection rate (I2)

0 500 1000 1500 2000

200

400

600

800

1000

Days

I2 w

ater

rat

e (m

3 /day

)


(d) DDHM injection rate (I2)

0 500 1000 1500 20000

200

400

600

800

1000

Days

I3 w

ater

rat

e (m

3 /day

)


(e) Prior injection rate (I3)

0 500 1000 1500 20000

200

400

600

800

1000

Days

I3 w

ater

rat

e (m

3 /day

)


(f) DDHM injection rate (I3)

0 500 1000 1500 20000

100

200

300

400

500

600

700

800

900

Days

I4 w

ater

rat

e (m

3 /day

)


(g) Prior injection rate (I4)

0 500 1000 1500 20000

100

200

300

400

500

600

700

800

900

Days

I4 w

ater

rat

e (m

3 /day

)


(h) DDHM injection rate (I4)

Figure 4.9: RML-DDHM results for WIR (I1 – I4)


0 500 1000 1500 20000

100

200

300

400

500

600

700

800

Days

P1

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

100

200

300

400

500

600

700

800

Days

P1

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

100

200

300

400

500

Days

P1

oil r

ate

(m3 /d

ay)



0 500 1000 1500 2000

0

100

200

300

400

500

Days

P1

oil r

ate

(m3 /d

ay)



0 500 1000 1500 20000

200

400

600

800

1000

Days

P2

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

200

400

600

800

1000

Days

P2

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

50

100

150

200

250

300

350

Days

P2

oil r

ate

(m3 /d

ay)



0 500 1000 1500 20000

50

100

150

200

250

300

350

Days

P2

oil r

ate

(m3 /d

ay)





0 500 1000 1500 20000

100

200

300

400

500

600

Days

P3

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

100

200

300

400

500

600

Days

P3

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

50

100

150

200

250

300

350

400

450

Days

P3

oil r

ate

(m3 /d

ay)



0 500 1000 1500 2000

0

50

100

150

200

250

300

350

400

450

Days

P3

oil r

ate

(m3 /d

ay)



0 500 1000 1500 20000

100

200

300

400

500

600

700

800

900

Days

P4

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

100

200

300

400

500

600

700

800

900

Days

P4

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

50

100

150

200

250

300

350

400

Days

P4

oil r

ate

(m3 /d

ay)



0 500 1000 1500 20000

50

100

150

200

250

300

350

400

Days

P4

oil r

ate

(m3 /d

ay)





0 500 1000 1500 20000

50

100

150

200

250

300

350

400

Days

P5

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 2000

0

50

100

150

200

250

300

350

400

Days

P5

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

50

100

150

200

Days

P5

oil r

ate

(m3 /d

ay)



0 500 1000 1500 20000

50

100

150

200

Days

P5

oil r

ate

(m3 /d

ay)



0 500 1000 1500 20000

0.5

1

1.5

2

2.5

Days

P6

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

0.5

1

1.5

2

2.5

Days

P6

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

1

2

3

4

5

6

Days

P6

oil r

ate

(m3 /d

ay)



0 500 1000 1500 20000

1

2

3

4

5

6

Days

P6

oil r

ate

(m3 /d

ay)





0 500 1000 1500 20000

50

100

150

200

250

300

Days

P7

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

50

100

150

200

250

300

Days

P7

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

20

40

60

80

100

120

140

160

180

Days

P7

oil r

ate

(m3 /d

ay)



0 500 1000 1500 20000

20

40

60

80

100

120

140

160

180

Days

P7

oil r

ate

(m3 /d

ay)



0 500 1000 1500 20000

1

2

3

4

5

6

7

8

9

Days

P8

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

1

2

3

4

5

6

7

8

9

Days

P8

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

1

2

3

4

5

6

7

8

9

Days

P8

oil r

ate

(m3 /d

ay)



0 500 1000 1500 20000

1

2

3

4

5

6

7

8

9

Days

P8

oil r

ate

(m3 /d

ay)





0 500 1000 1500 20000

50

100

150

200

250

300

350

400

Days

P9

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

50

100

150

200

250

300

350

400

Days

P9

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

50

100

150

200

250

300

Days

P9

oil r

ate

(m3 /d

ay)



0 500 1000 1500 20000

50

100

150

200

250

300

Days

P9

oil r

ate

(m3 /d

ay)



0 500 1000 1500 20000

0.5

1

1.5

2

2.5

3

3.5

4

Days

P10

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

0.5

1

1.5

2

2.5

3

3.5

4

Days

P10

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

1

2

3

4

5

6

7

8

9

Days

P10

oil

rate

(m

3 /day

)



0 500 1000 1500 20000

1

2

3

4

5

6

7

8

9

Days

P10

oil

rate

(m

3 /day

)





0 500 1000 1500 20000

50

100

150

200

250

300

350

400

Days

P11

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

50

100

150

200

250

300

350

400

Days

P11

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

50

100

150

200

250

300

Days

P11

oil

rate

(m

3 /day

)



0 500 1000 1500 20000

50

100

150

200

250

300

Days

P11

oil

rate

(m

3 /day

)



0 500 1000 1500 20000

100

200

300

400

500

600

700

800

900

Days

P12

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

100

200

300

400

500

600

700

800

900

Days

P12

wat

er r

ate

(m3 /d

ay)



0 500 1000 1500 20000

50

100

150

200

250

300

350

400

450

Days

P12

oil

rate

(m

3 /day

)



0 500 1000 1500 2000

0

50

100

150

200

250

300

350

400

450

Days

P12

oil

rate

(m

3 /day

)




4.3. SUMMARY 67

4.3 Summary

In this chapter, we applied the RML-DDHM framework for two more complicated

cases — a Gaussian model with 13 wells and a bimodal channelized model with 16

wells. The results indicate that the RML-DDHM framework is able to significantly

reduce the range of uncertainty for these cases. It is noteworthy that the true data

consistently fall within, or very near, the RML-DDHM results in these examples.

This suggests that RML-DDHM may indeed be useful for uncertainty assessment.


Chapter 5

Summary and Future Work

Summary

In this thesis, we developed a data driven history matching (DDHM) procedure for

reservoir forecasting and uncertainty quantification. In our DDHM procedure, no

model inversion is required. As a result, substantial computational savings can po-

tentially be achieved, and many of the complications (e.g., maintaining geological

realism of reservoir models) associated with model driven history matching (MDHM)

are circumvented.

Our procedure starts from the basic DDHM formulation used by Krishnamurti

et al. [9] for weather forecasting. Direct application of this approach, however, leads

to unphysical predictions due to the non-Gaussianity and nonlinearity of production

data. To resolve this, we introduced mapping operations, which provide mapped prior

production data that has better properties. Application of DDHM for a Gaussian

geological model illustrated the improvements that this treatment can provide.

To enable uncertainty quantification, we applied the randomized maximum like-

lihood (RML) approach within the context of DDHM. Gaussian and channelized

models were used for testing. Our RML-DDHM procedure was shown to provide

results in reasonably close agreement with those from a rejection sampling approach.

Two more complicated models (Gaussian model with 13 wells and channelized

model with 16 wells) were also used to test the RML-DDHM procedure. Substantial

reduction in prior uncertainty was achieved, and the RML-DDHM results were shown

to bracket the true data in most cases.

69

70 CHAPTER 5. SUMMARY AND FUTURE WORK

Future Work

In this thesis, we considered relatively simple cases. It will be of interest to extend and

apply the DDHM procedure to more complex systems such as fractured models. This

may be particularly useful because traditional MDHM approaches are not always well

suited for fractured systems. The use of DDHM for cases with varying well controls

should also be considered. For these cases, general mapping procedures should be

developed to avoid the need for case-by-case treatments (as were implemented in this

thesis).

It will also be of interest to test DDHM for cases where well controls change in

the prediction period. In such cases it will be useful to minimize the number of prior

models that must be simulated in order to reduce computational costs.

Nomenclature

Abbreviations

AD-GPRS Automatic Differentiation-based General Purpose Research Simulator

BHP bottom-hole pressure

DDHM data driven history matching

EnKF ensemble Kalman filter

MAP maximum a posteriori

MDHM model driven history matching

OPR oil production rate

PCA principal component analysis

PDF posterior distribution function

RML randomized maximum likelihood

RS rejection sampling

SVD singular value decomposition

WIR water injection rate

WPR water production rate

Variables

CD covariance of measurement error

Cd covariance of data d

CM covariance of model parameters

CV covariance of weights v

Cξ covariance of weights ξ

d simulation data containing both history-matching and prediction periods

71

72 NOMENCLATURE

d mapped simulation data

dobs observed data

dtrue unknown true data

ej simulation data for well j

ej mapped simulation data for well j

m reservoir model parameters

Nh number of observed data values

Nl number of principal components

Nm number of total prior models

Nt number of observed and future data values

R regularization term

tB breakthrough time of true model

tb breakthrough time

th end time of history-matching period

tini initial time for history matching

tt end time of prediction period

t mapped dimensionless time

v vector of weights

X matrix containing prior simulated data

Subscripts

F data corresponding to prediction period

H data corresponding to history-matching period

P predicted data

T data corresponding to both history-matching and prediction period

Greek

δ model error

ε measurement error

σ standard deviation

ξ regressed weights

Φ basis matrix

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