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Inverse Theory for Petroleum Reservoir Characterization and History Matching

This book is a guide to the use of inverse theory for estimation and conditional simulation of flow and transport parameters in porous media

It describes the theory and practice of estimating properties of underground petroleum reservoirs from measurements of flow in wells,

and it explains how to characterize the uncertainty in such estimates

Early chapters present the reader with the necessary background in inverse theory,

The book then goes on to develop physical explanations for the sensitivity of well data to rock or flow properties,

and demonstrates how to calculate sensitivity coefficients and the linearized relationship between models and production data

It also shows how to develop iterative methods for generating estimates and conditional realizations

Characterization of uncertainty for highly nonlinear inverse problems,

and the methods of sampling from high-dimensional probability density functions,

The book then ends with a chapter on the development and application of methods for sequentially assimilating data into reservoir models

This volume is aimed at graduate students and researchers in petroleum engineering and groundwater hydrology and can be used as a textbook for advanced courses on inverse theory in petroleum engineering

It includes many worked examples to demonstrate the methodologies,

Color figures that further illustrate the data in this book are available at www

org/9780521881517 Dean Oliver is the Mewbourne Chair Professor in the Mewbourne School of Petroleum and Geological Engineering at the University of Oklahoma,

where he was the Director for four years

Prior to joining the University of Oklahoma,

he worked for seventeen years as a research geophysicist and staff reservoir engineer for Chevron USA,

and for Saudi Aramco as a research scientist in reservoir characterization

He also spent six years as a professor in the Petroleum Engineering Department at the University of Tulsa

Professor Oliver has been awarded ‘best paper of the year’ awards from two journals and received the Society of Petroleum Engineers (SPE) Reservoir Description and Dynamics award in 2004

He is currently the Executive Editor of SPE Journal

His research interests are in inverse theory,

Albert Reynolds is Professor of Petroleum Engineering and Mathematics,

holder of the McMan chair in Petroleum Engineering,

and Director of the TUPREP Research Consortium at the University of Tulsa

He has published over 100 technical articles and one previous book,

and is well known for his contributions to pressure transient analysis and history matching

Professor Reynolds has won the SPE Distinguished Achievement Award for Petroleum Engineering Faculty,

the SPE Reservoir Description and Dynamics Award and the SPE Formation Award

He became an SPE Distinguished Member in 1999

Ning Liu holds a Ph

from the University of Oklahoma in petroleum engineering and now works as a Reservoir Simulation Consultant at Chevron Energy Technology Company

Dr Liu is a recipient of the Outstanding Ph

Scholarship Award at the University of Oklahoma and the Student Research Award from the International Association for Mathematical Geology (IAMG)

Her areas of interest are history matching,

Inverse Theory for Petroleum Reservoir Characterization and History Matching Dean S

Oliver Albert C

Reynolds Ning Liu



New York,



Cape Town,


São Paulo Cambridge University Press The Edinburgh Building,

Cambridge CB2 8RU,

UK Published in the United States of America by Cambridge University Press,

New York www

org Information on this title: www



Liu 2008 This publication is in copyright

Subject to statutory exception and to the provision of relevant collective licensing agreements,

no reproduction of any part may take place without the written permission of Cambridge University Press

First published in print format 2008

ISBN-13 978-0-511-39851-3

eBook (EBL)

ISBN-13 978-0-521-88151-7

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication,

and does not guarantee that any content on such websites is,

Al Reynolds dedicates the book to Anne,

Ning Liu dedicates the book to her parents and teachers

Dean Oliver dedicates the book to his wife Mary and daughters Sarah and Beth




The forward problem The inverse problem

Examples of inverse problems 2

Density of the Earth Acoustic tomography Steady-state 1D flow in porous media History matching in reservoir simulation Summary

Estimation for linear inverse problems

Characterization of discrete linear inverse problems Solutions of discrete linear inverse problems Singular value decomposition Backus and Gilbert method

Probability and estimation

69 73 78

Random variables Expected values Bayes’ rule


Descriptive geostatistics 5

Geologic constraints Univariate distribution Multi-variate distribution Gaussian random variables Random processes in function spaces

The maximum a posteriori estimate

Conditional probability for linear problems Model resolution Doubly stochastic Gaussian random field Matrix inversion identities

Optimization for nonlinear problems using sensitivities

Shape of the objective function Minimization problems Newton-like methods Levenberg–Marquardt algorithm Convergence criteria Scaling Line search methods BFGS and LBFGS Computational examples

Sensitivity coefficients

200 206

The Fr´echet derivative Discrete parameters


One-dimensional steady-state flow Adjoint methods applied to transient single-phase flow Adjoint equations Sensitivity calculation example Adjoint method for multi-phase flow Reparameterization Examples Evaluation of uncertainty with a posteriori covariance matrix

Quantifying uncertainty

Introduction to Monte Carlo methods Sampling based on experimental design Gaussian simulation General sampling algorithms Simulation methods based on minimization Conceptual model uncertainty Other approximate methods Comparison of uncertainty quantification methods

Recursive methods

Basic concepts of data assimilation Theoretical framework Kalman filter and extended Kalman filter The ensemble Kalman filter Application of EnKF to strongly nonlinear problems 1D example with nonlinear dynamics and observation operator Example – geologic facies

References Index

367 378


The intent of this book is to provide a rather broad overview of inverse theory as it might be applied to petroleum reservoir engineering and specifically to what has,

It has been strongly influenced by the geophysicists’ approach to inverse problems as opposed to that of mathematicians

In particular,

we emphasize that measurements have errors,

that the quantity of data are always limited,

and that the dimension of the model space is usually infinite,

so inverse problems are always underdetermined

The approach that we take to inverse theory is governed by the following philosophy

All inverse problems are characterized by large numbers of parameters (conceptually infinite)

We only limit the number of parameters in order to solve the forward problem

The number of data is always finite,

and the data always contain measurement errors

It is impossible to correctly estimate all the parameters of a model from inaccurate,

and inconsistent data,1 but reducing the number of parameters in order to get low levels of uncertainty is misleading

On the other hand,

we almost always have some prior information about the plausibility of models

This information might include positivity constraints (for density,

bounds (porosity between 0 and 1),

Most petroleum inverse problems related to fluid flow are nonlinear

The calculation of gradients is an important and expensive part of the problem

Because of the large cost of computing the output of a reservoir simulation model,

trial and error approaches to inverting data are impractical

Probabilisitic estimates or bounds are often the most meaningful

For nonlinear problems,

this is usually best accomplished using Monte Carlo methods

The ultimate goal of inverse theory (and history matching) is to make informed decisions on investments,

Good decisions can only be made if the uncertainty in future performance,

and the consequences of actions can be accurately characterized

This is part of the title of a famous paper by Jackson [1]: “Interpretation of inaccurate,


Other general references Several good books on geophysical inverse theory are available

Menke [2] provides good introductory information on the probabilistic interpretation of an answer to an inverse problem,

and much good material on the discrete inverse problem

Parker [3] contains good material on Hilbert space,

existence and uniqueness (for linear problems),

and functional differentiation

He does not,

get very deeply into nonlinear problems or stochastic approaches

Tarantola [4] comes closest to covering the material on linear inverse problems,

but has very little material on calculation of sensitivities

Sun [5] focusses on problems related to flow in porous media,

and contains useful material on the calculation of sensitivities for flow and transport problems

A highly relevant free source of information on inverse theory is the book by John Scales [6]

No single book contains a thorough description of the nonlinear developments in inverse theory or the applications to petroleum engineering

Most of the material that is specifically related to petroleum engineering is based on our publications

The choice of material for these notes is based on the observation that while many scientists and engineers have good intuition for the outcome of an experiment,

they often have poor intuition regarding inverse problems

This is not to say that they can not estimate some parameter values that might result in a specified response,

but that they have little feel for the degree of nonuniqueness of the answer,

or of the relationship of their answer to other answers or to the true parameters

We feel that this intuition is best developed through a study of linear theory and that the method of Backus and Gilbert is good for promoting understanding of many important concepts at a fundamental level

On the other hand,

the Backus and Gilbert method can produce solutions that are not plausible because they are too erratic or too smooth

introduce methods for incorporating prior information on smoothness and variability

One of the principal uses of these methods is to investigate risk and to make informed decisions regarding investment

For many petroleum engineering problems,

evaluation of uncertainty requires the ability to generate a meaningful distribution multiple of models

Characterization of uncertainty for highly nonlinear inverse problems,

and the methods of sampling from high-dimensional probability density functions are discussed in Chapter 10

Most history-matching problems in petroleum engineering are strongly nonlinear

Efficient incorporation of production-type data (e

) requires the calculation of sensitivity coefficients or the linearized relationship between model and data

This is the topic of Chapter 9

Although history matching has typically been a “batch process” in which all data are assimilated simultaneously,

the installation of permanent sensors in wells has increased the need for methods of updating reservoir models by sequentially assimilating data as it becomes available

A method for doing this is described in Chapter 11


If it were possible for geoloscientists and engineers to know the locations of oil and gas,

the locations and transmissivity of faults,

and the multi-phase flow properties such as relative permeability and capillary pressure at all locations in a reservoir,

it would be conceptually possible to develop a mathematical model that could be used to predict the outcome of any action

The relationship of the model variables,

describing the system to observable variables or data,

If the model variables are known,

usually by running a numerical reservoir simulator that solves a discretized approximation to a set of partial differential equations

This is termed the forward problem

Most oil and gas reservoirs are inconveniently buried beneath thousands of feet of overburden

Direct observations of the reservoir are available only at well locations that are often hundreds of meters apart

Indirect observations are typically made at the surface,

either at the well-head (production rates and pressures) or at distributed locations (e

In the inverse problem,

the observations are used to determine the variables that describe the system

Real observations are contaminated with errors,

so the inverse problem is to “solve” the set of equations dobs = g(m) +  for the model variables,

with the goal of making accurate predictions of future performance

The forward problem In a forward problem,

the physical properties of some system (system or model parameters) are known,

and a deterministic method is available for calculating the response or outcome of the system to a known stimulus

The physical properties are referred to as system or model parameters

A typical forward problem is represented by a differential equation with specified initial and/or boundary conditions

A simple example

of a forward problem of interest to petroleum engineers is the following steady-state problem for a one-dimensional flow in a porous medium:   d'k(x)A dp(x) = 0,

where A (cross sectional area to flow in cm2 ),

µ (viscosity in cp),

and pressure pe (atm) are assumed to be constant

The length of the system in cm is represented by L

The function k(x) represents the permeability field in Darcies

This steady-state problem could describe linear flow in either a core or a reservoir

For this forward problem,

which are assumed to be known,

The stimulus for the system (reservoir or core) is provided by prescribing q (the flow rate out the right-hand end) and p(0) (the pressure at the left-hand end),

which are assumed to be known exactly

The system output or response is the pressure field,

which can be determined by solving the boundary-value problem

The solution of this steady-state boundary-value problem is given by qµ p(x) = pe − A

If the emphasis is on the relationship between the permeability field and the pressure,

we might formally write the relationship between pressure,

and the permeability field as pi = gi (k)

This expression indicates that the function gi specifies the relation between the permeability field and pressure at the point xi

Forward problems of interest to us can usually be represented by a differential equation or system of differential equations together with initial and/or boundary conditions

Most such forward problems are well posed,

or can be made to be well posed by imposing natural physical constraints on the coefficients of the differential equation(s) and the auxiliary conditions

auxiliary conditions refer to the initial and boundary conditions

A boundary-value problem,

or initial boundary-value problem,

is said to be well posed in the sense of Hadamard [7],

if the following three criteria are satisfied: (a) the problem has a solution,

and (c) the solution is a continuous function of the problem data

It is important to note that the problem data include the functions defining the initial and boundary conditions and the coefficients in the differential equation


the problem data refers to pe ,

qµ/k(L)A and k(x)

If k(x) were zero in some part of the core,

then we can not obtain steady-state flow through the core and the pressure solution of Eq

the boundaryvalue problem of Eqs



if we impose the restriction that k(x) ≥ δ > 0 for any arbitrarily small δ then the boundaryvalue problem is well posed

If a problem is not well posed,

At one time,

most mathematicians believed that ill-posed problems were incorrectly formulated and nonphysical

We know now that this is incorrect and that a great deal of useful information can be obtained from ill-posed problems

If this were not so,

there would be little reason to study inverse problems,

as almost all inverse problems are ill posed

The inverse problem In its most general form,

an inverse problem refers to the determination of the plausible physical properties of the system,

or information about these properties,

given the observed response of the system to some stimulus

The observed response will be referred to as observed data

For example,

for the steady-state problem considered above,

an inverse problem could represent the problem of determining the permeability field from pressure data measured at points in the interval [0,

Note that measured or observed data is different from the problem data introduced in the definition of a well-posed problem

In both forward and inverse problems,

the physical system is characterized by a set of model parameters,

a model parameter is allowed to be either a function or a scalar

For the steady single-phase flow problem,

the model parameters can be chosen as the inverse permeability (m(x) = 1/k(x)),

cross sectional area A and length L

the model parameters could also be chosen as (k(x)A)/µ and L

If we were to attempt to solve Eq

we might discretize the permeability function,

and choose ki = k(xi ) for a limited number of integers i as our parameters

The choice of model parameters is referred to as a parameterization of the physical system

Observable parameters refer to those that can be observed or measured,

and will simply be referred to as observed data

For the above steady-state problem,

forcing fluid to flow through the porous medium at the specified rate q provides the stimulus and measured values of pressure at certain locations that represent observed data

Pressure can be measured only at a well location,

or in the case where the system represents a core,

at locations where pressure transducers are situated

Although the relation between observed data and model parameters is often referred to as the model,

we will refer to this relationship as the (assumed) theoretical model,

because we wish to refer to any feasible set of specific model parameters as a model

In the continuous

the model or model parameters may represent a function or set of functions rather than simply a discrete set of parameters

For the steady-state problem of Eqs


the boundary-value problem implicitly defines the theoretical model with the explicit relation between observable parameters and the model or model parameters given by Eq

The inverse problem is almost never well posed

In the cases of most interest to petroleum reservoir engineers and hydrogeologists,

an infinite number of equally good solutions exist

For the steady-state problem,

the general inverse problem represents the determination of information about model parameters (e


and L) from pressure measurements

As pressure measurements are subject to noise,

measured pressure data will not,

The assumed theoretical model may also not be exact

For the example problem considered earlier,

the theoretical model assumes constant viscosity and steady-state flow

If these assumptions are invalid,

then we are using an approximate theoretical model and these modeling errors should be accounted for when generating inverse solutions

For now,

we state the general inverse problem as follows: determine plausible values of model parameters given inexact (uncertain) data and an assumed theoretical model relating the observed data to the model

For problems of interest to petroleum engineers,

the theoretical model always represents an approximation to the true physical relation between physical and/or geometric properties and data

Left unsaid at this point is what is meant by plausible values (solutions) of the inverse problems

A plausible solution must of course be consistent with the observed data and physical constraints (permeability and porosity can not be negative),

but for problems of interest in petroleum reservoir characterization,

there will normally be an infinite number of models satisfying this criterion

Do we want to choose just one estimate

? Do we want to determine several solutions

we have a very definite philosophical approach to inverse problems,

one that is grounded in a Bayesian viewpoint of probability and assumes that prior information on model parameters is available

This prior information could be as simple as a geologist’s statement that he or she believes that permeability is 200 md plus or minus 50

To obtain a mathematically tractable inverse problem,

the prior information will always be encapsulated in a prior probability density function

Our general philosophy of the inverse problem can then be stated as follows: given prior information on some model parameters,

inexact measurements of some observable parameters,

and an uncertain relation between the data and the model parameters,

how should one modify the prior probability density function (PDF) to include the information provided by the inexact measurements

? The modified PDF is referred to as the a posteriori probability density function

In a sense,

the construction of the a posteriori PDF represents the solution to the inverse problem


one wishes to construct an estimate of the model (often,

the maximum a posteriori estimate) or realizations of the model by sampling the a posteriori PDF

The process of constructing a particular estimate

of the model will be referred to as estimation

the process of constructing a suite of realizations will be referred to as simulation

our emphasis is on estimating and simulating permeability and porosity fields

Our approach to the application of inverse problem theory to petroleum reservoir characterization problems may be summarized as follows

Postulate a prior PDF for the model parameters from analog fields,

We will often assume that the prior PDF is multi-variate Gaussian,

in which case the means and the covariance fully define the stochastic model

Formulate the a posteriori PDF conditioned to all observed data

Data could include both production data and “hard” data (direct measurements of the variables to be estimated) for the rock property fields

Construct a suite of realizations of the rock property fields by sampling the a posteriori PDF

Generate a reservoir performance prediction under proposed operating conditions for each realization

This step is done using a reservoir simulator

Construct statistics (e

variance) from the set of predicted outcomes for each performance variable (e

Determine the uncertainty in predicted performance from the statistics

In our view,

and most of our research effort has focussed either on step 3 or on issues related to computational efficiency including the development of methods to efficiently generate sensitivity coefficients

Note that if one simply generates a set of rock property fields consistent with all observed data,

but the set does not characterize the true uncertainty in the rock property fields (in our language,

does not represent a correct sampling of the a posteriori PDF),

steps 4 and 5 can not be expected to yield a meaningful characterization of the uncertainty in predicted reservoir performance

Examples of inverse problems

The inverse problems examples presented in this chapter illustrate the concepts of data,

Each of these concepts will be developed in much greater depth in subsequent chapters

The examples are all quite simple to describe and understand,

but several are difficult to solve

Density of the Earth The mass,

of the Earth are related to the density distribution,

(assuming mass density is only a function of radius) by the following formulas:  a r 2 ρ(r) dr,

8π I= 3

where a is the radius of the Earth

If the true density is known for all r,

then it is easy to compute the mass and the moment of inertia

In reality,

the mass and moment of inertia can be estimated from measurements of the precession of the axis of rotation and the gravitational constant

the density distribution must be estimated

The data vector consists of the “observed” mass and moment of inertia of the Earth: d'= [M

the superscript T on a matrix or vector denotes its transpose

) The relationship between the model variable and the theoretical data is   a 4πr 2 (2

the dimension of the model to be estimated is infinite,

while the dimension of the data space is just 2

Prior information might be a lower 6

Figure 2

The array of nine blocks with traveltime parameters,

and the six measurement locations for total traveltime,

A loose lower bound would be that density is positive

A reasonable lower bound with more information is that density is greater than or equal to 2250 kg/m3

Although it is easy to generate a model that fits the data exactly,

unless other information is available,

the uncertainty in the estimated density at a point or a radius is unbounded

Note also that the theoretical relationship between the density and the data in this example is only approximate as the Earth is not exactly spherical,

and there is no a priori reason to believe that the density is only a function of radius

Acoustic tomography One of the simplest examples that demonstrates the concepts of sensitivity,

and inconsistency is the problem of estimation of the spatial distribution of acoustic slowness (1/velocity) from measurements of traveltime along several ray paths through a solid body

For simplicity,

we assume that the material properties are uniform within each of the nine blocks (Fig

If t denotes the acoustic slowness of a homogeneous block,

and T denotes the time required to travel a distance D'within or across a block,

Consider a 3 × 3 array of blocks of various materials shown in Fig

Each homogeneous block is 1 unit in width by 1 unit in height

Measurements of traveltime have been made for each column and each row of blocks

If the slowness of the (1,

the total traveltime for a sound wave to travel across the first row of blocks,

Similar relations hold for the other rows and columns

measurements of traveltime are exact,

the entire set of relations between measurements and slowness in each block is T1 = t1 + t2 + t3 T2 = t4 + t5 + t6 T3 = t7 + t8 + t9 T4 = t1 + t4 + t7

T5 = t2 + t5 + t8 T6 = t3 + t6 + t9

Given measured values of Ti ,

the inverse problem is to determine information about the acoustic slownesses,

More specifically,

we may wish to determine the set of all solutions of Eq

is the vector of traveltime measurements,

T6 ]T ,

is the vector of slowness values given by m = [t1

is the matrix that relates the data to the model variables and is given by   1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0     0 0 0 0 0 0 1 1 1 G= (2

10) 

1 0 0 1 0 0 1 0 0   0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1

The reason for calling G the sensitivity matrix is easily understood by examining the particular row of G associated with a particular measurement

Note that there are as many rows as there are measurements

Each row has nine elements in this example,

The element in the ith row and j th column of G gives the “sensitivity” (∂Ti /∂tj ) of the ith measurement to a change in the j th model variable

the fourth measurement is only sensitive to t1 ,

As can be seen easily from Eq

5) or (2

Note when ∂Ti /∂tj = 0,

a change in the acoustic slowness tj will not affect the value of the traveltime Ti ,

thus we can find no information on the value of tj from the measured value of Ti

When we want to visualize the sensitivity for a particular measurement,

we often display the row in a natural ordering,

one that corresponds to the spatial distribution of model parameters

we let Gi denote the ith row of G and display G2 0 0 0 as: 1 1 1

This display is convenient as it indicates that the second traveltime 0 0 0 measurement only depends on the slowness values in the second row


G4 can 1 0 0 be displayed as: 1 0 0 ,

Of course,

when the models become very large,

we will not display all of the numbers

Instead we will use a shading scheme that shows the strength of the sensitivity by the darkness of the grayscale

Solutions Suppose that the values of acoustic slowness are such that the exact measurement of one-way traveltime in each of the columns and rows is equal to 6 units (i

Ti = 6 for all i)


a homogeneous model for which the slowness of each block is 2 will satisfy this data exactly,

with all tj = 2 and all Ti = 6,


it is easy to see that ˆ = [2 m

when all entries of the data vector are equal to 6

A little examination shows that the following models also satisfy the data exactly: 1 2 3

8 2 −4

Nonuniqueness The null space of G is the set of all real,

nine-dimensional column vectors m such that Gm = 0

It is easy to verify that each of the following models represent vectors in the null space of G,

1 −1 0

1 −1 0

0 1 −1

0 −1 1

0 1 −1

0 −1 1

In fact,

the four vectors represented by these four models represent a basis for the null space of G,

so any vector in the null space of G can be written as a unique linear combination of these four vectors

If v is any vector in the null space of G and m is a vector of acoustic slownesses that satisfies Gm = d'where d'is the vector of measured traveltimes,

then the model m + v also satisfies the data because G(m + v) = Gm + Gv = d

we can add any linear combination of models (vectors) in the null space of G to a model that satisfies the traveltime data and obtain another model which also satisfies the data

This acoustic tomography problem has an infinite number of models that satisfy the data exactly for certain data

As there are fewer traveltime data than model variables,

We show next,

that for other values of the traveltime data,

there are no values of acoustic slowness that satisfy Eq

No solution As measurements are always noisy,

let us assume that because of the inaccuracy of the timing,

the following measurements were made: Tobs = [6


despite the fact that there are fewer data than model parameters,

there are no models that satisfy this data

T2 should be the sum of the slowness values in the second row,

and T3 should be the sum of the slowness values in the third row

Thus T1 + T2 + T3 = t1 + t2 + · · · + t9

But T4 is the sum of slowness values in column one,

and similarly for T5 and T6 so if there are values of the model parameters that satisfy these data,

we must also have T4 + T5 + T6 = t1 + t2 + · · · + t9

From these results,

it is clear that in order for a solution to exist,

we must have T1 + T2 + T3 = T4 + T5 + T6 ,

but when the data contain noise this is extremely unlikely

For the data of Eq

T1 + T2 + T3 = 17


in this case one seeks a solution that comes as close as possible to satisfying the data

A reasonable measure of the goodness of fit is the sum of the squared errors,

(dobs,j − dj (m))2 = (dobs − Gm)T (dobs − Gm)

we have introduced notation that will be used throughout this book


dobs,j denotes the j component of the vector of measured or observed data (traveltimes in this example),

and dj denotes the corresponding data that would be calculated (predicted) from the assumed theoretical model relationship (Eq

O(m) denotes an objective function to be minimized and is defined by the first equality of Eq

The second equality of Eq

One solution that has the minimum data 2

From the last equality of Eq

it is clear that if m is a least-squares solution then so is m + v where v is a solution in the null space of G

similar to the case where data are exact,

an infinite number of solutions satisfy the data equally well in the least-squares sense

Steady-state 1D flow in porous media Here,

the steady-state flow problem introduced in Section 1

It is assumed that the cross sectional area A,

Although many other characteristics of the porous medium are also unknown (e

we will treat the permeability field as the only unknown

Let d(x) = pe − p(x)

Figure 2

A porous medium with constant pressure pe at the left-hand end,

constant production rate q at the right-hand end,

and Nd measurements of pressure at various locations along the medium

Nd where the xi s'define Nd distinct locations in the interval [0,

L] at which pressure measurements are recorded

If the inverse problem under consideration involves linear flow in a reservoir,

the xi s'would correspond to points at which wells are located


the steady-state problem could also represent flow through a core with the xi s'representing locations of pressure transducers

The di s'now represent pressure drops,


the data di for this problem will be referred to simply as pressure data

For linear flow problems,

it is convenient to define the model variable,

as inverse permeability m(x) =

With this notation,

ξ )m(ξ ) dξ,

Note that G is only nonzero in the region between the constant pressure boundary location and the measurement location,

so the data (pressure drop) are only sensitive to the permeability in that region

changing the permeability beyond the measurement location would have no effect on the measurement

Assuming pressure data,

are recorded at x1 < x2 < · · · < xNd ,

Nd where Gi (ξ ) = G(xi ,

In a general sense,

solving this inverse problem means determining the set of functions that satisfy Eq

If only a single pressure drop measurement,

the problem is to solve qµ d1 = pe − p(L) = A

L m(x) dx,

Clearly there is not a unique function that satisfies Eq

and u(x) is any function such that L u(ξ ) dξ = 0,

then the function m(x) + u(x) also satisfies Eq

Discretization A discrete inverse problem for the estimation of permeability in steady-state flow can be formulated in more than one way

By approximating the integral in Eq

23) or (2

a discrete inverse problem can be obtained

A second procedure for obtaining a discrete inverse problem would be to discretize the differential equation,

write down a finite-difference scheme for the steady-state flow problem of Eqs


There is no guarantee that these two approaches are equivalent

Most work on petroleum reservoir characterization is focussed on the second approach,

when observed and predicted data correspond to production data,

the forward problem is represented by a reservoir simulator

we consider the general continuous inverse problem,

and use a numerical quadrature formula to obtain a discrete inverse problem

In many cases,

the best choice of a numerical integration procedure would be a Gauss–Legendre formula (see,

since our purpose is only illustrative,

a midpoint rectangular rule is applied here to perform numerical integration

Let M be a positive integer,

x1/2 = 0

Then let xj +1/2 = xj −1/2 + x

Figure 2

Discretization of the porous medium for integration using the midpoint rectangular method

In this figure,

m(xi ) is the value of m(x) in the middle of the interval that extends from xi−1/2 to xi+1/2

Using the preceding partitioning of the interval [0,

defining the constant β by β=

qµ x,

and applying the midpoint rectangular rule [9] for integration,

For simplicity in notation,

it is again assumed that pressure data are measured at xri +1/2 ,

where the ri s'are a subset of {i}M i=1 and r1 < r2 < · · · < rNd

The pressure change data at xri +1/2 is denoted by dobs,i with corresponding calculated data represented by di for i = 1,

With this notation,

applying the midpoint rectangular integration rule to Eq

Now let d'denote the vector of calculated data given by d'= [d1 ,

dNd ]T ,

and let dobs denote the corresponding vector of observed (measured) pressure drop data

Also let G = [gi,j ] be the Nd × M matrix with the entry in the ith row and j th column defined by gi,j = β,

Then defining mi = m(xi ) for all i,

4) and (3

Solutions of Eq

In replacing m(x) by its values at discrete points,

the model has been reparameterized

To approximate m(x) from its values at discrete points would require interpolation


which corresponds to defining one permeability for each “gridblock” in the interval [0,

For the problem under consideration,

the discrete inverse problem is specified as dobs = Gm + ε,

where dobs is the vector of observed “pressure drop data” and ε represents measurement errors

The objective is to characterize the set of vectors m that in some sense satisfy or are consistent with Eq

In the case where pressure drop data is available at xi+1/2 for i = 1,

Nd = M,

G is a square Nd × Nd matrix which can be written as   1 0

0 1 1

G is nonsingular and the unique solution of Eq

If the number of data is fewer than the number of model parameters (components of m),

Nd < M,

As the number of equations is fewer than the number of unknowns,

the system of equations is said to be underdetermined


if the number of equations is greater than the number of unknowns,

Nd > M,

the problem is said to be overdetermined

A detailed classification of underdetermined,

overdetermined and mixed determined problems is presented later

Underdetermined problem Suppose the interval [0,

L] is partitioned into five gridblocks of equal size and pressure drop data dobs,1 is observed at x7/2 and dobs,2 is observed at x11/2 = L

Then Eq

dobs,1 m1 + m2 + m3 = 3 3β and dobs,2 m1 + m2 + m3 + m4 + m5 =

Clearly the preceding two equations uniquely determine the average value of the first three model parameters and the average value of all five model parameters,

but do not uniquely determine the values of the individual mi s

There are,

an infinite number of vectors m that satisfy Eq

Integral equation Many inverse problems are naturally formulated as integral equations,

In Chapter 1,

we considered a boundary-value problem for one-dimensional,


Here we assume that the constant flow rate q,

viscosity µ and cross sectional area A are known exactly,

where the constant C is defined by C = (qµ)/A,

q is the volumetric flow rate,

µ is the viscosity,

and A is the cross sectional area to flow

If the function pe − p(x) is also known at a measurement location x0 ,

The inverse problem is then to find a solution,

or characterize the solutions,

to find a model m(x) = k(x) which satisfies Eq

Stated this way the integral equation,

and hence the inverse problem,

This particular problem is somewhat atypical as it is possible to reformulate the problem as a linear inverse problem by defining the model as m(x) = 1/k(x)

and rewrite the integral equation as x pe − p(x) = C

Although for the physical problem under consideration,

m(x) must be positive for k(x) = 1/m(x) to represent a plausible permeability field,

here it is convenient to define the inverse problem as the problem of finding piecewise continuous real functions,

L] which satisfy Eq

whereas the subset of M consisting of all positive real-valued functions defined on [0,

L] is not a vector space

) The operator G defined on the model space by x [Gm](x) = C

for any constants α and β and any two models m1 (x) and m2 (x)   G αm1 + βm2 = αGm1 + βGm2

by replacing the parameter k −1 (x) by m(x),

we have converted the original nonlinear inverse problem (nonlinear integral equation) to a linear inverse problem

Also note Gm is a continuous function of x

Defining d(x) = pe − p(x),

Note the similarity to Eq

If the pressure change across the core,

d(L) = pe − p(L) = p(0) − p(L),

the inverse problem becomes to find models m(x) such that d(L) = [Gm](L),

where the linear operator G is now defined by L [Gm](L) = C 0

qµ m(ξ ) dξ = A

L m(ξ ) dξ

Note that G defined by Eq

History matching in reservoir simulation A major inverse problem of interest to reservoir engineers is the estimation of rock property fields by history-matching production data

we introduce the complexities,

The finite-difference equations for one-dimensional single-phase flow can be obtained from the differential equation,

t) − qδ(x − x0 ) = C2 φ(x)ct A ,

0) = pin ,

where pin is the initial pressure which is assumed to be uniform

The constants C1 and C2 which appear in Eq

In SI units,

both constants are equal to unity

we use oil field units in which case,

A has units of ft2 and represents the cross sectional area to flow which we assume to be uniform

µ in centipoise represents the fluid viscosity which we assume to be constant

k(x) in millidarcies represents a heterogeneous permeability field

φ(x) represents a heterogeneous porosity field

ct is the total compressibility in psi−1 and is assumed to be constant

δ(x − x0 ),

is used to model a production well at x0 produced at a rate q

The units of the delta function are ft−1

For simplicity,

we partition the reservoir into N uniform gridblocks of width x in the x direction,

let xi denote the center of the ith gridblock,

denote the right- and left-hand boundaries of gridblock i

The grid system is shown in Fig

where the circles represent the gridblock centers

We assume that a single producing well is located in gridblock k

Integrating Eq

and using the fact that the resulting integral of the Dirac delta function is equal to 1 gives     k(x)A ∂p k(x)A ∂p − C1 − qδi,k C1 µ ∂x (xi+1/2 ,t) µ ∂x (xi−1/2 ,t)  ∂p(x,

t) dx φ(x)ct A = C2 ∂t xi−1/2   ∂p = φi ct Ax ,

Figure 2

One-dimensional grid system

N and t > 0

the last equality assumes φ(x) and the time derivative of pressure are constant on the interval (xi−1/2 ,

xi+1/2 ) and equal to their values at the gridblock center

If this assumption is invalid then Eq


δi,k denotes the Kronecker delta function defined by  δi,k =

Note that Eq

A sequence of discrete times is defined using a constant time step,

by tn = tn−1 + t for n = 0,

If we consider Eq

we obtain the following finite-difference equation:  n   n  − pin ki+1/2 A pi+1 ki−1/2 A pin − pi−1 C1 − C1 − qδi,k µ x µ x   pin − pin−1 (2

N − 1 and n = 1,

At i = 1 and i = N,

we impose the no flow boundary conditions of Eq

the following two equations: k3/2 A p2n − p1n − qδ1,k = φ1 ct Ax C1 µ x

and n kN−1/2 A pNn − pN−1 − qδN,k = φN ct Ax −C1 µ x

pNn − pNn−1 t

The initial condition is imposed on the finite-difference problem by requiring that pi0 = pin

In general,

t) of the initial boundary-value problem specified by Eqs



exactly because of the approximations we have used in deriving the finite-difference equations,

for example in approximating partial derivatives by difference quotients

The expectation is that the solution,

57)– (2

tn ) if t and x are sufficiently small

Given the cross sectional area to flow,

the initial pressure and the flow rate,

the forward problem is to solve the system of finite-difference equations (Eqs


At the first time step,

As is usually done in reservoir simulation,

we now assume that permeability is constant on each gridblock,

Using the standard harmonic average to relate the permeabilities at a gridblock boundary to the permeabilities of the two adjacent gridblocks gives ki+1/2 =

2ki ki+1 ,

N − 1

A typical history-matching problem would be to estimate the permeability and porosity fields given the value of flow rate,

ct and observations of gridblock pressure at a few locations

Multiple solutions Using a numerical reservoir simulator,

we have generated a solution of the system of finite-difference equations given by Eqs


Table 2

Reservoir data

Number of gridblocks,

N Cross sectional area,

φ Permeability,

ft Well location Well production rate,

RB/D System compressibility,

9 2500 0

Pressure drop,

Pressure drop for one-dimensional single-phase flow example

Figure 2

Two permeability fields which honor the wellbore pressure data

Note that the “true” reservoir is homogeneous

Also note that the reservoir is produced by a single well located in gridblock 9

The wellbore pressure at the well in gridblock 9 was obtained by using a Peaceman [11] type equation to relate gridblock and flowing bottomhole wellbore pressure,

A plot of the wellbore pressure drop,

versus time for twenty days of production is shown in Fig

Figure 2

Both solutions match the wellbore pressure data of Fig

This example illustrates clearly that the inverse problem of determining the gridblock porosities and permeabilities from flowing wellbore pressure will not have a unique solution when the data are inaccurate and measurements are obtained at only a few locations


Figure 2

Two porosity fields which honor the wellbore pressure data

we have plotted the estimated value of permeability on each of the nine gridblocks,

versus i where i represents the gridblock index

The solid curve represents the first permeability field estimate and the dashed curve represents a second permeability field estimate

Each solution assumes that permeability ki is constant on the interval (xi−1/2 ,

the two permeability fields are quite different,

even though both honor the pressure data equally well


we can also reproduce the transient wellbore pressure drop shown in Fig

which again illustrates the nonuniqueness of the inverse problem

Summary The examples in this chapter would all have been infinite dimensional in their parameterization,

if a natural parameterization had been chosen

It was often necessary,

to discretize the system in order to solve the forward problem

That is typical for systems that are described mathematically by differential equations

Even with a reduced parameterization,

the inverse solutions were not unique

When the measurements contain noise (which is always the case),

there may be no solutions to the problem that match the data exactly

In the acoustic tomography example,

there were no solutions that honored the noisy data exactly,

but infinitely many that approximately honored the data equally well

The relationships of the data to the model variables varied from very simple weighted integrals for the relationship between mass of the Earth (data) and the mass density distribution (model),

nonlinear relationship between pressure (data) and permeability (model) for transient flow in a heterogeneous porous medium

5 Summary

One of the difficult features of petroleum inverse problems is that the relationship between measurements (water-cut,

seismic amplitude) and variables to be estimated (permeability,

fault transmissibility) is difficult to compute

For those cases where the solutions are nonunique or no exact solutions exist,

it is useful to relax the definition of a solution

It will sometimes be useful to identify a “best estimate” after carefully specifying the meaning of best

In some cases it might be the estimate with the fewest features not required by the data,

In any case,

it is also useful to provide an estimate of uncertainty,

either in the parameters or in some function of the parameters

Estimation for linear inverse problems

In this chapter,

the notions of underdetermined problems,

methods of constructing estimates,

sensitivities and resolution are explored for linear finitedimensional inverse problems

In petroleum reservoir characterization,

neither permeabilitie