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Petroleum

Reservoir

Characterization

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

- probability,
- and spatial statistics

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

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

- are discussed

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

- an extensive bibliography,
- and a selection of exercises

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

- cambridge

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,

- reservoir characterization,
- uncertainty quantification,
- and optimization

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

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,

- uncertainty forecasting,
- production optimization,
- and reservoir management

Oliver Albert C

Reynolds Ning Liu

CAMBRIDGE UNIVERSITY PRESS

Melbourne,

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

- cambridge

org Information on this title: www

- cambridge
- org/9780521881517 © D

Oliver,

Reynolds,

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

- hardback

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,

- or will remain,
- accurate or appropriate

- his wife and partner in life

Examples of inverse problems 2

- page xi

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

- 6 6 7 11 18 22

Estimation for linear inverse problems

- 25 33 49 55

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

Contents

Descriptive geostatistics 5

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

- 86 86 86 91 97 110
- 112 Production data Logs and core data Seismic data
- 112 119 121

- 127 131 137 141

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

- 143 146 149 157 163 167 172 180 192

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

200 206

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

- 210 217 223 228 232 249 254 261

- 270 274 286 301 319 334 337 340

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

- 347 348 350 353 355 358 359

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

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,

- in the past,
- been called history matching

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

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

and the data always contain measurement errors

- insufficient,

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

- permeability,
- and temperature),

bounds (porosity between 0 and 1),

- or smoothness

- it must be done efficiently

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

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,

- data aquisition,
- and reservoir management

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,

- insufficient,
- and inconsistent data

Preface

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

- inner products,
- functionals,

existence and uniqueness (for linear problems),

- resolution and inference,

and functional differentiation

He does not,

- however,

get very deeply into nonlinear problems or stochastic approaches

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

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

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

- therefore,

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

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

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

Efficient incorporation of production-type data (e

- pressure,
- concentration,
- water-oil ratio,

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

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

Introduction

the locations and transmissivity of faults,

- the porosity,
- the permeability,

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

describing the system to observable variables or data,

- is denoted g(m) = d

- outcomes can be predicted,

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

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

- seismic)

In the inverse problem,

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

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

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

- 1 Introduction

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,

- 1) dx µ dx for 0 < x < L,
- and dp qµ ,
- =− dx x=L k(L)A
- p(0) = pe

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

µ (viscosity in cp),

- q (flow rate in cm3 /s),

and pressure pe (atm) are assumed to be constant

The length of the system in cm is represented by L

- the model parameters,

which are assumed to be known,

- and k(x)

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),

- for example,
- by the boundary conditions,

which are assumed to be known exactly

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

we might formally write the relationship between pressure,

- at a location,

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

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,

- (b) the solution is unique,

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

- for the
- 2 The inverse problem
- boundary-value problem of Eqs

1)–(1

the problem data refers to pe ,

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

- 4) is not defined,

the boundaryvalue problem of Eqs

1)–(1

- 3) does not have a solution for q > 0

However,

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

- it is said to be ill 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

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

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

- where here,

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

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

- fluid viscosity,

cross sectional area A and length L

- however,

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

- 1) numerically,

we might discretize the permeability function,

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

and will simply be referred to as observed data

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

or in the case where the system represents a core,

at locations where pressure transducers are situated

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

- 1 Introduction
- inverse problem,

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

1)–(1

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

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

an infinite number of equally good solutions exist

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

1/k(x),

and L) from pressure measurements

measured pressure data will not,

- in general,
- be exact

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

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

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

- ? If so,
- which one

? Do we want to determine several solutions

- ? If so,
- and which ones
- ? As readers will see,

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

- in a practical sense,

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

- 2 The inverse problem

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

- and seismic data

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

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

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

Construct statistics (e

- histogram,

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

- cumulative oil production,
- water–oil ratio,
- breakthrough time)

- steps 2 and 3 are both vital,
- albeit difficult,

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

- uniqueness,
- and sensitivity

but several are difficult to solve

Density of the Earth The mass,

- and moment of inertia,

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,

- 1) M = 4π 0

8π I= 3

- r 4 ρ(r) dr,

where a is the radius of the Earth

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

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

- and the model variable,
- m = ρ(r),
- is the density
- (Throughout this book,

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

- 4) d= 8π 4 m dr
- r 0 3 Note that,
- in this example,

the dimension of the model to be estimated is infinite,

while the dimension of the data space is just 2

- 2 Acoustic tomography

and the six measurement locations for total traveltime,

- across the array
- bound on the density

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

- nonuniqueness,

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

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

- 1) and we only consider paths that are orthogonal to the block boundaries so that refraction can be ignored and the paths remain straight

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

- then T = tD

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

- 1) block is t1 ,
- the slowness of the (1,
- 2) block is t2 ,
- and the slowness of the (1,
- 3) block is t3 ,
- then T1 ,

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

- is given by T1 = t1 + t2 + t3

Similar relations hold for the other rows and columns

- 2 Examples of inverse problems

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

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

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

- 6) t1 t2 T1 1 1 1 0 0 0 0 0 0 t T 0 0 0 1 1 1 0 0 0 3 2 t 4 T 0 0 0 0 0 0 1 1 1 3
- 6) t5 = T4 1 0 0 1 0 0 1 0 0 t6 T5 0 1 0 0 1 0 0 1 0 t7 0 0 1 0 0 1 0 0 1 T6 t8 t9 With the notation commonly used in this book,
- 6) is written as d'= Gm,
- where the data,

is the vector of traveltime measurements,

- d'= [T1

- the model,

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

- and the sensitivity matrix,

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

- 2 Acoustic tomography

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

Each row has nine elements in this example,

- one for each model variable

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

- for example,

the fourth measurement is only sensitive to t1 ,

As can be seen easily from Eq

5) or (2

- ∂T4 /∂tj = 1 for j = 1,
- 7 and ∂T4 /∂tj = 0 otherwise

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

- see Fig

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

- when compared to Fig
- 1 shows clearly that 1 0 0 the fourth traveltime measurement is only sensitive to the slowness values of the first column of blocks

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,

- 6) is satisfied

it is easy to see that ˆ = [2 m

- is a solution of Eq
- for any real constant b,

when all entries of the data vector are equal to 6

- −2 −2 10

8 2 −4

- 2+a 2 2−a
- 2−a 2 2+a
- 2+b 2−b 2
- 2−b 2+b 2
- 2 Examples of inverse problems

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
- −1 1 0

1 −1 0

0 1 −1

0 −1 1

0 1 −1

0 −1 1

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,

- this is not surprising

- however,

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

- 5) indicates that T1 should be the sum of the slowness values in the first row,

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

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

- 3 Steady-state 1D flow in porous media

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

- 91 and T4 + T5 + T6 = 17
- so that with these data,
- 6) has no solution

Generally,

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

(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

- 7) in this example) for a given model variable,

- 16) follows from standard matrix vector algebra

- 044 mismatch is 2
- or equivalently,
- 944 ˆ = [2

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

the steady-state flow problem introduced in Section 1

- 1 is formulated as a linear inverse problem

It is assumed that the cross sectional area A,

- the viscosity µ,
- the flow rate q,
- and the end pressure pe in Eq
- 4) are known exactly

- mineralogy,
- grain size,
- porosity),

we will treat the permeability field as the only unknown

Let d(x) = pe − p(x)

- and di = d(xi ),
- 2 Examples of inverse problems

constant production rate q at the right-hand end,

and Nd measurements of pressure at various locations along the medium

- for i = 1,

L] at which pressure measurements are recorded

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

However,

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,

- or more generally,
- pressure changes

- for simplicity,

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) =

- 4) can be written as L d(x) =

- where G(x,
- for ξ ≤ x,
- for ξ > x

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

- di = d(xi ),

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

- 21) is replaced by the inverse problem L'qµ xi m(ξ ) dξ = Gi (ξ )m(ξ ) dξ,
- 23) di = A 0 0 for i = 1,

In a general sense,

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

- 23) given the values of the di s
- 3 Steady-state 1D flow in porous media

- d1 = d(L) is recorded at the right-hand end (x = L) of the system,

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

- for m(x)

- 25) since if m(x) satisfies this equation,

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

23) or (2

- 25) using numerical quadrature,

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

1)–(1

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

- however,

we consider the general continuous inverse problem,

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

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

- for example,
- chapter 18 of Press et al

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

- and x =

- and xj =
- xj −1/2 + xj +1/2 ,
- 2 Examples of inverse problems
- m(x1 ) m(x2 )
- m(x5 ) m(x4 ) m(x3 )
- x11/2

Figure 2

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

- for j = 1,

defining the constant β by β=

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

- 25) can be approximated by d1 = β

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,

applying the midpoint rectangular integration rule to Eq

- 23) (with i replaced by ri + 1/2) gives the approximation di = β
- m(xj ),
- for i = 1,

dNd ]T ,

- 3 Steady-state 1D flow in porous media

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,

- for j ≤ ri and gi,j = 0,
- for j > ri

- and using the notation of Eqs

4) and (3

- 33) can be written as d'= Gm,
- where G is an Nd × M matrix

- 23) are functions and as such represent elements of an infinite-dimensional linear space L2 [0,
- whereas,
- “solutions” of Eq
- 37) are vectors and are elements of a finite-dimensional linear space

the model has been reparameterized

- one could set m(x) = mi ,
- for xi−1/2 < xi+1/2 ,
- ki = 1/mi ,

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

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,

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

0 1 1

- 0 G = β
- 1 Note that G is a lower triangular matrix with all diagonal elements nonzero

- 38) is m = G−1 dobs

- then Eq
- 38) represents Nd equations in M unknowns

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

the system of equations is said to be underdetermined

Similarly,

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

- 2 Examples of inverse problems

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

- 38) becomes m1 m 2 dobs,1 1 1 1 0 0 =β (2
- 40) m3 ,
- 1 1 1 1 1 dobs,2 m4 m5 or,
- equivalently,

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

- in fact,

an infinite number of vectors m that satisfy Eq

- instead of matrix equations

In Chapter 1,

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

- single-phase,
- steady-state flow
- see Eqs

1)–(1

Here we assume that the constant flow rate q,

viscosity µ and cross sectional area A are known exactly,

- and rewrite Eq
- 4) as x pe − p(x) = C

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

q is the volumetric flow rate,

and A is the cross sectional area to flow

- then Eq
- 4) represents a Fredholm integral equation of the first kind [10]

or characterize the solutions,

- of the integral equation,

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

and hence the inverse problem,

- is nonlinear

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)

- 3 Steady-state 1D flow in porous media

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

- m(ξ ) dξ

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,

- defined on [0,

L] which satisfy Eq

- 43) and to define the model space M as the set of all positive piecewise continuous functions defined on [0,
- (M is a real vector space,

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

- m(ξ ) dξ,
- is a linear operator,

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

- 45) can be written as d(x) = [Gm](x)

Note the similarity to Eq

If the pressure change across the core,

d(L) = pe − p(L) = p(0) − p(L),

- is measured,

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

- 51) maps functions m(x) in the model space into the set of real numbers
- 2 Examples of inverse problems

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,

- using a single-phase,
- flow problem

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

- ∂p(x,
- t) ∂ k(x)A ∂p(x,

t) − qδ(x − x0 ) = C2 φ(x)ct A ,

- 52) C1 ∂x µ ∂x ∂t for 0 < x < L'and t > 0,
- ∂p(L,
- t) ∂p(0,
- t) = = 0,
- for all t > 0 (2
- 53) ∂x ∂x and p(x,

0) = pin ,

- for all t > 0,

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

- 52) depend on the system of units

In SI units,

both constants are equal to unity

we use oil field units in which case,

- 127 × 10−3 and C2 = 5
- 53) represents no flow boundaries at the ends of the system

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

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

- the Dirac delta function,

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

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

let xi denote the center of the ith gridblock,

- let xi+1/2 and xi−1/2 ,
- respectively,

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

where the circles represent the gridblock centers

Integrating Eq

- 52) with respect to x over the ith gridblock,
- from xi−1/2 to xi+1/2 ,

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 ,

- ∂t (xi ,t) xi+1/2
- 4 History matching in reservoir simulation

Figure 2

One-dimensional grid system

- for i = 1,

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

- 55) represents an approximation of Eq

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

- 0 for k = i,
- 1 for k = i

- 55) applies at any value of time

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

by tn = tn−1 + t for n = 0,

- where t0 = 0

If we consider Eq

- 55) at any t = tn > 0 and use standard Taylor series approximations for the spatial and time derivatives,

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

- 57) = φi ct Ax t for i = 2,

At i = 1 and i = N,

- respectively,

we impose the no flow boundary conditions of Eq

- 53) and obtain instead of Eq

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

- p1n − p1n−1 t

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

- 2 Examples of inverse problems
- for n = 1,

- the solution p(x,

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

52)–(2

- 54) will not satisfy the finite-difference system,

57)–(2

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

for example in approximating partial derivatives by difference quotients

- pin (i = 1,

57)– (2

- 60) will be close to p(xi ,

tn ) if t and x are sufficiently small

- rock and fluid properties,

the initial pressure and the flow rate,

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

57)–(2

- 59)) for pin ,
- given pin−1 ,

At the first time step,

- n = 1 and pin−1 = pi0 = pin

we now assume that permeability is constant on each gridblock,

- xi−1/2 < x < xi+1/2 ,
- with k(x) = ki for i = 1,

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 ,

- ki + ki+1
- for i = 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

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

57)–(2

- 59) for parameter values given in Table 2

Number of gridblocks,

- ft2 Porosity,

ft Well location Well production rate,

- psi−1 Fluid viscosity,
- cp Initial pressure,

9 2500 0

- 25 150 500 i=9 250 10−5 0
- 4 History matching in reservoir simulation

Pressure drop,

- 240 220 200 180 160 140 120 100 80 60 0
- days Figure 2

Pressure drop for one-dimensional single-phase flow example

- 104 Permeability,
- 2 4 6 8 Gridblock Index,

Two permeability fields which honor the wellbore pressure data

Note that the “true” reservoir is homogeneous

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,

- pwf (t)

- p = pin − pwf ,

versus time for twenty days of production is shown in Fig

- 6 shows two different permeability fields that were obtained as solutions to the history-matching problem,
- assuming that φ = 0
- 25 in all gridblocks

Both solutions match the wellbore pressure data of Fig

- 5 to within 0

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

- 2 Examples of inverse problems

Porosity

- 2 4 6 8 Gridblock Index,

Figure 2

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

- xi+1/2 )

the two permeability fields are quite different,

even though both honor the pressure data equally well

Interestingly,

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

- 5 to within 0
- 01 psi using k = 150 md in all gridblocks and either of the porosity fields 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

- however,

to discretize the system in order to solve the forward problem

- however,

the inverse solutions were not unique

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),

- to a highly complex,

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,

- pressure,

seismic amplitude) and variables to be estimated (permeability,

- porosity,

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,

- or the smoothest estimate

it is also useful to provide an estimate of uncertainty,

either in the parameters or in some function of the parameters

the notions of underdetermined problems,

- overdetermined problems,
- mixed determined problems,
- the null space,
- the generalized inverse,

methods of constructing estimates,

sensitivities and resolution are explored for linear finitedimensional inverse problems

neither permeabilitie

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