History matching problem in reservoir engineering

using the propagation back-propagation method

Pedro Gonzalez-Rodrıguez, Manuel Kindelan, Miguel Moscoso,

Oliver Dorn

Mathematics Department, Universidad Carlos III de Madrid, Avenida de laUniversidad 30, Leganes 28911, Spain

Abstract.This article describes the application of the adjoint method to the history matching

problem in reservoir engineering. The history matching problem consists in adjusting aset of parameters, in this case the permeability distribution, in order to match the dataobtained with the simulator to the actual production data in the reservoir. Severalnumerical experiments are presented which show that our method is able to reconstructthe unknown permeability distribution in a reliable and efficient way from syntheticdata provided by an independent numerical forward modelling code. An efficient andflexible regularization scheme is introduced as well, which stabilizes the inversion andenables the reservoir engineer to incorporate certain types of prior information intothe final result.

Submitted to: Inverse Problems

2

1. Introduction

History matching techniques are used in reservoir modelling for estimating unknown

properties of a reservoir, such as porosity and permeability, from measured data. The

data are usually taken at the production wells and might consist of pressure or flow data.

Proper characterization of the reservoir heterogeneity is a crucial aspect of any optimal

reservoir management strategy. It helps to better understand the reservoir behavior so

that its performance can be predicted and controlled with higher reliability.

History matching can be carried out either manually (by a trial-and-error process),

or automatically by computing a set of parameter values so as to minimize a prescribed

cost function. Generally, the reservoir simulator uses a spatial grid, and the history

matching process is used to determine the permeability and/or porosity in each grid cell.

This can be done manually when a small number of parameters have to be retrieved

from the observed production data. However, for large scale models, where more than

50,000 parameter values are unknown, this procedure is infeasible. Automatic strategies

for history matching were initially based on the perturbation method. This method

computes the single grid block sensitivity coefficients in order to evaluate the change

of an objective function under small perturbations of the permeability of each cell [29].

A direct method of obtaining these sensitivity coefficients requires a number of forward

solutions per iteration equal to the number of reservoir parameters to be determined.

In consequence, this approach is very time consuming for large geophysical models.

A major breakthrough was achieved by Jacquard and Jain [28]. They used an

electric circuit analog of a reservoir, to compute the sensitivity coefficients for pressure

to changes in the permeability map. The method was computationally efficient since

it required only one simulation per observation point. It was based on a linear, single

phase, reservoir model and it was later rigorously derived in terms of the diffusion

equation [9]. The resulting optimization problem was solved by a linear programming

procedure which computed the reservoir parameters including constraints.

The work by Chavent et al represented another major breakthrough in automatic

history matching [12]. They applied an adjoint technique from optimal-control theory to

this application. For the same linear, single phase, reservoir model as used in [28] they

derived a corresponding adjoint equation, and computed the sensitivity coefficients by

just solving two PDEs (the equation for the reservoir model, and the adjoint equation)

and some integrals involving the pressure and the adjoint variable.

Secondary oil recovery techniques involve the simultaneous flow of up to three fluid

phases. The water, oil and gas flows are governed by a set of coupled nonlinear partial

differential equations and therefore the derivations in [28, 9] had to be extended. Several

researchers have paid attenion to this and other coupled inverse problems [52, 8, 21, 34].

The combination of spatially dependent parameters and nonlinearity results in a very

challenging inverse problem. Wasserman et al. [54] modified the original work by

Chavent et al. [12] to treat multiphase effects approximately with a ’pseudo’ single-

phase model. They applied the adjoint method directly to the set of ordinary differential

3

equations resulting from the finite-difference discretized model. The adjoint method for

the case of a two-phase, non linear, reservoir model has been derived in [57] starting

from the discretized equations. Based on a set of adjoint operation rules, Sun and

Yeh [47] introduced a general procedure for deriving the adjoint state equations in a

straightforward way for general coupled inverse problems. As examples, they applied

their method to problems of saltwater intrusion and two-phase flow.

There are other more recent techniques which have been applied successfully in

reservoir engineering. We only want to mention two approaches in the following.

The first one is an alternative streamline approach which has been proposed in

[30, 49, 53]. A method is introduced in that work for estimating the sensitivity

coefficients semi-analytically with one single forward simulation, which significantly

reduces the numerical cost. This gives rise to a large and sparse, but ill-conditioned,

linear system. With appropriate regularization this system can be solved efficiently

[2, 27].

As second approach we mention the use of geostatistical methods [7, 23, 31] as

well as genetic algorithms [26, 42] in this application. These methods generate different

realizations of a geological model which is derived from well-log and seismic data, and

select the realization that best matches the pressure or production history. Although

accurate and fast simulators are available, the large number of realizations to be tested

usually makes this procedure computationally quite demanding.

Adjoint techniques, which we are focusing on in this paper, are particularly useful

in large scale inverse problems where relatively few independent experiments can be

performed for gathering data but many parameters need to be reconstructed. Since

typically only one experiment is performed in history matching due to the simultaneous

production process, the adjoint technique is therefore much faster in this application.

We mention that adjoint techniques have been applied recently with great success also

in other applications of medical and geophysical imaging. See for example the articles

[35, 3, 17, 51] dealing with examples from ultrasound tomography, optical tomography,

and microwave imaging with medical applications, or [25, 18, 19] describing applications

in geophysical monitoring or prospecting. More general information regarding adjoint

techniques for solving large scale inverse problems can be found for example in [37, 50].

One important difference between the history matching problem and these

applications is that in the history matching problem the experimental setup cannot

easily be changed in order to obtain independent data. As already mentioned,

typically only one field experiment is avalaible due to the production process. As a

consequence, the information content of the data is quite low, and the history matching

problem is highly ill-posed. Therefore, when developing reconstruction schemes for this

application, an important component will be the incorporation of efficient and flexible

regularization tools, as well as the capability to incorporate as much ’prior information’

as possible into the search for a good candidate for the parameter distribution. Prior

information can for example consist of results which have been obtained with different

physical experiments like seismic imaging, core analysis, well-logs, or general geological

4

information. Therefore, we will put a strong emphasis on the regularization tools which

we want to use in our reconstruction method. Good overviews of general regularization

techniques for inverse problems can be found for example in [20, 24, 36, 50].

In this paper, we derive the adjoint formulation as needed here for a two-phase

reservoir, starting directly from the partial differential equations of the nonlinear

reservoir model. The resulting method is a propagation back-propagation algorithm

for history matching which is used to estimate the permeability distribution inside the

reservoir based on the water flow extracted at a small number of production wells.

Starting from a homogeneous permeability distribution, we calculate the difference

between the computed and the given production data. These residuals are numerically

back-propagated into the reservoir by solving the corresponding adjoint equation, and

the correction to the guess is directly calculated from the result. The process is repeated

iteratively until a convergence criterion is satisfied.

As mentioned, a particular emphasis is put on the derivation and discussion of

flexible and useful ’regularization schemes’ for this method, which can be used for

stabilizing the reconstruction process and, in addition, for incorporating certain types

of prior information into the reconstruction. In contrary to the more standard approach

of Tikhonov-Phillips regularization, we will not change our cost functional for the

derivation, but instead will restrict the search for a minimum to a smaller subspace

of functions with certain properties. This has the advantage that, upon convergence,

only the data misfit has been minimized, which is our primary objective. For more

details we refer to the following sections 4 and 6.

For describing the flow dynamics in the reservoir (our basic ’forward model’), we

use a simplified Black-Oil model [48]. In order to avoid the so-called inverse crime, the

data is generated with a streamline method, while during the reconstruction process we

use an independent IMPES method to solve the equations (see Appendix B for details

of these two different forward modelling codes). Here our approach differs slightly from

the more standard approach of simply adding statistically random noise to the data

which have been created with the same simulator. We believe that, by using a different

simulator for creating data, we can incorporate some component of ’systematic noise’

which might render the data more realistic. Purely random noise often can easily be

removed (at least partially) by a simple filtering of the data, whereas systematic noise

might be harder to cope with. The noise content of our data using this strategy is

typically about 3% .

The paper is organized as follows. In section 2 we give a short introduction into our

setup of the reservoir characterization problem. In section 3 a mathematical description

of the underlying flow equations is given. Section 4 introduces the inverse problem and

gives a theoretical derivation of the basic algorithm which we use for the inversion. The

algorithm itself is outlined in section 5. In section 6 we propose an efficient regularization

scheme for the method. Various numerical experiments are presented in section 7 which

demonstrate the performance of the reconstruction method in different situations. In

section 8 we draw some conclusions and indicate some directions for future research.

5

0 200 400 6000

100

200

300

400

500

600mD

600

800

1000

1200

1400

1600

1800

(p)

x

x (p) 1

x

2

(p) 3 4

x (p)

x

1 (i)

0.61.2

0.61.2

0.61.2

1.22.4

0 100 200 300

51015

Time (Days)

Qw

× 1

06 (s−1

)

Producer 1

Producer 2

Producer 3

Producer 4

Injector 1

Figure 1. (a) Permeability distribution: 5-spot example; (b) extracted water flowsat the producers (shown in figure (a) with x’s) and injected water flow at the injector(shown in figure (a) with a circle).

In Appendix A a proof of Theorem 1 of the paper is given. In Appendix B we briefly

describe the two basic forward modelling codes (The IMPES method and the streamline

method).

2. Description of the problem

Our basic flow model (the ’Black-Oil model’) consists of partial differential equations

which govern the unsteady flow of different fluid phases in the reservoir medium. The

petroleum engineer often uses this model for understanding the dynamics of petroleum

reservoirs and petroleum production in order to design an efficient operational strategy.

We consider here the case of ’secondary recovery’ where water is injected through

several injection wells conveniently located in order to enhance oil production. The

behavior of the reservoir is usually influenced by many factors (permeability, porosity,

relative permeability, ...) which are never known precisely. Therefore, the engineer uses

the best values available and compares the results from the simulator with the field-

recorded histories at the wells. Generally, this history matching will show discrepancies

which the engineer tries to minimize by modifying one or several of the parameters

which define the reservoir (permeability, porosity, ...). Once the simulator matches the

production data, it is used to predict its future behavior and to design alternative plans

of operation of the reservoir. It should be pointed out that there is not a unique set of

parameters to match production data, so that a perfect match does not mean that the

reservoir is correctly characterized. In fact, if after a perfect match the simulator is used

to predict future behavior, the actual performance may differ from the predicted one so

that it is necessary to monitor periodically the predicted versus the actual performance

in order to update reservoir characterization.

We use inverse problems techniques to optimize this history matching process. For

simplicity, we consider the two-dimensional case and we select the distribution of rock

6

permeability throughout the reservoir as the parameter that we try to adjust. Therefore,

we assume that all the other parameters needed to define the reservoir behavior are

known.

The direct problem refers to the resolution of the equations describing the flow

within the reservoir assuming that the permeability distribution is known. For instance,

the left side of figure 1 shows a five-spot layout with an injector well (o) in the center

(location ~x(i)1 ) and four production wells (x) at the corners of a two-dimensional reservoir

(locations ~x(p)j , j = 1 . . . 4, being j = 1 the well in the upper left corner and numbered in

the clockwise direction). Also shown is the real permeability distribution in milli-Darcys

(mD). The water injected at the injection well displaces the oil in the reservoir towards

the production wells. Time resolution of the flow equations provides the time evolution

of pressure and flow at each point of the reservoir. Of particular interest is the oil and

water flow rate at each production well. The right hand side of figure 1 shows the time

history of water flow rate (Qw) at each well obtained by solving the direct problem.

Notice that water arrival occurs first at well four since it is surrounded by a region of

high permeability.

In the inverse problem we assume that the water flow rate at each well is known

but the permeability distribution is unknown. We will start with an initial permeability

guess (typically some constant distribution) and will iteratively modify the permeability

distribution until the actual water production rate at each well is matched by the

simulator.

3. The mathematical model

In order to model the time evolution of the flow in a reservoir we use a simplification

of the Black-oil model [48]. We consider two incompressible phases (water and oil) in a

porous medium where the effects of gravity and capillary pressure are neglected. Then,

the governing equations for the multiphase incompressible flow in a reservoir Ω ⊂ Rn

(n=2,3) can be written as

−∇ ·[T∇p

]= Q in Ω× [0, tf ] (1)

φ∂Sw

∂t−∇ · [Tw∇p] = Qw in Ω× [0, tf ] (2)

where p(~x, t) and Sw(~x, t) are the unknowns of the problem which represent the pressure

and the water saturation at position ~x and time t respectively. The water saturation

Sw measures the volume fraction of water. φ(~x) is the porosity, and T and Tw are the

transmissibilities, which are known functions which depend linearly on the permeability

K, the parameter to be reconstructed, and nonlinearly on Sw,

Tw = K(~x)Krw(Sw)

µw

; To = K(~x)Kro(Sw)

µo

; T = Tw + To . (3)

In (3), Krw(Sw), Kro(Sw), µw and µo denote the relative permeabilities and the

viscosities of each phase, respectively. Hereafter, the subindex ‘w’ stands for ‘water’,

7

while the subindex ‘o’ stands for ‘oil’. Q(~x, t) and Qw(~x, t) define the total flow and the

water flow at the wells, respectively. They are given by

Q = c T

Ni∑j=1

(p(i)wbj

− p)δ(~x− ~x(i)j ) + c T

Np∑j=1

(p(p)wbj

− p)δ(~x− ~x(p)j ) (4)

Qw = c T

Ni∑j=1

(p(i)wbj

− p)δ(~x− ~x(i)j ) + c Tw

Np∑j=1

(p(p)wbj

− p)δ(~x− ~x(p)j ) (5)

where ~x(i)j , j = 1, . . . , Ni, denote the locations of the Ni injector wells, ~x

(p)j , j = 1, . . . , Np,

denote the locations of the Np production wells, and p(i)wbj

, p(p)wbj

are the imposed well bore

pressures at the Ni injector wells and at the Np production wells, respectively. Here, c

is a constant that depends on the well model [14]. Since p(i)wbj

(p(p)wbj

) are larger (smaller)

than the reservoir pressure at the injector (production) wells, Q and Qw are positive

(negative) at the injector (production) wells.

Equation (2) is the conservation law for water in a porous medium and equation

(1) is obtained by combining the conservation laws for water and oil in order to

eliminate the time derivative term. It is assumed that the flow obeys Darcy’s law

(~ul(~x, t) = −K(~x)Krl(Sw)µl

∇p(~x, t), l = w, o) which defines the velocity of each phase in

the medium. Equations (1) and (2) are solved with the following initial and boundary

conditions:

Sw(~x, 0) = S0w(~x) in Ω , (6)

p(~x, 0) = p0(~x) in Ω , (7)

∇p · ~ν = 0 on ∂Ω , (8)

where ~ν is the outward unit normal to ∂Ω. The boundary condition (8) implies no flux

across the boundary.

Equations (1)-(8) define the direct problem for the dynamic production history at

the extraction wells. The properties of the porous media are given by K(~x) and φ(~x).

The properties of the fluids are defined by µw, µo, Krw(Sw), and Kro(Sw). The well bore

pressures p(i,p)wbj

are known functions of time at the well’s positions.

4. The inverse problem

4.1. A propagation back-propagation inversion method

In reservoir characterization, typically, one tries to estimate the permeability

distribution by matching production data. The porosity distribution and the relative

permeabilities are usually assumed known from core analysis [49, 53].

For the mathematical analysis of the above described problem, we need to specify

some function spaces that will be used in the formulation of the problem. These function

spaces will also become important when deriving our regularization tools. We will use

suitably chosen Hilbert spaces throughout our derivation. We will denote the space

8

of permeability distributions K by P , which is defined by P = L2(Ω), equipped with

the usual L2 inner product. (In section 6 we will introduce an additional space P

for the permeability distributions.) The set of measurement locations (’well-locations’)

will be denoted by Ω+ := ~x(p)1 , ~x

(p)2 , . . . , ~x

(p)Np. At each of these positions, the water

flow is measured during a time 0 ≤ t ≤ tf , such that the data space D is given by

D = (L2([0, tf ]))Np . Our forward operator M is given as

M : P −→ D

M [K] = Qw[K]|Ω+×[0,tf ] (9)

where Qw is obtained by solving the direct problem for a given permeability distribution

K (Eqs. (1)-(8)). For some guess K of the permeability, and given the measured data

G (water flow rate) at the production wells, we can furthermore define the residual

operator R : P −→ D by

R[K] = M [K]− G . (10)

Equation (10) describes the mismatch between these physically measured data and the

data corresponding to a guess K.

In the inverse problem, we ideally want to find a permeability distribution K in P

such that

R[K] = 0 . (11)

This equation has a solution in the situation where the data G are in the range of M .

Using real data, we cannot be sure whether this is the case. Therefore, we generalize

our criterion for a solution. Defining the least squares cost functional

J (K) =1

2‖R(K)‖2

L2, (12)

we are searching for a minimizer of this cost functional, which can be zero in the situation

where G ∈ range(M). Otherwise, it will be a positive number. A standard method for

finding a minimizer of the cost functional (12) is to start a search with some initial

guess K(0), and to find descent directions of (12) in each step of an iterative scheme.

Popular choices for descent directions are for example the gradient direction, conjugate

gradient directions, Newton- or Quasi-Newton directions (see for example [15, 38, 50]

for details). We will in the following derive update directions for minimizing (12) using

the formulation (11). These update directions will have the useful property that they

can easily be generalized in order to incorporate efficient regularization schemes in our

algorithm.

In order to find an ’update’ (or ’correction’) δK for our permeability K we linearize

the nonlinear operator R (assuming that this linearized operator R′[K] exists and is

well-defined) and write

R[K + δK] = R[K] + R′[K]δK + O(||δK||2). (13)

The linearized operator R′[K] is often called the Frechet derivative of R at K. (See for

example [16] and references therein for some formal derivations of Frechet derivatives

9

in different applications). It is also closely related to the ’sensitivity functions’ of the

parameter profile with respect to the data. Using (13) we want to look for a correction

δK such that R[K+δK] = 0. Neglecting terms of order O(||δK||2) in (13), this amounts

to solving

R′[K]δK = −R[K] . (14)

Certainly, due to the ill-posedness of our problem, this equation needs to be handled

with care. Treated as an ill-posed linear inverse problem, a classical solution of (14) will

be the minimum-norm solution

δKMN = −R′[K]∗ (R′[K]R′[K]∗)−1 R[K], (15)

where R′[K]∗ is the adjoint operator of R′[K] with respect to our chosen spaces P and

D [36]. In applications with very few data, this form has the useful property that it

avoids contributions in the solution which are in the (often non-empty) null-space of the

(linearized) forward operator R′[K]. Using (13) it can be verified by direct calculation

that

J (K + ωδKMN) = J (K) − ω‖R(K)‖2D + O(‖δKMN‖2

P ) (16)

such that (15) also is a descent direction of the least squares cost functional (12).

In our application the operator C = (R′[K]R′[K]∗)−1 is very ill-conditioned,

such that a regularized version needs to be used. This can be for example C =

(R′[K]R′[K]∗ + λI)−1 where λ is some regularization parameter and I is the identity

operator. Unfortunately, in practice both, C as well as C, are very expensive to calculate

and to apply to the residuals R. Typically, a direct calculation of the operator C would

require us to solve as many forward and adjoint problems as we have independent data

values. Iterative schemes (like the gradient or conjugate gradient method) for applying

this operator to a given vector are possible as well, but usually converge only slowly. In

addition, we also have the possibility to just use a rough approximation of it in each

step of the inversion, which is much easier and faster to do and might yield good results.

We will investigate these possibilities in our future research.

When using a very large regularization parameter λ, the contribution of

R′[K]R′[K]∗ can be neglected and we end up with essentially (i.e. up to the scaling

factor λ−1) calculating

δK = −R′[K]∗R(K). (17)

For this update direction we have

J (K + ωδK) = J (K) − ω‖R′[K]∗R(K)‖2P + O(‖δK‖2

P ) (18)

such that it is also a descent direction for (12).

We will use this update direction throughout the paper, with some important

modifications described below. Our goal will be to derive and test efficient schemes for

applying the adjoint linearized residual operator to given data (the basic propagation-

backpropagation scheme), and moreover to derive and evaluate a new regularization

10

scheme for this backpropagation technique. We will emphasize, however, that the

propagation-backpropagation scheme as well as the regularization scheme will also be

applicable directly to the update directions given in (15), which will be treated in our

future work.

A standard method for deriving regularization schemes is to explicitly try to

minimize a cost functional which incorporates, in addition to the usual least squares

data misfit, a Tikhonov-Phillips regularization term:

JTP (K) =1

2‖R(K)‖2

D +η

2‖K‖2

α (19)

where η > 0 is the regularization parameter and ‖ . ‖α indicates some norm or semi-

norm, e.g. ‖K‖α = ‖∇K‖L2 [22, 50]. Using this approach, the cost functional is

changed significantly with the goal of obtaining in a stable way a global minimizer. We

do not want to take this route, but prefer instead to keep working with the original

least-squares cost functional (12) which only involves the data fit. We will minimize

this cost functional by restricting the search to elements of a smaller function space,

which is an alternative form of regularization.

The regularization scheme will be derived and discussed in details in section 6.

In the following, we will present the basic structure of our inversion method, and we

will derive practical ways of applying the adjoint linearized residual operator R′[K]∗ to

vectors R in the data space D. This will lead us to the propagation-backpropagation

technique which is applied in this paper.

4.2. Computation of the operator R′[K]. The linearized problem.

Let us consider a small perturbation δK in the permeability distribution K that leads

to small perturbations W and q in the saturation and the pressure, respectively. Here

we assume that the pressure remains nearly unchanged so that ∇q is neglegible. This

is so because the pressure is a smooth function compared to the saturation. Using a

heuristic approach to derive an expression for R′, we introduce K + δK and Sw + W in

(2) and we neglect second order terms. Then, W solves the initial value problem

φ∂W

∂t−∇ · [∂Tw

∂Sw

W∇p]− ∂Qw

∂Sw

W =δK

KQw +∇ · [δK

KTw∇p] in Ω (20)

W (~x, 0) = 0 in Ω (21)

where Sw and p are the solutions of (1)-(8). From the value of W we derive the linearized

response of the data to a perturbation δK in the permeability distribution, which is given

by

R′[K]δK =∂Qw

∂Sw

W

∣∣∣∣Ω+×[0,tf ]

. (22)

11

4.3. Computation of the operator R′[K]∗. The adjoint problem.

Here, we derive an expression for the adjoint operator R′[K]∗ applied to a function ρ in

the data space. The operator R′[K]∗ is defined by

〈R′[K]δK, ρ〉D

= 〈δK,R′[K]∗ρ〉P

. (23)

We assume that the inner products in the parameter space P and in the data space D

are given by

〈f, g〉D

=

Np∑j=1

∫ tf

0

fj gj dt ; 〈A,B〉P

=

∫

Ω

A B d~x , (24)

where fj = f(~xpj, t) and gj = g(~xpj

, t), j = 1, . . . , Np, are time functions defined at

the production well positions ~xpj. We formulate the basic result of this section in the

following theorem, which is derived in Appendix A.

Theorem 1: Let ρ ∈ D be an arbitrary function in the data space. Then R′[K]∗ρ is

given by

R′[K]∗ρ =

∫ tf

0

(Tw

K∇p∇z − z

1

KQw

)dt (25)

where z is the solution of the adjoint equation

−φ∂z

∂t+

∂Tw

∂Sw

∇p∇z − (z −Np∑j=1

ρ δ(~x− ~x(p)j ))

∂Qw

∂Sw

= 0 in Ω (26)

z(~x, tf ) = 0 in Ω, (27)

and Sw and p are the solutions of (1)-(8).

Notice that Qw is nonzero only at the well locations. Therefore, when we assume in

the mathematical derivation of the theorem that the permeability is known directly at

the wells (a realistic assumption), the second term in (25) disappears and we only have

to evaluate the first term in order to calculate the update in the rest of the domain Ω.

This will be the approach we use in our numerical reconstructions.

Notice that, as typical for the adjoint scheme, the system (26), (27) physically

models some kind of backpropagation with respect to the linearized forward model. The

residuals are applied at the production wells as artificial injectors, and backpropagated

backward in time (notice the minus sign in front of the time derivative in (26) and the

prescribed final value conditions in (27), compared to a plus sign in (20) and initial values

in (21)) and in space by the system (26), (27). Equation (25) uses these backpropagated

fields to extract an update direction by combining forward and adjoint fields at each

location.

5. The basic algorithm

The basic inversion algorithm (without regularization) can be summarized as follows.

Assume that the n-th approximation K(n) to the true permeability distribution has been

obtained. Then:

12

(i) Compute the residual R[K(n)] = M [K(n)] − G on Ω+ × [0, tf ] by solving the direct

problem (1)-(8).

(ii) Apply ρ = R[K(n)], as computed in (i), at the production wells and ’backpropagate’

by solving the adjoint problem (26)-(27) for this ρ.

(iii) Combine the results of (i) and (ii) for calculating the update δK(n) from (25).

Let p be the pressure as calculated in (i), and let z be the solution of (26)-(27) as

calculated in (ii). Since we assume that K is known at the well locations, and Qw

is zero in the rest of the domain, (25) simplifies to

δK(n) =

∫ tf

0

Tw

K∇p∇z dt. (28)

(iv) The new approximation to the true permeability distribution is

K(n+1) = K(n) + ω δK(n),

where ω is some step-length to be chosen properly. An efficient scheme for

practically choosing ω (a variant of a line-search technique) will be explained in

section (7.1.2).

(v) Compute the residual R[K(n+1)] and proceed with step (i) until some convergence

criterion is achieved.

6. Regularization and smoothing.

6.1. Smoothing with function spaces.

We have presented above the basic algorithm which recovers L2 functions of permeability

from given data such that the misfit in the data is minimized. This procedure does not

incorporate explicit regularization (except of the stabilizing procedure incorporated in

the operator C). In some situations, it might be necessary or desirable to restrict the

search for permeability functions to a smaller subset of L2, for example of smoothly

varying functions. This might be so in order to regularize the reconstruction algorithm,

or in order to take into account some prior information or assumptions on the solution

we are looking for. For example, the reservoir engineer might know or assume that the

permeability distribution in some region is fairly smoothly varying. Or, he might only

have very few data available for the inversion, so that he wants to select a smoothly

varying profile as a regularized form of the reconstructed permeability distribution. This

can be easily done in our framework.

Instead of looking for permeability distributions in L2(Ω), let us assume now that

we require the permeability to be an element of the smaller subspace

H1(Ω) := m ∈ L2(Ω), ∂im ∈ L2(Ω) for i = 1, 2, 3 .

This Sobolev space is usually equipped with the standard norm

‖m‖1,1 :=(‖m‖2

L2+ ‖∇m‖2

L2

)1/2

13

and the standard inner product

〈m1,m2〉1,1 := 〈m1,m2〉L2+ 〈∇m1,∇m2〉L2

.

For reasons explained below, we will instead prefer to work with the equivalent norm

‖m‖α,β :=(α‖m‖2

L2+ β‖∇m‖2

L2

)1/2, α, β > 0

and its associated inner product

〈m1,m2〉α,β := α 〈m1,m2〉L2+ β 〈∇m1,∇m2〉L2

.

A proper choice of the weighting parameters α and β will allow us to steer the

regularization properties of our algorithm in an efficient and predictable way.

Let us denote the new parameter space H1(Ω), when equipped with the weighted

norm ‖ . ‖α,β, by P . When using this modified space in our algorithm, we also have

to adjust the operators acting on it, in particular the adjoint of the linearized residual

operator. This operator is now required to map from the data space D into P . Moreover,

the minimum norm solution of (14) is now taken with respect to the weighted norm

‖ . ‖α,β, which clearly gives us a different candidate. The necessary adjustments for our

algorithm can be done as follows.

Denote as before by R′[K]∗ζ the image of ζ ∈ D under application of the adjoint

linearized residual operator as calculated in section 4.3, considered as an operator

mapping from D into P = L2(Ω). Denote furthermore by R′[K]ζ its image under the

adjoint linearized residual operator with respect to the newly defined weighted inner

product, mapping into the smaller space P . With a straightforward calculation, using

the definitions of the two adjoint operators

〈R′[K]x, ζ〉D = 〈x,R

′[K]∗ζ〉P = 〈x, R

′[K]ζ〉P , (29)

it follows that

R′[K]ζ = (αI − β∆)−1 R′[K]∗ζ, (30)

where we supplement the inverted differential operator (αI − β∆)−1 by the boundary

condition ∇(R′[K]ζ) ·n = 0 on ∂Ω. The symbol I stands for the identity, and ∆ stands

for the Laplacian operator. (30) can be easily derived by applying Green’s formula to

the right hand side equality in (29).

In practice, the ratio γ = β/α (which can be considered being a ’regularization

parameter’) is an indicator for the ’smoothing properties’ of our scheme. The larger this

ratio, the more weight is put on minimizing the derivatives of our solution. Therefore,

by properly choosing this ratio, we can steer the smoothness properties of our final

reconstruction to a certain degree. In our numerical experiments, we will choose this

ratio once, when starting the algorithm, and keep it fixed during the iterations. The

other free parameter, say α, will be chosen in each individual step to scale the update

properly. In our numerical experiments, we choose α such that

‖R′[K]ζ‖L2 = ‖R′[K]∗ζ‖L2

14

is satisfied for the current update. This possibility of scaling the updates is the main

reason for keeping the parameter α throughout the calculations instead of simply putting

it to 1 right at the beginning. When testing and comparing the performance of different

regularization parameters γ it is practically useful (in particular for the line-search

method) that the order of magnitude of the calculated values of R′[K]ζ does not depend

too much on γ.

Notice also that the new search directions using this modified adjoint operator are

still descent directions for the least squares cost functional (12), as can be verified easily

by replacing P by P in (16) and (18).

Practically, the scheme is implemented as follows:

γ is fixed regularization parameter

Define Ψ = R′[K]∗ζ.

Solve (I − γ∆)ϕ = Ψ, ∇ϕ · n = 0 on ∂Ω.

Define Φ = ϕα

with α = ‖ϕ‖‖Ψ‖ (such that ‖Φ‖ = ‖Ψ‖)

Then we have (αI − β∆)Φ = Ψ, ∇Φ · n = 0 on ∂Ω, with β = αγ.

Put R′[K]ζ = Φ.

(31)

We mention that applying this regularization scheme amounts to applying the

postprocessing operator (αI − β∆)−1 to the updates calculated in the previous

’unregularized’ scheme. Therefore, the effect of the regularization is similar to filtering

the updates with a carefully designed (iteration-dependent) filtering operator.

In the following, we want to give an interesting additional interpretation of this

regularization scheme.

Define the cost functional

J (Φ) =a

2‖Φ‖2

L2+

b

2‖∇Φ‖2

L2+

c

2‖Φ−Ψ‖2

L2(32)

with Ψ = R′[K]∗ζ. Here, the third term penalizes the misfit between the unregularized

update direction δK = R′[K]∗ζ and the new candidate Φ, whereas the first two

terms penalize roughness of Φ. The gradient direction for this cost functional is

[(a+c)I−b∆]Φ−cΨ (where the Laplace operator is again understood to be accompanied

by the boundary condition ∇Φ · n = 0 on ∂Ω). Therefore, a necessary condition for the

minimum can be stated as

[(a + c)I − b∆]Φ = cΨ. (33)

Choosing c = 1, b = β ≥ 0 and a = α− 1 ≥ 0 this amounts to calculating

Φ = (αI − β∆)−1Ψ, (34)

which is equivalent to (30). Therefore, applying function space regularization as

described above can be interpreted as minimizing the cost functional (32) with

specifically chosen parameters a, b and c.

15

6.2. Smoothing with the heat kernel

An alternative (and slightly more ’ad hoc’) approach to regularization and smoothing

is to use well-known concepts from image processing. Denote Ψ = R′[K]∗ζ for some

residual vector ζ ∈ D. Then, we can convolve these (unregularized) updates Ψ calculated

by (15) with a Gaussian kernel of variance σ > 0

fσ(x) =1

4πσexp

(−|x|

2

4σ

)(35)

which produces the smoothed update

Φ = fσ ∗Ψ =

∫fσ(x− y)Ψ(y)dy. (36)

Practically, this can be done by solving the initial value problem for the heat equation

vt −∆v = 0 for t ∈ [0, τ ] (37)

v(0) = Ψ

on Ω with τ = σ and with suitably chosen boundary conditions, and putting

Φ = v(τ). (38)

Here, the smoothing time τ can be considered as a regularization parameter: for τ = 0 no

regularization takes place, whereas with increasing τ the updates become increasingly

smoothed. In image processing, this procedure (in a generalized form) is sometimes

referred to as ’defining a scale space’, with τ > 0 being the scale.

Although the scheme described above is only ’ad-hoc’, it turns out that a similar

scheme can be derived in a more rigorous way from the cost functional (32). This will

be demonstrated in the following.

Let us assume that we again want to minimize (32) for the above given choice of

the parameters c = 1, b = β ≥ 0 and a = α − 1 ≥ 0. Now we want to use a gradient

method for finding the minimum, starting with the initial guess Φ(0) = Ψ. Using the

gradient direction for (32) derived above, we get the iteration rule

Φ(n+1) = Φ(n) − θ[(αI − β∆)Φ(n) −Ψ

](39)

where θ is the (fixed) step-size in a given step n of the iteration. This can be written as

Φ(n+1) − Φ(n)

θ= β∆Φ(n) + (Ψ− αΦ(n)) (40)

which is just one step of a finite-difference time-discretization of a modified heat equation

(37)

vt − β∆v = (Ψ− αv) for t ∈ [0, τ ] (41)

v(0) = Ψ

with additional time-dependent heating source Ψ− αv and fixed time-step δt = θ.

16

The choice of the free parameter α in this iteration can be done according to the

same lines as described in the algorithm in section 6.1. This strategy leads to the

practical algorithm:

γ is fixed regularization parameter

Define Ψ = R′[K]∗ζ.

for n = 1, . . . , N :

ϕ(n+1) = ϕ(n) + θ[γ∆ϕ(n) + (Ψ− ϕ(n))

]

end for (Upon convergence we have ϕ(N) = (I − γ∆)−1Ψ)

Define Φ = ϕ(N)

αwith α = ‖ϕ(N)‖

‖Ψ‖ (such that ‖Φ‖ = ‖Ψ‖)Then we have (αI − β∆)Φ = Ψ, ∇Φ · n = 0 on ∂Ω, with β = αγ.

Put R′[K]ζ = Φ.

(42)

It turns out that this procedure has good regularization properties even if we choose N

fairly small (e.g. N = 5 or N = 10), which amounts to choosing the regularization time

τ in (41) small. In that case, we will not exactly calculate R′[K]ζ = (αI − β∆)−1 Ψ, but

we will have only a mildly smoothed form of Ψ = R′[K]∗ζ. Moreover, for small values of

N the additional heating term Ψ−Φ(n) is typically small (recall that we start the iteration

with Φ(0) = Ψ), such that we can safely neglect it and arrive at the regularization scheme

by the heat kernel (37). Although this yields only a very crude approximation to R′[K]ζ,

the results achieved with this scheme are usually quite satisfactory when used in each

step of our iterative scheme for solving the inverse problem.

We complete this section by mentioning, that, throughout this paper, we will

always use the scheme described in section 6.1 (regularization by function spaces) for

the regularization. However, most of our calculations have also been tested with the

alternative scheme described above using a small value of N , which also gave us good

results.

7. Numerical experiments

In our numerical experiments we use two different numerical schemes for the forward

modelling: The IMPES method and the streamline method. Both are described in

Appendix B. For the numerical solution of the adjoint problem (26)-(27) we have

implemented an explicit upwind finite difference scheme. We model a reservoir of

600×600 m2 which is discretized by a 25×25 uniform spatial grid. The typical time-step

in the discretization is between 2 hours and one day, and the reservoir is monitored for

a duration between 50 to 100 days. When using the streamline method, each cell of

the grid is intersected by at least one streamline. The parameters used to describe the

reservoir are φ = 0.213, µw = 8.2 × 10−4 Pa s, µo = 7.9 × 10−4 Pa s. As boundary

and initial conditions for the forward modelling we use S0w(x) = 0.187, p0 = 3000

psi, p(i)wb = 3500 psi, and p

(p)wb = 2000 psi ∀p at all wells. The relative permeabilities

are nonlinear functions of saturation which are shown in figure 2. For the value c in

formulas (4) and (5) we use c = 2πdx dy ln(

rdrw

)(see also [14, 39]) where rd = 0.14[dx2 +dy2]

12

17

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sw

Kr

Krw

Kro

Figure 2. Relative permeabilities versus saturation as used in the numericalexperiments.

and rw = 0.108 m is the well radius. Here, dx and dy denote the discretization lengths

of the grid in x and y direction.

In the following we show permeability estimations for several examples with two

different reference permeability distributions. In the first case we estimate a smooth

permeability distribution, whereas in the second example we estimate a permeability

distribution with sharp discontinuities. In both cases we use four injector wells and nine

production wells arranged as an array of so-called ’five-spot patterns’.

7.1. First example. Reconstruction of a smoothly varying function

We first investigate the reconstruction of the smoothly varying permeability distribution

which was already shown in figure 1. The extracted water flow G = Qw at the producers

is plotted in figure 3b with solid lines. In our initial experiment (sections 7.1.1 and

7.1.2), we compute the synthetic data by using the IMPES method and perform the

reconstruction by using the same forward modelling code (the IMPES method). In the

later section 7.1.3, and in all further numerical experiments presented in section 7.2, we

will use instead data which have been generated by the streamline method, whereas the

reconstruction is done by using the IMPES method. By doing this, we can avoid the

so-called ’inverse crime’, and also can investigate the performance of the algorithm in

the case of very noisy data.

7.1.1. Basic reconstruction without explicit regularization. The well configuration and

the estimated permeability distribution at the final iteration are shown in figure 3

a. Our initial model, a uniform permeability distribution corresponding to the mean

permeability value (1400 mD) serves to generate the initial data M [K(0)], plotted in

18

0 200 400 6000

200

400

600mD

600

1000

1400

1800

C

A

B D

(a)

Qw

× 1

06 (s−1

)

20 40 600 2.4

Time (Days)

(12,12)

(12,300)

(12,588)

(300,12)

(300,300)

(300,588)

(588,12)

(588,300)

(588,588)

(b)

2000

2000

mD

0

2000

200 400 6000

2000

A

B

C

D

(c)

20 40 600

2

4

Iterations

Flo

w r

esid

ual

(a.

u.)

(d)

Figure 3. Synthetic example with four injectors and nine producers: (a) Estimatedpermeability; (b) extracted water flows at the producers (solid lines, dot-dashed lines,and dashed lines represent the extracted water flow for the reference model, for theinitial model and for the estimated model, respectively); (c) cross sections (solid linesand dashed lines represent the reference permeability model and the estimated model,respectively); (d) normalized residuals at each iteration. The reference permeability isshown in figure 1 (a).

figure 3 b with dot-dashed lines. The calculated extracted flows at the producers after

100 iterations are shown with dashed lines. The agreement is very good, indicating

the validity of our approach. Note that the calculated responses match the ’true’

production data over almost the whole time history. The small differences might be

due to pressure effects that are not taken into account. Figure 3 d shows the root

mean square error in the measurements at each iteration. It decreases monotonically

until becoming stationary (up to small fluctuations) at a small value, after about 80

iterations. For comparison purposes, we plot in figure 3 c several cross sections of

the real permeability (solid lines) and of the estimated permeability (dashed lines).

Notice that no explicit regularization has been applied in this numerical experiment.

Accordingly, the reconstruction does not look very smooth. Moreover, a block structure

can be observed in the reconstruction which is due to the use of five-spot patterns in the

experimental setup. Each block reflects the flow from one injector to the corresponding

neighbouring producer.

19

In this example we have choosen the relaxation parameter ω to be constant.

Usually, its value is determined by trial and error. We have chosen it such that the

maximum of the relative update δK(0)/K(0) in the first iteration is 1% of the constant

permeability value of the initital guess.

7.1.2. Reconstruction with regularization and with a line search variant. In the next

example, we allow ω to be variable in order to speed up the convergence. We use a variant

of a line search technique for finding a suitable update. Our goal is to take a maximally

possible step which still decreases the mismatch between calculated and measured data.

Therefore, step (iv) in the algorithm is replaced by K(n+1) = K(n) + ω(n) δK(n),

n = 0, 1, 2, . . .. For starting this scheme, ω(0) is chosen as before. We introduce a

suitable factor γ > 1 for manipulating the relaxation parameter ω. If the residual

R[K(1)] decreases when applying the chosen value for ω(0), we accept this step, and set

ω(1) = γ ω(0) for the next iteration. Otherwise, we reject the update and try again with

ω(0) → ω(0)/γ. In this case, we continue reducing ω(0) until we find an ω(0) for which

the newly calculated residual R[K(1)] is reduced. This step is then accepted, and we

continue with the next iteration searching for a suitable ω(1), assuming as initial value

for the search the final value for ω(0). The algorithm stops if at least 15 consecutive

trials are rejected, i.e. if choosing ω(n+1) = γ−15ω(n) does not lead to a decrease of the

residual in some iteration step n. In our numerical experiments this scheme has shown

to reduce the total computational time for the reconstruction significantly. We note

here, that Bulishev et al. [6] have used a similar procedure for choosing the length of

the iterative step in a gradient method. Alternative line search techniques are possible,

and will be investigated in our future research.

Figure 4 a shows the estimated permeability after 20 of these modified iterations

when using the value γ = 1.5. Observe in figure 4 b that now the residual decreases

much faster. In addition to the line search technique, we have also applied to this

reconstruction the regularization scheme explained in section 6.1. Using here the same

well configuration and reference permeability distribution as in figure 3, it is apparent

that the estimated permeability distribution is now smoother than the one obtained in

figure 3 a.

7.1.3. Reconstruction from data created by an independent STREAMLINE method. In

the numerical experiments shown so far we have used the same forward modelling code

for creating the data as we have used for the reconstruction task, without any additional

noise added to these data. Certainly, in order to really evaluate the stability and

practical usefulness of a reconstruction algorithm, it is necessary to apply the algorithm

also to data corrupted with different types of noise. Noisy data with only random

Gaussian noise can be created by simply adding random numbers of a certain magnitude

to the previously considered noiseless data. In order to include some possible systematic

noise component into our investigation, we have decided not to take this route, but

instead to use a completely independent simulation technique for calculating our data.

20

0 200 400 6000

200

400

600mD

600

1000

1400

1800

(a)

20 60 800

2

4

6

8

Iterations

Flo

w r

esid

ual

(a.

u.)

(b)

Figure 4. Same as figure 3 but with variable ω and with regularization: (a) Estimatedpermeability; (b) normalized residuals at each iteration with variable ω (solid line) andwith constant ω (dashed line).

For this purpose, we use a streamline method that is not designed to produce data

with the highest accuracy. In fact, we have measured that the disagreement with data

obtained with the IMPES method is approximately 3 %. In summary, the streamline

created data are used as the input for our reconstruction scheme. Then, we use the

IMPES method throughout the iterations for carrying out the reconstruction task. We

believe that this is also an excellent test for evaluating the expected performance of the

code when applied to real data.

In order to regularize the scheme, we select a fixed ratio of β/α = c0 throughout

the iterations and apply the postprocessing operator (αI − β∆)−1 in each update as

described before. The results after 10 and 25 iteration steps are shown on the top left

and top right images of figure 5. In the bottom left image of this figure we have plotted

the evolution of the norm of the residuals during the iteration. It can be observed that

the residuals are continuously decreasing, as it is expected from the algorithm. In order

to get an indication of the actual level of noise contained in the data, we have also

plotted the L2-misfit between these data and the data we would get using the IMPES

method (the horizontal dashed line in the figure).

For the value c0 chosen here for our reconstruction we observe some form of semi-

convergence of the algorithm, as it often occurs in iterative methods applied to noisy

data. In order to illustrate this, we show in the bottom right image of figure 5 the

evolution of the L2-error of the reconstruction compared to the reference permeability

distribution. During the early iterations of the method this error obviously decreases.

Approximately at a point where the residuals hit the noise level, this error starts to

increase again, although the residuals are still decreasing. Our interpretation of this

phenomenon is that at this point the algorithm starts fitting the noise rather than the

data. The top left image of figure 5 shows the reconstruction at the point where the L2-

norm of the image error turns from decreasing to increasing (which can be considered as

the ’optimal stopping point for the algorithm’), and the top right image shows the final

21

0 200 400 6000

200

400

600mD

600

1000

1400

1800

(a)

0 200 400 6000

200

400

600mD

600

1000

1400

1800

(b)

5 10 15 20 250

2

4

6

8

Iterations

Flo

w r

esid

ual

(a.

u.)

(c)

5 10 15 20 250

5

10

15

20

Iterations

Err

or

in p

erm

eab

ility

(m

D)

(d)

Figure 5. Reconstruction of the smooth permeability distribution shown in figure 1from data created with an independent streamline method. Regularization is appliedwith a value β/α = c0. Top left: reconstruction after 10 iterations; top right:reconstruction after 25 iterations. Bottom left: evolution of residuals and actual noiselevel of data. Bottom right: evolution of L2-error of reconstruction.

reconstruction which we get when keeping the algorithm running until the residuals

do not decrease anymore. We see that the image obtained at the ’optimal stopping

point’ is a very good reconstruction of the reference permeability distribution, whereas

in the image obtained at the final iteration, noise-related artifacts have degraded the

reconstruction.

Certainly, in real applications the noise level of the data can only be estimated,

and the optimal stopping point must be chosen according to some appropriate criterion

(e.g. the ’Morozov discrepancy principle’). Moreover, it must be taken into acount that

the evolution of the L2-error of the reconstruction itself needs to be considered with

care. Since the problem at hand is vastly underdetermined, it does not really make

sense at all to speak about ’the correct solution’ without any further prior information

available. We want to point out the role of prior information in our scheme in the

following numerical experiment.

In order to investigate this semi-convergence property further, we have increased

in this experiment the ratio β/α to 10 times its previous value, i.e. β/α = 10c0. The

results of the reconstructions are displayed in figure 6. Now the residuals decrease

22

0 200 400 6000

200

400

600mD

600

1000

1400

1800

(a)

0 200 400 6000

200

400

600mD

600

1000

1400

1800

(b)

10 20 300

2

4

6

8

Iterations

Flo

w r

esid

ual

(a.

u.)

(c)

10 20 300

5

10

15

20

Iterations

Err

or

in p

erm

eab

ility

(m

D)

(d)

Figure 6. Reconstruction of the smooth profile shown in figure 1 from data createdwith an independent Streamline method. Regularization is applied with a valueβ/α = 10c0. Top left: reconstruction after 15 iterations; top right: after 37 iterations.Bottom left: evolution of residuals and actual noise level of data. Bottom right:evolution of L2-error of reconstruction.

about as much as in the previous situation. However, the L2-norm of the error in the

reconstruction (bottom right image of figure 6) does not show the semi-convergence

behavior here. The reconstruction error becomes practically stationary at some stage

of the algorithm and we get a good final reconstruction in a stable way. The reason for

this improvement is that now we have increased the smoothing property of the inversion

algorithm by increasing the ratio β/α. Since the reference permeability distribution is

fairly smooth, this amounts to putting more weight on correct prior information, such

that we arrive at a better reconstruction than before.

7.2. Second example. Reconstruction of a function with sharp discontinuities

In contrast to the previous situation where we tried to reconstruct a very smooth

unknown permeability distribution from the data, we will now investigate the

performance of the numerical scheme when the true permeability distribution is highly

discontinuous with large sharp edges of high contrast. The experimental setup is

the same as before. Figure 7 shows the true permeability distribution and the well

locations. The data have been calculated with the streamline method, and the numerical

23

0 200 400 6000

200

400

600mD

200

600

1000

1400

Figure 7. Reference permeability distribution for the second example. The figurealso shows the location of the four injectors ()and the nine producers (×).

reconstruction is done by using the IMPES method for forward modelling.

In this reconstruction, our initial guess is a constant permeability distribution of

1400 mD. Figure 8 shows the results of the experiment when using the ratio β/α = c0

(the same value as before) for the regularization at two different iteration steps of the

reconstruction. The lower left image in figure 8 shows the evolution of the residual

during the reconstruction, together with the theoretical ’noise level’ of the data. The

semiconvergence behavior of the reconstruction is hardly noticeable here. A reason might

be that the original profile is not as smooth as it was in the previous example. Certainly,

also here we can display the results after an intermediate stage of the reconstruction,

here after 19 iterations (top left image of figure 8), and at the final stage after about 70

iterations (top right image) when the iteration stops according to our stopping criterion.

It is difficult to say which one could be selected as the ’better’ reconstruction of the

original profile.

In the same way as in the previous example we want to compare the behavior

of the code when we change the ’regularization parameter’ to β/α = 10c0. As

mentioned before, this amounts to putting more weight on the smoothing properties

of the algorithm. Figure 9 shows the results in this situation. Again, the lower left

image shows the evolution of the residual norm during the iterations, together with the

theoretical ’noise level’ of the data. The bottom right image shows the evolution of the

L2-error in the reconstruction. In contrast to the previous situation where we tried to

recover the smooth profile, now the L2-error of the image does not stabilize when using

the higher value for β/α. Instead, we see in this situation a semi-convergence behaviour

for this larger value for β/α. Therefore, in a way, the behaviour of the algorithm seems

to be reversed compared to the previous smooth example. This makes sense since now

our reference permeability distribution is highly discontinuous, and enforcing a smooth

24

0 200 400 6000

200

400

600mD

500

1000

1500

2000

(a)

0 200 400 6000

200

400

600mD

500

1000

1500

2000

(b)

10 20 30 40 50 600

5

10

15

Iterations

Flo

w r

esid

ual

(a.

u.)

(c)

10 20 30 40 50 60 700

5

10

15

20

25

Iterations

Err

or

in p

erm

eab

ilty

(mD

)

(d)

Figure 8. Reconstruction of the non-smooth profile shown in figure 7 from datacreated with an independent Streamline method. Regularization is applied with avalue β/α = c0. Top left: reconstruction after 19 iterations; top right: after 70iterations. Bottom left: evolution of residuals and actual noise level of data. Bottomright: evolution of L2-error of reconstruction.

reconstruction is not the right way to proceed if this prior knowledge is available. Using

the smaller value for β/α = c0 could be the better choice in this situation. We have

displayed in the top left image of figure 9 the reconstruction at the point where the

L2-error in the image changes from decreasing to increasing (after 17 iterations), and in

the top right image the final reconstruction of the algorithm.

We want to mention at this point that, when it is known a priori that the

permeability distribution is highly discontinuous, or if it is desired for different

reasons to reconstruct a discontinuous profile, different regularization methods might

be particularly useful here. In particular the use of total variation regularization [1, 10],

Huber norm regularization [4], the use of a Mumford Shah functional [43] or a level set

based shape reconstruction scheme [19, 33, 45] would be appropriate tools to use. See

also [41] for a description (and alternative treatment) of such a situation in reservoir

engineering. We plan to investigate some of these regularization schemes in our future

research in order to compare their performances with the reconstructions shown here.

25

0 200 400 6000

200

400

600mD

500

1000

1500

2000

(a)

0 200 400 6000

200

400

600mD

500

1000

1500

2000

(b)

10 20 300

5

10

15

Iterations

Flo

w r

esid

ual

(a.

u.)

(c)

5 10 15 20 25 30 3518

20

22

24

26

28

Iterations

Err

or

in p

erm

eab

ility

(m

D)

(d)

Figure 9. Reconstruction of the non-smooth profile shown in figure 7 from datacreated with an independent Streamline method. Regularization is applied with avalue β/α = 10c0. Top left: reconstruction after 17 iterations; top right: after 36iterations. Bottom left: evolution of residuals and actual noise level of data. Bottomright: evolution of L2-error of reconstruction.

8. Conclusions

We have introduced in this paper a new reconstruction scheme for the history matching

problem which is based on an adjoint method. Starting from some initial guess, this

iterative scheme calculates successive corrections for the permeability distribution by

numerically back-propagating the residuals corresponding to the latest best guess. This

so-called ’adjoint scheme’ avoids the usually expensive calculation of large sensitivity

matrices in each step of the inversion. A line-search variant is used to find optimal steps

for the updates. In particular, new efficient regularization schemes have been derived

and tested which enable the reservoir engineer to incorporate prior information into the

inversion which stabilizes the reconstruction process.

The numerical experiments presented in this paper show that this algorithm is

able to reconstruct unknown permeability distributions in a fast and efficient way

from relatively few data which have been created by an independent forward modelling

scheme. In the case of very noisy data we observe a typical semi-convergence behavior.

However, it can be controlled to a certain degree by introducing correct prior information

26

into the reconstruction. In particular, for a smooth profile this leads to a significant

improvement of the results.

The presented scheme can easily be extended to a more realistic 3D situation.

Since it is based on repeated forward and adjoint calculations of the reservoir equations,

all that is needed are reliable forward and adjoint solvers in 3D. These solvers can be

integrated into the inversion code in a black-box fashion, such that the reservoir engineer

can easily switch back and forth between different forward modelling codes if necessary.

The same holds true for the two regularization schemes presented in this paper. Since

they have the form of easy to implement post-processing tools (applied in each step of

the iterative inversion), they can be used as black-box filters which can be modified and

exchanged independently from the rest of the algorithm in order to incorporate correct

or assumed prior information into the inversion.

In our future work we plan to investigate alternative regularization schemes, such

as level-set techniques, which are better suited for the presence of sharp discontinuities.

Appendix A: Proof of Theorem 1

In this appendix we prove Theorem 1. Let z be a function defined in Ω× [0, tf ]. Then,

using (20) and (23) we can write

〈δK, R′[K]∗ρ〉P

= 〈R′[K]δK, ρ〉D

+ C1 + C2 (43)

where

C1 =

∫

Ω

∫ tf

0

z

φ∂W

∂t−∇ · [∂Tw

∂Sw

W∇p]− ∂Qw

∂Sw

W

dt d~x (44)

C2 = −∫

Ω

∫ tf

0

z∇ · [δK

KTw∇p] +

δK

KQw

dt d~x (45)

Note that C1 + C2 = 0. Integrating by parts (45) and applying the divergence theorem

C2 =

∫

Ω

δK

∫ tf

0

(Tw

K∇p∇z − z

1

KQw

)dt d~x (46)

where (8) has been used. Thus, defining

R′[K]∗ρ =

∫ tf

0

(Tw

K∇p∇z − z

1

KQw

)dt (47)

we have C2 = 〈δK,R′[K]∗ρ〉P, if

〈R′[K]δK, ρ〉D

+ C1 = 0 . (48)

Note that z is not determined yet. We want to select z so that (48) is satisfied. In order

to find a formula for z we apply partial integration and the divergence theorem to (44).

Using (8) and (21)

C1 =

∫

Ω

z(~x, tf )φW (~x, tf ) d~x +

∫

Ω

∫ tf

0

W− φ

∂z

∂t+

∂Tw

∂Sw

∇p∇z − z∂Qw

∂Sw

dt d~x . (49)

27

We impose that z(~x, tf ) = 0 in (49) so the first integral vanishes and therefore (48) reads

〈R′[K]δK, ρ〉D

+

∫

Ω

∫ tf

0

W− φ

∂z

∂t+

∂Tw

∂Sw

∇p∇z − z∂Qw

∂Sw

dt d~x = 0 . (50)

On the other hand, using (22) and (24)

〈R′[K]δK, ρ〉D

=

Np∑j=1

∫ tf

0

∂Qw

∂Sw

Wjρj dt =

∫

Ω

∫ tf

0

∂Qw

∂Sw

Wρ

Np∑j=1

δ(~x− ~x(p)j )dt d~x , (51)

so (50) yields:

∫

Ω

∫ tf

0

W− φ

∂z

∂t+

∂Tw

∂Sw

∇p∇z − (z −Np∑j=1

ρ δ(~x− ~x(p)j ))

∂Qw

∂Sw

dt d~x = 0 . (52)

Therefore, by choosing z to be the solution of the following adjoint problem:

−φ∂z

∂t+

∂Tw

∂Sw

∇p∇z − (z −Np∑j=1

ρ δ(~x− ~x(p)j ))

∂Qw

∂Sw

= 0 (53)

z(~x, tf ) = 0 , (54)

where Sw and p is solution of (1)-(2) with boundary and initial conditions (6)-(8), we

have found a z which satisfies our requirements. With this, we have proven the theorem.

Appendix B: Numerical Methods

In this paper we have used two alternative methods to solve equations (1) and (2)

describing the direct problem. The first alternative is the IMPES method (IMplicit

Pressure Explicit Saturation) which is based on finite differences and was first proposed

by Sheldon et.al. [46, 48]. It is widely used in commercial reservoir simulators and

consists of solving at each time step equation (1) implicitly to derive the pressure

distribution, and then computing the saturation distribution at the new time step by

solving equation (2) with an explicit finite difference scheme (such as the upwind method)

where the values of transmissibilities, T , and pressure, p, are computed at the previous

time step.

The second alternative is the Streamline method, which instead of solving equation

(2) in the 2D or 3D space, solves it along one-dimensional streamlines. The method

became an attractive alternative for reservoir simulation after the work of Pollock

[40], which proposed a very efficient technique to compute the streamlines. The main

advantage of the method lies in its speed, since instead of computing the evolution of

saturation in a two dimensional or three dimensional grid, it is only necessary to solve a

set of one-dimensional problems to derive the evolution of saturation along streamlines.

Thus, it is a method very well suited for inverse problems, where many direct problems

have to be solved for convergence and, therefore, very efficient methods are needed in

order to solve realistic problems in three dimensions.

28

To derive the method it is useful to write equation (2) in terms of the total flow

velocity ~ut(~x, t) = −T (~x, t)∇p(~x, t). Introducing this variable in (2) and using (1) leads

to,

φ∂Sw

∂t+ ~ut∇

(Tw

T

)= Qw +

Tw

TQ (55)

This equation can be written along the streamlines using as independent variable the

time of flight,

τ =

∫ s

0

φ

|~ut| dσ (56)

where s represents distance along the streamline. Since the streamline has the direction

of the velocity ~ut, the derivative along a streamline can be obtained from the gradient

by,

∂

∂s=

~ut

|~ut| · ∇ (57)

Also, from (56),

∂

∂s=

φ

|~ut|∂

∂τ(58)

Thus,

φ∂

∂τ= ~ut · ∇ (59)

and equation (55) can be written as a one-dimensional conservation law in terms of the

variables time, t, and time of flight, τ , along the streamlines,

∂Sw

∂t+

∂(

Tw

T

)

∂τ=

1

φ

(Qw +

Tw

TQ

)(60)

The right hand side is zero except at injection and production wells.

Therefore, the procedure to solve the direct problem is the following;

(i) Compute the pressure at time tp by solving equation (1) with finite differences in a

2D or 3D grid.

(ii) From the pressure distribution compute the velocity distribution ~ut.

(iii) Use Pollock’s method to derive a set of streamlines (at least one streamline must

go through each grid cell).

(iv) Project the values of the saturation on each grid cell at time tp, in order to derive

the saturation distribution along each streamline.

(v) Along each streamline solve the conservation law (60) with time step ∆ts in order

to advance the saturation distribution along each streamline from time tp to time

tp + ∆tp.

(vi) Project the values of saturation along the streamlines at time tp + ∆tp in order to

compute the values of saturation at each grid cell.

(vii) Use the saturation values at each cell to compute the values of T at time tp + ∆tp.

29

(viii) If tp = tp + ∆tp is smaller than the total simulation time go to (i).

The method takes advantage of the fact that the pressure varies more slowly than the

saturation and, therefore, the time step for computation of the pressure, ∆tp, can be

taken much larger than the time step for saturation, ∆ts. When other effects, such as

gravity or capillarity forces, which can not be written in terms of τ are included, the

mapping between streamlines and grid cells has to be carried out at each time step, and

the method looses its efficiency.

Acknowledgments

We thank the anonymous referees for useful comments which improved the paper.

Funding for this work was provided by the Direccion de Tecnologıa y Soporte Tecnico,

Repsol-YPF. O. Dorn acknowledges support by the Ministerio de Ciencia y Tecnologia,

Spain, through a Ramon y Cajal grant.

References

[1] Acar R and Vogel C R 1994, Analysis of total variation penalty methods, Inverse Problems 10,1217–1229.

[2] Agarwal B and Blunt M 2003, A streamline-based method for assisted history matching appliedto an Arabian Gulf field, SPE paper 84462 , SPE ATCE, Denver, CO, Oct. 5-8.

[3] Arridge S R 1999 Optical tomography in medical imaging Inverse Problems 15 (2) R41–R93.[4] Ascher U and Haber E 2004 Computational methods for large distributed parameter estimation

problems with possible discontinuities, Inverse Problems, Design and Optimization Symposium2004, Rio de Janeiro, Brazil.

[5] Batycky R.P., Blunt M.J. and Thiele M.R. 1997, A 3-D field scale streamline-based reservoirsimulator, SPERE 246

[6] Bulishev A E, Souvorov A E, Semenov S Y, Svenson R H, Nazarov A G, Sizov Y E and Tatsi GP 2000 Three-dimensional microwave tomography. Theory and computer experiments in scalarapproximation Inverse Problems 16 863-875.

[7] Caers J 2002 .... SPE Annual Technical Conference and Exhibition Paper 77429.[8] Carrera J 1987, State of the art of the inverse problem applied to the flow and solute transport

equations, Groundwater Flow and Quality Modeling, NATO ASI Ser.[9] Carter R D, Kemp Jr L F, Pierce A C and Williams D L 1974 Performance Matching With

Constraints SPE Journal 14 187-196.[10] Chan T F and Tai X 2003, Identification of discontinuous coefficients in elliptic problems using

total variation regularization, SIAM J. Sci. Comput. 25 (3), 881–904.[11] Chardaire-Riviere C, Chavent G, Jaffre J and Liu J. 1990 Multiscale representation for

simultaneous estimation of relative permeabilities and capillary pressure SPE Annual TechnicalConference and Exhibition Paper 20501.

[12] Chavent G, Dupuy M and Lemmonier C 1975 History Matching by Use of Optimal Control TheorySPE Journal 15 74-86.

[13] Chu L Reynolds A C and Oliver D S 1995 Computation of Sensitivity Coefficients for Conditioningthe Permeability Field to Well-Test Pressure Data In Situ 19 179.

[14] Crichlow H B 1976 Modern Reservoir Engineering-A Simulation Approach, Prentice Hall, NewJersey.

[15] Dennis J E and Schnabel R B 1996 Numerical Methods for Unconstrained Optimization and

Nonlinear Equations, (Reprint in the SIAM Classics in Applied Mathematics Series No. 16).

30

[16] Dierkes T, Dorn O, Natterer F, Palamodov V and Sielschott 2002 Frechet derivatives for somebilinear inverse problems SIAM J. Appl. Math. 62 2029-113.

[17] Dorn O 1997 A transport-backtransport method for optical tomography Inverse Problems 141107-1130.

[18] Dorn O, Bertete-Aguirre H, Berryman J G and Papanicolaou G C 1999 A nonlinear inversionmethod for 3D electromagnetic imaging using adjoint fields Inverse Problems 15 1523–1558.

[19] Dorn O, Miller E L and Rappaport C 2000, A shape reconstruction method for electromagnetictomography using adoint fields and level sets, Inverse Problems 16, 1119–1156.

[20] Engl H W, Hanke M and Neubauer A 1996 Regularization of Inverse Problems (Kluwer AcademicPublishers: Mathematics and Its Applications Series No. 375).

[21] Ewing R, Lin T, and Falk R 1987 Inverse and ill-posed problems in reservoir simulation, Inverseand Ill-Posed Problem Notes and Reports on Mathematics in Science and Engineering AcademicPress 483-497.

[22] Ewing R E, Pilant M S, Wade J G and Watson A T 1995 Identification and Control Problemsin Petroleum and Groundwater Modeling, Control Problems in Industry (I. Lasciecka and B.Morton, eds.), Progress in Systems and Control Theory, Birkhauser, 119-149.

[23] Gomez-Hernandez J J, Sahuquillo A and Capilla J E 1997 Stochastic simulation of transmissivityfields conditional to both transmissivity and piezometric data - I: Theory; II: Demonstration ona synthetic aquifer J. Hydrology 203 162-188.

[24] Groetsch C W 1993 Inverse Problems in the mathematical sciences (Vieweg)[25] Haber E, Ascher U and Oldenburg D 2000 On optimization techniques for solving nonlinear inverse

problems Inverse Problems 16 1263–1280.[26] Harding T J, Radcliffe N J and King P R 1996 Optimization of Production Strategies using

Stochastic Search Methods SPE European 3-D Reservoir Modelling Conference Paper 35518.[27] He Z, Datta-Gupta A and Yoon S 2002, Strealime-based production data integration with gravity

and changing fields conditions, Soc. Petro. Eng. Journal 423-436.[28] Jacquard P and Jain C 1965 Permeability Distribution From Field Pressure Data SPE Journal 5

281-294.[29] Jahns H O 1966, A Rapid Method for Obtaining a Two-Dimensional Reservoir Description from

Well Pressure Response Data, Soc. Pet. Eng. J. 237 315-327.[30] Kulkarni K N and Datta-Gupta A 2000 Estimating Relative Permeability from Production Data:

A Streamline Approach SPE Journal 5 402-411.[31] Landa J L, Kamal M M and Jenkins C.D. 1996 ....SPE Annual Technical Conference and Exhibition

Paper 36511.[32] Landa J L and Home R N 1997 SPE Annual Conference paper 38653.[33] Litman A, Lesselier D and Santosa F 1998 Reconstruction of a two-dimensional binary obstacle

by controlled evolution of a level-set, Inverse Problems 14, 685–706.[34] Mishra S and Parker S C 1989, Parameter Estimation for Couple Unsaturated Flow and Transport,

Water Resour. Res. 25 385-396.[35] Natterer F and Wubbeling F 1995 A propagation-backpropagation method for ultrasound

tomography Inverse Problems 11 1225-32.[36] Natterer F 2001 The Mathematics of Computerized Tomography (Reprint in SIAM Classics in

Applied Mathematics Series No. 32).[37] Natterer F and Wubbeling F 2001 Mathematical Methods in Image Reconstruction, SIAM

Monographs on Mathematical Modeling and Computation, Philadelphia.[38] Nocedal J and Wright S J 1999 Numerical Optimization, (Springer: New York).[39] Peaceman D.W. 1978 Interpretation of Well-Block Pressure in Numerical Reservoir Simulation,

SPE paper 6893, Soc. Petro. Eng. Journal 183-194 (June 1978) Trans. AIME 253[40] Pollock D W 1988 Semianalytical Computation of Path Lines for Finite Difference Models, Ground

Water, 26 743-750.[41] Rahon D, Edoa P F and Masmoudi M 1997 Inversion of Geological Shapes in Reservoir Engineering

31

using Well tests and History Matching, Proceedings SPE 38656, San Antonio, Oct 5–6, 1997.[42] Romero C and Carter J N 2002 ... Developments in Petroleum Science 51 323-363.[43] Rondi L and Santosa F 2001 Enhanced Electrical Impedance Tomography via the Mumford-Shah

Functional, ESAIM: Control, Optimization and Calculus of Variations 6, 517–538.[44] Sagar R K Kelkar M G and Thomson L G 1995 Reservoir Description by Integrating Well-Test

Data and Spatial Statistics SPEFE (December) 267.[45] Santosa F 1996 A level-set approach for inverse problems involving obstacles, ESAIM: Control,

Optimization and Calculus of Variations 1, 17–22.[46] Sheldon J W, Harris C D and Bavly D 1960 A method for general reservoir behavior simulation

on digital computers, paper SPE 1521-G, presented at SPE 35th Annual Meeting, Denver, Oct.2-5, 1960.

[47] Sun N and Yeh W G 1990 Coupled inverse problems in groundwater modelling 1. Sensitivityanalysis and parameter identification Water resources research 26, 2507-2525.

[48] Thomas G W 1982 Principles of Hydrocarbon Reservoir Simulation, Prentice-Hall.[49] Vasco D M Yoon S and Datta-Gupta A 1999 Integrating Dynamic Data Into High-Resolution

Reservoir Models Using Streamline-Based Analytic Sensitivity Coefficients SPE Journal 4 389-399.

[50] Vogel C R Computational Methods for Inverse Problems (SIAM series ’Frontiers in AppliedMathematics’ 2002)

[51] Vogeler M 2003 Reconstruction of the three-dimensional refractive index in electromagneticscattering by using a propagation-backpropagation method Inverse Problems 19 739-753.

[52] Wagner B J and Gorelick S M 1987, Optimal Groundwater Quality Managment under ParameterUncertainty, Water Resour. Res. 23 1162-1174.

[53] Wang Y and Kovscek A R 2000 Streamline Approach for History Matching Production Data SPEJournal 5 353-362.

[54] Wasserman M L, Emanuel A S and Seinfeld J H 1975 Practical Applications of Optimal-ControlTheory to History-Matching Multiphase Simulator Models SPE Journal 15 347-355.

[55] Watson A T, Seinfeld J H, Gavalas G R and Woo P T 1980 History matching in two-phasepetroleum reservoirs SPEJ 21 521-530.

[56] Watson A T, Ewing R E, Pilant M S and Wade J.G. 1995 Control Problems in Industry, Birkhauser,Boston.

[57] Wu Z, Reynolds A C and Oliver D S 1999 Conditioning Geostatical Models to Two-PhaseProduction Data SPE Journal 4 142-155.