Applied Mathematics, 2011, 2, 165-180
doi:10.4236/am.2011.22019 Published Online February 2011 (http://www.SciRP.org/journal/am)
Reservoir Multiscale Data Assimilation Using the
Ensemble Kalman Filter
Santha R. Akella
Department of Earth and Planetary Sciences, Johns Hopkins University, Baltimore, USA
Received June 22, 201 0; revised November 11, 2010; accepted November 15, 2010
Abstract
In this paper we propose a way to integrate data at different spatial scales using the ensemble Kalman filter
(EnKF), such that the finest scale data is sequentially estimated, subject to the available data at the coarse
scale (s), as an additional constraint. Relationship between various scales has been modeled via upscaling
techniques. The proposed coarse-scale EnKF algorithm is recursive and easily implementable. Our numerical
results with the coarse-scale data provide improved fine-scale field estimates when compared to the results
with regular EnKF (which did not incorporate the coarse-scale data). We also tested our algorithm with var-
ious precisions of the coarse-scale data to account for the inexact relationship between the fine and coarse
scale data. As expected, the results show that higher precision in the coarse-scale data, yielded improved es-
timates.
Keywords: Kalman Filter, Reservoir Engineering, Uncertainty Quantification, Multiscale Data
1. Introduction
The principal objective of data assimilation methods [1]
is to combine the information provided by measured data
and a (numerical) forecast model to obtain an improved
estimate of the system state (and parameters). Unlike va-
riational methods which require availability of complex
adjoint models for data assimilation, the ensemble Kal-
man filter (EnKF) can be quickly implemented and one
can also obtain uncertainty estimates via error variance-
covariance propagation; see [2] and references therein for
further details. The EnKF is a sequential Monte Carlo
method based on Bayes theorem. The method is increa-
singly being used for estimating unknown model state
and parameters in various geological and hydrological
models [3].
One of the major problems in subsurface characteriza-
tion is the huge uncertainty in the knowledge of hydro-
carbon reservoir permeability and porosity. Since the
flow of hydrocarbons such as oil and gas through the
subsurface formation critically depends on the geological
rock properties, it is important to accurately know these
properties. This article focuses on methods to obtain
more accurate quantification of the reservoir rock prop-
erties using measured data. Broadly speaking, the meas-
ured data used for description of reservoir porosity and
permeability characterization consist of static and dyna-
mic data. Static data such as well logs, core samples can
resolve heterogeneity at a scale of a few inches or feet
with high reliability. However, dynamic data such as fra-
ctional flow (defined as the ratio of the injection fluid to
the total fluid produ ced at the production wells; or water
cut), pressure transient and tracer test data typically scan
the length scales comparable to the inter-well distances.
Additional dynamic data such as time-lapse seismic im-
ages [4] can provide improved spatial sampling, but at a
lower precision. A majority of p reviou s studies on uncer-
tainty quantification in reservoir performance forecasting
using EnKF have mostly dealt with integration of dy-
namic data (for e.g., [5-7]). However it is widely recog-
nized that integration of additional multiscale data could
further reduce the uncertainty (see [8,9] and references
therein). With that perspective, integration of data at
coarse-and fine-scales, is an important objective and is
addressed in this paper. We use the EnKF to estimate
fine-scale fields for subsurface characterization. Also,
our method could be generalized to other sequential data
assimilation methods such as particle filtering (where,
rather than updating the ensemble members model state,
we update the probability assigned to each ensemble
member based on model data misfit). The main reason
why we used EnKF in this paper is because it requires

S. R. AKELLA
166
fewer ensemble members than the particle filters, see [3]
and references therein for further details.
In this paper, apart from the water cut data, we consi-
der coarse-scale measured data as well. The coarse-scale
data is assumed to be permeability at some specified lev-
el of precision. The unknown variables: permeability, at
the fine-scale, are estimated using a modification to the
EnKF algorithm, linking the data at different scales via
upscaling. It is important to resolve fine-scale heteroge-
neity for various purposes such as, enhanced oil recove-
ry, environmental remediation, etc. The main idea behind
upscaling is to obtain an effective coarse-scale permea-
bility which yields the same average response as that of
the underlying fine-scale field, locally. Single phase flow
upscaling procedures for two phase flow problem have
been discussed by many authors; see e.g., [10,11] and
also Section 3.1. We will refer to our proposed v ariant of
EnKF as coarse-scale EnKF. Assimilation using dynam-
ic data, such as fractional flow data only, is therefore
referred to as regular EnKF. The coarse-scale permeabil-
ity data could be obtained either from geo logic consider-
ation or coarse-scale inversion of dynamic, fractional
flow data on a coarse grid as considered in [8,12]. This
coarse-scale, static data can be viewed as a constraint,
which is to be satisfied up to the prescribed variance for
obtaining the fine-scale estimates in every data assimila-
tion cycle. Upscaling methods relate the solution at the
finescale to the coarse-scale, therefore in the Kalman
filter context, it amounts to modeling a nonlinear obser-
vation operator. In our coarse-scale EnKF approach, we
use the measured data in batches, such that the estimate
with one data becomes a prior while assimilating the
other observation (see Section 3 for further details).
Though in this paper we used coarse-scale data at only
one scale, our approach can be easily generalized to as-
similate data at multiple scales by appropriately model-
ing the linkage between different scales. Also, our ups-
caling method is independent of the underlying fine-
scale field.
For the purpose of self-conten dness an d notationa l cla-
rity, we briefly review the governing equations, sequen-
tial data assimilation using the ensemble Kalman filter in
Section 2, which is followed by a description of the coar-
se-scale EnKF algorithm (Section 3). For our numerical
results (Section 4), we consider a five-spot pattern, with
the injection well placed in the middle of a rectangular
domain and four production wells located at the vertices
of the rectangle. A reference case is used to provide true
data, which is randomly perturbed to obtain synthetic
measurements. A comparison of the regular EnKF with
the coarse-scale EnKF (Sections 4.1 and 4.2 respectively)
shows that using coarse-scale permeability data (via
coarse-scale EnKF) significantly improves the fine-scale
estimates as well as future fractional flow prediction.
2. Background
2.1. Fine-Scale Model
In this paper, we consider two-phase flow in a subsurface
formation under the assumption that the displacement is
dominated by viscous effects. For simplicity, we neglect
the effects of gravity, compressibility, and capillary p res-
sure, although our proposed approach is independent of
the choice of physical mechanisms. Also, porosity will
be considered to be constant. The two phases will be re-
ferred to as water and oil (or a non-aqueous phase liquid),
designated by subscripts w and o, respectively. We write
Darcy’s law for each phase as follows:



=; =,
rj
jf f
j
kS
νκprS κpr h


 (1)





 

 

=,= ,
== ;
rwrorw w
wo rwwroo
wo f
kS kSkS
SfS
kS kS
S
νν νSκpr νS
t







 

(2)
The above descriptions are referred to as the fine-scale
model of the two-phase flow problem. Here
f
κ is the
(fine-scale) permeability of the medium,


S  is the
total mobility,
j

denotes phase viscosity, pr is the
pressure, h is the source term,

and S denote poros-
ity and water saturation (volume fraction), respectively.
2.2. Sequential Estimation using EnKF
Using dynamic measured data such as water cut, we can
sequentially estimate the unknown parameters (permea-
bility, porosity, etc.) and state v ariables such as pressure,
water saturation (two-phase flow) and production data at
well locations using the EnKF as discussed in [5,7,13].
Following these previous works, in th is paper we assume
that the only dynamic data available is water cut data,
and that porosity is known. The combined state-parameter
to be estimated are given by

=,,,.
T
fc
ln κ


Ψpr S W
Where


ln

is natural logarithm of permeability field
and c W denotes water cut; in order to distinguish ob-
served water cut from model predicted water cut, now
onwards we will denote the observed water cut data o
c
W,
by
y
.
The EnKF introduced in [14] is a sequential Monte
Carlo method where an ensemble of model states evolve
in state-space with mean as the best estimate and spread
of the ensemble as the error covariance, as summarized
in the following steps. Each of the ensemble members is

S. R. AKELLA
167
forecasted independently (in this work, we neglected mo-
deling errors),
 
1=,
ii
nn
F


ΨΨ
(3)
where


F
 is the forecast operator (Equations (1), (2)),
superscript

i denotes the th i ensemble member;
now onwards we will drop the time subscript. The en-
semble mean and covariance are defi ned as,

=1
1
=,
Nei
i
e
N
ΨΨ
(4)

1'',
1
T
f
e
N
 PAA (5)
where
 


12
'=,,,,
Ne
Abb b
 
=,
ii
bΨΨ and
e
N is the number of ensemble members. The observa-
tion vector for each ensemble member is given by,
 
=,
ii
t
Hν



yΨ (6)
where t


HΨ is the observed data from the truth and

i
ν represents observational errors, which are i.i.d.
samples [15] from a normal distribution with zero mean
and variance, .R We note that if only the water cut data
is being measured, the mapping from model-to-observa-
ti o n al space, H is trivially equal to


,000I since

=,,,.
T
c
ln κ 

Ψpr S W
The forecasted ensemble (Equation (3)) is updated by
assimilating the observed data,

  

,
iii i

 

ΨΨKy HΨ (7)
where K is the Kalman gain, given by
1
=.
fT fT 



KPHHPH R
Computationally efficient implementation of the EnKF
is discussed for e.g., in [2,16]. We use the above set of
corrected ensemble states,




0
e
N
i
i
Ψ in the simulation
model (Equation (3)) to predict until the next set of ob-
servational data is available.
3. Coarse-Scale Constrained EnKF
The EnKF presented so far, used only the dynamic, pro-
duction data (wat er cut) ,
y
with error =t ν





yHΨ,

,ν0R

to update the ensemble (Equation (7)). In
addition to ,
y
if we are also given static data (as men-
tioned in the Introduction), which is another set of inde-
pendently measured data, .z Assuming that the corres-
ponding measurement error is given by
=,
t
ω  zUΨ

,;ω0Q

:.UΨz Its like-
lihood is given by





1
1
exp .
2
T
z
p


 



zΨzUΨQzUΨ



(8)
If this static data z, corresponds to coarse-scale per-
meability data [12,8], then


=.U000Where :,
f
c
κκ
is a nonlinear mapping that maps the fine-scale permea-
bility field (
f
κ) to coarse-scale field (c κ) via an upscal-
ing procedure (e.g., [17,18]), details are provided in Sec-
tion 3.1. Note that by definition, the errors in water-cut
data, y and coarse-scale permeability, c κ data don’t
influence each other, since they are entirely differently
measured quantities.
Now, our goal is to obtain an estimate which is based
on both of the above dynamic and static data. The like-
lihood of y is given by





1
1
exp .
2
T
y
p






yΨyHΨRyHΨ



The probability distribution function (pdf) of the pre-
dicted ensemble,




1
1
exp ,
2
Tf
f
p


 



ΨΨΨPΨΨ


where Ψ and
f
P are the predicted ensemble mean
and covariance respectively (Equations (4) and (5)).
Then, using Bayes theorem, we obtain




 






|
,,
|, =,
,
=,
,
=.
p
p
pp
pp
pp
p
ppp


Ψy
Ψzy
Ψzy zy
zyΨΨzyΨΨ
zy
zΨyΨΨ

The last term in above equation implies that the two
independent data, y and z can be sequentially assimilated
in the following two steps. We first assimilate observa-
tion y to obtain an intermediate ensemble,




0
e
N
i
i

Ψ as
discussed in Section 2. 2.







exp ,
fy
ppyΨΨ  (9)
This intermediate ensemble and likelihood in Equation
(8), can then be combined to obtain the final estimate




0
e
N
i
i

Ψ.



 

,exp ,
fyz
ppΨΨzy  (10)
Therefore, in a least-squared sense, the final estimate
maximizes the posterior pdf


,
p
zyΨ which corres-

S. R. AKELLA
168
ponds to the minimum of =.
z
yf
  See Ap-
pendix A, for further details (where we show that the
solution


i
Ψ corresponds to the minimum of , for
any th i ensemble member). The coarse-scale EnKF
algorithm is detailed in Appendix B.
3.1. Upscaling Methods
In brief, the main idea behind upscaling of absolute
fine-scale permeability is to obtain effective coarse-scale
permeability for each coarse-grid block. Once the ups-
caled absolute permeability is computed, the original eq-
uations are solved on the coarse-grid, without changing
the form of relative permeability curves. This is an inex-
pensive calculation, since the pressure update involves
only solving the pressure equation on the coarse-grid,
and one can take larger time step for solving the trans-
port equation. In our numerical simulations, the fine-grid
is coarsened 10 times in each direction. These kinds of
upscaling techniques in conjunction with the upscaling of
absolute permeability have been used in groundwater
applications (see e.g., [18]).
The link between the coarse and the fine-scale per-
meability fields is usually nontrivial because one needs
to take into account the effects of all the scales present at
the fine level. In the past simple arithmetic, harmonic or
power averages have been used to link properties at var-
ious scales. These averages can be reasonable for low he-
terogeneities or for volumetric properties such as poros-
ity. For permeabilities, simple averaging can lead to in-
accurate and misleading results. In this paper we use the
flow-based upscaling methods using local solutions of
the equations [17,19 ].
First, we briefly describe flow based upscaling me-
thods. Consider the fine-scale permeability that is de-
fined in the domain with underlying fine grid as shown
in Figure 1. On the same graph we illustrate a coarse-
scale partition of the domain. To calculate the coarse-
scale permeability field at this level we need to deter-
mine it for each coarse block, c
. The coarse block per-
meability can be d efined both using the so lutions of local
or global problems. The main idea used to calculate the
coarse-scale permeability is that it should deliver the
same average response as that of the underlying fine-
scale problem, locally. The calculation of the coarse-
scale permeability based on local solutions is schemati-
cally depicted in Figure 1. For each coarse domain c

we solve the local problems




=0,
fj
κ   x (11)
with some coarse-scale boundary conditions.
One of such boundary conditions is given by =1
j

and =0
j

on the opposite sides along the direction
j
e
and no flow boundary conditions on all other sides, al-
ternatively, =
j
j
x

on .
c

 For these boundary condi-
tions the coarse-scale permeability is given by

=1 ,
cj lcfj l
c
κκdx 

 

eexe (12)
0


1




0div κx  
c

Figure 1. Schematic illustration of upscaling (not to scale): bold lines indicate a coarse-scale partitioning, while thin lines
show a fine-scale partitioning within coarse-grid cells. In this paper, we upscaled a 50 × 50 fine-grid to a 5 × 5 coarse-grid.

S. R. AKELLA
169
where
j

is the solution of Equation (11) with prescri-
bed boundary conditions. Various boundary condition can
have some influence on the accuracy of the calculations,
including periodic, Dirichlet, etc. These issues have been
discussed for e.g., in [19]. In particular, for determining
the coarse-scale permeability field one can choose local
domains that are larger than target coarse block, c

, for
Equation (11). Further Equation (12) is used in the do-
main c , where
j

are computed in the larger do-
mains with correct scaling (see [19]). This way one re-
duces the effects of the artificial boundary conditions
imposed on c  (for details see [19]).
The use of the local solutions Equation (11) for deter-
mining the permeability field at different scales gives
non-explicit relation for conditional distribution. We
denote by

the local operator that maps the local
fine-scale permeability field
f
κ into c κ, defined as
above. For our computations we assume

=,
cf
κκ

 (13)
where

are some random fluctuations that represent
inaccuracies in the coarse-scale permeability. In reality,
since we do not have the fine-scale field,
f
κ available,
it is difficult to characterize the exact (nature of the) er-
ror in upscaling. However, one of the sources of these
fluctuations are the errors associated with solving inv erse
problems on the coarse grid. The other source of the in-
accuracies of measured coarse-scale permeability is due
to the fact that the inversion on the coarse grid does not
take into account the adequate form of the coarse-scale
models. Here we assumed these errors to be normally
distributed (further details follow in Section 4.2).
4. Numerical Results
For our numerical tests with the coarse-scale EnKF algo-
rithm, we use a 50 50 fine grid (dimensionless do-
main size 50 50). We consider the coarse-scale per-
meability, which could be obtained by coarse-scale in-
version of fractional flow data on a coarse grid [12,20].
This coarse-scale field could be thought as static data,
which is to be honored as constraint (up to the data va-
riance) in Equation (8), hence we need to always assimi-
late it in our coarse-scale EnKF algorithm.
An initial ensemble with different permeability reali-
zations was generated using the sequential Gaussian si-
mulation (SGSIM)1 [22]. We specified a Gaussian vario-
gram model with a correlation length of 20 gridblocks in
the x-direction and 5 gridblocks in the y-direction; one of
the realizations is used as the reference field (depicted in
Figure 2). The fractional flow will be calculated based
on the fine-scale model in Section 2.1. Porosity (

) is
assumed to be equal to 0.15 for all grid blocks. For sim-
plicity, relative permeabilities, rj k are assumed to be
linear functions of water saturation (S):


=,
rw
kS S


=1 .
ro
kS S

One injection well at the center of the
field (injection rate: 71.4 m3/day) and four producing
wells at the four corners (all with equal rate of 17.85
m3/day) were considered. The model equations are sol-
ved with no flow boundary conditions, zero initial water
saturation, and discretizing the transport equation using
first order upwind finite volume method. In Figure 3, we
provide the predicted fractional flow for 256 initial en-
semble members along with the true fractional flow (ob-
tained from true permeability field).
To compare our proposed coarse-scale constrained
EnKF results with the regular EnKF we will use the fol-
lowing mean 2 L-norm error. Since we know the true
(fine and coarse-scale) field for our synthetic problem,
i.e., the true permeability field, denoting it by ,
true
κ the
error for any ensemble member is
 
=,=1,2,,.
ii
true
e
κκiNe
Consider the 2 L norm of the error for each member,
 
2
2=,
ii
j
j





ee using which we define the mean
2
L error as

2
=1
1
=,
Ne i
i
e
N
ee (14)
so that e gives us an indication of the distance of entire
ensemble from the true solution .
true
κ Since after every
Figure 2. Natural logarithm of 50 × 50 true permeability
field.
1For reservoir simulation applications, the SGSIM has been used [21,
13] for generating initial ensemble members. This approach yields
independent and identically distributed multivariate normal rando
m
fields (conditioned to well log data)

S. R. AKELLA
170
Figure 3. Fractional flow prediction with 256 initial ensemble members (no data assimilation); ensemble members (green
dots), ensemble mean (blue crosses) compared with true water cut data (red open circles).
observation, we have updated ensemble members, there-
fore we can monitor the variation of e over the time of
assimilation; the success of assimilation can therefore be
related to the decrease in e.
4.1. EnKF with Fractional Flow Data Only
We start with a presentation of results with regular
EnKF, assimilating only water cut data. Next we will
discuss results with the coarse-scale EnKF.
The water cut data from the reference field is assumed
to be available every 200 days, with mean zero and
standard deviation of 0.01 (therefore 1/24 =0.01 ,RI
where 4 I is unit matrix of size 44, since there are
four producing wells). The observed data is assumed to
be available up to 2400 days, hence we will perform as-
similation between 200 and 2400 days. A prediction
beyond interval of data assimilation, up to 4000 days is
also provided. We selected an ensemble of size 256 for
presenting our data assimilation results.
We assimilated the above described measured data,
and using the assimilated permeability field, in Figure 4
we plot the assimilated water cut data along with the true
data. Comparing with the initial forecast in Figure 3, we
observe that the assimilated ensemble better envelopes
the true data. We compare the initial permeability field
before assimilation (Figure 5(a)-(d)) for a few ensemble
members with the true field in Figure 2 and with those
obtained after assimilation in Figure 6(a)-(d); note that
the central, South East-North West channel is prominent
but the features at the South West and North East corners
are not well captured. Therefore assimilation of only
water cut data helps in identifying only some of the im-
portant features.
4.2. Coarse-Scale Constrained EnKF with
Fractional Flow and Coarse-Scale
Permeability Data
In addition to water cut production data, the coarse-scale

S. R. AKELLA
171
Figure 4. Water cut prediction using assimilated (regular EnKF for 2400 days) ensemble members, note the improved fit of
ensemble when compared to that in Figure 3.
permeability data, as described in Section 3.1 has been
used as additional measured data. Flow-based upscaling
of reference permeability field is used as a proxy for in-
verted coarse field. Following our previous notation, this
coarse-scale permeability data will be denoted by z
(Equation (8)). The mapping between state variables (at
fine-scale) and observations (at coarse-scale) is given by


=,U000


, denotes flow-based upscaling.
Exactly as in the previous section, we prescribed the
same frequency (of availability) and precision, R for the
fractional flow data. Since we use coarse-scale permea-
bility as additional data, it is to be assimilated whenever
we assimilate water cut data. A 5 × 5 coarse-scale data
with mean zero and variance, 25
=QqI (we will present
results with =4,2,1,0.5q and 0.1,) so that we can con-
sider the impact of coarse-scale data precision. In Figure
7 we plot the variation of mean 2 L error, e (Equation
(14)) with observation time, at the coarse-scale for dif-
ferent values of .q Figures 8(a) and (b) depict the cor-
relation between coarse-scale ensemble mean and true
fields for q = 4 and 0.1, respectively. As the precision of
coarse-scale data is increased, i.e., for smaller variance,
we observe a larger decrease in coarse-scale mean 2 L
error and higher correlation with true coarse-scale field
(correlation coefficient for =4,2,1,0.1q respectively
are 0.976, 0.992, 0.995, 0.999), b ecause smaller variance
Q implies more stricter coarse-scale data constraint in
Equation (8). Figure 9(a)-(d) depict the fractional flow
using the final permeability field after assimilation, for
different coarse-scale data precisions. Figures 7 and
8(a)-(b) show that the coarse-scale data is being more
accurately assimilated as it is made more precise. Also,
notice the improv ed fit of ense mble predictio n to the tru e
data, for more precise coarse-scale data; also when com-
pared to the regular EnKF results in Figure 4.
Now we discuss the results regarding fine-scale field.
In Figure 10 we plot the fine-scale mean 2
L error for
different values of ;q the coarse-scale EnKF yields
much lesser error than regular EnKF which assimilated
only fractional flow data. The correlation coefficient

S. R. AKELLA
172
(a) (b)
(c) (d)
Figure 5. Log permeabilities of a few i-th. initial ensemble members (before data assimilation); left-right, (a) i = 50, (b) 100, (c)
150, (d) 200.
(a) (b)
(c) (d)
Figure 6. Same as above, but after assimilating water cut data with regular EnKF.

S. R. AKELLA
173
Figure 7. Decrease in e computed at coarse-scale, as data (fractional flow and 5 × 5 coarse-scale permeability data at variance,
Q= I
q
) is assimilated using the coarse-scale EnKF algorithm.
Figure 8. Correlation between coarse-scale ensemble mean and true permeability after assimilation for low and high preci-
sion in coarse data; (a) and (b): q = 4, 0.1.

S. R. AKELLA
174
(a)
(b)

S. R. AKELLA
175
(c)
(d)
Figure 9. Same as in Figure 4, but using coarse-scale EnKF for data assimilation; clockwise, (a)-(d): q = 4, 2, 1, 0.1.

S. R. AKELLA
176
Figure 10. Same as in Figure 7, but at fine-scale, also shown is the error obtained with assimilation of fractional flow data only.
between fine-scale ensemble mean and true fields, after
assimilating using regular EnKF is equal to 0.409, while
with the coarse-scale EnKF for =4,2,1,0.1,q in that
order were 0.644, 0.652, 0.638 and 0.626; note higher
correlation with the coarse-scale EnKF. We observe that
higher precision, i.e. , lower q does not necessarily imply
least e or highest correlation, since highly precise
coarse-scale data is relatively more weighted than the
fractional flow data. Optimal value for the coarse-scale
data variance can be obtain ed by prio r calculation, which
will be addressed in a future study.
The final permeability field, for a few en semble mem-
bers after assimilating with coarse-scale EnKF, for
=1q is shown in Figure 11(a)-(d); all shown samples
seem to be more closer to the true field (Figure 2) than
those obtained with regular EnKF (Figure 6(a)-(d)). In
particular note that the low permeability region at the
North East and high permeability at the South West cor-
ners are well capt ured.
5 Conclusions
The EnKF is increasingly being used for subsurface cha-
racterization in various geological and groundwater ap-
plications to identify fine-scale state and parameters. So
far, various implementations have been based on using
dynamic, production data, such as water cut, well pres-
sures, etc, for sequential data assimilation. Only recently
dynamic data other than production data has been consi-
dered in the EnKF context ([23,24]), nevertheless the
observed data to be assimilated was assumed to be at the
finest scale. For a number of reasons, it is widely recog-
nized that usage of additional multiscale data could fur-
ther reduce the uncertainty at the fine scale. This is fur-
ther motivated by the increasing popularity of coarse-
scale modeling. In this light, here we proposed assimila-
tion of coarse-scale data along with water cut, production
data using coarse-scale EnKF. The modification to the
regular EnKF (assimilation of only water cut data) is
completely recursive and easily implementable. The re-
lation between fine and coarse scales has been modeled
via physics based upscaling, which could be thought of
as a nonlinear observation operator linking the coarse-
scale data to the unknown fine-scale variables. In addi-
tion, the proposed methodology could be used in any
other sequential dat a assimilation m ethod as wel l and al so

S. R. AKELLA
177
(a) (b)
(c) (d)
Figure 11. Same as in Figure 6, but assimilated using coarse-scale EnKF with q=1 for the variance of coarse-scale permeabil-
ity data.
also with any other upscali n g method.
The coarse-scale EnKF was tested and compared with
the regular EnKF for a 2D synthetic 50 50 heteroge-
nuous true field. We considered coarse-scale permeabil-
ity data as additional data on a 55 coarse grid. This
coarse-scale data was always assimilated along with wa-
ter cut data. The data variance was varied from low to
high, to study its impact on assimilated results. In all
cases, we observed that the assimilated, ensemble mean
coarse-scale field for all variances was highly correlated
to the true coarse-scale field. In addition, lower variance
in the coarse-scale data yielded higher correlation. The
water cut data was better h onored, both for higher preci-
sion of coarse data, and when compared with regular
EnKF. As for the fine-scale permeability field, the
coarse-scale EnKF yielded lesser error in an averaged
2
L norm, error taken w.r.t. the reference field. In addi-
tion, a few individual samples were picked to compare
the assimilated fields with different EnKF procedures;
experiment with coarse-scale permeability data provided
final samples which captured most closely the features in
the reference fine-scale field.
Though in our current paper we used only one coarse-
scale, the proposed method can be easily implemented to
integrate as many scales as required by the available data
and is independent of the underlying fine-scale field.
Based on our results we conclude that there could be a
singinificant improvement in subsurface characterization
if accurate and additional data at coarse-scales is availa-
ble. In future we plan to study the nature of errors in-
volved in coarse-scale data and upscaling and in-turn
their influence on the proposed coarse-scale EnKF.
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S. R. AKELLA
179
Appendix A: Two Step Coarse-Scale
Constrained Kalman Filter Estimate
From Section 3,
  
1
1
=,
2
Tf
f

ΨΨ PΨΨ
and




1
1
=.
2
T
y

yHΨRyHΨ
For notational simplicity we will denote Ψ
μ
as
μ
and denote
f
P by .B
Step 1 (minimize
f
y
):
First we minimize the sum, 1=.
f
y
 The gra-
dient2 of above quadratic cost functional with respect to
(w.r.t.) Ψ is given by
 


11
1=.
T

ΨBΨHRy HΨμ
Then the minimizer 
μ
, of 1  satisfies (we assume
H to be linear)
 
11
=0.
T
 BHRyH  μμμ
Rearranging the above equation we get,
111 1
=.
TT 



B HRHBHRy

μμ (15)
Note that the Hessian of 1  w.r.t. Ψ is given by
11T
BHRH and for linear quadratic cost functionals,
the Hessian inverse is equal to the error covariance ma-
trix. Therefore the error covariance matrix, B
 for 
μ
is given by 1
11
.
T 



BB HRH
 (16)
Step 2 (minimi ze
g
z
):
We use ,

μ
B
 in



1
1.
2
T
g

 ΨBΨ


μ
μ
   
1
1
=.
2
T
zQ
zUΨzUΨ
Therefore the minimum , 
μ
of
g
z
 satisfies
 
11
11
ˆ.
TT






B UQUBUQz

 μμ
Using Equations (16) and (15) we can rewrite above as

1
11 1
111
.. (14)
ˆ
.
TT
TT
rh s ofeqation

 

 

 
B
BHRHUQU
BHRyUQz



   
μ
μ
It is trivial to show that ˆ
μ
also satisfies
=0.
fyz



Ψ
Therefore the two step method to obtain the final es-
timate ˆ
μ
, gives the same results as a one shot approach
of minimizing
f
yz

.
Appendix B: The Coarse-Scale EnKF
Algorithm
Run the simulation model up to a particular observation
time for entire ensemble to get predicted samples:



=1 ,
Ne
i
i
Ψ
 


12
=,,, .
Ne
AΨΨ Ψ
1) Step 1: Using measured water cut data y with vari-
ance ,R get updated ensemble:



=1 ,
Ne
i
i
Ψ
 Step 1.1 Find ensemble mean (Equation (4)), .Ψ
 Step 1.2 Subtract deviation from mean
 



12
'=,,,,
Ne
Abb b

=.
ii
bΨΨ
 Step 1.3 Apply H to each column of 'A to get
='.SHA i.e., simply pick the water cut deviations
in 'A.
 Step 1.4 for =1,2, ,,
e
iN
Sample


i.i.d. ,.
i
ν0R
 
=,
ii
νyy




12
12 =,,, ,
Ne
νν νR
 



12
=,,, ,
Ne
Ddd d
 
=;
ii i
c
dyW

i
c
W
is predicted water cut for each ensemble member.
end for
 Step 1.5 Compute SVD 1/2=.
L
R



SRX X
Get ˆ

retaining first few singular values which
explain most variability in ,  corresponding left
singular vectors: .
L
X
 Step 1.6 Update ensem ble: Eqnuation ( 7),
 







12
,,,
e
N
AΨΨ Ψ,

2
ˆˆ ˆ.
TT
LL

 
AAASX XD
2) Step 2: Using coarse-scale data z with variance Q,
get updated ensemble:




=1 .
Ne
i
i
Ψ
 Step 2.1 Compute coarse-scale ensemble prediction:



,1,2,,.
i
i
e
iNuUΨ
 Step 2.2 Coarse-scale mean:

=1
1
'= .
Ni
e
i
e
Nuμ
 Step 2.3 Coarse-scale deviations:




12
'=,, ,,
Ne
Ssss
 
='.
ii
su
μ
 Step 2.4 Repeat Step 1.4, using coarse-scale mea-
2We note in passing that B and R are covariance matrices and are posi-
tive definite by construction, and hence for our derivation purposes, are
formally invertible.

S. R. AKELLA
180
surement. for =1,2, ,,
e
iN
Sample


i.i.d.,.
i
ω0R
 
=,
ii
ωzz
 


12
1/2=,,, ,
Ne
ωω ωQ



12
'=,, ,,
Ne
Ddd d
 
=.
iii
dzu
end for
 Step 2.5 Compute SVD 1/2 '=.
L
R



SQ XX
Get ˆ  and
L
X as in step 1.5
 Step 2.6 Compute fine-scale mean:

=1
1
'= .
Ni
e
i
e
N
μΨ

 Step 2.7 Compute fine-scale deviations:
 


12
=,,,,
Ne

Abbb
 
ii
bΨ

μ
.
 Step 2.8 Update ensem ble:
 





12
,,, ,
Ne



AΨΨΨ



2
ˆˆ ˆ
TT
LL
 
 AAA SXXD .
Remark 1:
Note that steps 2.6 and 2.7 in above algorithm approx-
imate the intermediate fine-scale error covariance


1.
1
T
f
e
N
 PAA
Remark 2:
Steps 2.1-2 . 3 a cc omplish3
'= .

SUA
Note that the above algorithm is independent of the
choice of upscaling procedure and also, we can use the
same algorithm for different kinds of coarse-scale ob-
served data (if available).
Remark 3:
Note tha t the above coar se-scale cons trained EnKF algo -
rithm can be readily extended to incorporate data at mul-
tiple coarse scales, with appropriate upscaling procedure
in .U To elaborate, if we had another independent data
at a scale different from ,z we use the estimates
(




=1 .
Ne
i
i
Ψ) obtained using ,z as intermediate solution,
repeat Step 2 to assimilate the data at another scale.
3As noted in [2], this approach of accounting for nonlinear observations
operator U, works well, as long as U is weakly nonlinear and a mono-
tonic function of model variables Ψ.