Engineering, 2011, 3, 538-548
doi:10.4236/eng.2011.35063 Published Online May 2011 (
Copyright © 2011 SciRes. ENG
Updating Geologic Models Using Ensemble Kalman Filter
for Water Coning Control
Cesar A. Mantilla, Sanjay Srinivasan, Quoc P. Nguyen
The University of Texas at Austin, Austin, USA
Received February 5, 2011; revised March 15, 2011; accepted April 19, 2011
This study investigated the feasibility of updating prior uncertain geologic models using Ensemble Kalman
filter for controlling water coning problems in horizontal wells. Current downhole data acquisition technol-
ogy allows continuous updating of the reservoir models and real-time control of well operations. Ensemble
Kalman Filter is a model updating algorithm that permits rapid assimilation of production response for res-
ervoir model updating and uncertainty assessment. The effect of the type and amount of production data on
the updated geologic models was investigated first through a synthetic reservoir model, and then imple-
mented on a laboratory experiment that simulated the production of a horizontal well affected by water con-
ing. The worth of periodic model updating for optimized production and oil recovery is demonstrated.
Keywords: Ensemble Kalman Filter, Reservoir Model Updating, Oil Production Optimization
1. Introduction
The productive life of a well completed over an active
aquifer is strongly affected by water coning. The evolu-
tion of the water cone is driven by non-uniform draw-
down pressure near the wellbore. The irreversibility of
water coning mandates its early detection, modeling and
control before and/or during water breakthrough. Recent
development in technologies for in-situ monitoring using
pressure and temperature distributed sensors allow ac-
quisition of real-time production data that carry valuable
information about reservoir properties and the water
conning process in the vicinity of the well. The acquired
data can be used to maximize oil-water ratio through
model based dynamic optimal control of pressure draw-
down along a production well and/or inflow allocation
among the wells draining the same reservoir.
Water coning is a local phenomenon governed by the
gradient of the flow potential in the vicinity of the well.
Since the flux variations in the far-field have a relatively
insignificant influence on the characteristics of coning, a
full reservoir model is not required to control this local
phenomenon. Muskat [1] presented an analytical solution
for the critical flow rate of a well in a homogeneous and
isotropic reservoir below which a stable water cone is
formed. The velocity of cone propagation is partially
determined by reservoir permeability as shown in (1).
tot c
hPlngbh P
 
 (1)
h is Cone height from original water-oil contact, ΔP is
the pressure difference from the tip of the cone to the
wellbore, µ is the oil viscosity, k the average permeabil-
ity, b is the distance from the well to original water-oil
contact, Δρ is the oil-water density difference, g is the
gravity force, and Pc is the capillary pressure.
Several analytical and numerical correlations that pre-
dict water coning performance accounting for more
complex geometries and heterogeneity have been devel-
oped. A comprehensive literature survey on water coning
was published by Alikhan and Ali [2]. Despite the exten-
sive work done on water coning modeling, most of the
existing models for this process are deterministic in the
sense that the reservoir properties (such as permeability k
in (1)) are assumed to be known. However, in most cases,
the knowledge of reservoir properties is limited, and un-
certainty in model parameters causes deviations in the
prediction of these physical quantities. Updating of res-
ervoir properties based on acquired dynamic data is cru-
cial. Conventional history matching techniques may not
be appropriate for this purpose since they are computa-
tionally expensive and time-consuming for real-time
integration of data. Ensemble Kalman filter has demon-
strated to be a rapid tool for subsurface process identifi-
cation and control. Ensemble Kalman filter (EnKF) has
been applied for water flooding optimization ([3,4]), un-
certainty assessment in reservoir description [5], history
matching [6], seismic incorporation [7], and other areas.
In this paper, we present an iterative form of EnKF
used in conjunction with dynamic process optimization.
The effect of data type and its diversity on the adequacy
of EnKF is first investigated through a synthetic case of
water coning, and then demonstrated on a laboratory
flow model. The results from this study provide a guide-
line for development of an effective monitoring system
for real-time system characterization. The integration of
such a monitoring system into an optimal process control
system is also demonstrated through the lab-scale water
conning problem.
A schematic flow chart summarizing the system model
updating procedure is presented in Figure 1. This pro-
cedure is connected to an optimization module as part of
a closed-loop feedback control system. The system,
which is composed of the reservoir, wells and surface
facilities is modeled using a reservoir simulator. Conven-
tionally, the geologic model is constructed using petro-
physical data such as permeability and porosity obtained
from well logs and/or well tests. However, the inherent
uncertainty in the distribution of these properties
throughout the reservoir results in uncertain predictions
of reservoir performance. EnKF enables a rapid geologic
model updating for short-term or local problems such as
water coning, and thus allow real-time optimal control of
operating conditions.
A brief description of ensemble Kalman Filter is pre-
sented here; a detailed derivation of EnKF is presented
by Gu and Oliver [4]. Denote NR as the numbers of
equally probable realizations of the distribution of an
uncertain reservoir property (i.e. permeability k in this
study) and NB as the number of gridblocks in the reser-
voir model. The ensemble of NR realizations is generated
from the prior information of the reservoir using sequen-
tial Gaussian simulation. Each realization is input into
the reservoir simulator and run for one time step (ti),
producing a set of NRes production data, denoted by the
vector d (including oil, water, gas rates, etc.). The co-
variance between the NRes responses and the NB grid
block permeability can be calculated using the NR reali-
zations. This covariance matrix forms the basis for the
Kalman Gain matrix KG. The model represented by the
vector K is updated using the following linear model:
new oldref
where dref is the vector containing the set of production
data of the real or reference reservoir, Knew is the vector
with the updated variable corresponding to the ith realiza-
tion, Kold
i is the vector with the prior variable of the ith
realization, di is the vector with the set of production data
of the ith realization. The covariance matrix between the
permeability values at each location and the production
responses from the ensemble responses is formed. Then,
each element of the covariance matrix is standardized by
the variance of the production responses of the ensemble
to compute the Kalman Gain matrix.
Figure 1. Model Updating flow chart.
Copyright © 2011 SciRes. ENG
It is expected that the ensemble of updated models
matches better with the true data. However, in practice
when the updated models are again processed through
the flow simulator, the corresponding responses continue
to exhibit deviation from the measurements. This mis-
match reflects the inadequacy of the covariance-driven
updating scheme for capturing the non-linearity of the
transfer function (simulator) model. In order to achieve a
better match, the updating procedure is continued itera-
tively until a tolerable deviation is obtained. The whole
process is then repeated for the next time step (tk+1).
Once the deviation of updated model responses from the
measured data is acceptable, it is transferred to the opti-
mization module, and the well settings are re-evaluated
to maximize the production for the remaining time. The
wells are operated with the new operating conditions and
the monitoring/data acquisition step is continued.
The type and amount of data included in the responses
(d) plays an important role in the update of the uncertain
variable (K). More data assimilated not necessarily im-
prove the estimation. If the responses are highly corre-
lated to each other, as in the case of two downhole pres-
sure sensors located too close to each other, they can
bring redundant information with the natural noise asso-
ciated to a measurement without improving the estima-
tion. On the other hand, if few data are used, it becomes
difficult to update the model properly. This type of sensi-
tivity analysis would be useful to choose the best loca-
tion of the sensors that improves the updating process
without being redundant.
In this case, the types of data used are oil and water
rates. Intuitively, each type of data carries information
from different parts of the reservoir, for example water
rates are correlated to properties of an associated aquifer.
In other cases, data can be uncorrelated to the spatial
variations in reservoir properties and hence model up-
dated including such data will not necessarily improve
the predictability of the model. A sensitivity analysis on
the type of data is recommended to select the responses
according to the model updating objectives.
2. Synthetic Flow Model with a Horizontal
2.1. System Description
A three dimensional rectangular synthetic reservoir was
created to run sensitivity cases on the type and amount of
data to be used for updating. Since water coning is a lo-
cal phenomenon unaffected by the pressure conditions in
the remaining reservoir, the near wellbore area is as-
sumed surrounded by a constant pressure boundary con-
dition. This is simulated by placing a set of oil injectors
replenishing the oil produced by the producers. The grid
is composed of 25 × 20 × 20 blocks in x, y and z direc-
tions respectively. Local grid refinement is performed
near the well. A constant porosity value equal to 0.1 is
set. A horizontal well is completed in the middle of the
layer and spans a length of six grid blocks in the x direc-
tion for a total length of 240 ft. The dimensionless well
length (Lwell/Lreservoir) is 0.1.
The well has 3 segments whose center points are
regularly spaced 80 ft apart. The well production is con-
strained by the bottomhole pressure in the toe (P1), and
the bottomhole pressure at the other two segments (P2
and P3) is calculated from frictional pressure gradient.
An active aquifer is beneath the oil reservoir, and the
oil and water rate data for each segment are used for up-
dating the prior geological model. The reservoir property
to be updated is the permeability value in all grid-blocks
which is spatially distributed using a semi-variogram
with a range of 15 grid-blocks in x. The average perme-
ability of the reference reservoir is 100 mD.
An initial suite of 40 realizations was generated using
sequential Gaussian simulation sampling from a log-
normal distribution with mean 180 mD and standard de-
viation 50 mD. The permeability field was updated in the
normal (log-transformed) space, so that it is guaranteed
to be positive after back-transformation to the original
space. As shown by Evensen [8], the distribution of the
variable of interest in Ensemble Kalman Filter should be
Normal for the updating equation to work properly.
2.2. Results and Discussion Synthetic Flow
Water broke in the well after 726 days of net oil produc-
tion, initially through segment P2. Water arrives first to
the central perforation because the apex of the cone is
centered within the well path, as observed on the satura-
tion map shown in Figure 2. Once water breaks into P2,
the oil rate decreases as consequence of the reduction in
the relative permeability to oil in that grid-block (Figure
3); the other two segments maintain the oil rate for a
short time until water makes its way to those grid blocks.
The total duration of the simulation was 1500 days. Al-
though the reservoir simulator reports data every week,
for simplicity only 5 time steps of 300 days each were
used to update the model. Before water breakthrough the
model was updated by only using oil rates. Subsequently
water rates were also included as part of the sensitivity
Effect of data type. Oil and water rates are the two
types of data available to update the model. The objec-
tive is to select the best data wo cases were to update. T
Copyright © 2011 SciRes. ENG
Figure 2. Profile of the water cone resulting in breakthrough at 726 days.s
02004006008001000 1200 1400 1600
Oil Rate [m3/day]
Time (days)
Oil Rate P1
Oil Rate P2
Oil Rate P3
0200 400 600 800100012001400160
Water Rate [m 3/day]
Ti me ( da y s )
Water Rate P1
Water Rate P2
Water Rate P3
Figure 3. Oil production rates corresponding to the reference reservoir. Water breaks through initially in the well segment P2
and later in P1 and P3. Note that the reduction in the oil rate in P1 and P3 is less severe.
analyzed: in Case 1, water and oil rates from the 3 per-
forations were included in the objective function; and
Case 2 only the oil rate from the 3 perforations were in-
cluded, the water rate was excluded from the objective
function. Oil and water rates from each well segment
enter in the updating equation independently, and the
model is updated until each segment rate (water and oil)
satisfies the specified tolerance. In order to have a fair
comparison, the total oil and water rates from the well
(adding the three production points) are compared re-
gardless whether they are used to update the model or
not. The ultimate goal is to predict the total production of
the well, not partial rates.
Predictability. The accuracy of the updated models in
terms of the predictability of oil and water rates from
Cases 1 and 2 are compared in Figure 4. Figure 4(a)
shows the predicted oil rates for time step 3 (900 days),
at this time both cases had the same predicted oil and
water rates since in time steps 1 and 2 previous time
steps they both used only oil rates to update because wa-
ter had not breakthrough yet. After updating at 900 days,
Case 2 resulted with oil rates closer to the reference be-
cause its objective was only to reduce the mismatch the
oil rates as opposed to Case 1, where the objective was to
simultaneously reduce the mismatch in both oil and wa-
ter rates. In Case 1, the minimization of mismatch of
total (oil + water) rates causes some compromise to be
made in honoring the oil rates alone.
Although Case 2 matched closer the oil rate, the mis-
match in predicted water rate was higher than in Case 1
as shown in Figure 4(b). This is because Case 2 did not
included water rates in the updating equation. In sum-
mary, Case 2 predicts better oil rates while Case 1 pre-
dicts water rates. However, Figure 5(a) shows that the
Copyright © 2011 SciRes. ENG
02004006008001000 1200 1400 1600
Water Rate [m
Ti me ( d a y s )
Upda tedmodelusingonlyoilrate
Upda tedmodelusingoilandwaterrates
Reference pointsfor updating
02004006008001,000 1,200 1,400 1,600
Oil Rate [m
Time (days)
Referenc ewithtolerance
Updatedmodelusingonlyoilrat e
Reference pointsforupdating
(a) (b)
Figure 4. Prediction of (a) oil and (b) water rates with a model updated using both oil and water rates as compared to that
using only oil rate.
02004006008001000 1200 1400 1600
Oil Rate [m
Time (days)
Reference pointsforupdating
02004006008001000 1200 1400 1600
Wat er R ate (m
Time (days)
Upda t ed modelusingonlyoi lrate
Upda t ed modelusingoilandwaterrates
Reference pointsforupdating
(a) (b)
Figure 5. The match in (a) oil and (b) water rates over the entire simulation duration using a model updated using only the oil
rate data as compared to that using total fluid rate data.
overall performance of both cases in terms of predicting
the oil rate was not significantly different over the entire
simulation duration. On the other hand, Figure 5(b) also
indicates that the mismatch in terms of water rate be-
comes progressively worst for Case 2. This has drastic
consequences in terms of deriving and implementing an
optimum control scheme.
Estimation of Permeability. The estimated permeabil-
ity from Cases 1 and 2 is then compared to identify res-
ervoir regions where the water rate improves the model
updating. Figure 6 compares the deviation of updated
model corresponding to Case 2 against the reference and
indicates that the highest error in permeability prediction
occurs in the aquifer and the water invaded zone because
these directly affect the flow of water. On the other hand,
Case 1 yields a better estimation of permeability in the
water zone as shown in Figure 6(b).
Geological consistency in the updated models. The
experimental variogram is a representation of the spatial
variability and geological consistency exhibited by the
updated model. Figure 7 compares the updated experi-
mental variogram from Cases 1, 2 and the reference; the
range of the variogram in x for the reference permeability
distribution was 15 gridblocks. For each case, the per-
meability values of all realizations were averaged, and
the variogram of the averaged ensemble was compared
with the reference variogram. In Case 1, where water
rates improved the estimation of permeability in the wa-
ter zone, and the variogram is similar to the reference
variogram. In comparison, the updated model for Case 2
exhibits a variogram that did not reach a sill. This lack of
variogram reproduction and the associated difference in
spatial characteristics of the models is what is responsi-
ble for the poor predictive capability of the models, es-
pecially in the Case 2 where the permeability in the wa-
ter zone was poorly estimated.
Copyright © 2011 SciRes. ENG
(a) (b)
Figure 6. (a) Error map computed as model minus the reference for Case 1 . (b) Error map for Case 2. Results show that the
permeability water zone was better estimated in Case 1.
Variogram 
Lag distance (number of gridblocks)
Case 1: updatin g with oil
an d water rates
Case 2: udatin g only with
oil rate
Refe r e nc e
Figure 7. Variogram reproduction of the updated model
indicates that Case 1 had similar characteristics as the ref-
erence, while the variogram for Case 2 is non-stationary.
3. Laboratory 2D-Flow Model
3.1. Description of the Experiments
A sand pack with glass beads and two layers of different
grain sizes, and consequently two permeability layers
was created. The top layer was packed with 0.3 to 0.43
mm beads corresponding to the low permeability layer
and the bottom layer with 1.4 - 1.7 mm beads, corre-
sponding to the high permeability layer. The total poros-
ity was measured while the initial filling and assumed
constant (25.9%) after packing and shaking the apparatus.
The main elements of the laboratory model are indicated
in Figure 8. Oil was replenished from 6 injection points
at the top of the reservoir at constant pressure maintained
with air in a tank with decane (oil phase). These injection
Figure 8. The laboratory 2D-flow model prior to packing
the second layer of glass beads. A horizontal well spans
through the center of the porous medium with two seg-
ments independently ope rated.
points provided a constant pressure boundary. Red dyed
water was injected laterally to simulate an active aquifer
at approximately the same constant pressure as the oil
injection pressure. Oil flows through the low permeabil-
ity layer whereas water flows through the high perme-
ability layer, this was designed to promote water coning
in the laboratory. A horizontal well, located inside the
high permeable layer close to the interface with the low
permeability layer, spans through the center of the res-
ervoir with two independently segments controlled using
electronically actuated inflow control valves
The main characteristics of a two-layer 2D reservoir
with an active aquifer and a horizontal well at the center
were simulated. Before conducting the experiments, the
average permeability of each layer was estimated with a
Copyright © 2011 SciRes. ENG
steady-state experiment. The estimated average perme-
ability was 1100 and 6700 mD for the top and the bottom
layer respectively. The permeability of each layer was
also calculated using Carman-Kozeny’s equation result-
ing in 700 mD and 10,700 mD. Though this estimation is
not accurate because values for porosity and tortuosity
were assumed, they are in the same range of the experi-
mental measurement. With this prior information the
initial ensemble of realizations was generated using se-
quential Gaussian simulation in each layer separately.
The glass pack was modeled in CMG’s IMEX using the
same dimensions as the laboratory prototype. The
boundary conditions were (1) constant pressure oil injec-
tion at the top, (2) constant pressure water injection from
the bottom left side and (3) the producers were pressure
constrained according to the data recorded by the pres-
sure transducers (Figure 9). Nine measurements of oil
and water rates were collected over a 35-minute period
for each well segment (Table 1). Pressure data recorded
by the transducers from each segment at the times listed
in Table 1 were used as the pressure constraints for the
horizontal well in the simulation. Oil and water rates
were measured by recording the weight and volume of
the total liquids produced from each well segment.
In the laboratory prototype, oil production generated a
pressure gradient towards the well, forming a uniform
water cone with the tip at the center of the porous me-
dium, which eventually broke through close to the center
of the well. Figure 10(a) shows that water production
began after seven minutes of clean oil production. The
water cone invaded the production port P2 more severely
than P1. After 17 minutes, water and oil rates became
steady until the end of the experiment. Pressure data re-
corded by the transducers from each segment at the times
listed in Table 1 were used as the pressure constraints
Table 1. Measured rate and pressure data during the water
coning experiment
TimeOil P1 Oil P2Wat.P1 Wat.P2 Press.P1Pres.P2
min cm3/min cm3/min cm3/min cm3/min KPa KPa
7 339 188 1 0 128.06122.84
10.5213 302 15 10 127.99122.54
14 231 225 22 38 127.43122.16
17.5245 210 23 43 127.42121.86
21 240 210 23 46 126.88121.70
24.5250 205 26 45 126.77121.81
28 250 200 26 46 126.49121.58
31.5255 205 27 47 126.32121.42
35 260 205 28 44 126.32121.35
for the horizontal well in the simulation. Oil and water
rates were measured by recording the weight and volume
of the total liquids produced from each well segment.
3.2. Updating the Sand-Pack Flow Model
Fifty realizations were generated using sequential Gaus-
sian simulation conditioned to the average permeability
measured for each layer. Similar to the synthetic example,
two cases were run. In Case 1 both water and oil rates
from P1 and P2 were used to update the model, and in
Case 2 only the oil rates were used. A model was ac-
cepted as updated if the mismatch in the prediction of the
water and oil rates was less than 3 cm3/min. After the
ensemble was updated, the simulator predicted the oil
and water rates of each well segment for the next time
step. At each time step, the error in the predicted oil rate
over the set of NR realizations was calculated as:
12 1
1NR 2
ooo o,io
ˆˆ ˆˆ
Eqq qq
 
High Perm LayerLow Perm Layer
In je c tio n
P1 P2
and flow
* Not dr awn t o scale
Figure 9. Laboratory porous media simulated in CMG. Loc ation of injection and production wells is indicated.
Copyright © 2011 SciRes. ENG
0510 15 20 25 30 35 40
Water Rate [cm
Production Port 1
Pr o duc t i o n po r t 2
0510 15 20 25 30 35 40
Oil Rate [cm
Ti me [ m i n]
Pr o duc t i on Po r t 1
Pr o duc t i on po rt 2
(a) (b)
Figure 10. Measured (a) swater and (b) oil rates for each well segment P1 and P2.
Error in Oil Rate [cm
Ti me S t e
Oil & Water Rates
Oil Rate
Error in Water Rate [cm
Ti me S t e p #
Oil & Water Rates
Oil Rate
(a) (b)
Figure 11. The total (a) oil and (b) water rates were compared with the actual oil rate at each time step. Updated models
with only oil rates (Case 2) resulted in better oil rate prediction than Case 2. Better water rate prediction was obtained in
Case 1 than Case 2.
For total water rate:
12 1
1NR 2
www w,iw
ˆˆ ˆˆ
Eqq qq
 
The error in the net rate is:
neto w
EEE (5)
q is the measured rate of phase j. Similar to
the synthetic case, Case 2 showed less error in the pre-
diction of oil rates than Case 1. This is because in Case 1,
the objective is to reduce the error in both water and oil
rates at the expense of increased mismatch in oil rates
(Figure 11). Another comparison was looking at the er-
ror in the net production (5). The net production can be
weighted according to the price of oil and water treat-
ment and included as an objective function for optimiza-
tion. Figure 12 shows similar errors in the prediction at
initial times, but at later times the first case seems to re-
duce the error systematically while the mismatch for the
second case continues deviated. This is because the sec-
ond case does not explicitly consider the water produc-
Error in Net Rate [cm
Ti me S te
Oil & Water Rates
Oil Rate
Figure 12. The overall error in the prediction of the net
production is similar for both cases at early times. How-
ever, at later times the first case gives better predictions in
the net production than the se cond one.
Copyright © 2011 SciRes. ENG
tion rate as an objective function component.
The permeability maps obtained after updating Cases
1 and 2 are compared in Figure 13. As in the synthetic
case, the water rates carried information about the region
invaded by water. The permeability distribution in the
water leg is therefore updated in the first case where the
water rate was included as an objective function compo-
nent. In Case 2, no major changes occurred in that zone.
The streaks of permeability observed in the map for
the first case can be interpreted as local heterogeneities
created due to compaction/settling of the coarse glass
beads. The resulting permeability map from Case 2 indi-
cates that either the zone invaded by water was very ho-
mogeneous (contrary to the Case 1 result) or the oil rate
does not contain enough information for updating the
model in that region. The synthetic example indicated
that the latter explanation is more probable. However,
the prediction of streaks of high permeability close to the
well needs further investigation and visualization schemes.
4. Optimal Control from Updated Model
Once all the permeability realizations are updated using
(2), the most probable permeability field or the mean of
all realizations is computed. The optimum control set-
tings are developed for this most probable realization.
This can be posed as an optimization problem as follows:
Find the set of controls (u) that maximizes the cumula-
tive oil production minus the cumulative water produc-
tion from the current time to the terminal time subject to
the water coning equations as constraints. Denoting the
objective function as J:
 
Tbt T
oo ooww
 t
qo is the oil rate, qw is the water rate, T is the final time,
and Tbt is the breakthrough time. ωo and ωw are weights
assigned to the oil and water production rates respec-
tively, according the revenue or cost associated to them.
The breakthrough time Tbt can be obtained by integrating
the velocity equation found in [9], which requires
knowledge of the updated average permeability k around
the well. The water oil ratio after breakthrough can be
obtained from correlations from literature such as Bour-
nazel and Jeansen’s [10] correlation. For water conning
problem in the laboratory 2D-flow model, the specified
values of ωo and ωw are 0.1 and 1, respectively. These
values are arbitrary. The objective function integrates the
time discrete oil and water rates from initial time to a
pre-set terminal time. This function can be rapidly evalu-
ated without using the flow simulator by calculating the
integral from time zero to the breakthrough time and
from that time to the terminal time. The breakthrough
time is the numerical integral of the velocity expression
given by Farmen et al. [9]:
ln 2
bh P
kD bh
 
Every time step, after the model is updated, the objec-
tive function is evaluated and maximized for the remain-
ing time using Newton-Raphson method. The average
permeability required for the correlations is calculated
using the most probable permeability field and the con-
trol flow rate qo is assumed constant for the remaining
time. Iterative methods require multiple function evalua-
tions, for that reason is convenient to have an analytical
solution that can be evaluated without running the reser-
voir simulator.
4.1. Model Based Optimization of Production
The value of the data assimilation and model updating is
capitalized only when the well operating conditions are
adjusted to maximize the production of clean oil. This
implementation assess the importance of reservoir model
updating for establishing optimum control of well pro-
duction and to demonstrate the complete feedback loop.
In this example the flow rate of a vertical well in a 2D
Figure 13. (a) Permeability model obtained constrained only to oil rate (left), and (b) final permeability map of one realiza-
tion obtained constrained to the measured oil and water rates (right).
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Figure 14. Cost Function plot. The initial flow rate was ad-
justed such that the cost function was maximized
laboratory size reservoir is optimized as the model is
updated. The porous medium is divided into 50x100
gridblocks (0.5 cm × 1 cm). The reference permeability
values were drawn from a log-normal pdf with 340 mD
as mean, and then distributed in the gridblocks using
sequential Gaussian simulation forming thin layers. Two
water injector wells maintain a constant pressure bound-
ary at the left-hand side of the box; an oil injector is per-
forated in the top 4 gridblocks part of the model, and a
water injector is perforated in the bottom zone. These
injectors are placed to mimic constant pressure condi-
tions at the far boundary. On the right edge of the model,
a producer well is perforated in the top oil saturated in-
terval and produces at a constant liquid rate. This flow
rate constraint is the controllable variable used to opti-
mize the cost function. The initial saturation and pressure
distribution is assumed to be known. The reference pro-
duction data are the bottom-hole pressure, and the water
cut of the producer over a pre-determined time interval.
The well is assumed to be under constant oil rate control.
After breakthrough, water cut increases until it reaches
a plateau at steady state. The correlation for water cut
prediction after breakthrough proposed by Bournazel and
Jeanson [10] was employed here to evaluate it as func-
tion of time. Both breakthrough time and water cut cor-
relations facilitated the fast calculation of the cost func-
tion without using the reservoir simulator. A gradient
based optimization routine can be easily implemented to
find the optimum liquid rate that maximizes the cost
function J. The initial liquid rate for the reference case is
set as the optimum one, based on the initial guess of the
permeability distribution.
The cost function plot shown in Figure 14 was con-
structed by directly running the flow simulator to the
final time using different constant flow rates. The initial
flow rate (10.47 cm3/min) was calculated to maximize
Figure 15. The total liquid rate control was dynamically
updated after each time step was assimilated. The final cost
function was maximized.
the objective function using the breakthrough time and
water-oil ratio correlations according to the initial aver-
age permeability, as indicated in Figures 14 and 15.
However, since the permeability of the model was up-
dated after each time step, a new optimal flow rate was
recalculated. After time step 2, the flow rate was recal-
culated and corrected towards the true optimal. After
water breakthrough at time step 3, the rate was moved
closer to the true optimal rate. The rate correction at the
last time step was minimal because the average perme-
ability of the updated model did not change significantly.
This dynamic rate control resulted in a maximization of
the objective function as shown in Figure 15. The devia-
tion from the optimum flow rate is due to two reasons:
first, the approximations made for calculating the break-
through time and water cut evolution using correlations
and second because of the residual uncertainty in the
average permeability. Nevertheless, the flow rate was
effectively guided towards the real optimum rate
5. Conclusions and Further Research
Oil wells affected by water coning can be managed ef-
fectively with the aid of rapid model updating scheme
such as Ensemble Kalman Filter in combination with
optimal control. The sensitivity analysis on the type and
amount of data used to update models shows that models
are better updated when both oil and water rates are as-
similated using Ensemble Kalman Filter, especially in
the aquifer zone. This approach was implemented in a
laboratory experiment, confirming the important role of
the water rates to update the model in the water invaded
zone. Finally, the value added by this feedback control
scheme was capitalized when the operating conditions of
a vertical well affected by water coning were re-evalu-
ated after updating prior geological models, using ana-
lytical correlations of breakthrough time and water-oil
ratio for a fast evaluation and optimization of the objec-
tive function.
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