7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Estimation of Multiphase Flow Rates in a Horizontal Wellbore Using
an Ensemble Kalman Filter
A. Gryzlov*, W. Schiferli** and R.F. Mudde*
MultiScale Physics Department, Delft University of Technology, Delft, 2628 BW, The Netherlands
** Flow Control, TNO Science and Industry, Delft, 2600 AD, The Netherlands
A. Gryzlov@tudelft.nl, Wouter.Schiferli@tno.nl and R.F.Mudde tudelft.nl
Keywords: ensemble Kalman filter, driftflux modelling, wellbore flow, softsensing
Abstract
A new approach for realtime monitoring of horizontal wells, which is based on data assimilation concepts, is presented. Such
methodology can be used when the direct measurement of multiphase flow rates is unfeasible or unavailable. In that case one
can use socalled multiphase softsensors. Such sensors estimate phase flow rates using data from conventional sensors in
combination with a dynamic multiphase flow model (Leskens et al., 2008), potentially eliminating the need for complex
downhole instrumentation.
The realtime estimator proposed is an ensemble Kalman filter employing a dynamic driftflux model of the wellbore flow
and information from several downhole pressure sensors with a single measurement of the gas velocity and holdup. By means
of simulation examples it has been shown that the proposed algorithm operates quite accurately both with artificial
measurements and data generated by the OLGA simulator.
Introduction
The growing demand for hydrocarbon production has
resulted into improved oilfield management with various
control and optimization strategies (Glandt, 2003, Jansen et
al., 2008). These strategies in turn strongly rely on the
efficiency of downhole equipment which is used to obtain
realtime oil and gas production rates with sufficient spatial
and temporal resolution. In particular, multiphase
flowmeters installed downhole can improve the production
of long horizontal wells by allocating the zones of oil, gas
and water inflow. However, existing multiphase meters are
expensive, inaccurate or accurate only within a limited
operating range and therefore such monitoring is unrealistic.
To overcome these problems one can use socalled
multiphase softsensors, i.e. to estimate flow rates from
conventional meters, such as downhole pressure gauges, in
combination with a dynamic multiphase flow model.
Despite the variety of softsensing techniques (which are
also referred to as data assimilation methods), one can note
two principal approaches. Variational data assimilation,
which is based on the minimization of a cost function within
a certain time interval and sequential methods or filtering
when the state of the system is updated every time instant
data becomes available. One way to solve these sequential
data assimilation problems is to use Kalman filtering
(Kalman, 1960). This method, which was originally
developed for linear models, has got numerous extensions
(Jazwinski, 1970, Evensen, 1994 and Julier et al., 2000) to
deal with nonlinearity, which is the case for most industrial
processes.
Although there are numerous applications of softsensing
techniques in oil and gas industry, they mainly deal with the
estimation of reservoir properties (Naevdal et al., 2003,
Evensen et al., 2007). The range of wellbore flow
application includes gaslift wells (Bloemen et al., 2004)
and underbalanced drilling (Lorentzen et al., 2001). Also,
the Kalman filter has been used for tuning the parameters of
two phase flow models (Lorentzen et al., 2003). Leskens et
al. (2'"'1) considered the simultaneous estimation of
downhole oil, water and gas flow rates from downhole
pressure and temperature measurements in a single well.
This approach has been extended by de Kruif et al. (2'" i',
to the multilateral well case both for the twophase (oil and
gas) and threephase (oil, gas and water) cases. Softsensor
based on the ensemble Kalman filter has been used for a gas
cone allocation in a multizone completion well by Gryzlov
et al. (2'" I' ). However, it has only been performed with a
homogeneous noslip model, which is only valid for a
dispersed bubble flow regime. To be able to use the model
over a wider range of flow conditions, the model should be
modified.
Despite the variety of applications considered, little
attention has been given to production instability problems.
More specific, long horizontal wells with a continuous
inflow from a reservoir to a wellbore require the use of
softsensing techniques for the gas breakthrough prediction.
Gas coning is a phenomenon where the gasoil contact of a
reservoir moves towards a producing well (see Figure 1).
At a certain moment the gasoil contact will reach the well
and gas breakthrough can happen causing a large gas
influx. Consequently, the gas phase may start to dominate
production making the well uneconomical. In order to
handle or prevent this, several strategies are available.
However, the most convenient countermeasure is to isolate
gas producing zones of a wellbore by means of inflow
control valves (ICV). The purpose of a softsensor is to
provide these valves with information of the downhole
flow rate distribution in order to decide where
breakthrough is occurring. The relevant ICV can then be
closed.
I
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Where H is the liquid volume fraction, pg is the gas
density, pi is the liquid density, t denotes time and s denotes
the coordinate along the length of the pipe. 4D and Dg are the
mass sources representing the inflow from a reservoir to the
pipe. These sources are normally time dependent.
Although the continuity equations have been written for
each phase it is common to write the momentum equation
for the mixture.
3 6 2
(p ) + (Hp,u, + (1 H)p ,u)
dt as
Where pm is the mixture density defined by
p
S, (3)
as
pm = Pg(1 H)+ p1H (4)
A frequently used model for frictional losses in the
momentum equation has the form
S 2d
With the mixture velocity um defined as
Inflow Control Valves (ICV))
Figure 1: Schematic view of a horizontal well.
In this work estimates of downhole flow rates in a
horizontal well are obtained using a transient driftflux
model, together with the ensemble Kalman filter.
Furthermore, the influence of model error and
measurement noise on the quality of estimates is studied.
This paper is organized as follows. First, brief descriptions
are given of both the wellbore flow model and the
computational setup for softsensing. Next, an overview of
available softsensing methods is presented and the
ensemble Kalman filter, which is applied in this paper, is
explained in more detail. Then, different softsensing
solutions are analyzed by means of simulations. At the end
of the paper, the conclusions are given.
Formulation of the inverse problem
A model describing onedimensional twophase flow in
pipes consists of nonlinear partial differential equations
describing mass and momentum conservation for each
phase. This model is obtained from crosssectional
averaging of the NavierStokes equations and replacing
diffusion terms by empirical correlations.
The simplified mass conservation equations are
(pH)+(pHUg) = (1)
at as
(p (1
)
H))+(p ( H)u,) = q
as
u, =ug(1H)+uH (6)
Here d is the pipe diameter and A is the friction factor,
which is a function of the Reynolds number and pipe
roughness k. In this study the Techo formula is used, where
A is given explicitly
1.964 In (Re)3.822 k 2
= 0.868In + (7)
Re d 3.71)_
Here Re is the Reynolds number defined as
Re = umdp/ 1/u (8)
with the mixture viscosity /m calculated in terms of liquid
volume fraction and gas pg and liquid p1 viscosities
mt = tg (1 H)+ p, H (9)
The gas is treated as a compressible phase with a
corresponding equation of state given in the form
P, = f(p) (10)
Using the driftflux approach the actual gas velocity ug is
correlated with a mixture velocity um using two
parameters.
S= Cu + ud (11)
Co is referred to as the distribution parameter and it
accounts for the effects of the nonuniform distribution of
both velocity and concentration profiles; ud is the drift
velocity of gas, and accounts for the local relative velocity
between the two phases.
The closure of the problem is given by the following
boundary conditions.
p(s = L,t) p, p(s, t = 0) = po (12)
u(s = 0,t) = u, u(s,t = 0) = u~ (13)
H(s = O,t) = Hf,H(s,t = 0)= Hf (14)
Here the subscripts infand out refer to inflow and outflow
crosssection of the pipe respectively.
The computational setup for the inverse problem is shown
in Figure 2. It should be noted here, that only the
horizontal part of the well is being modelled, and the
outflow measurements are assumed to be available directly
at the outflow crosssection of the horizontal part.
liquid/gas mixture
Inflow from reservoir liquid/gasmixture
(liquid or gas) (t),,(t) (t
Pressure measurements
(downhole)
Flow rate measurements
(surface)
Figure 2: Scheme of the computational setup for
softsensing.
For the soft sensing purposes the augmented state vector is
introduced:
S= [p, Uig H, 4 g, ,, (15)
Here i indicates the number of the cell defined by the
numerical method, which is discussed in the next section.
It is assumed that several downhole pressure measurements
are available. Moreover, outflow information about flow
rates is also known, giving the following measurement
vector:
y = [ Uo, Ho T (16)
Finally, the data assimilation problem can be formulated as
follows: with the measurements (16) and the flow model
(1)(14) available the components of the augmented state
vector must be estimated.
Due to a lack of experimental data, a set of synthetic
measurements has been used as a source for softsensing.
First, a twin experiment concept has been implemented.
Here the same mathematical model was used both for
generating measurements with predefined inflow
distribution and the inverse modelling, when missing
dynamic variables are estimated by means of the
softsensing algorithm. In order to mimic the situation of
testing the softsensor with a "reallife" data, simulation
results from the commercially available flow simulator
OLGA were used in the second test case.
Numerical Scheme
Statespace form of the model equations
Due to the nonlinearity of the given equation system a
numerical solution is needed in order to solve it for the
dependent variables.
For the discretization of the simulation domain a staggered
grid approach has been used, meaning that a different grid
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
for the continuity and momentum equation has been
employed. Afterwards, the mass and momentum equations
are integrated over different control volumes. Any solution
procedure can be applied for solving the nonlinear system
of algebraic equations. Finally, the model can be written in
the following statespace notation (Crassidis, 2004):
x,= f(xk1,U ); (17)
Here Uk1 is the model input representing the inflow from
reservoir to wellbore. xkl is the state vector evaluated on the
previous time step. Using the natural set of variables, the
state vector can be written as
x=[pu H]T; (18)
Here p, Ug and H are the vectors, representing pressure
velocity and liquid volume fraction related to the spatial
grid.
Ensemble Kalman filtering
One way to solve estimation problems via the sequential
data assimilation algorithm is by using the Kalman filter
equations. The Kalman filter is a stochastic recursive
estimator, which estimates the values of model states and
unknown input by integrating in realtime measurements in
a mathematical model. Due to its straightforward numerical
implementation and recursive nature, the Kalman filter
algorithm is very well adapted to online model calibration.
Kalman filtering was initially developed for linear
dynamical systems. Although several extensions of the
Kalman filter exist for nonlinear system, here the ensemble
Kalman filter (EnKF) is used (Evensen, 1994). The most
straightforward extension for the nonlinear systems is the
extended Kalman filter, and the main difference is in how
the error covariance matrix is calculated. In the extended
Kalman filter approach it is performed using a linearized
model. The ensemble Kalman filter calculates the
approximation of the covariance matrix using an ensemble
of model forecasts.
In order to initialize the filter the initial ensemble is
generated. Here a mean value of the initial state vector Xo
and a corresponding covariance matrix Qo is required. The
mean value of the initial ensemble should be a good
estimate of the true initial state. The members of the
ensemble are generated randomly according to a Gaussian
distribution. Thej 'th member of the ensemble is defined as
X" = X + Woj (19)
With an EnKF the augmented state vector, which also
contains the inflow input, is estimated in a recursive manner
through the following two steps:
1) The forecast step, which consists in running the forward
model one time step forward for each member of the
ensemble. This leads to
X, f (x1,, )+w k, (20)
where , is a Gaussian zero mean white noise with the
corresponding covariance matrix Qk representing the model
error. This noise is only added to components of the state
vector, which produce the most uncertainty in a simulation.
These are in this case the inflow sources DoI and 9g.
Using the calculated forecast of ensemble states, the error
covariance matrix can be calculated using the covariance
matrix of the ensemble. The mean value of the ensemble is
given by
= X1,j (21)
And the error covariance matrix is then calculated as
Pk = L (L')T
With
1
kf N [(x
> )(Xf, X ).(X, X)] (23)
where N is the number of members in the ensemble.
2) The analysis step, which takes into account
measurements. The errors in the measurements are assumed
to be statistically independent with known variances. This
leads to a diagonal covariance matrix for the measurement
errors. As it has been pointed out in Burgers et al. (1998), it
is necessary to define new measurements for the proper
error propagation. Therefore, a new observation vector is
introduced for each member of the ensemble
Yk, = Mk *Xk, + vk, (24)
here Mk is the measurement matrix and vk; is the
measurement noise generated from a normal distribution
with zero mean and covariance matrix Rk.
The Kalman gain is then calculated as follows
Kk = Pk M[ (MkP M[k + Rk )1 (25)
The analyzed state for each member of the ensemble is
given by
X" = X +K (y,j MkX, )
The mean value of the analyzed ensemble is
fX Xt ,(27)
The unknown inflow parameters are updated at each time
step measurements are available and extracted from the
augmented state vector. The analyzed error covariance
matrix, from which the estimation error of the inflow
parameters can be defined, is then approximated by
Pk = (I KMk )Pk (28)
An important issue with the use of the EnKF is the size of
the ensemble. Based on the experience of data assimilation
for largescale atmospheric models (Houtekamer and
Mitchel, 1998), 100 ensemble members have been chosen
for the ensemble Kalman filtering. The optimal size of the
ensemble is, however, not known and it is a subject for
future research.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Implementation issues
One problem of implementation of the EnKF algorithm is
related to the different scale of variables included in the
augmented state vector. While the pressure is of order 107
the liquid velocity is of order 10' and liquid volume
fraction is of order 10. Due to the fact that computation of
the Kalman gain and calculation of the corrected estimates
is based on a substantial amount of matrix algebra, the
truncation error raised by this order of magnitude
heterogeneity may lead to poor filter performance or even
its divergence.
In order to prevent this, the variables in the state vector
should be adjusted in such a way that all of them are of the
same order of magnitude. This can be performed using the
nondimensional state vector given in a form
K =[P,S1^, ^ CD, ]T (29)
Here denotes the nondimensional type of state variables,
which are all of order 100. For the variable 0 its
nondimensional counterpart is given by
0 = 01/0 (30)
Here the subscript ref denotes reference value used for
nondimensionalization. The reference values can be taken
from the available measurements and they are updated
every time step when new measurements become available.
For this specific problem the reference values can be
defined as follows
p, M = pT ; = lt ; H;, = HM (31)
Prref P out ; Uf ou = ref out
D, = ," TP" .S.(1 H^) (32)
D, = ref L P S 1 Hl (33)
Here the superscript M denotes the relationship to a
measurement vector (16). Although the density is not
available directly from the measurements, the estimate of
the measured density may be obtained from the outflow
pressure and equation of state (10). The same procedure is
applied to the measurement vector
g =[p igotHo]T
Results and Discussion
Soft sensing under measurement error
A first test case considered uses a twin experiment concept.
Here the same mathematical model is used for generating
the measurements with predefined inflow distribution. This
study deals with twophase liquid/gas flow and the details
of the initial data are given in Table 1. The sketch of the
simulation domain is given in Figure 3. The inflow profiles
are given only as a reference since they are unknown and
have to be estimated via the proposed data assimilation
procedure.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
The results of the simulation are given in Figures 45.
Figure 4 shows the comparison between the estimated and
true gas velocity distributions along the pipe length.
Velocity is used to allocate the zones where a fluid is
entering or leaving the wellbore. In order to identify the
type of fluid, the distribution of the estimated holdup is
required. It is depicted in Figure 5. The results are given for
three time instants 30 minutes, 40 minutes and 50 minutes.
Since the pressure is available continuously from the
measurements it is not depicted as a softsensing result.
Figure 3: Computational setup for softsensing.
Initially the well produces a mixture of liquid and gas with
a total flow rate of 2 kg/s. After 20 minutes of production,
gas is injected in two locations of the wellbore. In the third
spot, located in the middle of the simulation domain liquid
is leaving the pipe. Such a crossflow scenario is possible
due to heterogeneity of reservoir properties.
The inflow sources provide a linear increase of liquid flow
rate for 30 minutes of production with a constant value of 2
kg/s each. The amount of liquid leaving the wellbore is
decreasing linearly in the same time interval, with a
maximum value of 2 kg/s.
The softsensor has been tested using the following
measurement layout. The number of pressure measurements
was taken equal to number of grid nodes obtained from the
discretization. The velocity and liquid volume fraction
measurements are located at the last grid block of the
simulation domain.
Table 1: Initial data for the numerical experiments
Quantity Value
Pipe diameter, m 0.05
Pipe length, m 100
Liquid density, kg/m3 1000
Liquid viscosity, Pa s 0.001
Gas reference density, kg/m3 11.93
Gas viscosity, Pa s 1.82 10
Inflow liquid rate F1, kg/s 1.98
Inflow gas rate F kg/s 0.02
x1,m 15
x2, m 45
x3, m 75
Absolute roughness, m 0
Distribution parameter, [] 1.2
Bubble drift velocity, m/s 0
The Kalman filter initialization is based here on the outflow
values of gas velocity and holdup, which are assumed to be
know due to metering. Since all the pressure measurements
are available, pressure is initialized from the current
pressure distribution. The synthetic measurements
representing downhole pressure and liquid outflow flow
rate are generated using equations (1)(12). A zero mean
white Gaussian noise is then added to represent the
uncertainty in measurements.
Table 2: Measurement noise used in simulations
Uncertainty in pressure measurements 0.5%
Uncertainty in outflow velocity measurements 1%
Uncertainty in holdup measurements 1%
SIUVV ll lU . True, t1800s
1 { )* Estimated, t1800s
True,t=2400s
SEstimated, t2400
S ..... True, t=3000s
1 . Estimated,t=3000s
0 20 40 60 80 100
L,m
Figure 4: Comparison of estimated and true gas velocity.
075 Flow mete
07
..... t.
06    
S55    
05  True,t=1800s
Flow direction E True t 2400s
045 Truet=2400s
0 4 Estimated, t2400s
04 * True, t=3000s
S*Estimated, t3000s
035 
0 20 40 60 80 100
L,m
Figure 5: Comparison of estimated and true holdup.
The results show that the proposed softsensor, for the
given simplified setup, is very well capable for
reproducing the gas velocity and holdup distributions
along the considered well part, even under a certain
measurement error. Therefore, it is capable to detect
multiple fluid sources as it is depicted in the figures.
Estimated liquid velocity is obtained from (11) and
therefore it is as accurate as predicted gas velocity. In that
simulation case no attention has been paid to the flow
regime. In reality, the flow may be described by the
driftflux model only in a limited range of flow conditions.
Since both measurement generation and softsensing are
based on the same mathematical formulation, input data
does not have to be physically realistic: any input set of
parameters would result in accurate softsensor
performance. It should also be noted here that the added
measurement error is the main source of the mismatch
between estimated and true values. In principle,
performing noisefree simulations would result in perfect
estimation, however, only if the number of ensemble
members is sufficient enough.
Soft sensing under model error
The second study provides an assessment of the influence of
the model error on the softsensing estimation results. A
similar softsensing setup was used as depicted in Figure 2
for case study 1. An important difference, however, was that
the "true" well was not the same as the model used in the
softsensor. The true wellbore measurements were obtained
from the commercially available simulator OLGA. This was
done to assess the inevitable effect of the model error on the
softsensing estimation results. Here both transient gas and
liquid sources are present in a computational setup. Liquid
is injected in the first part of the pipe, while a gas source is
present close to its outflow crosssection. This situation is a
rough approximation of the gas breakthrough scenario. The
scheme of the simulation domain is given in Figure 6.
Due to differences between the flow model used in OLGA
simulator and the softsensor developed, one can point at
the following sources of the model error:
 Friction factor correlation
 Fluid properties
 Simulation grid
 Mathematical model
0.04,
J [ 1200 3000 r'
Figure 6: Computational setup for softsensing. Test case 2.
Estimated holdup is given in Figure 7. Figures 8 and 9
represent the comparison between the obtained and true
estimated velocities of gas and liquid. A particularly
important modelling assumption for performing OLGA
simulations was to keep a slug flow regime, since the model
used for softsensing is accurate enough for that type of
multiphase flow. This was possible using the same set of
input parameters, as for the test case 1. The OLGA
simulations were performed with 10 grid nodes, where the
source term for liquid has been defined in the third grid
block, and for gas in eighth grid block. This consequently
led to a softsensing setup with 10 available pressure
measurements.
Results obtained are not as accurate as for the
twinexperiment. However, it is still possible to allocate
easily zones of liquid or gas inflow. A displacement of the
estimated profiles with respect to the true ones is observed.
This can be explained by the use of a different grid in the
OLGA simulator and different interpolation of the flow
variables between grid nodes and edges.
Obtained estimated velocity profiles, both for liquid and gas
phases, are quite accurate despite the fact that the OLGA
solver is based on a rigorous twofluid model, whereas the
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
softsensor employs driftflux formulation. Driftflux
parameters seem to be key values to provide accurate
performance of a realtime estimator. For the cases, when
they are not well defined, it is reasonable to include them
into the state vector and estimate these parameters online
from the data available, though it might increase the size of
the state vector and dramatically decrease the robustness of
the method.
This leads to a conclusion that for the estimation purposes
complex flow modelling is not required. The dynamic flow
model used as a softsensor may be simple enough to
perform simulations in realtime, though it still should
capture the main physics of the flow.
08 ..   T.. . ........
08 T
075  ,i__ Flow meter
or * 
065
SOLGA, 1800s
06   Estimated, t1800s
I  OLGA, t2400s .
!0I Estimated, t2400s i\
055 I  OLGA, t3000 
Ii * Estimated, t3000s
05     
0 20 40 60 80 100
L,m
Figure 7: Estimated holdup distribution for the OLGA
data.
 OLGA, t=1800s
6 Tr Estimated, t1800s F r
 OLGA, t=2400s Flow meter
Estimated, t2400s
OLGA, t=3000s 
5  Estimated, t3000 
Flow dire tion / .
E4  
A   s
3 I
2      
U .,i'
0 20 40 60 80 100
L, m
Figure 8: Estimated gas velocity distribution for the
OLGA data.
4 T OLGA, t1800s ..... . ......... .. .....
i X Estimated, t=1800s i __
 OLGA, t=2400s I / !
35 
S Estimated, t=2400s
OLGA, t3000s
3  Estimated, t=3000 __ 
E 25   n  
SI ;// 1. "1" I^
2  Flow i to     
15 Flow d ion LL,
Flo i wf i Flow meter
1 .... I ... . 4 ....... . ..... .... .. .
0 20 40 60 80 100
L,m
Figure 9: Estimated liquid velocity distribution for the
OLGA data.
Conclusions
By means of two case studies, the possibilities of
multiphase softsensors have been discussed. The proposed
realtime estimator is based on the ensemble Kalman filter
approach and requires as the input a dynamic model of the
pipe flow together with pressure measurements available
downhole and one holdup and velocity measurement at the
outflow.
It has been shown, that for a driftflux flow formulation it is
possible to obtain the estimations of the gas velocity and
holdup along a well and to allocate the inflow of certain
fluids in specific location along it.
The results indicate that the proposed method is quite stable
for a certain range of wellbore operational conditions, and
capable of taking into account measurement and model
error.
Acknowledgements
This work has been supported by ISAPP knowledge center,
which is a joint research project of Shell, TNO and Delft
University of Technology.
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