Data-driven soft sensor of downhole pressure for a

gas-lift oil well

Bruno O.S. Teixeira† Walace S. Castro† Alex F. Teixeira‡ & Luis A. Aguirre†

†Department of Electronic Engineering, Universidade Federal de Minas Gerais (UFMG), Belo Horizonte,

MG, Brazil; (e-mail: brunoot@ufmg.br)

‡Research and Development Center (CENPES), Petroleo Brasileiro S.A. (Petrobras), Rio de Janeiro, RJ,

Brazil

Abstract

Downhole pressure is a key variable in the operation of gas-lift oil wells. However, main-taining and replacing downhole sensors is a challenging task. In this context, we design andimplement a data-driven soft sensor to estimate online the downhole pressure based on other(seabed and platform) available measurements. Such application is based on a two-step pro-cedure. In the first step, discrete-time black-box and grey-box NARX models are identifiedoffline and indepenently using historical data. Both polynomial and neural models are ob-tained. In the second step, recursive predictions of these multiple models are combined withcurrent measured data (of variables other than the downhole pressure) by means of an in-teracting bank of unscented Kalman filters. In doing so, a closed-loop model prediction isperformed. Three issues are investigated in this paper concerning: i) the usage of a filterbank rather than a single filter approach, ii) the availability of seabed variables as inputsof the models compared to the case where only platform variables are available, and iii) theemployment of grey-box models in the filters. Experimental results along 7 months of testsindicate that such closed-loop scheme improves estimation accuracy and robustness comparedto the free-run model prediction or to the use of a single unscented Kalman filter. The methodemployed in this paper can also be applied to other soft sensing applications in industry.

Keywords: soft sensors; system identification; Kalman filtering; interacting multiple models;permanent downhole gauge (PDG); offshore oil platform.

1 Introduction

In the last two decades, soft sensors have been increasingly applied in process industry as an alter-native to traditional hardware instruments. Applications range from oil industry [Domlan et al.(2011),Fujiwara et al.(2012)], chemical processes [Gjerkes et al.(2011), Jin et al.(2012)] to metallurgicalindustry [Li and Jiang(2011), Wu et al.(2011)], to name a few. Roughly speaking, soft sensors arepredictive mathematical models that infer the values of a given process variable from measure-ments of other process variables [Fortuna et al.(2007)]. Though the range of applications coveredby soft sensors is broad, their most dominant application field is the online prediction of processvariables which are measured only at low sampling rates, using expensive or unreliable instruments,or through offline analysis. Other important areas of application include process monitoring andprocess and sensor fault detection [Kadlec et al.(2009)].

Thus, modeling is the keypoint for soft sensor development. Different modeling approachesmay be employed. Two classes of soft sensors can be distinguished, namely, model-driven and data-

driven [Kadlec et al.(2009)]. The former is based on first-principle models, while the latter usesdata-driven back-box or grey-box identified models. First-principle models are generally developedfor the planning and design of industry plants and are not recommended for soft sensor development

1

(a)

(b)

Figure 1: Comparative perspective of soft sensors concerning modeling approaches. In (a), predictive FIRmodels yield open-loop estimates while, in (b), the combination of predictive FIR models with measure-ments of auxiliary variables and observation models by means of a state estimator works as a closed-loopestimation scheme.

[Kadlec et al.(2009)]. Also, industrial processes are described by nonlinear phenomena; therefore,nonlinear models should be the natural choice. However, nonlinear modeling is not a trivial task[Aguirre and Letellier(2009)]. Alternatively, multiple linear models are often employed in softsensor applications [Jin et al.(2012), Domlan et al.(2011), Li and Jiang(2011)]. Nonlinear black-box models have been increasingly used in soft sensor applications; especially neural networks[Fujiwara et al.(2012), Wu et al.(2011)]. To the best of our knowledge, grey-box modeling is notoften applied to soft sensor development [Sbarbaro et al.(2008)]. Finally, it is important to pointout that soft sensors often employ finite impulse response (FIR) models. Alternatively, infiniteimpulse response (IIR) models can be combined with FIR models by means of state estimators,resulting in a closed-loop prediction scheme; see Figure 1.

Gas-lift is a technology to produce oil and gas from low pressure oil wells. The gas-lift flowrate is determinant in the well productivity and affects the flow dynamic stability. Also, its valuestrongly depends on the downhole pressure, which, therefore, must be monitored. According to[Nygaard et al.(2006)], downhole pressure is the most important variable to describe the dynamicsof a oil well. However, maintaining and replacing permanent downhole gauge (PDG) sensors isa challenging task, especially in deepwater oil wells [Eck et al.(1999)]. Also, sensor prematurefailure often happens. Actually, if at all available, downhole measurements are generally unreliable[Aamo et al.(2004)]. In this context, soft-sensor techniques are promising alternatives to monitorthe downhole variables.

2

In [Aamo et al.(2004), Nazari et al.(2009), Nygaard et al.(2006), Bloemen et al.(2006)], model-driven soft sensors are investigated to monitor downhole variables in gas-lifted oil wells. However,a natural shortcoming arises in such cases, related to the need of obtaining physical parame-ters of the corresponding mass-balance-based nonlinear models. Such issue can be circumventedby employing a bank of locally linear models, for which it is easier to estimate the parameters.[Jahanshahi et al.(2008)] investigate the fuzzy combination of local linear models from simulateddata. In all aforementioned applications of soft sensing techniques to gas-lift oil wells, closed-loopprediction schemes are used. Except for [Nazari et al.(2009)], in all cases above, the extendedKalman filter is used to assimilate measurements of other variables to the predictions of the IIRfirst-principle models.

In this paper, we employ a two-step procedure to estimate online the downhole pressure ofan actual deepwater gas-lift oil well. Indeed, our approach can be applied to other soft sensingproblems. Our procedure characterizes a data-driven soft sensor. First, discrete-time nonlin-ear autoregressive with exogenous inputs (NARX) polynomial models and multilayer perceptron(MLP) neural networks are identified offline using experimental data as in [Aguirre et al.(2005)].Two kinds of models are obtained: IIR process models and FIR/IIR observation models. Dif-ferent configurations of inputs and outputs and of model structures are tested. Each model isbuilt completely independent from the others. Second, an interacting multiple model (IMM) fil-ter bank [Mazor et al.(1998), Bar-Shalom et al.(2001)] is employed, with each unscented Kalmanfilter (UKF) [Julier and Uhlmann(2004)] of the bank combining a different pair of process andobservation models. That is, local nonlinear “closed-loop” models are combined to yield improveddownhole pressure estimates compared to the free-run simulation of a single (open-loop) model ora single UKF. Note that, in our approach, the downhole pressure is assumed to be known onlyduring the system identification step. For practical applications, this is the case after downholesensor installation, when such sensors are more reliable [Eck et al.(1999)].

We therefore investigate three relevant issues regarding closed-loop data-driven soft sensing.First, we evaluate what is the gain of using a filter bank rather than a single filter approach.Second, we evaluate the impact of using seabed auxiliary variables in the models compared to thecase of using only platform variables since measurements of the former are not always available inoffshore oil wells. Finally, we assess the impact of using grey-box models, for which steady-statedata are used to improve modeling [Barbosa et al.(2011), Teixeira and Aguirre(2011)], comparedto the case where only black-box models are used. Thus, it is important to clarify that this paperdoes not aim at providing an comprehensive comparison among soft sensor techniques, but it ratherreports a well-succeeded experience of developing and implementing a soft sensor in an industrialframework.

This paper is outlined as follows. Section 2 briefly describes the process under investigation.Section 3 presents the two steps of the methodology employed. Then, sections 4 and 5 discussesthe experimental results. Finally, concluding remarks are presented in Section 6. A preliminaryversion of this paper appears as [Teixeira et al.(2012)].

2 Process description

To produce oil from low pressure and/or deepwater oil wells, gas-lift technology is often employed.Figure 2 presents a simplified diagram of a gas-lift oil well and Table 1 lists some of the processvariables often measured. Except for TT1, TT4 and FT4, all variables listed in Table 1 are usedto build models in this work. The PDG sensor provides measurements of PT1 and TT1.

The process is summarized as follows. Pressured gas from the gas-lift header at the platform(instruments tagged by 4) is injected through annulus between tubing and casing string until itreaches an orifice valve located in the lower part of the tubing (in the fluid reservoir at seabed).

3

The fluid density is then reduced such that the reservoir pressure is high enough to transport themultiphase mixture of oil, gas, water to the platform. In the seabed, a set of valves and adaptersknown as wet christmas tree (PT2 and TT2) control the production flow from seabed to theplatform. In the platform, a shutdown valve (PT3a) is available to interrupt the production during

TT3 PT3

TT4 PT4

TT1

PT1

PT2

TT2

Wet Christmas Tree

seabed

choke valve

platform

produce (liquid)(Qliq+Qg)

manifoldproduction

injected gas (Qginj)

Header

Gas Lift

(a)

TT3

PT1

TT1

TT2

PT2

FT4

PT3

PT4a PT4TT4

Header

Gas!lift

Gas!lift Choke

SDV (on/off) Choke valve

SDV (on/off)

(b)

Figure 2: Simplified P&ID diagram of a gas-lifted oil well, where TT refers to temperature transmitterand PT refers to pressure transmitter. In (a), there is an overall view, while, in (b), the platform is shownin more details. The numbers 1 (downhole) and 2 (wet christmas tree) accounts for seabed variables, while3 (production) and 4 (gas lift) accounts for platform variables. Flow direction is 4-1-2-3. The downholevariables are measured close to the reservoir outlet.

4

an emergency situation and a choke production (PT3 and TT3) valve regulates the production flowrate at the platform. Different flow dynamics are achieved depending on the values gas-lift (PT4and PT4a) and downhole (PT1) pressure.

Table 1: Process variables used to obtain models for the gas-lift oil well. Tags corresponds to thecodes shown in Figure 2.

Tag Description UnitsPT1 Downhole pressure kgf/s2

TT1 Downhole temperature ◦CPT2 Wet christmas tree pressure kgf/s2

TT2 Wet christmas tree temperature ◦CPT3a Pressure before shutdown valve kgf/cm2

PT3 Pressure before production choke valve kgf/cm2

TT3 Temperature before production choke valve ◦CPT4a Pressure before gas-lift shutdown valve kgf/cm2

TT4 Temperature before gas-lift shutdown valve ◦CFT4 Instantaneous gas-lift flow rate m3/hFV4 Gas-lift valve position %PT4 Pressure after gas-lift choke valve kgf/cm2

3 Methodology

3.1 Problem statement

Consider the nonlinear dynamic system given by

xk = f(xk−1, ufk−1) + wk−1, (1)

yk = h(xk, uhk) + vk, (2)

where f : Rn×Rpf −→ Rn is the process model and h : Rn×Rph −→ Rm is the observation model,

xk ∈ Rn is the state vector, yk ∈ Rm are measured outputs, uk�=

�ufk−1

uhk

�∈ Rp, p = pf + ph,

are known inputs, wk−1 ∈ Rn is the zero-mean process noise with covariance Q and vk ∈ Rm

is the zero-mean measurement noise with covariance R. Our goal is to obtain state estimatesxk|k and corresponding covariance P xx

k|k that approximate E [xk] and E [(xk − E [xk])(xk − E [xk])T],

respectively, where E is the expected value. To accomplish that, a two-step procedure is employedas illustrated by Figure 3.

The first step is the system identification step, for which it is assumed that a set of dynamicaldata {uk, xk, yk}, k = 1, . . . , N, is known. Indeed, historical data is recovered from a plant informa-tion management system (PIMS). Using only these offline data, a set of black-box identified modelsf i, i = 1, . . . ,Mf , and hj , j = 1, . . . ,Mh, are built independently from each other and rewrittenin state space as described in Section 3.2. We also obtain a grey-box model by using steady-statedata {ul, xl, yl}, l = 1, . . . , L, together with dynamical data. It is not assumed that such modelsare globally valid. Here, Qi and Rj account for the covariance of the one-step-ahead simulationerror of f i and hj , respectively. We use the same methodology employed in [Aguirre et al.(2005)].

The second step is the filter bank step, which is illustrated in figures 3 and 4 and describedin Section 3.3. It is assumed that {us

k, ysk}, s = 1, . . . ,M, are known for all k > 0 together with

5

Figure 3: Diagram of the two-step methodology used in this work to develop a soft sensor for downholepressure.

a set of M ≤ MfMh state-space model pairs {fs, hs}. For each pair {fs, hs}, state estimatesxsk|k with covariance P xx,s

k|k are recursively obtained using M UKFs [Julier and Uhlmann(2004),

Arasaratnam and Haykin(2009)] running in parallel. Then, these estimates are combined by anIMM filter bank, yielding xk|k and P xx

k|k for all k > 0 [Bar-Shalom et al.(2001)]. Note that x isassumed to be known only during the system identification step for a time interval of duration N .Other nonlinear filters and filter bank algorithms could be used to implement the methodologyillustrated by Figure 3.

Next, we present an overview of the methods used in this work for system identification andstate estimation. For details of the algorithms, the reader is referred to the references.

3.2 Black-box and grey-box modeling

In the system identification step, we aim at building models f i and hj that approximate (1) and(2), respectively, from historical data. Next, we present the methodology used to separately obtainboth f i and hj , for i = 1, . . . ,Mf and j = 1, . . . ,Mh.

Let g be the multi-input single-output NARMAX (MA stands for moving average) model givenby

zk = g (zk−1, . . . , zk−nz , qk,1, . . . , qk,p, ek, θ) , (3)

where zk ∈ R is the output, qk,l =�µk,l . . . µk−nql ,l

�T

, l = 1, . . . , p, is the lth vector of delayed

values of the input µl, p is the number of inputs, θ ∈ Rnθ is the unknown parameter vector,

ek�= [ξk, . . . , ξk−nr ]

T

is the residual vector, and nz, nql , and nr are the maximum lags allowedto z, ql, and ξ, respectively.

We consider two cases. First, g is assumed to be polynomial with degree �, such that (3) canbe expressed as the regression

zk = ψT

k−1θ + ξk, (4)

6

where ψk−1 ∈ Rnθ is the regression vector given by affine and nonlinear combinations of delayedsystem outputs zk−1, . . . , zk−nz , inputs µk,1, . . . , µk−nq1 ,1

, . . . , µk,p, . . . , µk−nqp ,p, and residualsξk−1, . . . , ξk−r+1. The problems of structure selection and parameter estimation are solved using,respectively, the error reduction ratio and orthogonal extended least squares [Billings et al.(1989)].Recall that the delayed values of ξ are used in (4) only to mitigate bias in parameter estimation;that is, if we set ek = 0 in g, then we obtain a NARX polinomial model.

Second, g is assumed to be a committee machine model of feedforward multilayer perceptron(MLP) neural networks. An ensemble of ten neural networks is built using the bagging (bootstrapaggregating) procedure [Breiman(1996)] for ensemble learning applied to data collected at differentinstants of time and operating points. The simple average is used to combine the neural modelpredictions. Each neural model is set with ten nodes in the hidden layer with hyperbolic tangentactivation function and with linear output layer. All MLP models have the same regressors set.Also, we combine both dynamical and steady-state data to obtain a grey-box MLP model whosestatic function is closer to the one produced by steady-state data. To achieve so, we employ themulti-objective evolutionary algorithm described in [Barbosa et al.(2011)].

Varying model structure in (3) and the dynamical data set used to estimate its parameters,different process and observation models can be obtained. According to Figure 3, note that anymodel of the type (1)-(2) can be employed in the methodology investigated. Polynomial and MLPmodels are chosen for convenience. Next, we show how to write the NARMAX model given by (3)as either f i (1) or hj (2) (state-space model).

3.2.1 Process Model

Set [zk zk−1 . . . zk−nz+1]T

= xk such that nz = n and [qk−1,1 . . . qk−1,p]T

= ufk−1 such that

nq1 + . . .+ nqp = pf . Then, we obtain

xk = f i(xk−1, ufk−1) + wk−1

as

x1,k

x2,k,...

xn,k

=

g(x1,k−1, x2,k−1, . . . xn,k−1, ufk−1, 0, θ)

x1,k−1...

xn−1,k−1

+

w1,k−1

w2,k−1...

wn,k−1

, (5)

where xl,k is the lth entry of xk and w1,k−1 is the one-step-ahead simulation error of g withcovariance σ2

wi such that we set Qi = σ2wiIn.

3.2.2 Observation Model

Likewise, in (3), set zk = yk such that m = 1, qk,1 = xk and [qk,2 . . . qk,p]T

= uh,1k such that

nq1 = n and nq2 + . . . + nqp = ph,1, where

�uh,1k

uh,2k

��= uh

k such that ph,1 + ph,2�= ph,

[zk−1 zk−2 . . . zk−nz ]T

= uh,2k such that nz = ph,2, ek = 0. Then, we obtain

yk = hj(xk, uhk) + vk

asyk = g(uh,2

1,k , . . . uh,2nz,k

, xk, uh,1k , 0, θ) + vk, (6)

where uh,2l,k is the lth entry of uh,2

k and vk is the one-step-ahead simulation error of g with covariance

Rj . Recall that the model g used in (6) is different from its counterpart in (5). If we set nz ≥ 1 in(6), then hj is actually an IIR model; otherwise hj is a FIR model.

7

3.3 Filter bank

3.3.1 Unscented Kalman filter

In this work UKF is chosen to perform state estimation. Figure 1b illustrates how it combinesuncertain information from stochastic models and noisy measurements. To achieve that, first,initialize the filter with the state estimate xu

0|0 and corresponding covariance P xx,u0|0 . UKF is a

two-step recursive state estimator, whose forecast step is given by

{xuk|k−1, P

xx,uk|k−1, y

uk|k−1, P

yy,uk|k−1, P

xy,uk|k−1} =

Forecast

�xuk−1|k−1, P

xx,uk−1|k−1, u

fk−1

T

, uhk

T

, f, h,Qu, Ru

�, (7)

and whose data-assimilation step is given by

{xuk|k, P

xx,uk|k , ηuk|k−1} = DataAssimilation

�xuk|k−1, P

xx,uk|k−1, yk, P

yy,uk|k−1, P

xy,uk|k−1

�, (8)

where ηuk|k−1 = yk − yuk|k−1 is the innovation with covariance P yy,uk|k−1 and cross-covariance P xy,u

k|k−1.One-step ahead prediction is performed during forecast using the process model and measuredinputs, while correction is performed during the data-assimilation step using observation modeland measured outputs.

The detailed equations of UKF are not presented here for convenience but they are found in[Julier and Uhlmann(2004)]. The UKF parameters were set as in [Arasaratnam and Haykin(2009)].

Rather than linearizing models f (1) and h (2) in (7) and (8), respectively, UKF propagates anensemble of 2n sigma points through such nonlinear models. For notational convenience, we referto (7) and (8) as the function

{xuk|k, P

xx,uk|k , ηuk|k−1, P

yy,uk|k−1} = UKF

�xuk−1|k−1,

P xx,uk−1|k−1, [u

fk−1

T

uhk

T

]T

, yk, f, h,Qu, Ru

�. (9)

3.3.2 Interacting multiple model bank

The IMM filter bank is illustrated in Figure 4. IMM runs in parallel M Kalman filters andcombines theirs estimates based on the innovation of each filter. That is, the filter whose innovationrepresents a small output error estimate receives a larger weight. In doing so, switching dynamicsis suboptimally tracked along time.

To achieve that, IMM is a three-step recursive algorithm. To start, each sth filter is initializedwith xs

0|0 and P xx,s0|0 , and with the initial combination weight γs

0 . Next, the model switching

probabilities pr|s, r, s = 1, . . . ,M , are set as the probabilities of the model {fr, hr} at time k − 1be switched to model {fs, hs} at time k.

In the first step, the input {x0sk−1|k−1, P

xx,0sk−1|k−1} to the sth filter, s = 1, . . . ,M , is obtained

from an interaction of the M filters by mixing the previous estimates {xsk−1|k−1, P

xx,sk−1|k−1} with

the mixing weights 0 ≤ ωr|sk−1 ≤ 1, r, s = 1, . . . ,M . That is, each filter is driven with different

initial conditions.In the second step, a bank of M Kalman filters run in parallel. Here, we choose the UKF

algorithm such that, for all s = 1, . . . ,M ,

{xsk|k, P

xx,sk|k , ηsk|k−1, P

yy,sk|k−1} = UKF

�x0sk−1|k−1, P

xx,0sk−1|k−1,

usk, y

sk, f

s, hs, Qs, Rs) , (10)

8

Figure 4: Diagram of the filter bank approach, in which estimates xsk|k, s = 1, . . . ,M, obtained from

M UKFs running in parallel are combined. Each UKF uses a different combination of process fs andobservation hsmodels and is initialized by a different prior estimate xs

k−1|k−1 at each time step. Like in aclosed-loop system, the error between the estimated output ys

k|k−1 and the measured output ysk is used to

correct the prediction of the process model xsk|k−1 in each filter. The estimates of each filter are combined

to yield the final filter bank estimate.

9

where usk, y

sk, Q

s, and Rs are the parameters of the process model fs and observation model hs

used in the sth UKF. Each UKF uses a different pair of models {fs, hs}. Note that, in additionto the estimate xs

k|k with covariance P xx,sk|k , it is necessary to return the innovation ηsk|k−1 and

covariance P yy,sk|k−1.

In the third step, the M estimates are combined using the combination weights 0 ≤ γsk ≤ 1,

s = 1, . . . ,M .The detailed equations of IMM are not presented here for shortness but they are found in

[Bar-Shalom et al.(2001), pp. 453–460].

4 Experimental results: System identification

For the gas-lifted oil well illustrated in Figure 2, different configurations of models are obtained.Table 2 summarizes the main features of the 4 process models, f i, i = 1, 2, 3, 4, and 3 observationmodels, hj , j = 1, 2, 3, that were built.

Both NARX polynomial and NARX MLP models were built from historical data. All variableslisted in Table 1 were sampled at T = 1min, as the fastest sampling rate provided by the PIMS.Linear and nonlinear autocorrelation analysis [Aguirre(2005)] indicated that such value was ap-propriate. Figure 5a shows the time series of the downhole pressure PT1 in a period of 7 months,ending up with the PDG failure. PT2, TT3, and FT4 are shown in Figure 5b,c,d. For convenience,time series of all variables are divided into 6 windows: Data11, Data12, Data2, Data4, Data22,and Data5.

10

0 1000 2000 3000 4000 50000

20

40

60

80

100

120

140

160

180Data11 Data12 Data2 Data4 Data22 Data5

Time (hr)

PT

1 k (kg

f/cm

2 )

0 1000 2000 3000 4000 50000

20

40

60

80

100

120

140

160

180Data11 Data12 Data2 Data4 Data22 Data5

Time (hr)

PT

2 k (kg

f/cm

2 )

(a) (b)

0 1000 2000 3000 4000 5000!20

!10

0

10

20

30

40

50

60Data11 Data12 Data2 Data4 Data22 Data5

Time (hr)

TT3 k (C

)

0 1000 2000 3000 4000 50000

0.5

1

1.5

2

2.5

3

3.5

4x 10

4

Data11 Data12 Data2 Data4 Data22 Data5

Time (hr)

FT

4 k (m

3 /h)

(c) (d)

Figure 5: Time series of the (a) downhole pressure PT1, (b) wet christmas tree pressure PT2, (c) temper-ature before production choke valve TT3 and (d) gas-lift flow rate FT4 with duration of about 7 monthsand sampling time of 1 min. For convenience, such data were divided (and labeled) into 6 windows, namely,Data11, Data12, Data2, Data4, Data22, and Data5. Dynamical data for modeling are selected from Data2,while steady-state data (for grey-box modeling) are taken from both Data2 and Data4. At about hour4800, the downhole sensor fails permanently.

11

Tab

le2:

Con

figu

ration

ofpolyn

omialan

dMLP

NARX

mod

elsidentified

bythetags

f(process

mod

el)or

h(observation

mod

el).

Polyn

omialmod

elsarecharacterizedby

degree�,whileneuralmod

elsby

thenu

mber

ofnon

linearnod

esdin

thehidden

layer.

Theneural

networks

areallcommitteemachinemod

els,

each

onecompou

nded

of10

feedforw

ardMLP

mod

els.

Inputs

andou

tputs

aredefined

from

theprocess

variab

leslisted

inTab

le1.

σ2 wan

dσ2 vaccountsforon

e-step-aheaderrorvarian

ceof

each

mod

elduringvalidation.nqlan

dnz

correspon

dsto

themax

imum

delaysof

each

inputor

output,

respectively.It

isim

portant

topoint

that,fortheMLP

mod

els,

nqldoes

not

necessarily

meansthat

allpossible

delayed

regressors

from

k=

1to

k=

nqlareused.In

fact,fornql=

150,

regressors

withdelays1,

100,

and150wereused;fornql=

136,

regressors

withdelays1,

42,an

d136wereused;an

dfornql=

22,regressors

withdelays1,

5,an

d22

wereused.

Processmodels

Observationmodels

Tag

type

�/d

inputsµl

nql

outputz

nz

σ2 w

Tag

type

�/d

inputsµl

nql

outputz

nz

σ2 v

f1

poly

3FV4

2PT1

20.0814

h1

poly

1PT1

2PT2

20.1386

f2

poly

3FV4

5PT1

50.0489

h2

poly

2PT1,T

T2

2,2

PT2

20.1302

f3

MLP

10

FT4,P

T4

150,150,

PT1

30.0192

h3

MLP

10

PT1

3TT3

30.2698

PT3,T

T3,P

T3a

22,22,3

f4

MLP

10

FT4,P

T4

136,136,

PT1

31.4723

PT3,T

T3,P

T3a

22,22,22

12

Polynomial models f1, f2, h1 and h2 were built using only 2250 samples (37.5 hours) of Data2. The neural models f3 and h3 were built using 10000 samples (50 windows of 600 samples spreadalong Data 2) so that different operating points are used. These samples were chosen to include thetime intervals for which changes in the operating points are observed. At this point, it is importantto recognize the existence of outliers in the dynamical data shown in Figure 5. However, since themodeling techniques of Section 3.2 are run offline (i.e., batch algorithms), the modeling data werechosen to not include outliers. Finally, the grey-box neural model f4 was built from 4150 samplesof dynamical data of Data2 plus the steady-state data shown in Figure 6 that was obtained fromobserving Data2 and Data4 as described by [Abreu al.(2012)]. Note that Data 2 has operatingpoints not covered by Data 4.

Different configurations of inputs and outputs (and corresponding maximum delays) are set foreach model. Recall that the polynomial models h1 and h2 are based on process variables fromboth seabed and platform, while h3 uses only platform variables. For each model, the one-step-ahead error variance during validation, σ2

w or σ2v , is calculated. These values are used to set noise

covariances Q and R as indicated in Section 3.2. Details on the modeling step can be found in[Teixeira et al.(2012), Abreu al.(2012)]. Next, for convenience, we present the obtained processand observation models as NARX models rather than in state space. They can be rewritten in thestate-space form following the procedure detailed in sections 3.2.1 and 3.2.2.

4.1 Process Models

We built two polynomial process models. f1 is given by

zk =1.6780 zk−1 − 0.6523 zk−2 − 2.9730×10−4 z2k−2−4.3333×10−6 µ2

k−2zk−1 + 2.1669×10−4 µk−2µk−1 + ξk,(11)

and f2 is given byzk =1.9455 zk−1 − 1.0706 zk−2 + 0.1683 zk−5

−0.9848×10−3z2k−3 − 0.3838×10−6µk−2µk−4z−1+0.5636×10−5z3k−1 + ξk.

(12)

Both models have input µk = FV4k as the gas-lift valve position and output zk = PT1 as thedownhole pressure. Other features of these polynomial models are found in Table 2.

Model validation for f1 and f2 is shown in Figure 7. The mean-absolute-percentual error(MAPE) of (11) and (12) are 2.12% and 1.77%, respectively. Note that, though the free-runsimulation of such models can track the nonlinear gain variations, they cannot reproduce thesevere slugging flow dynamics. Indeed, gas-lift flow does not have a direct influence on it, as dothe seabed topography and piping configuration. Therefore, other input variables may be used todrive models that reproduce such flow dynamics, such as the production temperature TT3.

Aiming at finding process models that may reproduce the slugging flow dynamics as well asthe nonlinear gain variations, two neural models are identified: f3 and f4. To achieve that, otherprocess variables are set as inputs as follows. f3 and f4 are both represented by

zk = g(zk−1, zk−2, zk−3, µk−1,1, µk−a,1, µk−b,1,µk−1,2, µk−a,2, µk−b,2, µk−1,3, µk−5,3, µk−22,3,µk−1,4, µk−5,4, µk−22,4,µk−1,5, µk−5,5, µk−22,5, µk−1,5, µk−c,5, , µk−d,5) + ξk,

(13)

with inputs as the gas-lift flow rate µk,1 = FT4k, gas-lift pressure µk,2 = PT4k, productionpressure µk,3 = PT3k, production temperature µk,4 = TT3k, and the shut-down valve (SDV)

pressure µk,5 = PT3ak, and with output zk = PT1k as the downhole pressure. For f3, the input

13

2000 3000 4000 5000 6000 7000 8000 9000 10000 1100065

70

75

80

85

90

95

FT4 (m3/h)

PT

1 (k

gf/c

m2 )

Figure 6: Steady-state data with FT4 as input and PT1 as output. Experimental data (•) are obtained

from Data2 and Data4. The static curve of model f4, which was identified using such steady-state data,is given by (�). For comparison, (+) indicates the static curve of f4 without the use of steady-state dataduring identification. Adapted from: [Abreu al.(2012)].

delays are a = 100, b = 150, c = 5, and d = 22; while, for f4, we have a = 42, b = 136, c = 2,and d = 3. Model f4 is a grey-box model for which steady-state data were used to improve globalbehavior.

Model validation for f3 and f4 is also shown in Figure 7, with MAPE indices respectively givenby 0.72% and 1.67%. Also, the static curve of model f4 is shown in Figure 6.

4.2 Observation Models

We built two polynomial observation models, h1 and h2, and one neural observation model, h3. h1

is the affine model given by

zk = 1.7128 zk−1 − 0.90191 zk−2

+0.1184µk−1 + 1.0701 + ξk,(14)

whose input µk = PT1k is the downhole pressure, while h2 is the multi-input model given by

zk = 1.6205 zk−1 − 0.8190 zk−2 + 0.15296µk−1

−0.10697×10−2µk−2,1µk−2,2

+0.15138×10−2µ2k−1,2 + ξk,

(15)

whose inputs are the downhole pressure µk,1 = PT1 and the christmas tree temperature µk,2 =TT2. In both cases, the output zk = PT2k is the christmas tree pressure. Finally, we obtain theneural model h3

zk = gh(zk−1, zk−2, zk−3, µk−1, µk−2, µk−3) + ξk, (16)

14

1.3 1.35 1.4 1.45 1.5 1.55

x 104

74

76

78

80

82

84

86

88

k

PT

1k (

kgf/cm

2)

Figure 7: Validation of process models f1 (blue dashed), f2 (red dot-dashed), f3 (cyan dotted) and f4

(green dotted) by free-run simulation. Measured data is in black solid line. Oscillations are due to severeslugging in flow dynamics.

whose input µk = PT1 is the downhole pressure and output zk = TT3 is the production tempera-ture.

Model validation for (14)-(16) is shown in Figure 8. The corresponding MAPE indices aregiven by 1.17%, 0.87%, and 4.69%, respectively. Such indices suggests that the oscillations inthe christmas tree pressure PT2 can be more precisely explained by the downhole pressure PT1compared to the production temperature TT3. However, measurements of PT2 are not alwaysavailable in gas-lift oil wells, justifying the need for an observation model like h3.

5 Experimental results: State estimation

In order to recursively estimate the downhole pressure (PT1), we now use the identified modelspresented in Section 4. We test five closed-loop schemes: combining pairs of models (individually)i) {f1, h1}, ii) {f2, h2}, iii) {f3, h3}, and iv) {f4, h3} using UKF, and v) combining the four pairsof models using an IMM filter bank. Henceforth, these five schemes will be respectively referred asUKF1, UKF2, UKF3, UKF4 and IMM. Recall that downhole pressure (PT1) measurements arenot required to perform downhole pressure estimation by none of the aforementioned schemes.

These four pairs of models are chosen because they yield better performance compared to theother possible combinations (not shown) of process and observation models. Moreover, note that

15

(a)

(b)

Figure 8: Validation of NARX observation models (a) polynomial h1 (14) and h2 (15) and (b) MLP h3

(16) by free-run simulation. Here, validated output is known only until k = 13700 in order to better zoomin the severe slugging oscillations.

16

1600 1620 1640 1660 1680 1700 1720 1740 1760 1780 180076

78

80

82

84

k

PT

1k (

kgf/cm

2)

measured UKF1 UKF2 UKF3 UKF4 IMM

3.572 3.573 3.574 3.575 3.576 3.577 3.578

x 104

65

70

75

80

k

PT

1k (

kgf/cm

2)

4.225 4.23 4.235 4.24 4.245 4.25

x 104

60

80

100

120

k

PT

1k (

kgf/cm

2)

1600 1620 1640 1660 1680 1700 1720 1740 1760 1780 18000

0.2

0.4

0.6

0.8

k

!k

3.572 3.573 3.574 3.575 3.576 3.577 3.578

x 104

0

0.5

1

k

!k

4.225 4.23 4.235 4.24 4.245 4.25

x 104

0

0.5

1

k

!k

UKF1 UKF2 UKF3 UKF4

(a) (b)

Figure 9: (a) Downhole pressure estimation using UKF1, UKF2, UKF3, UKF4, and IMM, and (b)the corresponding weights γk to yield IMM estimates. Different windows of Data12 are shown ineach plot to indicate that relative performance depends on the operating point. In the top plot,UKF2 has the best performance (RMSE indices are, respectively, given by 2.03, 0.69, 0.70, 1.02,0.77). In the middle plot, IMM yields the smallest RMSE (0.66, 0.66, 0.64, 3.06, 0.48), UKF3 hasthe best performance (3.33, 5.62, 2.02, 4.46, 2.44) in the bottom plot.

only polynomial models are used in UKF1 and UKF2; while, neural networks are used in UKF3and UKF4. Also, seabed variables are used only in UKF1 and UKF2. That is, for convenience,models are paired up based on the mathematical representation and the employed input variables.Since all models are somewhat combined by the filter bank, their pairing is not the most criticalissue. For scheme v), we set pr|s based on the relative uncertainty of each model, that is,

pr|s =

��Mj=1 σ

2w,j

�− σ2

w,s

2(M − 1)�M

j=1 σ2w,j

+

��Mj=1 σ

2v,j

�− σ2

v,s

2(M − 1)�M

j=1 σ2v,j

;

see Table 2 for numerical values of each model uncertainty. For this application, IMM was notoversensitive to different values of pr|s (not shown).

5.1 Single filter versus filter bank

The exploration of gas-lift oil wells is typically characterized by a time-varying and nonlinearprocess. Depending on the downhole and gas-lift pressure, different flow dynamics may be observed.Therefore, it is expected that finding a single pair of models that fits well the dynamics throughouttime is not an easy task. Recall that the closed-loop scheme already adds robustness to variationsdue time and operating point. At this point, the following question arises: does using a filter bankrather than a single filter pay off?

To illustrate the aforementioned point, the schemes UKF1-UKF4 were tested. On the one hand,UKF1 and UKF2 employ “simpler” models but using seabed process variables as measurements.On the other hand, UKF3 and UKF4 employ more “elaborated” models but using only platformvariables. Figure 9 illustrates how the relative performance of the algorithms under investigationvaries depending on the operating point. Figure 10 shows the RMSE index for each scheme along

17

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

1

2

3

4

5

6

7

8

9

10

11

Time (hr)

RM

SE

(kg

f/cm

2)

Data 11 Data 12 Data 2 Data 4 Data 22 Data 5

2.35

2.25

3.39

2.78

1.76

UKF1 UKF2 UKF3 UKF4 IMM

Figure 10: RMSE index (kgf/cm2) calculated for each one of the 6 windows of data (#11, #12,#2, #4, #22, #5). Each point in the plot is the average RMSE for an interval of one week. Thehorizontal solid lines determines the average RMSE for the period of 7 months, while the verticaldashed lines determines the windows of data.

Table 3: Number of intervals per window of data for which a given algorithm yielded the smallestRMSE index. Each one of the 6 windows of data (#11, #12, #2, #4, #22, #5) was dividedinto smaller intervals (whose number is shown in the last row) according to the different operatingpoints observed. Therefore, the intervals defined here are different from those in Figure 10.

#11 #12 #2 #4 #22 #5 TotalUKF1 0 0 0 8 2 2 12UKF2 4 3 0 1 1 0 9UKF3 1 8 4 1 2 0 16UKF4 0 0 0 1 1 0 2IMM 2 3 3 6 4 0 18Total 7 14 7 17 10 2 57

the 7-month test. Accordingly, Table 3 enumerates how many times a given scheme yields the bestRMSE value for each one of the 6 data sets.

There is no evident dominance of a single filter compared to the others. For instance, UKF3seems to be superior for data windows 12 and 2, while UKF1 outperforms the other single schemesfor data windows 4, 22, and 5; see Table 3.

Still based on Table 3, we see that IMM yields the best RMSE in 32% of the intervals (18 outof 57), while the single UKF1 almost ties such performance reaching 28%. If we only look at thesenumbers, we may mistakenly conclude that using a filter bank does not pay off. However, it isinteresting to note that when IMM is not the best choice (regarding RMSE), it is the second bestchoice with RMSE close to the best algorithm; see Figure 10. That is, combining different pairsof models to build a soft sensor is a more efficient and stable scheme. For instance, the average

18

of RMSE for IMM is 1.76kgf/cm2, while the second best average is 2.25kgf/cm2 using UKF1.Moreover, we observe that the RMSE for IMM has a smaller standard deviation.

Concerning computational processing time, tests performed in a PC with processor Intel(R)Core(TM) i7-2670QM, CPU @ 2.20GHz, 6 GB DDR3 running Matlab indicated that it takes about83 ms to run one iteration of IMM and 0.7 ms to run one iteration of UKF1. Recall that data aremeasured at every 1 min. For a C++ implementation, these processing times were about 3 timeslarger.

5.2 Black-box versus grey-box modeling

UKF3 and UKF4 use the same observation model h3 but different process models f3 and f4.Recall that f4 is built using both dynamical and steady-state data by means of grey-box modeling[Nepomuceno et al.(2007)]. In doing so, global performance is improved. Now we investigate whatis the advantage of employing a grey-box model.

Figure 11 shows the results of the 7-month estimation of downhole pressure for both UKF3 andUKF4. Figure 10 compares the performance of UKF3 and UKF4.

The relative performance of the RMSE indices for these algorithms alternates throughout time.For the data tested, the average RMSE improved from about 3.39 kgf/cm2 (UKF3) to 2.78 kgf/cm2

(UKF4). Also, UKF4 seems to yield a more steady performance.UKF3 beats UKF4 for data windows 11, 12, and 2; however, it loses performance starting at

Data4. Note that the data windows 4, 22 and 5 have operating points that were not presentin the identification data of model f3 (namely, Data2). Recall that when steady-state datafrom Data 2 was used to build the grey-box model f4, different operating points (not presetin the dynamical data) were considered, adding generality to the final model. In accordance with[Aguirre and Letellier(2009)], this explains the improved performance of f4 for the data windows4, 22 and 5.

5.3 Platform versus seabed process variables

Finally, we investigate what is the cost of using only platform process variables compared to thecase in which seabed (specifically, wet christmas tree) variables are also assumed to be measured.From Figure 10, one can see that UKF1 and UKF2 (that is, filters that use seabed measurements)yield smaller RMSE indices compared to UKF3 and UKF4.

Figure 12 shows the results of the 7-month estimation of downhole pressure. UKF3 representsthe case where only platform variables are available, while IMM uses both platform and seabedvariable. The average RMSE for UKF3 is 3.39 kgf/cm2. Compared to IMM, for which RMSE 1.76kgf/cm2 (Figure 10), this increase is RMSE is not significant. Actually, if we consider that PT1varies in the range 60-110 kgf/cm2 (see Figure 12), using seabed variables improved accuracy inabout 3.3%. That is, measuring seabed variables seems not to be critical for monitoring downholepressure. It is important to point out that, at some time instants (for instance, close to PDGfailure), UKF3 performance is much worse than IMM (for which accuracy improvement is about14%). These results suggest that if seabed measurements are not available, then building more“elaborated” models (such as those employed in UKF3 and UKF4) may pay off.

6 Concluding Remarks

The problem of designing a data-driven soft sensor to estimate the downhole pressure in gas-liftedoil wells is investigated in this paper. Most soft sensors developed for gas-lift oil wells (reportedin the literature) are model-driven and, thus, require the knowledge of physical parameters. We

19

0 500 1000 1500 2000 2500 3000 3500 4000 4500 500040

50

60

70

80

90

100

110

120

130

140

Time (hr)

PT

1k (

kgf/

cm2)

Data11 Data12 Data2 Data4 Data22 Data5

measured UKF3 UKF4

Figure 11: Downhole pressure estimation using UKF3 and UKF4 during 7 months of tests. Dif-ferent from UKF3, UKF4 has a process model built using grey-box techniques.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 500040

50

60

70

80

90

100

110

120

130

140

Time (hr)

PT

1k (

kgf/

cm2)

Data11 Data12 Data2 Data4 Data22 Data5

measured IMM UKF3

Figure 12: Downhole pressure estimation using UKF3 and IMM during 7 months of tests. Differentfrom IMM, UKF3 uses only platform process variables.

20

employ a two-step procedure. First, NARX polynomial and NARX MLP neural models are builtusing historical data collected from an actual oil well. Second, recursive predictions of downholepressure by such models are combined with measured data of other variables (for instance, christmastree or production variables) by means of a interacting multiple model filter bank. In each filterof the bank, an unscented Kalman filter performs a closed-loop model prediction.

Experimental results indicate that such closed-loop scheme improves estimation accuracy com-pared to the free-run model prediction. Also, employing a filter bank pays off compared to thecase in which a single filter is used. More importantly than improving accuracy, the bank ap-proach adds robustness regarding outliers in measured data and time-varying dynamics. That is,if multiple (closed-loop) models are used, then “simpler” models can used. Conversely, if a single(closed-loop) model is used, its global performance has more impact on the estimation accuracy.For instance, when we used a grey-box model for which steady-state data was used to improveglobal performance, the estimation accuracy of the corresponding filter also improved. For thedata tested, RMSE decreased from about 3.39 kgf/cm2 to 2.78 kgf/cm2. Interestingly, we ob-served that, compared to the performance of the single filters, the performance of the filter bankis always better or at least close to the performance of the most accurate filter.

Finally, from the process perspective, we evaluated the advantage of measuring seabed processvariables and using such measurements in state estimation. For the data tested, RMSE decreasedfrom about 3.39 kgf/cm2 to 1.76 kgf/cm2.

Acknowledgements

This work was partially supported by Petrobras and Brazilian agencies FAPEMIG and CNPq. Theauthors are grateful for the assistance and cooperation of Leandro F. Abreu, Lucas P. Gomes, andProf. Bruno H.G. Barbosa during the development of this research.

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