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Transient Fluid Temperature Estimation in Wellbores
1st IFAC Workshop on Thermodynamic Foundations of Mathematical Systems Theory July 13-16, 2013. Lyon, France
Transient Fluid Temperature Estimation in Wellbores Javad Abdollahi, Stevan Dubljevic Department of Chemical & Materials Engineering, University of Alberta, Edmonton, Canada, T6G 2V4 (Tel: 780-248-1596; e-mail: [email protected]) Abstract: Knowledge of transient temperature distribution in wellbore fluid is critical for the pressure transient tests during draw-down and shut-in processes. In absence or limitation of measurement devices installed on the wellbore, temperature estimation through soft sensors can properly fulfill the absence of these measurement devices. In this work, the heat transfer model for wellbore fluid is reduced to a set of hyperbolic PDEs and applying the method of characteristics and calculation of Riemann invariants, the observability of the temperature distribution is discussed for the governing set of hyperbolic PDEs. In case of oil well, the governing equations are reduced and discretized using method of characteristics and then reformulated in order to implement the observer. The developed model is used for Luenberger observer and Kalman filter synthesis using well-head temperature measurements. The observer is synthesized based on the reduced linear model and is examined on the high fidelity finite difference model of the wellbore fluid flow. The provided transient temperature estimation is enough accurate and reliable for on-line and real-time implementation in wellbore fluid. 1. INTRODUCTION Real-time knowledge of temperature distribution in wellbore fluid is crucial for density profile prediction. Analysis of transient temperature along with transient pressure tests are necessary for monitoring and maintenance of flow lines and facilities. During the drawdown and shutin, transient temperature and fluid flow occurs, while the fluid dissipates heat to the surrounding. The heat transfer process changes the temperature which affects the density, momentum and fluid pressure. In the state-of-theart completed and realized wellbores, the temperature profile along the wellbore is measured using Distributed Temperature Sensors (DTS) [Wang et al., 2008]. Although DTS systems provide high resolution temperature measurements, the technology is new and not available for already existing mature oil and/or gas wells. The realtime transient temperature estimation and monitoring using available wellhead temperature measurements can substitute DTS systems by soft sensors developed based on dynamic system model. Temperature distribution modeling and prediction in wellbore fluid has attracted attention of petroleum engineering community [Hasan and Kabir, 1994, Ramey JR., 1962, Hasan and Kabir, 2012, Izgec et al., 2007, Kabir et al., 1996a, Hasan et al., 2009, 2005, Spindler, 2011]. The temperature modeling approaches are mostly founded on the work of Ramey JR. [1962], where the wellbore model is assumed to be surrounded by an infinite reservoir and heat transfer takes place through the wellbore to the surroundings. Hasan and Kabir [1994] improved the model to more realistic finite radius reservoir and considered the heat diffusion from the formation to the wellbore. The transient temperature distribution in wellbore occurs when there is a change in production rate manifested as 978-3-902823-40-3/2013 © IFAC
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shut-in, restarting or change of production schedule. Along this line, due to the compressible flow nature, the temperature transients produce a transient momentum, heat and flow change, see [Hasan and Kabir, 2012, Kabir et al., 1996a]. Kabir et al. [1996b] developed a transient model for coupled reservoir/wellbore temperature distribution and Hasan et al. [2005] decoupled the heat transfer equation and momentum equations by neglecting the afterflow in the wellbore. In addition to above contributions, Spindler [2011] provided an analytical solution for approximated transient temperature model and also extended the model to include heat conduction in axial direction. Due to the long length of the oil/gas wells, the modelling of the transport phenomena (momentum transport and heat transfer) leads to distributed parameter model equations which can not be easily approximated by a single lumped system equation. The transport of petroleum is governed by momentum and heat transfer equations given as one dimensional hyperbolic partial differential equations. Since there are limited approaches for control and estimation of hyperbolic processes [Ray, 1981], a major part of recent contributions is focused on simulation of existing temperature models and in best case on coupling existing models with reservoir equations, see [Izgec et al., 2007, Kabir et al., 1996a, Hasan et al., 2005]. Since the wellbore temperature model is given by the first order one dimensional hyperbolic PDE system, the infinite-dimensional nature of distributed parameter processes makes the estimation procedure a challenging task. State estimation and control of systems governed by partial differential equations (PDEs) is a mature area of research activity, see [Curtain and Zwart, 1995, Bensoussan et al., 2007]. Despite the fact that boundary control strategies for parabolic PDEs are well developed, a state estimation of parabolic PDEs is less developed and is of 10.3182/20130714-3-FR-4040.00018
2013 IFAC TFMST July 13-16, 2013. Lyon, France
interest. In particular, for parabolic systems, Xu et al. [1995] discuss an observer design for dissipative bilinear systems with weak error convergence to zero. The Luenberger observer is implemented by Vries et al. [2010] for synthesis of observer for the Strum-Lioville systems and also by Li and Xu [2011] for a higher order PDE describing the model of rotating body-beam system. Along the same line, Harkort and Deutscher [2011] have focused on observer based controller synthesis for Riesz-spectral systems. Although there is a significant number of contributions on parabolic PDEs, there are quite a few contributions on state estimation and observer design for hyperbolic systems [Wu and Liou, 2001, Vazquez et al., 2011, Di Meglio et al., 2012]. Namely, Wu and Liou [2001] have demonstrated output-regulation for a set of hyperbolic systems, while the backstepping method is used by Vazquez et al. [2011] for the state estimation of 2 × 2 linear hyperbolic system. In this work, a set of hyperbolic partial differential equations arising from the heat transfer, fluid motion and continuity balance equations in wellbore is introduced as relevant model of the transport processes taking place in oil/gas wellbores. The method of characteristics and the Riemann invariants are utilized in evaluating the model features. In particular observability condition is explored within the observer design and limitations associated with the model are discussed. The model is simplified for incompressible oil flow in wellbore and reduced to the heat transfer model given by a one dimensional hyperbolic PDE. Subsequently, the Luenberger observer and Kalman filter are used for on-line estimation of transient temperature distribution in the wellbore fluid. The hyperbolic temperature model is reformulated to standard linear time invariant form using method of characteristics and then the observer is designed for real-time temperature estimation. Motivated by realistic set of available measurements, the state estimation is based on the temperature measurement available at the wellhead (point measurement), and the estimated transient temperature profile is provided. The temperature estimation is also examined by assuming stochastic uncertainties in the model and in the measurement and the state reconstruction is realized by standard Kalman filter. In the ensuing sections, we provide a description of the transient temperature and momentum dynamics model and discuss the characteristics and Riemann invariants. The following section deals with the model description and reduction. In section 3, the method of characteristics is used to reformulate the model to a suitable matrix form for state construction and estimation purpose. Finally, the numerical simulation results are provided along with Luenberger observer and stochastic state estimation synthesis for transient temperature.
dimensional coupled hyperbolic equations. The following schematic and geometry of the well used for the modelling is shown in Fig.1. ∂T ∂T (1 + CT ) +v = v[Tei − T ]LR ∂t ∂z Af CJ ∂p ∂p +[ + ][ +v ] mf cf Jgc Jgc ∂t ∂z Af k f Aw kw CJ ∂ 2 T +[ + ] mf cf Jgc mw cw Jgc ∂z 2 4 µf Af ∂v + ( )2 (1) 3 mf cf Jgc ∂z ∂v ∂v 1 ∂p 4 ∂2v +v =− − g sin θ + µf 2 ∂t ∂z ρ ∂z 3 ∂z ∂ρ ∂ρ ∂v +v +ρ =0 ∂t ∂z ∂z
(2) (3)
where T and Tei are the fluid temperature and earth formation temperature, p, ρ and v are the pressure, density and fluid velocity along the well, z and t represent depth of the well measured from well bottom and time. All other parameters are shown and described in Table 1. In the case of oil flow which can be considered incompressible, the equations of motion, continuity and heat transfer equation can be reduced to energy balance equation and this special case can be considered as a subset of general wellbore models. However for the gas wells, the momentum, energy and continuity conservation equations should be considered and solved simultaneously in order to obtain the evolution of the physical variables within well. In order to model the wellbore behaviour the comprehensive coupled momentum, continuity and heat transfer model including conductive terms, viscous terms and higher order terms is necessary and it is given by Eqs.1-23, see [Spindler, 2011]. The fluid is assumed to flow along the constant diameter well and there is no disturbance in earth formation temperature. Namely, the heat conduction through the fluid and between the fluid and the well surrounding are main heat transport modes. The model also accounts for the work of fluid pressure and viscous forces, given in Eq.1. In general one needs to account for the coupling among Eqs.1-2-3, however the case of the incompressible fluid flow (for example the oil flow within wellbores characterized by the artificial lift - sucker rod pumps) can allow for the
2. FLUID FLOW AND HEAT TRANSFER MODEL IN WELLBORE Fluid flow in a wellbore can be considered as one dimensional flow in a constant radius pipe. Depending on the flowing fluid (e.g. oil or gas), the fluid flow can be considered as compressible or incompressible flow. The general governing equations (momentum, continuity and heat transfer) are given in Eqs.1-2-3 as a set of one 109
θ Fig. 1. Schematic of wellbore geometry used for modeling [Hasan and Kabir, 2012].
2013 IFAC TFMST July 13-16, 2013. Lyon, France
model reduction since the main model feature becomes constant velocity flow from the wellbore bottom to the surface. We provide a brief explanation for compressible flow in the following paragraph. The oil flow will be discussed in details in the following sections. Compressible flow in a wellbore consist of energy balance along with continuity and momentum equations in one dimensional fluid flow driven by the non-artificial lift are given by Eqs.1-2-3, see Chorin and Marsden [1993], Hasan et al. [2005]. In general, one needs to solve equations Eqs.1-2-3 which are in addition coupled with the gas state 29γg p , [Hasan et al., 2005]. The equation in the form ρ = ZRT gas state equation can be substituted into Eqs.1-2-3 and the system can be rewritten in a compact form as: ∂u ∂u + A(u, z, t) = B(u, z, t) (4) ∂t ∂z where u is the vector of states, u = [T, v, ρ]T . The matrices A(u, z, t) and B(u, z, t) are given as: v 0 1+CT 0 ZR ZR T v 29γ (5) A(u, z, t) = 29γ g g ρ 0 ρ v v 1+CT (Tei − T )LR B(u, z, t) = (6) −g sin θ 0 with the assumption that dominant behaviour is given by convective transport modes so that conduction and viscosity can be neglected. 2.1 Model Reduction In the case of oil wells one can consider the Eq.1 and Eqs.23 can be neglected in order to reduce the model. The assumption is that the transient fluid velocity time constant is much smaller with respect to transient temperature time constant and the heat equation can be transferred in nondimensionalized form given as: Table 1. Physical and numerical parameters [Spindler, 2011] P
Description
value
Unit
g gc J L gG Tei LR θ dti dw ρw CT kw cw ρ0 m0 v0 µf CJ kf cf
Gravitational acceleration Conversion factor Conversion factor Length of wellbore Geothermal gradient Initial temperature Relaxation distance Wellbore angle Inside diameter of tube Diameter of wellbore Wellbore density Thermal storage constant Wellbore conductivity Wellbore specific heat Fluid density Linear mass density Steady state velocity Fluid viscosity Joule-Thomson coefficient Fluid conductivity Fluid specific heat
32.2 32.2 778.1693 5000 0.02 100 5.0605 × 10−4 π/2 3/12 20/12 0.30634 3 2 1 55.042 2.7019 1.2719 13.09 1.9185 × 10−2 0.1/3600 0.947
f t/s2 lbm.f t/lbf.s2 f t.lbf /BT U ft o F/f t oF − − ft ft lbm/f t3 − BT U/s.f t.o F BT U/s.f t.o F lbm/f t3 lbm/f t3 f t/s lbm/f t.s o F 3 /BT U BT U/s.f t.o F BT U/lbm.o F
110
∂ T¯ ∂ T¯ + v¯ = v¯[Tei − T¯] ¯ ∂t ∂ z¯ π KE 1 KE ∂ p¯ ∂ p¯ + ζ 2[ + ][ + v¯ ] 2 2 ¯ 2 (1 + CT ) HE m ¯f (1 + CT ) HEJT ∂ t ∂ z¯ π 2 CEf 1 Aw CEw 1 ∂ 2 T¯ + ζ (1 + CT )[ + CT ] 8 KE mf Af KE m ¯ w ∂ z¯2 8 ζ V E 1 ∂¯ v + ( )2 (7) 3 1 + CT HE m ¯ f ∂ z¯ where variables with bar are non-dimensionalized variables and all other parameters and constants are given in Tables 1 and 2. The variables are non-dimensionalized as T = KT¯, z = Z z¯ and t = T t¯, where, K, Z and T are constant factors which are defined as follow, see Spindler [2011]: 1 + CT gG sinθ 1 K= ; Z= ; T = v0 LR LR LR ζ is the measure of wellbore aspect ration and all other constant parameters are given in Table 2. Considering the magnitude of coefficients in Eq.7, namely, ζ = 10−4 , VE VE −15 ζ HE = 10−14 , ζ KE = 10−8 and ζ 2 KE negligible HE = 10 with respect to the fist term in right hand side, the equation is simplified to scalar hyperbolic heat transfer equation in the wellbore. The reduced model of Eq.7 is used for the observer design in the ensuing section and is given as: ∂ T¯ ∂ T¯ + = −T¯ − z¯ + T¯ei (8) ∂ t¯ ∂ z¯ where T¯, z¯ and t¯ are non-dimensionalized temperature, height and time. The boundary condition at the well bottom is given by: T |z=0 = T¯eiwh (9) and initial temperature distribution given as: Teiwh gG T |t=0 = + (LLR − z) (10) K KLR the system output is the temperature measurement at wellhead, given as: Y = T |z=L (11) The temperature wellbore model Eqs.8 along with the boundary condition at bottomwell Eq.9 is used for temperature estimation in oil wells with incompressible fluid. Finally, the general conservation equations provided in section II are used to develop models that can be used for sythesis of Luenberger observer and Kalman filter for transient temperature profile estimation of oil/gas in the wellbore. In the case of the oil well, the governing equations are reduced to a scalar hyperbolic PDE. In next section, by utilizing the method of characteristics, the hyperbolic PDE with characteristic lines is discretized and reformulated to provide a form of linear time invariant system suitable for the observer and/or filter synthesis. 3. OBSERVER DESIGN The model of wellbore fluid temperature is given by the hyperbolic infinite dimensional partial differential Eq.8, which can be reduce to a finite dimensional dynamic model by using the method of characteristics. Method of characteristic is used to reduce the hyperbolic partial differential equation to finite dimensional approximation model by reformulating the hyperbolic PDE in the large number of ODEs. In particular, using the method
2013 IFAC TFMST July 13-16, 2013. Lyon, France
Table 2. Energy constants used for nondimensionalization [Spindler, 2011] Energy constant
Definition
Potential energy, P E
sin θ m0 g L
Heat energy, HE
m0 gGLsin θ cf Jgc
Kinetic energy, KE
1 m v2 2 0 0 Af gGLsinCθJgc R J kf v0 cf LR 1 uf v0 Af 2 dti kw v0 cw LR
R
R
Heat energy, Joule-Thomson effect, HEJT Fluid conduction energy, CEf Viscous energy, V E Wellbore conduction energy, CEw
of characteristics, the change of variables transfers the PDE to a set of ordinary differential equations along the characteristic lines. For the Eq.8, characteristic lines are ξ = z − t and applying this transformation, the equation along characteristic lines is given as: ∂T (ξ, t) = −T (ξ, t) − ξ + t + Tei (12) ∂t which is an ordinary differential equation along the characteristic lines, ξ = constant. In order to obtain the system evolution, the time and space is discretized according to the characteristic lines’ slope. Assuming h = ∆z and k = ∆t for length and time discretization, where h = k, the length and time domain will be discretized as zm = mh and tn = nk, see Fig.2. The model equation Eq.8 is written in matrix form as Eq.13, where the boundary condition at the well bottom, T |z=0 is assumed to be constant: T1 0 0 ... 0 0 T1 T2 1 − k 0 . . . 0 0 T2 . . . . .. . . . . . . ... = . .. (13) . .. T 0 0 . . . 0 0 TM −1 M −1 TM n+1 0 0 ... 1 − k 0 TM n (1 − k)T0 − hk + dk −2hk + dk .. + . −(M − 1)hk + dk
temperature evolution observability. The observability of the set of hyperbolic systems for compressible fluid flow in wellbore is different due to coupling between energy, momentum and continuity equations. There are three sets of characteristic lines corresponding to each conservation equation. The matrices A and B in Eq.4 are functions of states and time-varying. This hyperbolic system can be solved using method of characteristics, however, there are three different sets of characteristics of Eq.4 and moreover, these characteristics are not straight lines due to time varying coefficients. The conservation equations are fully coupled and due to offdiagonal terms in the matrix A(u, z, t), the method of characteristics employed for incompressible flow can not be applied directly. In other words, the states can not be defined as a function of a single variable (e.g. ξ = z − vt). In order to define and calculate characteristic lines, the eigenvalues of matrix A(u, z, t) should be integrated as follow: dz = λi (u) (17) Ci : dt where Ci is the set of characteristics corresponding to ith eigen value of matrix A. Along the characteristics of the hyperbolic system, Riemann invariants, f (z, t), can be calculated by solving the following equation for f (z, t): ∂f ∂f AT =λ (18) ∂u ∂u By calculating eigen vectors of AT , Riemann invariants, i f (z, t), are calculated as ∂f ∂u = wi , where fi and wi are th the i Riemann invariant and ith eigen vector of matrix AT , respectively [Chorin and Marsden, 1993]. The observability of a hyperbolic system is defined with respect to characteristics. A hyperbolic set of equations is observable if the characteristics intersect the measurement line as time passes [Ray, 1981]. It can be demonstrated that the eigenvalues of matrix A(u, z, t) for gas flow Measurement path ξ = −h
(14)
ξ=0
Nk ξ=h
−M hk + dk where n is the time step and M is the number of spatial discretization points. The Eq.13 is in standard discrete linear time invariant (LTI) system form given as: Tn+1 = ATn + B (15) Yn = CTn (16) where C = [0, 0, , . . . , 1]. The model given by Eqs.15-16 is used for Luenberger observer design. The characteristics of the temperature hyperbolic model for the oil wellbore are parallel lines due to the constant velocity of oil flow in the well. The temperature measurement at wellhead can guarantee the observability of the system. The measurement line is shown in Fig.2 and as it can be seen the characteristic lines intersect the measurement line located at wellhead, z = L = M h. It can be observed, as time elapses the information from all discretization points along the wellbore domain reach the L which implies that the wellhead temperature measurement guarantees the 111
(N − 1)k
k tn = 0 zm = 0
h
(M − 1)h L = M h
Fig. 2. Characteristic lines and the discretization scheme, the characteristic lines are parallel and the time and space discretization is done according to the slope of characteristics. The measurement sensor located at wellhead determines the measurement path shown in figure.
2013 IFAC TFMST July 13-16, 2013. Lyon, France
The hyperbolic system Eqs.1-2-3 is simplified and reduced to linear discrete time model Eqs.15-16. The transient temperature estimation can be seen as designing an observer for the system while the state variables are the fluid temperatures at different well depth. As mentioned in section III, the discretized linear system is observable and the transient temperature can be estimated using temperature measurements at wellhead. The Luenberger observer is given as:. Tˆn+1 = ATˆn + B + L(Yn − C Tˆn ) (19) where en = Yn − C Tˆn is the error between the wellhead measurements and model predictions and L is the observer gain determined using pole placement to obtain desired pre-specified eigenvalues of error dynamics. The earth (formation) temperature is the driving heat source and is assumed to be a constant linear function of depth during the process, while in realistic situation, there are uncertainties and inevitably the measurement noise is present in the measured data. In order to overcome the uncertainties and the influence of measurement noises, Kalman filter is utilized and examined. The proposed methods, Luenberger observer and Kalman filter, for transient temperature estimation are implemented on a high fidelity Finite Difference Model (FDM). The finite difference model takes into account all the higher order and neglected terms in Eq.1 along with disturbances in the model. Numerical simulation of the wellbore fluid flow and heat transfer is performed using explicit Euler integration. The sampling time of ∆t = 1min and space discretization of ∆z = 10f t are used. The parameters used for numerical simulation are given in Table 1.
200
Bottomhole, z = 150 f t 180 o
4. RESULTS AND DISCUSSION
The estimation results and efficiency of the observers are compared to the actual temperature in Figs.4,5. Fig.3 shows the steady state solution to the finite difference method in presence of disturbances along with the estimated steady state profiles using Luenberger and Kalman filter. Time evolution of the transient temperature at three different locations (near bottomhole, middle of the well and near the wellhead) are shown in Fig.4. The state estimation accuracy increases near the bottomhole. The Dirichlet boundary condition at the bottomhole makes the estimation more accurate in depths close to the bottomhole, however the Kalman filter is more favourable in points close to the wellhead, where it compensates the measurement noises and the disturbances in the process model. The spatial transient temperature profile and the estimation error is given in Fig.5. Due to hyperbolic nature of the process, the resulting linear system presented by Eqs.13-14 has only zero eigenvalues. Zero eigen-values for discrete linear system implies that one can not design an observer with faster error dynamics than the system dynamics itself, however, slower estimation can provide enough accuracy for temperature estimation. The smallest distinctive numerically possible eigen-values are assigned such that the fastest error dynamic is achieved. The slower error dynamic leads to a delay in estimation but will not affect the efficiency of the estimation.
Temperature ( F)
in wellbore have not the same sign (two positive and one negative) which implies two different direction of information flow. Thus, the observability of the hyperbolic system, Eqs.1-2-3, will be guaranteed only by placing sensors at both the bottomhole and wellhead positions.
160
Middle, z = 2500 f t
140
Near-wellhead, z = 4850 f t 120
100
Finite Difference Method Leunberger observer Kalman filter
210 Finite Difference Method Leunberger observer Kalman Filter Steady state with disturbance
200
80 0
5
10
15
20
Time (hours)
o
Temperature ( F)
190
Fig. 4. Transient temperature estimated by Luenberger observer and Kalman filter along with actual temperature evolution at different depths.
180 170 160
5. CONCLUSION
150
In this paper, incompressible and compressible flow in wellbore is investigated and the observability condition for hyperbolic set of equations of fluid motion and heat transfer model is discussed. The observability condition is provided as a condition on sensor placement in the wellbore. It is shown that wellhead temperature measurement is enough to guarantee observability, however for compressible flow temperature measurements are required at both the wellhead and bottomhole to satisfy observability
140 130 0
1000
2000
3000
4000
5000
Well Depth − z (feet)
Fig. 3. Steady state temperature profile - comparison with estimated steady state temperature profiles 112
2013 IFAC TFMST July 13-16, 2013. Lyon, France
Temperature (o F)
200 180 160 140 120 100 25
0
20
1000
15
2000
10
(a)
3000 5
4000 0
5000
Length (ft)
Estimation Error (o F)
Time (hours)
10 0 −10 −20
−30 5000 4000
25
3000
(b)
20 15
2000
10
1000
Length (ft)
0
5 0
Time (hours)
Fig. 5. (a) Transient temperature evolution estimated by Luenberger observer; (b) Estimation error. condition. Using the method of characteristics the hyperbolic equation of oil flow in wellbore is reformulated as a discrete linear dynamic, then Luenberger observer synthesis with point temperature measurement at wellhead is presented. Numerical simulation results show efficiency of transient temperature estimation. In addition, it is shown that there is a trade off in choosing the estimation method. Kalman filter has higher efficiency at depths close to wellhead where noisy measurement appears while Luenberger observer is more suitable for estimation of temperature near the bottomhole, where Dirichlet boundary condition is present. REFERENCES A. Bensoussan, G. D. Prato, M. Delfour, and S. Mitter. Representation and Control of Infnite Dimensional Systems. Springer, 2007. A. J. Chorin and J. E. Marsden. A Mathematical Introduction to Fluid Mechanics. Texts in Applied Mathematics. Springer, 1993. R. F. Curtain and H. Zwart. An Introduction to InfiniteDimensional Linear Systems Theory, Texts In Applied Mathematics. Springer, 1995. 113
F. Di Meglio, R. Vazquez, M. Krstic, and N. Petit. Backstepping stabilization of an underactuated 3 × 3 linear hyperbolic system of fluid flow equations. In American Control Conference (ACC), pages 3365 –3370, 2012. C. Harkort and J. Deutscher. Finite-dimensional observerbased control of linear distributed parameter systems using cascaded output observers. International Journal of Control, 84(1):107–122, 2011. A. R. Hasan and C. S. Kabir. Aspects of wellbore heat transfer during two-phase flow. SPE Production & Facilities, 9(3):211–216, 1994. A. R. Hasan and C. S. Kabir. Wellbore heat-transfer modelling and applications. Journal of Petroleum Science and Engineering, 86-87:127–136, 2012. A. R. Hasan, C. S. Kabir, and D. Lin. Analytic wellbore temperature model for transient gas-well testing. SPE Reservoir Evaluation & Engineering, 8(3):240–247, 2005. A. R. Hasan, C.S. Kabir, and X. Wang. A robust steadystate model for flowing-fluid temperature in complex wells. SPE Production & Operations, 24(2):269–276, 2009. B. Izgec, C. S. Kabir, D. Zhu, and A. R. Hasan. Transient fluid and heat flow modelling in coupled wellbore/reservoir systems. SPE Reservoir Evaluation & Engineering, 10(3):294–301, 2007. C. S. Kabir, A. R. Hasan, G. E. Kouba, and M. Ameen. Determining circulating fluid temperature in drilling, workover, and well control operations. SPE Drilling & Completion, 11(2):74–79, 1996a. C. S. Kabir, A.R. Hasan, D.L. Jordan, and X. Wang. A wellbore/reservoir simulator for testing gas wells in high-temperature reservoirs. SPE Formation Evaluation, 11(2):128–134, 1996b. X. Li and C. Xu. Infinite-dimensional Luenberger-like observers for a rotating body-beam system. Systems & Control Letters, 60(2):138–145, 2011. H. J. Ramey JR. Wellbore heat transmission. Journal of Petroleum Technology, 14(4):427–435, 1962. W. H. Ray. Advanced process control. McGraw-Hill chemical engineering series. McGraw-Hill, 1981. R. P. Spindler. Analytical models for wellbore temperature distribution. SPE Journal, 16(1):125–133, 2011. R. Vazquez, M. Krstic, and J. M. Coron. Backstepping boundary stabilization and state estimation of a 2 × 2 linear hyperbolic system. In CDC-ECE, pages 4937– 4942. IEEE, 2011. D. Vries, K. J. Keesman, and H. Zwart. Luenberger boundary observer synthesis for Sturm-Liouville systems. International Journal of Control, 83(7):1504–1514, 2010. X. Wang, J. Lee, B. Thigpen, G. Vachon, S. Poland, and D. Norton. Modelling flow profile using distributed temperature sensor (dts) system. In Intelligent Energy Conference and Exhibition, pages 637–647, 2008. W. Wu and C. Liou. Output regulation of two-time-scale hyperbolic pde systems. Journal of Process Control, 11 (6):637–647, 2001. C. Z. Xu, P. Ligarius, and J. P. Gauthier. An observer for infinite-dimensional dissipative bilinear systems. Computers & Mathematics with Applications, 29(7):13–21, 1995.
Transient Fluid Temperature Estimation in Wellbores Javad Abdollahi, Stevan Dubljevic Department of Chemical & Materials Engineering, University of Alberta, Edmonton, Canada, T6G 2V4 (Tel: 780-248-1596; e-mail: [email protected]) Abstract: Knowledge of transient temperature distribution in wellbore fluid is critical for the pressure transient tests during draw-down and shut-in processes. In absence or limitation of measurement devices installed on the wellbore, temperature estimation through soft sensors can properly fulfill the absence of these measurement devices. In this work, the heat transfer model for wellbore fluid is reduced to a set of hyperbolic PDEs and applying the method of characteristics and calculation of Riemann invariants, the observability of the temperature distribution is discussed for the governing set of hyperbolic PDEs. In case of oil well, the governing equations are reduced and discretized using method of characteristics and then reformulated in order to implement the observer. The developed model is used for Luenberger observer and Kalman filter synthesis using well-head temperature measurements. The observer is synthesized based on the reduced linear model and is examined on the high fidelity finite difference model of the wellbore fluid flow. The provided transient temperature estimation is enough accurate and reliable for on-line and real-time implementation in wellbore fluid. 1. INTRODUCTION Real-time knowledge of temperature distribution in wellbore fluid is crucial for density profile prediction. Analysis of transient temperature along with transient pressure tests are necessary for monitoring and maintenance of flow lines and facilities. During the drawdown and shutin, transient temperature and fluid flow occurs, while the fluid dissipates heat to the surrounding. The heat transfer process changes the temperature which affects the density, momentum and fluid pressure. In the state-of-theart completed and realized wellbores, the temperature profile along the wellbore is measured using Distributed Temperature Sensors (DTS) [Wang et al., 2008]. Although DTS systems provide high resolution temperature measurements, the technology is new and not available for already existing mature oil and/or gas wells. The realtime transient temperature estimation and monitoring using available wellhead temperature measurements can substitute DTS systems by soft sensors developed based on dynamic system model. Temperature distribution modeling and prediction in wellbore fluid has attracted attention of petroleum engineering community [Hasan and Kabir, 1994, Ramey JR., 1962, Hasan and Kabir, 2012, Izgec et al., 2007, Kabir et al., 1996a, Hasan et al., 2009, 2005, Spindler, 2011]. The temperature modeling approaches are mostly founded on the work of Ramey JR. [1962], where the wellbore model is assumed to be surrounded by an infinite reservoir and heat transfer takes place through the wellbore to the surroundings. Hasan and Kabir [1994] improved the model to more realistic finite radius reservoir and considered the heat diffusion from the formation to the wellbore. The transient temperature distribution in wellbore occurs when there is a change in production rate manifested as 978-3-902823-40-3/2013 © IFAC
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shut-in, restarting or change of production schedule. Along this line, due to the compressible flow nature, the temperature transients produce a transient momentum, heat and flow change, see [Hasan and Kabir, 2012, Kabir et al., 1996a]. Kabir et al. [1996b] developed a transient model for coupled reservoir/wellbore temperature distribution and Hasan et al. [2005] decoupled the heat transfer equation and momentum equations by neglecting the afterflow in the wellbore. In addition to above contributions, Spindler [2011] provided an analytical solution for approximated transient temperature model and also extended the model to include heat conduction in axial direction. Due to the long length of the oil/gas wells, the modelling of the transport phenomena (momentum transport and heat transfer) leads to distributed parameter model equations which can not be easily approximated by a single lumped system equation. The transport of petroleum is governed by momentum and heat transfer equations given as one dimensional hyperbolic partial differential equations. Since there are limited approaches for control and estimation of hyperbolic processes [Ray, 1981], a major part of recent contributions is focused on simulation of existing temperature models and in best case on coupling existing models with reservoir equations, see [Izgec et al., 2007, Kabir et al., 1996a, Hasan et al., 2005]. Since the wellbore temperature model is given by the first order one dimensional hyperbolic PDE system, the infinite-dimensional nature of distributed parameter processes makes the estimation procedure a challenging task. State estimation and control of systems governed by partial differential equations (PDEs) is a mature area of research activity, see [Curtain and Zwart, 1995, Bensoussan et al., 2007]. Despite the fact that boundary control strategies for parabolic PDEs are well developed, a state estimation of parabolic PDEs is less developed and is of 10.3182/20130714-3-FR-4040.00018
2013 IFAC TFMST July 13-16, 2013. Lyon, France
interest. In particular, for parabolic systems, Xu et al. [1995] discuss an observer design for dissipative bilinear systems with weak error convergence to zero. The Luenberger observer is implemented by Vries et al. [2010] for synthesis of observer for the Strum-Lioville systems and also by Li and Xu [2011] for a higher order PDE describing the model of rotating body-beam system. Along the same line, Harkort and Deutscher [2011] have focused on observer based controller synthesis for Riesz-spectral systems. Although there is a significant number of contributions on parabolic PDEs, there are quite a few contributions on state estimation and observer design for hyperbolic systems [Wu and Liou, 2001, Vazquez et al., 2011, Di Meglio et al., 2012]. Namely, Wu and Liou [2001] have demonstrated output-regulation for a set of hyperbolic systems, while the backstepping method is used by Vazquez et al. [2011] for the state estimation of 2 × 2 linear hyperbolic system. In this work, a set of hyperbolic partial differential equations arising from the heat transfer, fluid motion and continuity balance equations in wellbore is introduced as relevant model of the transport processes taking place in oil/gas wellbores. The method of characteristics and the Riemann invariants are utilized in evaluating the model features. In particular observability condition is explored within the observer design and limitations associated with the model are discussed. The model is simplified for incompressible oil flow in wellbore and reduced to the heat transfer model given by a one dimensional hyperbolic PDE. Subsequently, the Luenberger observer and Kalman filter are used for on-line estimation of transient temperature distribution in the wellbore fluid. The hyperbolic temperature model is reformulated to standard linear time invariant form using method of characteristics and then the observer is designed for real-time temperature estimation. Motivated by realistic set of available measurements, the state estimation is based on the temperature measurement available at the wellhead (point measurement), and the estimated transient temperature profile is provided. The temperature estimation is also examined by assuming stochastic uncertainties in the model and in the measurement and the state reconstruction is realized by standard Kalman filter. In the ensuing sections, we provide a description of the transient temperature and momentum dynamics model and discuss the characteristics and Riemann invariants. The following section deals with the model description and reduction. In section 3, the method of characteristics is used to reformulate the model to a suitable matrix form for state construction and estimation purpose. Finally, the numerical simulation results are provided along with Luenberger observer and stochastic state estimation synthesis for transient temperature.
dimensional coupled hyperbolic equations. The following schematic and geometry of the well used for the modelling is shown in Fig.1. ∂T ∂T (1 + CT ) +v = v[Tei − T ]LR ∂t ∂z Af CJ ∂p ∂p +[ + ][ +v ] mf cf Jgc Jgc ∂t ∂z Af k f Aw kw CJ ∂ 2 T +[ + ] mf cf Jgc mw cw Jgc ∂z 2 4 µf Af ∂v + ( )2 (1) 3 mf cf Jgc ∂z ∂v ∂v 1 ∂p 4 ∂2v +v =− − g sin θ + µf 2 ∂t ∂z ρ ∂z 3 ∂z ∂ρ ∂ρ ∂v +v +ρ =0 ∂t ∂z ∂z
(2) (3)
where T and Tei are the fluid temperature and earth formation temperature, p, ρ and v are the pressure, density and fluid velocity along the well, z and t represent depth of the well measured from well bottom and time. All other parameters are shown and described in Table 1. In the case of oil flow which can be considered incompressible, the equations of motion, continuity and heat transfer equation can be reduced to energy balance equation and this special case can be considered as a subset of general wellbore models. However for the gas wells, the momentum, energy and continuity conservation equations should be considered and solved simultaneously in order to obtain the evolution of the physical variables within well. In order to model the wellbore behaviour the comprehensive coupled momentum, continuity and heat transfer model including conductive terms, viscous terms and higher order terms is necessary and it is given by Eqs.1-23, see [Spindler, 2011]. The fluid is assumed to flow along the constant diameter well and there is no disturbance in earth formation temperature. Namely, the heat conduction through the fluid and between the fluid and the well surrounding are main heat transport modes. The model also accounts for the work of fluid pressure and viscous forces, given in Eq.1. In general one needs to account for the coupling among Eqs.1-2-3, however the case of the incompressible fluid flow (for example the oil flow within wellbores characterized by the artificial lift - sucker rod pumps) can allow for the
2. FLUID FLOW AND HEAT TRANSFER MODEL IN WELLBORE Fluid flow in a wellbore can be considered as one dimensional flow in a constant radius pipe. Depending on the flowing fluid (e.g. oil or gas), the fluid flow can be considered as compressible or incompressible flow. The general governing equations (momentum, continuity and heat transfer) are given in Eqs.1-2-3 as a set of one 109
θ Fig. 1. Schematic of wellbore geometry used for modeling [Hasan and Kabir, 2012].
2013 IFAC TFMST July 13-16, 2013. Lyon, France
model reduction since the main model feature becomes constant velocity flow from the wellbore bottom to the surface. We provide a brief explanation for compressible flow in the following paragraph. The oil flow will be discussed in details in the following sections. Compressible flow in a wellbore consist of energy balance along with continuity and momentum equations in one dimensional fluid flow driven by the non-artificial lift are given by Eqs.1-2-3, see Chorin and Marsden [1993], Hasan et al. [2005]. In general, one needs to solve equations Eqs.1-2-3 which are in addition coupled with the gas state 29γg p , [Hasan et al., 2005]. The equation in the form ρ = ZRT gas state equation can be substituted into Eqs.1-2-3 and the system can be rewritten in a compact form as: ∂u ∂u + A(u, z, t) = B(u, z, t) (4) ∂t ∂z where u is the vector of states, u = [T, v, ρ]T . The matrices A(u, z, t) and B(u, z, t) are given as: v 0 1+CT 0 ZR ZR T v 29γ (5) A(u, z, t) = 29γ g g ρ 0 ρ v v 1+CT (Tei − T )LR B(u, z, t) = (6) −g sin θ 0 with the assumption that dominant behaviour is given by convective transport modes so that conduction and viscosity can be neglected. 2.1 Model Reduction In the case of oil wells one can consider the Eq.1 and Eqs.23 can be neglected in order to reduce the model. The assumption is that the transient fluid velocity time constant is much smaller with respect to transient temperature time constant and the heat equation can be transferred in nondimensionalized form given as: Table 1. Physical and numerical parameters [Spindler, 2011] P
Description
value
Unit
g gc J L gG Tei LR θ dti dw ρw CT kw cw ρ0 m0 v0 µf CJ kf cf
Gravitational acceleration Conversion factor Conversion factor Length of wellbore Geothermal gradient Initial temperature Relaxation distance Wellbore angle Inside diameter of tube Diameter of wellbore Wellbore density Thermal storage constant Wellbore conductivity Wellbore specific heat Fluid density Linear mass density Steady state velocity Fluid viscosity Joule-Thomson coefficient Fluid conductivity Fluid specific heat
32.2 32.2 778.1693 5000 0.02 100 5.0605 × 10−4 π/2 3/12 20/12 0.30634 3 2 1 55.042 2.7019 1.2719 13.09 1.9185 × 10−2 0.1/3600 0.947
f t/s2 lbm.f t/lbf.s2 f t.lbf /BT U ft o F/f t oF − − ft ft lbm/f t3 − BT U/s.f t.o F BT U/s.f t.o F lbm/f t3 lbm/f t3 f t/s lbm/f t.s o F 3 /BT U BT U/s.f t.o F BT U/lbm.o F
110
∂ T¯ ∂ T¯ + v¯ = v¯[Tei − T¯] ¯ ∂t ∂ z¯ π KE 1 KE ∂ p¯ ∂ p¯ + ζ 2[ + ][ + v¯ ] 2 2 ¯ 2 (1 + CT ) HE m ¯f (1 + CT ) HEJT ∂ t ∂ z¯ π 2 CEf 1 Aw CEw 1 ∂ 2 T¯ + ζ (1 + CT )[ + CT ] 8 KE mf Af KE m ¯ w ∂ z¯2 8 ζ V E 1 ∂¯ v + ( )2 (7) 3 1 + CT HE m ¯ f ∂ z¯ where variables with bar are non-dimensionalized variables and all other parameters and constants are given in Tables 1 and 2. The variables are non-dimensionalized as T = KT¯, z = Z z¯ and t = T t¯, where, K, Z and T are constant factors which are defined as follow, see Spindler [2011]: 1 + CT gG sinθ 1 K= ; Z= ; T = v0 LR LR LR ζ is the measure of wellbore aspect ration and all other constant parameters are given in Table 2. Considering the magnitude of coefficients in Eq.7, namely, ζ = 10−4 , VE VE −15 ζ HE = 10−14 , ζ KE = 10−8 and ζ 2 KE negligible HE = 10 with respect to the fist term in right hand side, the equation is simplified to scalar hyperbolic heat transfer equation in the wellbore. The reduced model of Eq.7 is used for the observer design in the ensuing section and is given as: ∂ T¯ ∂ T¯ + = −T¯ − z¯ + T¯ei (8) ∂ t¯ ∂ z¯ where T¯, z¯ and t¯ are non-dimensionalized temperature, height and time. The boundary condition at the well bottom is given by: T |z=0 = T¯eiwh (9) and initial temperature distribution given as: Teiwh gG T |t=0 = + (LLR − z) (10) K KLR the system output is the temperature measurement at wellhead, given as: Y = T |z=L (11) The temperature wellbore model Eqs.8 along with the boundary condition at bottomwell Eq.9 is used for temperature estimation in oil wells with incompressible fluid. Finally, the general conservation equations provided in section II are used to develop models that can be used for sythesis of Luenberger observer and Kalman filter for transient temperature profile estimation of oil/gas in the wellbore. In the case of the oil well, the governing equations are reduced to a scalar hyperbolic PDE. In next section, by utilizing the method of characteristics, the hyperbolic PDE with characteristic lines is discretized and reformulated to provide a form of linear time invariant system suitable for the observer and/or filter synthesis. 3. OBSERVER DESIGN The model of wellbore fluid temperature is given by the hyperbolic infinite dimensional partial differential Eq.8, which can be reduce to a finite dimensional dynamic model by using the method of characteristics. Method of characteristic is used to reduce the hyperbolic partial differential equation to finite dimensional approximation model by reformulating the hyperbolic PDE in the large number of ODEs. In particular, using the method
2013 IFAC TFMST July 13-16, 2013. Lyon, France
Table 2. Energy constants used for nondimensionalization [Spindler, 2011] Energy constant
Definition
Potential energy, P E
sin θ m0 g L
Heat energy, HE
m0 gGLsin θ cf Jgc
Kinetic energy, KE
1 m v2 2 0 0 Af gGLsinCθJgc R J kf v0 cf LR 1 uf v0 Af 2 dti kw v0 cw LR
R
R
Heat energy, Joule-Thomson effect, HEJT Fluid conduction energy, CEf Viscous energy, V E Wellbore conduction energy, CEw
of characteristics, the change of variables transfers the PDE to a set of ordinary differential equations along the characteristic lines. For the Eq.8, characteristic lines are ξ = z − t and applying this transformation, the equation along characteristic lines is given as: ∂T (ξ, t) = −T (ξ, t) − ξ + t + Tei (12) ∂t which is an ordinary differential equation along the characteristic lines, ξ = constant. In order to obtain the system evolution, the time and space is discretized according to the characteristic lines’ slope. Assuming h = ∆z and k = ∆t for length and time discretization, where h = k, the length and time domain will be discretized as zm = mh and tn = nk, see Fig.2. The model equation Eq.8 is written in matrix form as Eq.13, where the boundary condition at the well bottom, T |z=0 is assumed to be constant: T1 0 0 ... 0 0 T1 T2 1 − k 0 . . . 0 0 T2 . . . . .. . . . . . . ... = . .. (13) . .. T 0 0 . . . 0 0 TM −1 M −1 TM n+1 0 0 ... 1 − k 0 TM n (1 − k)T0 − hk + dk −2hk + dk .. + . −(M − 1)hk + dk
temperature evolution observability. The observability of the set of hyperbolic systems for compressible fluid flow in wellbore is different due to coupling between energy, momentum and continuity equations. There are three sets of characteristic lines corresponding to each conservation equation. The matrices A and B in Eq.4 are functions of states and time-varying. This hyperbolic system can be solved using method of characteristics, however, there are three different sets of characteristics of Eq.4 and moreover, these characteristics are not straight lines due to time varying coefficients. The conservation equations are fully coupled and due to offdiagonal terms in the matrix A(u, z, t), the method of characteristics employed for incompressible flow can not be applied directly. In other words, the states can not be defined as a function of a single variable (e.g. ξ = z − vt). In order to define and calculate characteristic lines, the eigenvalues of matrix A(u, z, t) should be integrated as follow: dz = λi (u) (17) Ci : dt where Ci is the set of characteristics corresponding to ith eigen value of matrix A. Along the characteristics of the hyperbolic system, Riemann invariants, f (z, t), can be calculated by solving the following equation for f (z, t): ∂f ∂f AT =λ (18) ∂u ∂u By calculating eigen vectors of AT , Riemann invariants, i f (z, t), are calculated as ∂f ∂u = wi , where fi and wi are th the i Riemann invariant and ith eigen vector of matrix AT , respectively [Chorin and Marsden, 1993]. The observability of a hyperbolic system is defined with respect to characteristics. A hyperbolic set of equations is observable if the characteristics intersect the measurement line as time passes [Ray, 1981]. It can be demonstrated that the eigenvalues of matrix A(u, z, t) for gas flow Measurement path ξ = −h
(14)
ξ=0
Nk ξ=h
−M hk + dk where n is the time step and M is the number of spatial discretization points. The Eq.13 is in standard discrete linear time invariant (LTI) system form given as: Tn+1 = ATn + B (15) Yn = CTn (16) where C = [0, 0, , . . . , 1]. The model given by Eqs.15-16 is used for Luenberger observer design. The characteristics of the temperature hyperbolic model for the oil wellbore are parallel lines due to the constant velocity of oil flow in the well. The temperature measurement at wellhead can guarantee the observability of the system. The measurement line is shown in Fig.2 and as it can be seen the characteristic lines intersect the measurement line located at wellhead, z = L = M h. It can be observed, as time elapses the information from all discretization points along the wellbore domain reach the L which implies that the wellhead temperature measurement guarantees the 111
(N − 1)k
k tn = 0 zm = 0
h
(M − 1)h L = M h
Fig. 2. Characteristic lines and the discretization scheme, the characteristic lines are parallel and the time and space discretization is done according to the slope of characteristics. The measurement sensor located at wellhead determines the measurement path shown in figure.
2013 IFAC TFMST July 13-16, 2013. Lyon, France
The hyperbolic system Eqs.1-2-3 is simplified and reduced to linear discrete time model Eqs.15-16. The transient temperature estimation can be seen as designing an observer for the system while the state variables are the fluid temperatures at different well depth. As mentioned in section III, the discretized linear system is observable and the transient temperature can be estimated using temperature measurements at wellhead. The Luenberger observer is given as:. Tˆn+1 = ATˆn + B + L(Yn − C Tˆn ) (19) where en = Yn − C Tˆn is the error between the wellhead measurements and model predictions and L is the observer gain determined using pole placement to obtain desired pre-specified eigenvalues of error dynamics. The earth (formation) temperature is the driving heat source and is assumed to be a constant linear function of depth during the process, while in realistic situation, there are uncertainties and inevitably the measurement noise is present in the measured data. In order to overcome the uncertainties and the influence of measurement noises, Kalman filter is utilized and examined. The proposed methods, Luenberger observer and Kalman filter, for transient temperature estimation are implemented on a high fidelity Finite Difference Model (FDM). The finite difference model takes into account all the higher order and neglected terms in Eq.1 along with disturbances in the model. Numerical simulation of the wellbore fluid flow and heat transfer is performed using explicit Euler integration. The sampling time of ∆t = 1min and space discretization of ∆z = 10f t are used. The parameters used for numerical simulation are given in Table 1.
200
Bottomhole, z = 150 f t 180 o
4. RESULTS AND DISCUSSION
The estimation results and efficiency of the observers are compared to the actual temperature in Figs.4,5. Fig.3 shows the steady state solution to the finite difference method in presence of disturbances along with the estimated steady state profiles using Luenberger and Kalman filter. Time evolution of the transient temperature at three different locations (near bottomhole, middle of the well and near the wellhead) are shown in Fig.4. The state estimation accuracy increases near the bottomhole. The Dirichlet boundary condition at the bottomhole makes the estimation more accurate in depths close to the bottomhole, however the Kalman filter is more favourable in points close to the wellhead, where it compensates the measurement noises and the disturbances in the process model. The spatial transient temperature profile and the estimation error is given in Fig.5. Due to hyperbolic nature of the process, the resulting linear system presented by Eqs.13-14 has only zero eigenvalues. Zero eigen-values for discrete linear system implies that one can not design an observer with faster error dynamics than the system dynamics itself, however, slower estimation can provide enough accuracy for temperature estimation. The smallest distinctive numerically possible eigen-values are assigned such that the fastest error dynamic is achieved. The slower error dynamic leads to a delay in estimation but will not affect the efficiency of the estimation.
Temperature ( F)
in wellbore have not the same sign (two positive and one negative) which implies two different direction of information flow. Thus, the observability of the hyperbolic system, Eqs.1-2-3, will be guaranteed only by placing sensors at both the bottomhole and wellhead positions.
160
Middle, z = 2500 f t
140
Near-wellhead, z = 4850 f t 120
100
Finite Difference Method Leunberger observer Kalman filter
210 Finite Difference Method Leunberger observer Kalman Filter Steady state with disturbance
200
80 0
5
10
15
20
Time (hours)
o
Temperature ( F)
190
Fig. 4. Transient temperature estimated by Luenberger observer and Kalman filter along with actual temperature evolution at different depths.
180 170 160
5. CONCLUSION
150
In this paper, incompressible and compressible flow in wellbore is investigated and the observability condition for hyperbolic set of equations of fluid motion and heat transfer model is discussed. The observability condition is provided as a condition on sensor placement in the wellbore. It is shown that wellhead temperature measurement is enough to guarantee observability, however for compressible flow temperature measurements are required at both the wellhead and bottomhole to satisfy observability
140 130 0
1000
2000
3000
4000
5000
Well Depth − z (feet)
Fig. 3. Steady state temperature profile - comparison with estimated steady state temperature profiles 112
2013 IFAC TFMST July 13-16, 2013. Lyon, France
Temperature (o F)
200 180 160 140 120 100 25
0
20
1000
15
2000
10
(a)
3000 5
4000 0
5000
Length (ft)
Estimation Error (o F)
Time (hours)
10 0 −10 −20
−30 5000 4000
25
3000
(b)
20 15
2000
10
1000
Length (ft)
0
5 0
Time (hours)
Fig. 5. (a) Transient temperature evolution estimated by Luenberger observer; (b) Estimation error. condition. Using the method of characteristics the hyperbolic equation of oil flow in wellbore is reformulated as a discrete linear dynamic, then Luenberger observer synthesis with point temperature measurement at wellhead is presented. Numerical simulation results show efficiency of transient temperature estimation. In addition, it is shown that there is a trade off in choosing the estimation method. Kalman filter has higher efficiency at depths close to wellhead where noisy measurement appears while Luenberger observer is more suitable for estimation of temperature near the bottomhole, where Dirichlet boundary condition is present. REFERENCES A. Bensoussan, G. D. Prato, M. Delfour, and S. Mitter. Representation and Control of Infnite Dimensional Systems. Springer, 2007. A. J. Chorin and J. E. Marsden. A Mathematical Introduction to Fluid Mechanics. Texts in Applied Mathematics. Springer, 1993. R. F. Curtain and H. Zwart. An Introduction to InfiniteDimensional Linear Systems Theory, Texts In Applied Mathematics. Springer, 1995. 113
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