METHOD AND APPARATUS FOR ESTIMATING DOWN-HOLE PROCESS VARIABLES OF GAS LIFT SYSTEM

Abstract

A method and apparatus for estimating a down-hole annulus pressure, and/or a down-hole tubing pressure, and/or a down-hole lift gas rate through a gas injection valve of gas lift system, which comprises at least one gas lift choke and at least one gas injection valve and at least one production choke. A state estimator is proposed to obtain the values of these down-hole process variables. The estimator contains a number of local estimators and an interpolation module.

Claims

1. A state estimator system for determining down-hole process parameters for an artificial gas lift system, the artificial gas lift system comprising a well casing which connects a surface equipment with an oil reservoir; a tubing inserted inside said well casing and forming an annulus as a space between said well casing and said tubing; a gas lift choke installed between a source of a high-pressure gas and said annulus; a gas injection valve installed between said annulus and a volume inside said tubing; a production choke installed between a volume inside said tubing and oil processing facilities above the surface; the state estimator system comprising: a) an electronic receiver configured to receive a plurality of input parameters measured at an upper end of the artificial lift system; b) N local estimators where N is greater than or equal to two, each providing based on the plurality of input parameters an 11.sup.th vector signal representative containing estimates of a down-hole annulus pressure about a reference point of an i-th local estimator, and/or a down-hole tubing pressure about the reference point of an i-th local estimator, and/or a lift gas rate through a said gas injection valve about the reference point of an i-th local estimator; and c) an interpolation module that incorporates estimates provided by all local estimators into a single estimate to determine estimated values of a down-hole annulus pressure, of a down-hole tubing pressure, and of a lift gas rate (LGR) through said gas injection valve, in steady and transient modes; based on the input parameters measured at the upper end of the artificial gas lift system; and d) an output device configured to generate an output associated with the determined estimated values.

2. A state estimator system as recited in claim 1, wherein the parameters measured at the upper end of the artificial gas lift system comprise a vector signal representative of a gas lift choke opening value and/or a production choke opening value; and/or a vector signal representative of a measured annulus pressure and/or a measured surface tubing pressure.

3. A state estimator system as recited in claim 1, wherein N reference points are selected as equilibrium points in a nonlinear model describing dynamics of an artificial gas lift system, for selected values of a gas lift choke and a production choke.

4. A state estimator system as recited in claim 1, wherein each of N local estimators comprises a first adder, and/or a second adder, and/or a third adder, and/or a nonlinear module, and/or a linear module, and/or a fourth adder.

5. A state estimator system as recited in claim 4, wherein the first adder produces a 1.sup.st vector signal representative containing a deviation of a gas lift choke opening and a deviation of a production choke opening, by subtracting a 2.sup.nd vector signal representative containing a gas lift choke opening and a production coke opening about a reference point of an i-th local estimator from a 3.sup.rd vector signal representative containing a gas lift choke opening and a production choke opening.

6. A state estimator system as recited in claim 4, wherein the second adder produces a 4.sup.th vector signal representative containing a deviation of a surface annulus pressure and a deviation of a surface tubing pressure, by subtracting a 5.sup.th vector signal representative containing a surface annulus pressure and a surface tubing pressure about a reference point of an i-th local estimator from a 6.sup.th vector signal representative containing a measured surface annulus pressure and a measured surface tubing pressure.

7. A state estimator system as recited in claim 4, wherein the third adder produces a 7.sup.th vector error signal representative by subtracting an 8.sup.th vector signal representative containing an estimate of a deviation of a surface annulus pressure and an estimate of a deviation of a surface tubing pressure from a 4.sup.th vector signal representative containing a deviation of a surface annulus pressure and a deviation of a surface tubing pressure.

8. A state estimator system as recited in claim 4, wherein the nonlinear module further contains a relay nonlinearity, produces a 9.sup.th vector control signal representative based on a said seventh vector error signal representative as per formula: v.sub.i=h.Math.sign e.sub.i where v.sub.i is an i-th vector control signal, h is an amplitude of the relay, h=1 in a preferred embodiment, and e.sub.i is an i-th vector error signal.

9. A state estimator system as recited in claim 4, wherein the linear module realizes computation of a 10.sup.th vector signal representative containing an estimate of a deviation of a down-hole annulus pressure, an estimate of a deviation of a down-hole tubing pressure and an estimate of a deviation of a lift gas rate through a said gas injection valve (6), according to equations; {circumflex over ({dot over (x)})}=A{circumflex over (x)}+Bũ+Lv, and ŷ=C{circumflex over (x)}+Dũ where AεR.sup.3×3, BεR.sup.3×3, CεR.sup.1×3, and DεR.sup.3×2 are matrices computed as ${{{{A=\frac{\partial f}{\partial x}.Math.}_p,B=\frac{\partial f}{\partial u}.Math.}_p,C=\frac{\partial y}{\partial x}.Math.}_p,\mathrm{and}.Math..Math.D=\frac{\partial y}{\partial u}.Math.}_p$ where f is the right-hand side of the equation {dot over (x)}=f(x,u) describing the dynamics of the gas lift system, y is a vector signal containing a deviation of a surface annulus pressure and a deviation of a surface tubing pressure, p is an i-th equilibrium point, LεR.sup.3×2 is a matrix containing constant coefficients, {circumflex over (x)} is a vector containing an estimate of a deviation of a down-hole annulus pressure, an estimate of a deviation of a down-hole tubing pressure and an estimate of a deviation of a lift gas rate through a said gas injection valve, if is a vector containing a deviation of a controller command of a gas lift choke opening and a deviation of a controller command of a production choke opening.

10. A state estimator system as recited in claim 4, wherein the fourth adder produces an 11.sup.th vector signal representative containing estimates of a down-hole annulus pressure about a reference point of an i-th local estimator, a down-hole tubing pressure about a reference point of an i-th local estimator, and a lift gas rate through a said gas injection valve about a reference point of an i-th local estimator, by adding a 10.sup.th vector signal representative containing an estimate of a deviation of a down-hole annulus pressure, an estimate of a deviation of a down-hole tubing pressure and an estimate of a deviation of a lift gas rate through a said gas injection valve, and a 12.sup.th vector signal representative containing a down-hole annulus pressure, a down-hole tubing pressure, and a lift gas rate through a said gas injection valve about a reference point of an i-th local estimator.

11. A state estimator system as recited in claim 1, wherein an interpolation module incorporates a said 11.sup.th vector signal representative containing estimates of a down-hole annulus pressure about a reference point of an i-th local estimator, a down-hole tubing pressure about a reference point of an i-th local estimator, and a lift gas rate through a said gas injection valve about a reference point of an i-th local estimator, from all local estimators into a single estimate.

12. A state estimator system as recited in claim 11, wherein the interpolation module produces a single estimate based on the following steps; Selection of two local estimators with reference points closest to the current operating point of a gas lift system, Computation of said single estimate from said two estimates based on the closeness of the actual operating point to reference points of these said two estimates; wherein the computation of a said single estimate is done per formula: ${\hat{x}}_f=\{\begin{array}{c}{\hat{x}}_i.Math..Math.\mathrm{if}.Math..Math.y\le r_i\\ \alpha .Math.{\hat{x}}_i+\left(1-\alpha \right).Math.{\hat{x}}_{i+1}.Math..Math.\mathrm{if}.Math..Math.r_i.Math..Math.y\le r_{i+1}\\ {\hat{x}}_{i+1}.Math..Math.\mathrm{if}.Math..Math.y\ge r_{i+1}\end{array}$ where {circumflex over (x)}.sub.i is an estimate of an i-th local estimator, {circumflex over (x)}.sub.i+1 is an estimate of an i-th+1 local estimator, {circumflex over (x)}.sub.f is a single estimate from all local estimators, r.sub.i is an i-th reference point of an i-th local estimator, r.sub.i+1 is an i-th+1 reference point of an i-th+1 local estimator, α is an interpolation coefficient and 0α1, which preferably is equal to 0.3.

13. A state estimator system according to claim 1, wherein the estimated values of a down-hole annulus pressure, of a down-hole tubing pressure, and of a lift gas rate through said gas injection valve are outputted to control the artificial gas lift system.

14. A method for determining down-hole process parameters for an artificial gas lift system, comprising the following steps: measuring an annulus pressure and/or a surface tubing pressure of the gas lift system; operating a state estimator to perform the steps of: a) determining a vector signal representative of a gas lift choke opening value and/or a production choke opening value and/or a vector signal representative of the measured annulus pressure and/or the measured surface tubing pressure as input values; b) providing a vector signal representative containing estimates of a down-hole annulus pressure about a reference point of an i-th local estimator, and/or a down-hole tubing pressure about a reference point of an i-th local estimator, and/or a lift gas rate through said gas injection valve about a reference point of an i-th local estimator by means of N local estimators where N is greater than or equal to two; and c) incorporating the estimates provided by all local estimators into a single estimate by means of an interpolation module to determine estimated values of a down-hole annulus pressure, and/or of a down-hole tubing pressure, and/or of a lift gas rate through said gas injection valve; and generating an output associated with the determined estimated values.

15. The method according to claim 14, further comprising the step of controlling the artificial gas lift system by using the outputted estimated values of a down-hole annulus pressure, and/or of a down-hole tubing pressure, and/or of a lift gas rate through said gas injection valve.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0039] FIG. 1 shows a schematic of an artificial gas lift system with measurements of surface annulus pressure and surface tubing pressure.

[0040] FIG. 2 shows a structure of a state estimator using measurements of surface annulus pressure and surface tubing pressure in a nonlinear model describing dynamics of an artificial gas lift system.

[0041] FIG. 3 shows an illustration of interpolation based on selection of two estimates and computation of a single estimate.

[0042] FIG. 4 shows an illustration of an exemplary single estimate comprising N reference points and their respective estimates around said reference points.

DETAILED DESCRIPTION OF THE INVENTION

[0043] FIG. 1 shows a schematic of an artificial gas lift system 16 comprising a well casing 1 which connects a surface equipment with an oil reservoir 2; a tubing 3 inserted inside said well casing 1 and forming a space between said well casing 1, and said tubing 3, hereafter referred to as an annulus 5; a gas lift choke 4 installed between a source of a high-pressure gas and said annulus 5, the gas lift choke is mounted on the gas injection flow-line for controlling the injection of lift gas; a gas injection valve 6 installed between said annulus 5 and a volume inside said tubing 3; and during normal operation the gas is usually injected at a deepest feasible point in a well; in this case an annulus 5 is filled with pressurized gas; a production choke 7 installed between a volume inside said tubing 3 and oil processing facilities above the surface, the production choke 7 is mounted in said tubing 3 for controlling the production rate; a gas-conduit may be equipped with a device for measurement of surface annulus pressure 11; and a well head may be equipped with a device for measurement of surface tubing pressure 12. Currently, measurement of a down-hole annulus pressure 8, of a down-hole tubing pressure 9, and of a lift gas rate (LGR) 10 are technically difficult to realize due to high pressures and impossibility to do proper calibration of these measurement devices. However, it is possible to estimate these quantities through surface measurements and properly designed algorithms.

[0044] FIG. 2 shows a structure of a state estimation system 40 comprising a first adder 15, a second adder 19, a third adder 23, a nonlinear module 25, a linear module 27, and a fourth adder 30; using measurements of surface annulus pressure and surface tubing pressure in a nonlinear model describing dynamics of an artificial gas lift system. The first and second adders in particular each may include an electronic receiver configured to receive a plurality of input parameters measured at an upper end of the artificial lift system. A nonlinear control-oriented model of the gas lift system 16 was used along with the state estimator, which is given in the form of Ordinary Differential Equations (ODEs); the nonlinear model is derived based on mass balance equations in an annulus 5 and a tubing 3; based on this principle, the three states are the masses present in certain control volumes, the two control volumes in the model are an annulus 5 and a tubing 3; the masses considered are m.sub.ga, m.sub.gt, m.sub.lt; where m.sub.ga is the mass of gas in an annulus 5, m.sub.gt is the mass of gas in a tubing 3, and m.sub.lt is the mass of liquid in a tubing 3, the dynamics of the state variables are modeled as follows:

{dot over (m)}.sub.ga=w.sub.ga−w.sub.gi

{dot over (m)}.sub.gt=w.sub.gi−w.sub.gp

{dot over (m)}.sub.lt=w.sub.lt−w.sub.lp

[0045] Where w.sub.ga is the mass flow rate of gas into an annulus 5 through a gas lift choke 4, w.sub.gi is the mass flow rate of the injected gas from an annulus 5 into a tubing 3 through a gas injection valve 6, w.sub.gp is the mass flow rate of gas from a tubing 3 through a production choke 7, w.sub.lr is the mass flow rate of liquid (oil and water) from a reservoir 2 into a tubing 3 and w.sub.lp is the mass flow rate of liquid produced from a tubing 3 through a production choke 7; an algebraic equation describing the relation between the down-hole annulus pressure 8 and down-hole tubing pressure 9 and the mass flow rate of gas through the choke. One possible equation to be used is:

[00001]$w_{\mathrm{ga}}=A_{\mathrm{ga}}.Math.c_{\mathrm{ga}}.Math.P_m.Math.\sqrt{\frac{g.Math..Math.\gamma }{{\mathrm{RT}}_a}.Math.{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{\gamma }}}.Math.\Psi \left(\frac{P_a}{P_m}\right)$

[0046] Here, w.sub.ga is the mass flow rate of gas through the choke, A.sub.ga is the area of the valve's orifice, c.sub.ga is the discharge coefficient, g is the gravity, γ is the isentropic coefficient of natural gas, R is the universal gas constant, T.sub.a is the temperature at the annulus, Ψ is a flow function, P.sub.a is the surface annulus pressure, and P.sub.m is the pressure at the gas manifold; one possible equation for the mass flow rate of gas from the annulus into the tubing through the injection valve to be used is:

[00002]$w_{\mathrm{gi}}=A_{\mathrm{iv}}.Math.c_{\mathrm{iv}}.Math.P_{\mathrm{ai}}.Math.\sqrt{\frac{g.Math..Math.\gamma }{{\mathrm{RT}}_a}.Math.{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{\gamma }}}.Math.\Psi \left(\frac{P_{\mathrm{ti}}}{P_{\mathrm{si}}}\right)$

[0047] Here, w.sub.gi is the mass flow rate of gas from the annulus into the tubing through the injection valve, A.sub.iv is the area of the valve's orifice, c.sub.iv is the discharge coefficient, g is the gravity, γ is the isentropic coefficient of natural gas, R is the universal gas constant, T.sub.a is the temperature at the annulus, Ψ is a flow function, P.sub.ti is the down-hole tubing pressure, and P.sub.ai is the down-hole annulus pressure.

[0048] An algebraic equation describes the relation between the down-hole annulus pressure 8 and down-hole tubing pressure 9 and the mass flow rate of liquid from the reservoir into the tubing through the perforations valve. One possible equation to be used is:

[00003]$w_{\mathrm{lr}}={\rho }_l.Math.N_{\mathrm{lr}}.Math.A_{\mathrm{lr}}.Math.c_{\mathrm{lr}}.Math.\sqrt{\frac{P_r-P_{\mathrm{ti}}}{g_l}}$

[0049] Here, ρ.sub.l is the density of the liquid (oil), A.sub.ir is the equivalent area of the total orifice available for oil flow, c.sub.lr is the discharge coefficient, N.sub.lr is a constant added for units conversion, g.sub.1 is the specific gravity of the liquid, P.sub.r is a constant pressure in the reservoir, and P.sub.ti is the down-hole tubing pressure.

[00004]$w_{\mathrm{lp}}=\frac{q_{\mathrm{mp}}.Math.{\rho }_m}{\lambda +1},.Math.\mathrm{where}$$q_{\mathrm{mp}}=N_{\mathrm{pc}}.Math.A_{\mathrm{pc}}.Math.c_{\mathrm{pc}}.Math.\sqrt{\frac{P_p-P_s}{g_m}}$${\rho }_m=\frac{\lambda +1}{\lambda .Math.\frac{{\mathrm{RT}}_t}{P_p.Math.\mu }+\frac{1}{{\rho }_l}}$$g_m=\frac{{\rho }_m}{{\rho }_{\mathrm{water}}}$$\lambda =\frac{m_{\mathrm{gt}}}{m_{\mathrm{lt}}}$

[0050] Here, ρ.sub.m is the density of the mixture (oil and gas), q.sub.mp is the volumetric flow of the mixture, R is the universal gas constant, T.sub.t is the temperature in the tubing, μ is the molar mass of the gas, g.sub.m is the specific gravity of the mixture, and λ is the ratio of the mass of gas to the mass of liquid in the tubing, P.sub.p is the surface tubing pressure, and P.sub.s is the pressure at the output to the production choke. An algebraic equation for the mass flow of gas through the production valve is computed from the mixture flow and a known ratio λ:

w.sub.gp=λw.sub.lp

[0051] The flow function Ψ used for compressible fluid of the gas lift system is given by:

[00005]$\Psi \left(\frac{P_r}{P_s}\right)=\{\begin{array}{cc}1& \mathrm{if}.Math..Math.\frac{P_r}{P_s}\le {\beta }_c\\ \sqrt{\frac{2}{\gamma -1}.Math.{\left(\frac{\gamma +1}{2}\right)}^{\frac{\gamma +1}{\gamma -1}}}.Math.\sqrt{{\left(\frac{P_r}{P_s}\right)}^{\frac{2}{\gamma }}-{\left(\frac{P_r}{P_s}\right)}^{\frac{\gamma +1}{\gamma }}}& \mathrm{if}.Math..Math.\frac{P_r}{P_s}\ge {\beta }_c\end{array}$

[0052] Here,

[00006]${\beta }_c={\left(2/\gamma +1\right)}^{\frac{\gamma }{\gamma -1}}$

is the critical pressure ratio, P.sub.s is the pressure inside the volume the gas is leaving and the pressure P.sub.r signifies the pressure inside the volume the gas is entering. The mass flow rate functions depends on the pressure. The pressure distribution in the well is described in Hussein et al. (2015) by the flowing set of mass dependent nonlinear functions:

[00007]$P_p=\frac{m_{\mathrm{mix}}.Math.g}{A_t\left(e^{\frac{\mathrm{ah}}{b}-\frac{m_{\mathrm{mix}}.Math.g}{\mathrm{Ab}}}-1\right)}$$P_{\mathrm{ti}}=\frac{m_{\mathrm{mix}}.Math.{\mathrm{ge}}^{\frac{\mathrm{ah}}{b}-\frac{m_{\mathrm{mix}}.Math.g}{\mathrm{Ab}}}}{A_t\left(e^{\frac{\mathrm{ah}}{b}-\frac{m_{\mathrm{mix}}.Math.g}{\mathrm{Ab}}}-1\right)}$$P_a=\frac{m_{\mathrm{ga}}.Math.g}{A_a\left(e^{\frac{g.Math..Math.\mu }{{\mathrm{RT}}_a}.Math.h}-1\right)}$$P_{\mathrm{ai}}=\frac{m_{\mathrm{ga}}.Math.{\mathrm{ge}}^{\frac{g.Math..Math.\mu }{{\mathrm{RT}}_a}.Math.h}}{A_t\left(e^{\frac{g.Math..Math.\mu }{{\mathrm{RT}}_a}.Math.h}-1\right)}$

[0053] Here, a=g(λ+1)ρ.sub.l, b=λRT.sub.tρ.sub.l/μ, h is the height of a tubing 3, A.sub.t is the cross-sectional area of a tubing 3, T.sub.t is the temperature in a tubing 3, μ is the molar mass, T.sub.a is the temperature in an annulus 5, and A.sub.a is the cross-sectional area of an annulus 5. The first two pressure equations depict a tubing pressure based on mass mixture of gas and liquid m.sub.mix, where P.sub.p is a surface tubing pressure 12, and P.sub.ti is a down-hole tubing pressure 9. The last two pressure equations represent an annulus pressure based on mass of gas m.sub.ga, where P.sub.a is a surface annulus pressure 11, and P.sub.ai is a down-hole annulus pressure 8.

[0054] A generic nonlinear model of a gas lift system is given as follows:

{dot over (x)}=f(x,u)

y=g(x,u)

Here, x=[x.sub.1 x.sub.2 x.sub.3] is a state vector comprised of the state variables x.sub.1=m.sub.ga, X.sub.2=m.sub.gt, x.sub.3=m.sub.lt, y=[y.sub.1 y.sub.2].sup.T is the output vector comprised of two measured process variables y.sub.1=P.sub.a is a surface annulus pressure, y.sub.2=P.sub.p is a surface tubing pressure and u=[u.sub.1 u.sub.2] is a control vector comprised of the controller commands defining the required value openings: u.sub.1=l.sub.1 is a gas lift choke opening, u.sub.2=l.sub.2 is a production choke opening. The process variables available for measurement are the surface flow and surface pressures. Down-hole flow, down-hole pressure, and down-hole masses of a gas lift system can only be estimated using a state estimator. The use of a state estimator 50 would eliminate the necessity of down-hole state measurements that are considered difficult to achieve due to the harsh environment associated, which include limited accessibility and high down-hole pressures.

[0055] Gas and liquid dynamics in an artificial gas lift are described by a third order nonlinear equation along with the dynamics of an actuator, valves, and controllers. These dynamics result in a high order nonlinear system with delays, which is very complex to analyze and control. Linearization of a nonlinear process at specific points provides a linear approximation of the process dynamics at these points. Therefore, the approach we propose involves linearization of a gas lift system in multiple selected operating points. A real physical system cannot reveal arbitrary combinations of the states, due to various physical constraints. Consequently, all system's state trajectories are concentrated within a specific boundary in the state space. The state corresponding to an equilibrium point is given by x.sub.o:={x|0=f(x, u.sub.o)}, which is created artificially by setting the control signal to the plant to a specific constant value given as u.sub.o. Through varying these constant control inputs, various (multiple) equilibrium points are enforced. All system motions that occur in practice belong to a domain around a specific equilibrium point. If the state trajectory moves closer to another equilibrium point, this new equilibrium point can be considered as a new reference with its own domain. Each domain can be described by a certain linearized model obtained by linearization of the original nonlinear equations at a specific equilibrium point. As a result, as it is shown in FIG. 4, a set of state estimators for deviations 31-1, 31-2 from a specific equilibrium point 41-1, 41-2, based on linearized models with linearization at these specific equilibrium points, is designed. The steady state can be artificially obtained by means of setting a specific controller output. It can be assumed that a linearized model at a certain equilibrium point 41-1, 41-2 is valid in the vicinity of that specific equilibrium point. Therefore, a sliding mode observer is designed for each linear model, and then these estimators are incorporated in such a way that it provides proper estimation 33 for the original nonlinear system. Hammadih et al. (2015) proposed a design methodology for computing a gain matrix for each sliding mode observer for a gas lift system.

[0056] In the structure of a state estimation system 40, the first adder 15, the second adder 19 and the fourth adder 30 are used to enforce multiple equilibrium points 41. The state estimator 40 uses a 3.sup.rd vector signal representative 13 containing a gas lift choke opening and a production choke opening of gas lift system 16 to calculate a 1.sup.st vector signal representative 20 containing a deviation of a gas lift choke opening and a deviation of a production choke opening, by subtracting a 2.sup.nd vector signal representative 14 containing a gas lift choke opening and a production choke opening about a reference point 41 of an i-th local estimator 40 from the 3.sup.rd vector signal representative 13; along with a 6.sup.th vector signal representative 17 containing a measured surface annulus pressure and a measured surface tubing pressure from gas lift system 16 to calculate a 4.sup.th vector signal representative 21 containing a deviation of a surface annulus pressure and a deviation of a surface tubing pressure, by subtracting a fifth vector signal representative 18 containing a surface annulus pressure and a surface tubing pressure about a reference point 41 of an i-th local estimator 40 from the 6.sup.th vector signal representative 17.

[0057] FIG. 3 shows an illustration of interpolation comprising N local estimators 40-1-40-N and an interpolation module 32 that incorporates estimates 31-1 provided by local estimator 1, 31-2 provided by local estimator 2, up to 31-N provided by local estimator N; and produces a single estimate 33 through the use of the interpolation module 32 based on a selection of two estimates and computation of a single estimate. Hammadih et al. (2015) proposed the idea of using multiple linearized models in sliding mode observer design. The incorporation of the two estimates is based on the output measurement y. The main condition that governs the interpolation function is that the total sum of coefficients α.sub.i and α.sub.i+1 is 1 and the total sum of coefficients β.sub.k and β.sub.k+1 is 1. Provided that ζ=0.3 where ζε[0,1], Δ.sub.1i=y.sub.1−r.sub.1i, Δ.sub.2(i+1)=r.sub.1(i+1)−y.sub.1, Δ.sub.1k=y.sub.2−r.sub.2k, and Δ.sub.2(k+1)=r.sub.2(k+1)−y.sub.2. When the set point is below 30% of the output range y.sub.1 and below 30% of the output range y.sub.2, the coefficients α.sub.i and β.sub.k will account for the main contribution, while the contribution from the coefficients α.sub.i+1 and β.sub.k+1 is zero. When the set point is below 30% of the output range γ.sub.1 and above 70% of the output range y.sub.2, the coefficients α.sub.i and β.sub.k+1 will account for the main contribution, while the contribution from the coefficients α.sub.i+1 and β.sub.k is zero. When the set point is above 70% of the output range y.sub.1 and below 30% of the output range y.sub.2, the coefficients α.sub.i+1 and β.sub.k will account for the main contribution, while the contribution from the coefficients α.sub.i and β.sub.k+1 is zero. When the set point is above 70% of the output range γ.sub.1 and above 70% of the output range y.sub.2, the coefficients α.sub.i+1 and β.sub.k+1 will account for the main contribution, while the contribution from the coefficients α.sub.i and β.sub.k is zero. When the output measurement lies exactly in the center of both ranges and between the four set points r.sub.1i, r.sub.1(i+1), r.sub.2k and r.sub.2(k+1), the coefficients would have equal contribution. Consequently, the coefficients α.sub.A and α.sub.B are obtained as follows:

[00008]$\mathrm{If}.Math..Math.\frac{{\Delta }_{1.Math.i}}{r_{1.Math.\left(i+1\right)}-r_{1.Math.i}}\le \zeta .Math..Math.\mathrm{and}.Math..Math.\frac{{\Delta }_{1.Math.k}}{r_{2.Math.\left(k+1\right)}-r_{2.Math.k}}\le \zeta ,\mathrm{then}.Math..Math.{\alpha }_i=1,{\alpha }_{i+1}=1-{\alpha }_i,{\beta }_k=1.Math..Math.\mathrm{and}.Math..Math.{\beta }_{k+1}=0$$\mathrm{If}.Math..Math.\frac{{\Delta }_{1.Math.i}}{r_{1.Math.\left(i+1\right)}-r_{1.Math.i}}\le \zeta .Math..Math.\mathrm{and}.Math..Math.\frac{{\Delta }_{1.Math.k}}{r_{2.Math.\left(k+1\right)}-r_{2.Math.k}}\ge \left(1-\zeta \right),\mathrm{then}.Math..Math.{\alpha }_i=1,{\alpha }_{i+1}=1-{\alpha }_i,{\beta }_k=0.Math..Math.\mathrm{and}.Math..Math.{\beta }_{k+1}=1$$\mathrm{If}.Math..Math.\frac{{\Delta }_{1.Math.i}}{r_{1.Math.\left(i+1\right)}-r_{1.Math.i}}\ge \left(1-\zeta \right).Math..Math.\mathrm{and}.Math..Math.\frac{{\Delta }_{1.Math.k}}{r_{2.Math.\left(k+1\right)}-r_{2.Math.k}}\le \zeta ,\mathrm{then}.Math..Math.{\alpha }_i=0,{\alpha }_{i+1}=1-{\alpha }_i,{\beta }_k=1.Math..Math.\mathrm{and}.Math..Math.{\beta }_{k+1}=1$$\mathrm{If}.Math..Math.\frac{{\Delta }_{1.Math.i}}{r_{1.Math.\left(i+1\right)}-r_{1.Math.i}}\ge \left(1-\zeta \right).Math..Math.\mathrm{and}.Math..Math.\frac{{\Delta }_{1.Math.k}}{r_{2.Math.\left(k+1\right)}-r_{2.Math.k}}\ge \left(1-\zeta \right),\mathrm{then}.Math..Math.{\alpha }_i=0,{\alpha }_{i+1}=1-{\alpha }_i,{\beta }_k=0.Math..Math.\mathrm{and}.Math..Math.{\beta }_{k+1}=1$$\mathrm{Else},{\alpha }_i=\frac{\left(\left(1-\zeta \right)-\frac{{\Delta }_{1.Math.i}}{r_{1.Math.\left(i+1\right)}-r_{1.Math.i}}\right)}{\left(1-2.Math.\zeta \right)},{\alpha }_{i+1}=1-{\alpha }_i,{\beta }_k=\frac{\left(\left(1-\zeta \right)-\frac{{\Delta }_{1.Math.k}}{r_{2.Math.\left(k+1\right)}-r_{2.Math.k}}\right)}{\left(1-2.Math.\zeta \right)}.Math..Math.\mathrm{and}.Math..Math.{\beta }_{k+1}=1-{\beta }_k.$

[0058] Then, the final estimation is selected based on the nearest distance to the specific sliding mode observer. In a preferred embodiment, the nonlinear model of the gas lift system 16 is linearized at specific linearization points. These points are assumed to comprise the whole operating range of the gas lift system. The resulting linearization provides linear approximation of the system at these equilibrium points. In the linearization of the model, the derivations of the linearized state space matrices of the nonlinear gas lift system are obtained using the Jacobian matrix approach in Hussein et al. (2015).

[0059] Linearization of the given generic nonlinear model of the gas lift system at the point x.sub.o, u.sub.o yields:

{tilde over ({dot over (x)})}=A{tilde over (x)}+B{tilde over (u)}

{tilde over (y)}=C{tilde over (x)}+Dũ

[0060] Here, where {tilde over (x)}=x−x.sub.o, {tilde over (y)}=y−y.sub.o, and ũ=u−u.sub.o. Derivation of the four state space matrices A, B, C, and D was computed using the Jacobian matrix. The state matrix of the linear system is written in term of the three state variables m.sub.ga, m.sub.gt, m.sub.lt, as the following:

[00009]${A=\frac{\partial f}{\partial x}.Math.}_p=\left[\begin{array}{ccc}\frac{\partial w_{\mathrm{ga}}}{\partial m_{\mathrm{ga}}}-\frac{\partial w_{\mathrm{gi}}}{\partial m_{\mathrm{ga}}}& \frac{\partial w_{\mathrm{ga}}}{\partial m_{\mathrm{gt}}}-\frac{\partial w_{\mathrm{gi}}}{\partial m_{\mathrm{gt}}}& \frac{\partial w_{\mathrm{ga}}}{\partial m_{\mathrm{lt}}}-\frac{\partial w_{\mathrm{gi}}}{\partial m_{\mathrm{lt}}}\\ \frac{\partial w_{\mathrm{gi}}}{\partial m_{\mathrm{ga}}}-\frac{\partial w_{\mathrm{gp}}}{\partial m_{\mathrm{ga}}}& \frac{\partial w_{\mathrm{gi}}}{\partial m_{\mathrm{gt}}}-\frac{\partial w_{\mathrm{gp}}}{\partial m_{\mathrm{gt}}}& \frac{\partial w_{\mathrm{gi}}}{\partial m_{\mathrm{lt}}}-\frac{\partial w_{\mathrm{gp}}}{\partial m_{\mathrm{lt}}}\\ \frac{\partial w_{\mathrm{lr}}}{\partial m_{\mathrm{ga}}}-\frac{\partial w_{\mathrm{lp}}}{\partial m_{\mathrm{ga}}}& \frac{\partial w_{\mathrm{lr}}}{\partial m_{\mathrm{gt}}}-\frac{\partial w_{\mathrm{lp}}}{\partial m_{\mathrm{gt}}}& \frac{\partial w_{\mathrm{lr}}}{\partial m_{\mathrm{lt}}}-\frac{\partial w_{\mathrm{lp}}}{\partial m_{\mathrm{lt}}}\end{array}\right]$

[0061] Where all the derivatives are computed using the formulas provided in Hussein et al. (2015).

[0062] The input matrix of the linear system is related to the control action which is the percentage of opening of the valve l. Using the Jacobian matrix, the matrix is given by:

[00010]${B=\frac{\partial f}{\partial u}.Math.}_p=\left[\begin{array}{c}\frac{\partial w_{\mathrm{ga}}}{\partial l}-\frac{\partial w_{\mathrm{gi}}}{\partial l}\\ \frac{\partial w_{\mathrm{gi}}}{\partial l}-\frac{\partial w_{\mathrm{gp}}}{\partial l}\\ \frac{\partial w_{\mathrm{lr}}}{\partial l}-\frac{\partial w_{\mathrm{lp}}}{\partial l}\end{array}\right]$

[0063] Where all the derivatives are computed using the formulas provided in Hussein et al. (2015).

[0064] The output matrix of the linear system is related to the flow of gas into the annulus w.sub.ga. Thus, the matrix is given by:

[00011]${C=\frac{\partial y}{\partial x}.Math.}_p=\left[\begin{array}{ccc}\frac{\partial w_{\mathrm{ga}}}{\partial m_{\mathrm{ga}}}& \frac{\partial w_{\mathrm{ga}}}{\partial m_{\mathrm{gt}}}& \frac{\partial w_{\mathrm{ga}}}{\partial m_{\mathrm{lt}}}\end{array}\right]$

[0065] Where all the derivatives are computed using the formulas provided in Hussein et al. (2015).

[0066] The feedforward matrix of the linear system is given as follows:

[00012]${D=\frac{\partial y}{\partial u}.Math.}_p=\left[\frac{\partial w_{\mathrm{ga}}}{\partial l}\right]$

[0067] Where all the derivatives are computed using the formulas provided in Hussein et al. (2015).

[0068] Using higher number of linearization points and acquiring higher number of linearized models result in a better (on average) approximation of the nonlinear system due to the smaller (on average) distances between the actual state vector and the value of the state vector at a nearest equilibrium point. The principle of interpolating sliding mode observer is introduced by Hammadih et al. (2016), which is applied in this case on a ball and beam system for estimation of the slope of the beam from the measurement of the ball position.

[0069] The state estimation system and the various components thereof, including each of the N local estimators, the referenced component adders and the linear and non-linear modules, and the interpolation modules, and like or related processing components, may be implemented as any suitable electronic control or processing device. Such components may constitute or include suitable hardware, firmware, and various combinations thereof. Such components further may include control circuitry that is configured to carry out overall functions and operations of the state estimator system. The control circuitry may include an electronic processor, such as a CPU, microcontroller or microprocessor. Among their functions, to implement the features of the present invention, the control circuitry and/or electronic processor may execute program code stored on a non-transitory computer readable medium, such as any conventional computer memory device. The computer code may be executed for determining and outputting the estimated values as described above. It will be apparent to a person having ordinary skill in the art of computer programming, and specifically in application programming for electronic control devices for oil well control, how to program the state estimator components to operate and carry out logical functions associated with the present invention. Accordingly, details as to specific programming code have been left out for the sake of brevity. Also, while the code may be executed by control circuitry in accordance with an exemplary embodiment, such computational functionality could also be carried out via dedicated hardware, firmware, software, or combinations thereof, without departing from the scope of the invention.

[0070] The processor devices further may include any suitable output devices for generating signals associated with the calculations. The output devices may comprise electronic circuitry or other processing devices comparably as described above, for outputting electronic signals associated with the calculated values, which may then be interpreted by additional downstream processing devices or interface devices (e.g., displays, speakers, and other visual, audio, tactile, and/or other sensory indicators). For example, the adders 30 may constitute output devices comparably as described above that output signals associated with the estimated values 31 for use by the interpolation module 32. Similarly, the interpolation module 32 may include output processor devices comparably as described above that generate output signals associated with the single estimate 33 for use by the other interface devices

[0071] Although the invention has been shown and described with respect to a certain embodiment or embodiments, it is obvious that equivalent alterations and modifications will occur to others skilled in the art upon the reading and understanding of this specification and the annexed drawings. In particular regard to the various functions performed by the above described elements (components, assemblies, devices, compositions, etc.), the terms (including a reference to a “means”) used to describe such elements are intended to correspond, unless otherwise indicated, to any element which performs the specified function of the described element (i.e., that is functionally equivalent), even though not structurally equivalent to the disclosed structure which performs the function in the herein illustrated exemplary embodiment or embodiments of the invention. In addition, while a particular feature of the invention may have been described above with respect to only one or more of several illustrated embodiments, such feature may be combined with one or more other features of the other embodiments, as may be desired and advantageous for any given or particular application.