Dynamic plunger lift model for deliquification of shale gas wells

Computers and Chemical Engineering 103 (2017) 81–90

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Computers and Chemical Engineering journal homepage: www.elsevier.com/locate/compchemeng

Dynamic plunger lift model for deliquification of shale gas wells Arun Gupta a,∗ , Niket S. Kaisare b,1 , Naresh N. Nandola a a b

ABB Corporate Research Center,Bhoruka Tech Park, Whitefield Main Road, Bangalore 560048, India Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600036, India

a r t i c l e

i n f o

Article history: Received 15 January 2016 Received in revised form 1 March 2017 Accepted 8 March 2017 Available online 11 March 2017 Keywords: Hybrid state model Artificial lift Plunger lift Upstream oil and gas Simulation Periodic process

a b s t r a c t This paper presents first principles model for operation of a plunger lift system in natural gas wells. The model consists of pressure and flow dynamics of fluids in annulus and the central tubing sections of the well, and dynamics of plunger fall and rise in the tubing. System dynamics switch on shutting or opening the production valve, and with autonomous events related to plunger motion. A nonlinear hybrid model is realized with nine states, switching though six stages (modes) of operation. This is the first instance of modeling complex system of plunger lift using a standard hybrid system model (HSM) framework. The resulting model is used to present insight into plunger lift operation, including an efficient simulation of multiple plunger cycles and analysis of effect of uncertainties on the well behavior. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Shale reservoirs are classified as unconventional reservoirs due to the low permeability, low porosity, rapid decline in production rate and liquid loading at the well-bottom. Recent advances in horizontal drilling and hydraulic fracturing have made extraction of shale gas economically viable. While liquids are initially produced as a mist with the flowing gas, they are removed at later stages using Artificial Lift to prevent accumulation at the well bottom and ensuring gas flow without hindrance. Various artificial lift alternatives may be broadly categorized into two groups: those that use well’s own energy and those that use external energy to deliquefy the wells (Kaisare et al., 2013). Plunger Lift is an intermittent artificial lift technique that uses well’s own energy to de-liquefy the well. A metal rod, called plunger, is introduced in a well to efficiently deliquefy an intermittently operated well. Plungers are classified as either conventional (such as dual pad, brush type etc.) or flow-through plungers (such as two piece plungers) having different fall and rise characteristics (Lea et al., 2008). Since conventional plungers are more commonly used for deliquification, this work focuses on developing a mathematical model for conventional plunger lift systems.

∗ Corresponding author. E-mail address: [email protected] (A. Gupta). 1 NSK was with ABB Corporate Research Center when the technical work in this paper was completed. http://dx.doi.org/10.1016/j.compchemeng.2017.03.005 0098-1354/© 2017 Elsevier Ltd. All rights reserved.

A conventional plunger lift is a cyclic process that involves actuating the production valve to open or close position. When the production valve is closed, the plunger falls down entire height of the well due to gravity; when opened, pressure difference across the plunger causes it to rise to the surface, along with liquids accumulated at the well bottom. Plunger lift models can be broadly categorized into static and dynamic models. Static models find a semi-empirical correlation between the variables of interest for a particular part or stage of the process. The Foss and Gaul (1965) pressure and flow rate calculations by Turner et al. (1969)) and Coleman et al. (1991) are examples of semi-empirical static models. These models are primarily developed to calculate the operating condition in a given well rather than to simulate the plunger lift process. While static models are approximate, they have been used to give some estimate of the conditions to open or close the production valve. For example, Foss and Gaul (1965) calculate the minimum pressure that is required to ensure the plunger surfaces with the liquid slug. The model by Lea (1982) is perhaps the first dynamic model for plunger rise stage of plunger lift cycle. Lea solved a force balance on rising plunger, allowing the plunger velocity to vary as it rises through the tubing. This information was used to improve the prediction of threshold pressure required to ensure plunger arrival at the surface as previously calculated by Foss and Gaul (1965). Plunger rise is the critical part of lift cycle as the success of plunger lift operation is characterized by amount of liquid lifted and the arrival velocity during the plunger rise stage. Thus, several plunger lift models focus on the dynamics of plunger rise with the liquid

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Nomenclature Well parameters H Height of the well (m) dt Tubing diameter (m) Area of tubing, annulus (m2 ) At , Aa Cv Flow coefficient (kg/s.Pa−0.5 ) Cd Drag co-efficient Constants and fluids information Gravitational acceleration (m/s2 ) g R Gas constant (J/mol/K) Mg Avg. molecular weight (kg/mol) Reservoir parameters GLR Gas to liquid ratio (m3 /m3 ) Cres Reservoir flow constant (various) Pres Reservoir pressure (Pa) Other variables Pressure, (i) = l, t, c (Pa) P(i) T Average temperature (K) g , l Gas, liquid density (kg/m3 ) Compressibility factor (–) Z Fres Flow from reservoir (kg/s) Fluid flow in/out annulus (kg/s) Fann Ftub Fluid flow in tubing (kg/s) Gas flow rate to prod. line (kg/s) Fg,out ε Tubing surface roughness (m) f Friction factor (–) Reynolds’ number (–) Re

slug (Avery and Evans, 1988; Marcano and Chacin, 1994; Baruzzi and Alhanati, 1995). Avery and Evans (1988) developed a computational model to simulate the plunger rise phase of the process. The model of Marcano and Chacin (1994) captured liquid fallback, whereas Baruzzi and Alhanati (1995) used their model to determine the minimum pressure buildup required to ensure plunger arrival with high production rate. Tang and Liang (2008) used dynamic model of plunger lift, including both plunger rise and fall stages, to optimize well production. They assumed pressurization (when the well is shut-in) and depressurization (when the well is flowing) to follow simple polynomial or first-order functions of time (Tang and Liang, 2008). These models are able to simulate a particular stage of the cycle, however, not a full cycle or multiple cycles in sequence. Another approach to calculate the plunger fall and rise characteristics are to use measurements from highly instrumented wells, such as acoustic sensors and/or smart plunger. Becker et al. (2006) used acoustic measurements of the plunger position and liquid levels in a well, whereas Chava et al. (2010, 2008) used measurements from a smart plunger (plunger that has been instrumented with additional sensors) to model and optimize plunger lift cycle. While these measurements provide valuable insights into the plunger lift process dynamics, obtaining them is often an expensive exercise. Gasbarri and Wiggins (2001, 1997) built on the earlier works by various authors to develop a dynamic model for all stages of the plunger lift cycle. They discretized the tubing into n-control volumes and solved for coupled pressure and flow equations within each volume to calculate the surface measurements. Maggard et al. (2000), Maggard (2000) built upon their model and coupled it with a detailed reservoir simulator to account for conditions in tight gas wells. Though these models provide a dynamical evolution of states for plunger lift cycle, they are computationally intensive, and

Fig. 1. Schematic of plunger lifted well. A typical shale gas well is about 3 km deep.

are not suitable for applications such as model-based control, optimization, monitoring and comprehensive analysis of the process. Therefore, there is a need to develop a comprehensive model for plunger lift that captures the complex nonlinear dynamics of pressurization and flow of natural gas and liquids in the well and dynamics of plunger motion; as well as transitions in the system dynamics due to opening/closing of production valve (manipulated event) and plunger arrival at the surface or well-bottom (autonomous events). Hybrid systems approach for modeling provides a unified framework that can efficiently handle these discrete events along with continuous dynamics. However, the challenges in modeling plunger lift system into a standard hybrid system modeling framework lies in identifying different modes of the systems and dynamics in each mode (including continuous dynamics and discrete states), their evolution based on defined events in plunger lift (i.e. conditions for switching from one mode to another), and synchronization of state variables between various modes (i.e., state transition during switching). In this work, we have formulated the plunger lift process as a hybrid state model (HSM) (Buss et al., 2002; Nandola and Bhartiya, 2008), which enables to define and capture different stages of plunger lift process in relation to discrete manipulation of production valve and continuous states. After describing plunger lift process in the next section, a transient model is developed within the standard HSM framework in Section 3, followed by presentation of simulation results showing key signatures of a conventional plunger lift cycle in Section 4. Section 5 summarizes broad conclusions about the proposed model. 2. Plunger lift process An oil and/or gas well consists of concentric tubes that run several thousand feet below the surface (see Fig. 1 for a schematic). The outer tube, called the casing, is typically 4–6 inches in diameter. The inner tube, called the tubing, is about 2 inches in diameter. The gap between casing and tubing is referred to as annulus. Fluids from the reservoir flow through fractures (natural and induced) into the well through perforations at the bottom of the casing. An instrumented plunger lifted well, as shown in Fig. 1, has pressure sensors, gas flow sensor and a plunger arrival sensor connected

A. Gupta et al. / Computers and Chemical Engineering 103 (2017) 81–90

Fig. 2. Various stages and events in a single plunger lift cycle. Valve close and open are decisions taken by a controller or an operator, whereas autonomous events are shown as boxes.

83

pressures in the tubing and casing increase due to net gas flow from the reservoir to the well (see Fig. 3). The plunger stays at the plunger-seat located at the well bottom, with liquids accumulating on top of the plunger. The production valve is then opened signaling the start of plunger rise stage. The gas above the plunger flows out, resulting in steep drop of tubing pressure and a relatively slower drop in casing pressure as shown in Fig. 3. The casing pressure translated to the pressure at the bottom of the plunger causes the plunger-slug subsystem to rise through the tubing, with the plunger forming a temporary seal between the gas below and liquids above it. After certain time, the liquid slug surfaces at the well-head, indicating the start of slug arrival stage. Slug arrival is marked by a sharp peak in the tubing pressure and flowrate as in Fig. 3. The plunger continues to rise in the tubing till it reaches the catcher or lubricator. The plunger stays in the catcher and gas continues to flow from the tubing to the line through production valve; this is known as the after-flow stage. In this stage, the tubing and casing pressures continue to fall due to net gas out from the system as in Fig. 3. The valve remains open during plunger rise, slug arrival and after-flow stages, as shown by thick empty arrows in Fig. 2, leading to net gas flow from well to the production line. To summarize, the plunger lift process can be divided into six stages (modes) that occur cyclically: i) plunger fall (gas) stage, ii) plunger fall (liquid) stage, iii) build-up stage, iv) plunger rise stage, v) slug-arrival stage, and vi) after-flow stage. Discrete decisions of closing/opening the valve, and discrete autonomous events related to plunger motion trigger switching between these stages/modes. The dynamics of gas pressurization and flow, and plunger motion are governed by continuous variables, leading to hybrid dynamical nature of the system. Reservoir dynamics and well-bottom conditions (which are unmeasured) strongly influence plunger lift operation. 3. Modeling of plunger lift system

Fig. 3. Typical pressure and flow signature of a conventional plunger lift cycle.

to a controller (Remote Terminal Unit). The arrival sensor detects plunger arrival at the surface, and thus measures the time taken for it to move from well-bottom to the surface (called arrival time). All these measurements are available at the surface. Plunger lift is a cyclic process in which the plunger travels through the depth of the well and carries liquid slug to the surface during its upward journey. Fig. 2 shows various stages in the operation of a single plunger-lift cycle, while Fig. 3 shows typical profiles of surface pressure and flow-rate measurements. The decision boxes (diamond boxes) in Fig. 2 represents the controller/operator actions of opening and closing the production valve and rectangular boxes represent autonomous events (relating to plunger motion). A plunger lift cycle is considered to start when the production valve is just closed with the plunger at the surface. The gas flow stops, and the plunger falls due to gravity. This stage is called the plunger fall (gas) stage. The plunger reaches the liquid accumulated at the bottom; the plunger now falls through the liquid column, called as plunger fall (liquid) stage. Eventually, the plunger reaches the plunger seat, which marks the start of the build up stage. The gas and liquids from the reservoir continue to flow into the well during the entire cycle, which is affected by bottomhole pressure (Pwf ) and reservoir dynamics. The production valve remains closed during the plunger fall and build up stages and the

Plunger lift operation exhibits hybrid character and is controlled by binary manipulated input. Various representation of hybrid systems such as an unified framework for nonlinear hybrid systems proposed by Branicky et al. (1998), Branicky (1995), mixed logical dynamical (MLD) system (Bemporad and Morari, 1999), piece wise affine (PWA) system (Heemels et al., 2001), multiple partially linearized (MPL) model (Nandola and Bhartiya, 2008), etc. are proposed in system theoretic literature. In this work, the hybrid state model (HSM) framework of Buss et al. (2002) is used to represent plunger-lift model in a formal nonlinear hybrid system framework. 3.1. Background of hybrid state model representation A model for hybrid system involves both continuous and discrete variables (Branicky et al., 1998). In HSM (Buss et al., 2002), the continuous states, x evolve by flow-field F as: dx / 0; ∀j = 1, 2, . . ., ns = F (x, q, u, v) if sj (x, q, u, v) = dt

(1)

 n ᏾nx are continuous states, q ∈ 0, 1 q  nv nu

In the above equation, x ∈

are binary states, u ∈ ᏾ are continuous inputs and v ∈ 0, 1 are discrete inputs. Various events result in transition of the system among finite modes in the HSM model. Since the plunger lift system operates in six distinct stages or modes, Eq. (1) will be written as: dx  = / 0; ∀j = 1, 2, . . ., ns ql fl (x, q, u, v) if sj (x, q, u, v) = dt 6

(2)

l=1

There are six binary state variables, one of which takes value 1 (corresponding to the current mode) while the others are 0. Thus,

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A. Gupta et al. / Computers and Chemical Engineering 103 (2017) 81–90

the equality



ql = 1 holds. This ensures that when the system is

l

The measurements include casing, tubing and line pressures at the surface, production flow rate, and arrival time:



in particular stage or mode, only the corresponding binary variable is active. In the HSM framework, Buss et al. (2002) propose evolution of discrete states is determined by combination of discrete events. In plunger lift, these discrete events are the ones represented in Fig. 2. An event is said to occur when sj (·) changes its values from positive to negative or negative to positive. So long as the sign of sj (·) remains same, the system remains in the same mode. The time at which an event generating functions take the value 0 is the time at which transition is said to occur. Upon transition, states of system are reset based on the map, ˇ (·):

y = Pc



L(j) =

x (t + )



q (t + )

= ˇj (x, q, u, v) if sj (x, q, u, v) = 0

(3)

where x (t + ) and q (t + ) represents state set immediately after occurrence of an event. For example, when the system transitions from m to mode (m + 1) at time t, the mth element of vector q is set to 0, (m + 1)th element is set to 1, other elements of q (t + ) remain unchanged at 0; whereas, continuous states, x (t + ) may be set based on the physics of plunger lift during transition. The measured output from the model are given by: y (t) = g (x, q, u, v)

(4)

The rest of this section is devoted to developing the hybrid state model for plunger lift. The state variables, inputs and disturbances are described in the next subsection; the subsequent subsection describes flow-fields fl (·); this is followed by description of event generating and transition functions sj (·) and ˇj (·); followed by the output model.

Plunger lift operation consists of six different stages in a cycle and one binary manipulated input. Thus, HSM model with six modes is governed by six different continuous dynamics (or flowfields). As shown in Fig. 4(a), the well is divided into three sections: Annulus is considered as a one section, whereas the tubing is divided into two dynamic sections, tubing above the plunger and tubing below the plunger. Mass of gas and liquid in each of the three sections are defined as state variables. Plunger position and velocity are defined as state variables to keep track of plunger movement during rise and fall stages. Thus, the state vector common to all modes of the lift process has following variables:



x = mga

mla

mgtt

mltt

mgtb

mltb

Xp

Vp

Ar

T

(5)

where, mg and ml represent mass of gas and liquid, respectively; subscripts a, tt, tb represent annulus, tubing above and tubing below the plunger, respectively; Xp is plunger location (from well bottom); Vp is the plunger velocity (upward velocity is positive); and Ar is a state variable introduced to determine arrival time. Recall that arrival time is the time taken by the plunger to reach the surface after the production valve is opened (i.e., total time the system spends in stages 4 and 5). Six binary states are defined to determine the mode of plunger lift. When the system is in lth mode (stage), ql = 1 and the remaining five elements of binary vector q are zero. For example, in plunger rise stage (mode 4), q4 = 1 and all other elements are 0. There are no continuous inputs (nu = 0) and a single binary input that determines whether the production valve is open (v = 1) or closed (v = 0).

Pl

Fout

A¯ r

T

(6)

In addition to the state variables and measurements, other intermediate variables are also computed in the model (see Fig. 4). These include height of liquid columns (L), flow rate from the reservoir (Fg,res , Fl,res ) and pressure at the top and bottom of the plunger (Ppt , Ppb ), bottom-hole pressure (Pwf ). Liquid is present only at the base of every sub-section of well. Hence, height of liquid columns in the annulus and tubing subsections is given by: ml(j) l A(j)

, with (j) = a, tb, or tt

(7)

where, l is density of liquid and ml is the mass of the liquid column in annulus or tubing. The reservoir flow is modeled using Inflow Performance Relationship (IPR) curve (Vogel, 1968). This can be replaced with complex reservoir models. The average compressibility (Z) and temperature (T) is used to calculate pressure in each sub-section of well. Using this method, pressures at the top and bottom of a gas column of height Lg are related by: Pbottom = Ptop e˛Lg

(8)

Mg g ˛= ZRT

(9)

where, Mg is the average molecular weight. In case of the tubing above plunger, with the production valve open, the gas flows from the production line at the rate of Fout . Maggard et al. (2000), showed that the above equation can be modified as: 2 2 2 Pbottom = Ptop e2˛Lg + b2 Fg,out . (e2˛Lg − 1)

b2 =

3.2. Model states and assumptions

Pt

8g

(10) (11)

2 ˛2 dt5

Gas properties used in the above calculations are dependent on the depth. Temperature increases linearly along the depth (x) of the well from the well-head: Tx = T0 (1 + Tx)

(12)

Starling-Ellington correlation is used to compute viscosity (Perry and Green, 2007), whereas the Breggs-Brill approximation for the Sterling-Katz relationship is used for calculating the compressibility factor, Z (Perry and Green, 2007). 3.3. Model equation for continuous states The balance equations for the nine state variables (x) and one binary input variable (v), are developed as six staged process. As described in Section 2, these stages include: plunger fall (gas), plunger fall (liquid), build-up, plunger rise, slug arrival, and after flow stages. The mass, momentum and force balance equations constitute flow-fields of the six modes of the HSM (corresponding to the six stages of plunger lift cycle) are presented next. 3.3.1. Reservoir and production flow rates Vogel’s Inflow Performance Relationship (IPR) (Vogel, 1968) is used in this work as the reservoir model to obtain the flow of fluids from the reservoir to the well-bore:



2 2 Fgres = gstd · Cres Pres − Pwf

Flres =

Fgres GLR

n

(13) (14)

A. Gupta et al. / Computers and Chemical Engineering 103 (2017) 81–90

85

Fig. 4. Plunger lift schematic showing (a) state variables (b) & (c) measurements and intermediate variables.

The reservoir flow is thus a function of pressure at well bottom, Pwf , and reservoir pressure, Pres . The flow from the reservoir splits into annulus and tubing, we represent flow into the tubing as Ftub and flow into the annulus as Fann . Note that due to large reservoir pressures, fluids only flow from reservoir to the well-bore through casing perforations; back-flow into the reservoir does not happen. Since Fres is the only flow into the entire well, we have:

where, Mg is the average molecular weight of gas. The bottom-hole flowing pressure Pwfa is calculated as pressure exerted by the gas as well as liquid hold-up at the bottom of annulus:

Fres = Fann + Ftub

(15)

dmgann = Fgann dt

(16)

dmlann = Flann dt

or equivalently: Fgann = Fgres − Fgtub Flann = Flres − Fltub Completion at the well-bottom ensures that flow takes place only from annulus to tubing, and not from tubing to annulus. The net flow into the tubing, Ftub happens so as to equalize the pressures at the bottom of the tubing and annulus. Since reservoir flow is known from Eqs. (13) and (14), flow into the annulus can be obtained from Eq. (16). Gas flows from the well tubing to the production line through an on-off valve. Standard valve equation is used to model the flow across the production valve:

Fgout =

⎧ ⎪ ⎨ ⎪ ⎩

g .Cv .Pt

2g Cv



ifPt ≥ 2Pl (17)

(Pt − Pl )Pl

ifPl < Pt < 2Pl

3.3.2. Mass balance on annulus The annulus contains gas column of height (H − La ) on top of liquid column of height La . Height of the liquid column is given by Eq. (7). Since annulus is closed at the top, the pressures at the surface and bottom of the gas column (i.e., top of the liquid column) are related as per Eq. (9). These two pressures are the surface casing pressure, Pc , and pressure at the top of the liquid column, Pcb . Pc is a function of the mass of gas in annulus section of the well. Thus: Zc RT Pc = mga Aa (H − La ) Mg

(18)

Pcb = Pc .e˛(H−La )

(19)

Pwfa = Pcb + La L g

(20)

Since the only flow into or out of the annulus is Fann , the balance equations on the annulus (in all modes) are given by: (21)

Note that depending on Eq. (16), Fann may be either positive or negative. For example, during plunger rise stage (mode represented by q4 = 1), the tubing depressurization due to flow to the production line may result in Ftub exceeding Fres and net out-flow from the annulus. 3.3.3. Mass balance on tubing sections with valve closed We first consider the case of tubing mass balance with valve closed (first three stages). During plunger fall and build-up stages, mass balance on the tubing sections follow similar reasoning as that of casing. The net mass in tubing is mgtt + mgtb and Ltb is the level of liquid at the bottom





Pt = mgtb + mgtt .

Zt RT At (H − Ltb ) Mg

(22)

Ptb = Pt .e˛(H−Ltb )

(23)

Pwft = Ptb + Ltb L g

(24)

With the production valve closed, net mass in the tubing changes only due to net inflow, Fgtub . Mass of gas in the two sections distributes based on the plunger location, i.e.: mgtb =



Xp mgtt + mgtb H

Differentiating mgtt + mgtb



with

(25) respect

to

time,

and

noting

that

= Fgtub , we get

Xp d mgtb = Fg + Fg,p H tub dt

(26)

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A. Gupta et al. / Computers and Chemical Engineering 103 (2017) 81–90

H − Xp d Fgtub − Fg,p mgtt = H dt

(27)

Using Eq. (25), the flow of gas past the falling plunger is given by:



Fg,p = mgtt + mgtb

Vp

(28)

H

The first term above partitions the tubing inflow into the two sections, whereas the second term captures the fact that mass of gas in the two tubing sections changes due to the downward motion of the plunger (with velocity Vp ). When the plunger reaches the liquid i.e., plunger fall (liquid), since there is no gas below the plunger and Fg,p is not relevant and mass of liquid varies with the plunger velocity,

to get the value of Ppt , Pt and Fgout . For tubing section below the plunger the pressure are given by following equations. Ppb = mgtb



Zt RT



At Xp − Ltb Mg

(38)

Ptb = Ppb .e˛(Xp −Ltb )

(39)

Pwft = Ptb + Ltb L g

(40)

b) Slug arrival rise stage During the slug arrival stage, gas mass above plunger is zero and liquid above the plunger exits through the production valve. Since the liquid slug rises with the same velocity as the plunger, liquid flow rate depends on the plunger velocity as:

mltb = Fltub − Vp l At

(29)

Flout = Vp At l

(41)

mltt = Vp l At

(30)

d mgtt = 0 dt

(42)

d m = −Flout dt ltt

(43)

As the plunger reaches the well bottoms (i.e. after-flow stage), the gas and liquid below the plunger are non-existent and liquid above the plunger increases due to flow from reservoir to tubing, Fltub . 3.3.4. Mass balance on tubing sections with valve open a Plunger rise stage

c) After-flow stage

During plunger rise, the plunger travels upwards along with the liquid slug on the account of pressure difference above and below the plunger plus slug (referred as plunger). The gas flows out of the tubing section above the plunger whereas the section below is getting pressurized by gas flowing into the tubing. Since plunger forms a barrier between the two sections, there is no flow of gases from bottom to top section of the tubing. Thus, mass balance for the two tubing sections is given by: d mgtt = −Fgout dt

(31)

d mgtb = Fgtub dt

(32)

During the plunger rise stage, liquids from reservoir and annulus flow into the tubing, thus changing the amount of liquid at the bottom of the tubing, mltb . In ideal plunger lift, the liquid above the plunger does not change. However, in case of liquid leakage past the rising plunger, loss due to leakage can be modeled as a first order process: Fleak = kleak mltt

(33)

d m = −Fleak dt ltt

(34)

d m = Fltub + Fleak dt ltb

(35) 



Note that in an ideal plunger lift, Fleak = 0, ml = 0 and ml = Fltub . tt

tb

Following equations calculate the pressure at the top and bottom of the plunger along with pressure at the top and bottom of the tubing. For tubing section above the plunger, the gas mass hold up and pressure profile is given by following nonlinear equations. Pt = mgtt 2 Ppt

=



Zt RT



2

+b

.Fg2out

During after-flow plunger is stationary in catcher or lubricator at well head. Thus, equations for tubing above the plunger are  redundant (i.e., mtt = 0). Tubing section below the plunger now has inflow through the well-bottom and outflow through the production valve. Pressures in the tubing are related to state variables as per the pressure-flow coupling of Eqs. (11) and (12). Thus, the outflow part of the model is similar to the gas flow above the plunger in the plunger rise stage. The overall balance equations are given by: d mgtb = Fgtub − Fgout dt

(44)

d m = Fltub dt ltb

(45)

The pressure profile is calculated using the flowing gas equation in the tubing section below the plunger. Pt = mgtb

(e

− 1)

(37)

where, Ppt is the pressure at plunger top, Fgout is the gas flowing out of tubing to line. The three equations, mass hold up (36), pressureflow relation (37) and flow Eq. (17) are to be solved simultaneously

(46) (47)

Pwft = Ptb + Ltb L g

(48)

3.3.5. Plunger force balance a) Plunger position and velocity during plunger fall (gas and liquid) Plunger fall velocity can be either modelled as the drag model or orifice model (Nadkrynechny et al., 2013). A detailed orifice flow based model shows plunger quickly reaches the terminal velocity and falls at constant velocity thereafter till it reaches the liquid at bottom of well (Nadkrynechny et al., 2013). Based on these results, the velocity of plunger fall is given by

Vp = 2.˛(H−Xp −Ltt )

Zt RT At (H − Ltb ) Mg

2 Ptb = Pt2 .e2.˛(H−Ltb ) + b2 . (e2.˛(H−Ltb ) − 1)

(36)

At H − Xp − Ltt Mg

Pt2 .e2.˛(H−Xp −Ltt )

The balance equations for tubing sections below the plunger remain same as before.

⎧ C ⎪ ⎨ √g (in stage 1) ⎪ ⎩ √C (in stage 2) l



Aa where, C = Cd . . At

2Mp .g At

(49)

(50)

A. Gupta et al. / Computers and Chemical Engineering 103 (2017) 81–90

87

Table 1 State evolution functions fl used in the plunger lift model as HSM. dx/dt is represented as x . The expressions for individual terms in the table below are given in Section 3.3. Stage

Plunger fall (gas)

Plunger fall (liquid)

Build-up

Plunger rise

Slug arrival

After-flow

Mode m’g a m’l

q1 = 1 Fgann Flann

q2 = 1 Fgann Flann

q3 = 1 Fgann Flann

q4 = 1 Fgann Flann

q5 = 1 Fgann Flann

q6 = 1 Fgann Flann

m’gtt m’l

H−Xp H

0

Fgtub Vp l At

Fgtub Fltub

−Fgout −Fleak

0 −Flout

0 0

Fgtub Fltub + Fleak

Fgtub Fltub

Fgtub − Fgout Fltub

a

tt

Fgtub − Fg,p

X

m’g tb m’l

p F + Fg,p H gtub Fltub

0 Fltub − Vp l At

0 0

Xp’

− √C

− √C

0

Vp

Vp

0

0 0

0 0

0 0

V˙ p 1

0 1

0 0

tb

l

g

Vp’ A’r

Plunger fall dynamics are typically less important than plunger rise, because the flow of fluids into the well-bore are not affected by plunger fall dynamics. Typical values of the constant for various plunger types are reported in (Nadkrynechny et al., 2013), which uses data from smart plungers. b) Plunger position and velocity during plunger rise





Ppb − Ppt − Pfric At mp + mltt

−g

(51)

where, is the frictional force between the liquid slug of length Ls and the tubing wall is: Pfric =

1  V 2f 2 l p

L  s

(52)

dt

  7 0.9  1 ε  = −4log10 0.27 + dt

f

j(Current Mode)

Transition function sj (·)

1–Plunger fall (gas)

Xp −

2–Plunger fall (liquid) 3–Build up

Xp 1 − v

Re

In the above, Re is the Reynolds number and ε is the tubing roughness.

ml

tb

l At

H − Xp +

4–Plunger rise

The dynamic model for plunger rise stage is based on the one developed by Lea (1982), which uses a force balance across the plunger. The pressure just above (Ppt ) and just below the plunger (Ppb ) are calculated using tubing pressure equations. The plunger force balance yields: V˙ p =

Table 2 Event generating functions, sj (x, v). Only the condition sj (·) = 0 is tested when the model is in jth mode.

H − Xp

5–Slug arrival 6–After flow

ml

tt



l At

v

3.4. Measurement equations For the measurement function y = g (x, q, u, v), given below, calculates casing pressure, tubing pressure, line pressure, flow rate into the sales line, and plunger arrival time.



y = Pc

Pt

Pl

F

Ar

T

(54)

Casing pressure is calculated using Eq. (18). The tubing pressure, Pt uses Eq. (22) for plunger fall (gas and liquid) and build up modes, Eq. (36) for plunger rise mode, Eq. (38) for slug arrival mode and Eq. (46) for after-flow mode. The line pressure Pl is a measured disturbance and is input to the model. Flow rate of gas, F, is calculated using Eq. (17). Plunger arrival time, which is the time spent in plunger rise and slug arrival modes for each cycle, is calculated using Eq. (53). Then, after-flow stage begins and cycle repeats when the valve is closed. 3.5. Mode transition criteria

3.3.6. Arrival time Arrival time is the time taken for plunger to reach the surface after the production valve is opened. The state is reset to Ar = 0 when the system enters stage 6 (After-flow stage).

 dAr = dt

1 when q4 = 1 or q5 = 1

(53)

0 otherwise

Thus, Eq. (53) simply calculates the time spent in modes 4 and 5.

3.3.7. Summary of state evolution equations Table 1 summarizes the state evolution function as per Eq. (2) for the six modes of the plunger lift model. The differential equations are derived for each stage using Eq. (13) to Eq. (53). The corresponding binary variables q (t) are also shown. Note that exactly one element of the vector q (t) is positive. Six columns of Table 1 give the six functions, fl (·).

The switching between the modes or stages is determined by event generating function sj (x, q, u, v), as summarized in Table 2. The six modes in plunger lift occur in succession, i.e. 1 → 2. . . → 6 → 1. Therefore, only the condition sl (·) = 0 is tested th in the lth mode and it signals transition to (l + 1) mode (except mode 6 transitions to mode 1). The following physical events in plunger lift operation determine mode transitions in the plunger lift model: • Plunger fall (gas) stage starts when the production valve is closed, i.e., when the binary input v switches to 0. • Plunger fall (liquid) stage starts when the plunger reaches liquid level in tubing • Build-up stage starts when the plunger reaches well-bottom • Plunger rise stage starts when the production valve is opened • Slug arrival stage starts when the plunger + liquid-slug reaches the surface • After-flow stage starts when plunger reaches the surface

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Table 3 Well parameters for simulation. Well parameters Depth Tubing ID Tubing OD Casing ID Valve, Cv

Plunger parameters 10,000 ft 1.995 in 2.125 in 4.850 in 1.64 e-7

Reservoir (IPR) PRes CRes n GLR

Mass Diameter Fall, Cd Rise, ε

3.62 kg 1.9 in 0.1019 0.0457

Fluid properties 4.3 MPa 2.58 e-15 1 2.52

␳l Twh ␮l Mg

1.06 288 K 4e–4 N/m 0.0184

Mathematical abstraction of these conditions in the form of event generation functions is summarized in Table 2. Note that as written in Table 2, sl (x) is positive in the lth mode. Transition functions, ˇj (·) determine how the state variables are set after a transition. For most of the cases, the state variables take the same values as before, i.e.:



x t + = x (t) except the final state, Ar , which keeps track of arrival time, is reset to 0 at the beginning of each cycle. Likewise, when the model transitions from mode m to next mode, the corresponding discrete variable qm (t + ) = 0, whereas qm+1 (t + ) = 1 (except in mode 6, wherein, q1 (t + ) = 1). This completes the model formulation of plunger lift in HSM framework. The model developed is modular and flexible to include more complex reservoir and plunger dynamics. The framework is extendible to other periodic processes in upstream oil and gas as well. 4. Results and discussions ®

All the simulations reported here are performed using MATLAB ® − R2013b on an Intel core – i5 1.90 GHz system with 8 GB RAM ® and 64-bit Windows 7 as operating system. Time required for simulating a single cycle is about 1–2 mins and the time required for simulating one day of plunger lift operation is about 6–10 mins. The simulation time includes the model initialization, solving the HSM model using MATLAB’s ode45 and plotting the results. The opportunity of using this model for online control and monitoring applications is apparent due to the quick simulation time. Two simulation scenarios are presented, representative of a typical shale gas well in North America: First, a single cycle is discussed which captures the key trends and signature of a plunger lift cycle; followed by a full day operation showing multiple cycles in continuum. The former is run without any disturbances to analyze an ideal plunger cycle. Random walk type input disturbance is added in the latter to highlight the well behavior when trends from previous cycle affect the response of next cycles. Table 3 lists the value of various model parameters. 4.1. Model simulation – single cycle Fig. 5 shows the simulated data from a single plunger lift cycle. A sampling time of 10 s is selected for plotting the data. A constant line pressure is chosen to showcase the pressure and flow profile signatures. The cycle starts in plunger fall (gas) stage (q1 = 1), with valve just being closed and completes at end of stage-6 when valve is about to be closed. The total simulation time is 80 min, in which for the first 60 min the valve remains closed and is kept open for the remaining 20 min. The plunger takes about 13 min to reach the well-bottom. Thus, transition from stage-2 to

stage-3 occurs around 13 min which is not visible in pressure or flow profile (Fig. 5); similar to a real well trend. During the start of the cycle, when the valve is just closed, the tubing pressure rises quickly and after some time, (about 4 min) the bottom-hole pressure beneath casing and tubing nearly equalizes and the pressure increases in both casing and tubing on account of reservoir flow. Thus, in the first four minutes, Ftub exceeds Fres so that there is a net flow out of the annulus according to Eq. (16). The casing pressure increases faster than that of tubing during shut-in period. The difference between casing and tubing pressure (during well shut-in) represents liquid hold-up in the tubing section. This is also evident in Fig. 6 as the liquid height rises in tubing in proportion to the difference between casing and tubing pressures. Unlike the models available in the literature, our model accurately capture liquid hold-up in the annulus. The liquid hold-up in annulus is counterproductive to plunger lift operation and a good estimate can be used to calculate optimal operating conditions. This provides important insight into the well dynamics since the model reliably predicts the well-bottom conditions that govern the overall production characteristics. At 60 min, the production valve is opened and gas starts to flow from tubing to production line. This leads to rapid fall in tubing pressure allowing enough pressure differential across plungerslug to start moving upwards. During plunger rise (60–72.83 min), instantaneous plunger velocity is calculated by the force balance across the plunger and is shown in Fig. 6. The plunger quickly accelerates to a velocity of about 2.5 m/s in first 30 s and accelerates relatively slowly till the plunger reaches well head. The velocity profile is a typical representation of the plunger arrival velocity, however, it can deviate and plunger can decelerate as well during latter half of the arrival stage, depending of slug height and differential pressure across the plunger. This model captures the plunger velocity profile, which is important for analyzing plunger cycle performance, without any expensive instrumentation. High instantaneous arrival velocity (at well head) can damage the wellhead equipment and reduces plunger life, whereas low (nearing zero) instantaneous arrival velocity can often lead to missed arrival and hence loss of production. During the plunger rise and slug arrival stages, fluids from annulus flow rapidly to the tubing which is marked by drop in casing pressure (Fig. 5) and liquid height in annulus section. The liquid hold-up in tubing increases rapidly during the initial part of plunger rise stage (Fig. 6, 60–64 min), as liquid move from both annulus and reservoir to the tubing. When the plunger-slug arrives at the well head, slug arrival stage beings. Slug arrival is shortest stage in the plunger cycle, lasting for less than 10 s in this particular case. A peak in the flow rate and tubing pressure (Fig. 5) at 72.83 min is a characteristic signature of slug arrival stage. Liquid holdup in the tubing also falls (Fig. 6) because the liquid slug is produced at the surface. The distinct slug arrival signature marked in well data is often used by field engineers to analyze plunger lift performance. Since this model has detailed treatment of well dynamics, coupled with reservoir behavior and production valve characteristics, it is able to reproduce this signature reliably. The importance of gas and liquid dynamics on plunger performance is highlighted by this model. At completion of slug arrival stage, plunger reaches the well head and the after-flow stage begins. During after-flow, gas continues to flow from tubing to production line. The production valve is then closed at 80 min and the cycle repeats. 4.2. Model simulation – one day operation In order to access the systems response to a given control setting (production valve open and close conditions), the cyclic process must maintain a state continuum from cycle to cycle, enabling the simulation for longer time periods. The model is simulated for one

A. Gupta et al. / Computers and Chemical Engineering 103 (2017) 81–90

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Fig. 5. Pressure and Flow profile of single plunger lift cycle.

Fig. 6. Arrival velocity and liquid loading profile in a single plunger lift cycle. Liquid loading in the tubing includes both the liquid at well-bottom and slug on top of the plunger.

Fig. 7. Pressure and Flow data for one day simulation with significant variation in production line pressure.

day with 60 min of valve close and 20 min of valve open per cycle, resulting in 18 complete cycles. The line pressure variation disturbance is generated using a filtered variable step random walk process. Fig. 7 shows the pressure and flow signals from the simulation. The solid (green) line closest to the abscissa of the top-panel is the line pressure (measured disturbance), which increases by over 40% in the single-day operation. Such pressure surges are common onfield. The casing and bottom-hole pressure increases with increase in line pressure, showing the effect of higher back pressure on the

well. The constant timer operation of the well does not correct for this disturbance. Consequently, the net gas production (area under the flow-rate curves) falls in later cycles. Liquid height in tubing and annulus are plotted in Fig. 8 (bottom). The liquid loading is minimum when the plunger just arrives at the surface, producing liquid slug. The dotted (magenta) line is included to show an increasing trend in liquid loading at the well bottom. The effect of liquid hold-up is that the bottom-hole pressure increases and reservoir flow to the well reduces, and thereby reducing the net production. Fig. 8, also shows reduction in average arrival velocity

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A. Gupta et al. / Computers and Chemical Engineering 103 (2017) 81–90

Fig. 8. Velocity (top) and liquid hold-up (bottom) for one day simulation with surge in line pressure. The dotted (magenta) line is a linear fit to the liquid height at plunger arrival during multiple cycles.

which is an effect of increased slug size. This is consistent with our expectation, as with increase in line pressure, production drops and plunger arrival velocity reduces. Furthermore, the velocity profile changes and model is able to capture the effect of instantaneous line pressure, slug size, and casing pressure on the plunger arrival velocity. This model provides much richer information on plunger lift operation in presence of external disturbances. 5. Conclusions A hybrid state model (HSM) for a plunger lift system was developed in this work. The model captures all the six stages of plunger lift operation as six modes in the HSM framework. Evolution of continuous states is challenging because of plunger motion through the entire 10,000 feet of well-depth, coupled with pressure and flow dynamics above and below the plunger. This is further complicated by various binary events in the system, owing to valve opening/closing and plunger motion in the system. This work demonstrated the ability of the proposed model to simulate the full system operation with multiple cycles to evaluate the performance of plunger lift. It reproduces typically observed signatures of plunger lift system quantitatively, and captures uncertainties in reservoir characteristics, well-bore and production line through various model parameters and disturbances. The model can be tuned using standard surface measurements and reservoir performance parameters. Exemplary simulation for a single plunger lift cycle under ideal conditions show that the model is able to capture all the essential features of plunger lift cycle. Stable operation with optimal production rate can be achieved by managing the liquids at the well-bottom. Disturbances in line pressure were introduced to analyze the well response in presence of external disturbances typically observed in the field. The well behavior under model uncertainties and line pressure surge shows all the qualitative signatures similar to that of an on-field plunger lifted well. Trends of plunger velocity, well-bottom pressure and liquid levels in the well were predicted without the need of expensive downhole equipment. This work provides a comprehensive and standardized plunger-lift model for simulation, which can be employed for control, optimization and wellhead monitoring solutions. References Avery, D.J., Evans, R.D., 1988. Design optimization of plunger lift systems. SPE Int. Meet. Petrol. Eng. SPE-17585, 367–380. Baruzzi, J.O.A., Alhanati, F.J.S., 1995. Optimum plunger lift operation. SPE Production Operations Symposium, SPE-29455-MS.

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