May 26, 2011 - Modeling, Simulation, and Optimal Control of Oil Production under Gas ... because one needs access to reservoir simulator code and ...
SPE 143520 Modeling, Simulation, and Optimal Control of Oil Production under Gas Coning Conditions Agus Hasan, SPE, NTNU; Bjarne Foss, SPE, NTNU; Svein Ivar Sagatun, SPE, Statoil ASA; Bjørn Peter Tjøstheim, Statoil ASA; Atle Svandal, Statoil ASA; Cato Hatland, Statoil ASA
Copyright 2011, Society of Petroleum Engineers This paper was prepared for presentation at the SPE EUROPEC/EAGE Annual Conference and Exhibition held in Vienna, Austria, 23–26 May 2011. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.
Abstract Gas coning is a tendency of the gas to impel the oil downward in an inverse cone contour toward the well perforations. Once the gas reaches the well, gas production will dominate the well flow and the oil production will hence significantly decrease. From an economical and operational standpoint this condition is undesirable since the gas price is much lower than the oil price, and the gas handling capacity often is a constraint. Therefore, there is an incentive to maximize oil production up until gas breakthrough. In this paper, the gas coning process in a gas oil reservoir completed with a single horizontal well is analytically modeled, simulated, and analyzed applying a nonlinear control approach. The model which describes the interaction between the well and the reservoir may be cast into a boundary control problem of the porous media equation with two boundary conditions; a Neumann boundary condition describing no flow at the outer boundary of the reservoir, and a nonlinear boundary condition describing the well production rate. A well rate controller for the boundary control problem is designed using the Lyapunov method. The controller holds some formal performance guarantees and requires information on the gas oil contact at the well heel only. Further, the controller has a tuning parameter which can be used to maximize a suitable performance measure. The controller is evaluated using a detailed ECLIPSE simulator of a gas coning reservoir. Simulation results show significant improvement of production profit of the proposed method compared to a conventional method which usually uses a constant rate up until gas breakthrough. Introduction Optimization of the trade-off between oil and gas production is an important issue in reservoir management. The use of secondary recovery techniques such as gas lift and waterflooding, and EOR techniques such as surfactant injection has been proven successful to increase the oil production significantly. Those techniques are now supported by the growing application of smart well technologies. A smart well is usually equipped with several valves that can be regulated over the time of production. Questions regarding how to operate these valves can be partially answered using optimal control theory, in particular when it is combined with the adjoint method; see , , and . Adjoint based optimization can also be used to determine optimal well placement  and for history matching . Optimal control theory combined with data assimilation form a closed-loop reservoir management. A comprehensive summary of the closed loop reservoir management concept may be found in . Although providing solutions in a relatively short time in an efficient way, the adjoint method is difficult to implement. This is because one needs access to reservoir simulator code and implements the algorithm there. An alternative way which can be done is by creating a mathematical model which is simpler but can be used to explain the same physical process. This model will then be referred to as a proxy model and serves as a representative of a complex model which is usually contained in a reservoir simulator. A proxy model may be derived from the basic principle of physics such as mass conservation and Darcy law. Since the proxy model is simpler than the high-fidelity model, a proxy model is certainly easier than working with the actual one. Therefore, instead of using the full model, in this paper a proxy model will be used for design and analysis purposes. Further, the results will be tested using a complex reservoir dynamics which is usually contained in a tested and well known reservoir
simulator, e.g. ECLIPSE, MoReS, GPRS. The model presented here is where a reservoir, which is equipped with a horizontal production well and located in the middle of thin oil rim, tends to experience gas coning (Fig. 1).
Figure 1. A gas-oil reservoir equipped with a horizontal well. The oil and the gas will be perforated along the horizontal well from the heel toward the toe. Due to pressure difference, the gas coning will be occurs first time in the well heel.
Typically, a well is produced with a constant oil rate with constant gas oil ratio (GOR) in the subcritical phase (before gas breakthrough). The presence of gas coning in the production wells may reduce the oil production drastically since the mobility of the gas is much higher than the mobility of the oil, thus inside the production well the gas flow will be more dominant than the oil flow. The sharp decline in the oil rate will be followed by a sharp increase in the well head pressure. As oil price is higher than the gas price, this condition is not desirable. Moreover, the gas handling capacity is also often a constraint. Therefore, there is an incentive to produce such wells in their subcritical phase, i.e. before the GOC reaches the horizontal well perforation. The gas coning model was first proposed by Muskat in 1937 , in his book the model equation was derived from the thermodynamic relation under isothermal expansion. Using a different approach, Konieczeck arrives at the same equation . In addition, in his article he introduced the boundary conditions at the outer boundary of the well and at the well heel. Mjaavatten et. al. , based on Konieczek’s work, developed a mathematical model that could predict the gas coning behaviour. Despite a simple model structure and short computation time, the accuracy of the predictions has been good. Further, the model forms the basis of the GORM (Gas Oil Ratio Model) computer program that has been used for production planning and optimization at the Troll field since 2003. The use of control theory in the gas coning problem using a linear proxy model was first put forward by Sagatun in . In his article, he formulates an optimal oil production problem as a boundary control problem. His work became the basis for further studies by Hasan et. al.  where the controller was designed using the backstepping method. Other approaches on gas coning control may be found in  and . The first section of this paper will be used to show how the proxy model can be derived from the basic laws of physics. A control theory approach called Lyapunov Method, which is used to designing an optimal rate controller, will be presented in the second section of this paper. Optimal in this sense is to provide a high oil recovery in the fastest time. This method was proposed in  and can be applied for nonlinear gas coning model. A numerical example will be done using ECLIPSE simulator. The proxy model which has been derived may be used to represent a more complex dynamical model, in this case a model which is used in ECLIPSE simulator. The last section will be closed by conclusions. Reservoir Model Consider Fig. 1, here oil and the gas are trapped by overlying rock formations with zero permeability. Since the density of the gas is lower than the density of oil, the gas remains on the top of the reservoir. In order to produce the oil, a reservoir is completed with a single horizontal production wells located in the middle of the reservoir. The horizontal part of the well in Fig. 1 is perforated between the well heel and the well toe to allow inflow of the reservoir fluid. For the oil to flow inside the horizontal well, the pressure at the well heel should be lower than the pressure at the well toe. This implies the pressure drawdown between the reservoir and the well heel is bigger than the pressure drawdown between the reservoir and the well toe. Hence, the GOC at the well heel will be lower than towards the well toe, i.e. the first gas breakthrough will occur at the well heel. Because of this reason, the analysis may be reduced to a problem that only involves the reservoir dynamics at the well heel. Thus, one may consider Fig. 2.
Figure 2. GOC at the well heel. Due to symmetry one half of the reservoir is displayed. The well is placed at
The following derivation may be found in , , and . Lets define denotes the difference between them as Δρ =
ρ g − ρo .
pressure gradient in the oil column is given by
as the oil density and
z = 0.
as the gas density, and
Assuming hydrostatic equilibrium in the vertical direction, the
∂p ∂h = −Δρ g ∂x ∂x
where g is the gravity constant. Here p is defined as the oil pressure and h ( x, t ) is defined as the gas oil contact. Assuming laminar oil flow inside the reservoir, the velocity is given by the following Darcy’s law
K ∂p μ ∂x
μ denotes the oil viscosity. Further, assuming isotropic conditions with respect to porosity and permeability, the permeability tensor K reduces to the horizontal permeability k h = k . Substituting Here K denotes the permeability tensor while
(1.1) into (1.2), yields
kg Δρ ∂h μ ∂x
Let φ be the effective porosity (i.e., the volume fraction occupied by the movable oil). The mass rate change per unit area is given by
∂ ∂ ( hφ ) = (δ h ) ∂t ∂x
Assuming constant in porosity, permeability, density, and viscosity, (1.3) and (1.4) yields
∂h kg Δρ ∂ ⎛ ∂h ⎞ = h ∂t μφ ∂x ⎜⎝ ∂x ⎟⎠
(1.5) may be considered as the porous media equation and has been used to model the ground water movement . This type of equation also has been appeared in the early work of Muskat to model the single phase of gas flow through a homogeneous porous media . Next, assuming no flow boundary at the external boundary of the reservoir, yields the Neumann boundary condition
∂h ( 0, t ) = 0 ∂x
Accounting for the half well symmetry, the well production rate at the well heel is given by
u (t ) =
h ( L, t )
∂h ( L, t ) ∂x
Further, (1.5)-(1.7) can be made dimensionless by the following variable transformations
h x kg Δρ μ , x= , t =t , u =u L L 2kg Δρ L μφ L
This yields a set of partial differential equation with two boundary conditions
∂h ∂ ∂h ( x, t ) = ⎛⎜ h ( x, t ) ( x, t ) ⎞⎟ ∂t ∂x ⎝ ∂x ⎠ ∂h ( 0, t ) = 0 ∂t ∂h h (1, t ) (1, t ) = u ( t ) ∂t
Note that for brevity the bars denoting dimensionless parameters are omitted. In (1.8), the well rate u ( t ) usually is chosen as the control variable. Hence (1.8) is a boundary control problem of the porous media equation. This problem was first put forward by Sagatun , however, the analysis is still limited to the linear model. Well Model The well is a type of smart well and modeled as in Fig. 3. The downhole control valves enable the splitting up of the well into a number of segments such that they can be controlled individually.
Figure 3. Smart horizontal well, consisting of three segments, each having a down-hole control valve .
The frictional pressure drop over a length L of tubing is calculated using the following formula
2 fL ρ v 2 d
where f denotes the Fanning’s friction factor. d is the tubing inner diameter while v is the fluid velocity. The Fanning friction factor depends on the Reynold’s number. For laminar flow
while for turbulent flow
⎛ 6.9 ⎛ e ⎞10 9 ⎞ 1 ⎟ = − 3.6 log10 ⎜ + ⎜ Re ⎜⎝ 3.7d ⎟⎠ ⎟ f ⎝ ⎠
Here e denotes the absolute roughness of the tubing. Well Rate Control Design In the previous section, it was mentioned that there is an incentive to produce as much oil as possible in their subcritical phase, i.e. before the GOC reaches the horizontal well perforation. Thus, the question on how to produce maximal oil in a fastest way may be translated into a question on how to construct the control input u ( t ) such that the oil thickness h ( x, t ) → 0 in asymptotically or exponentially. In other words, the controller is designed such that it either exponentially or asympotically
stabilizes the system to the origin h ( x, t ) ≡ 0 as t increases. An exponentially stable property is useful when depletion using the control u ( t ) in (1.8) is combined with more advanced production strategies involving injection. Moreover, it is necessary to have a stability property of all the components of a larger automation system comprising of automatic control of the reservoir, wells, flow lines and production system . Further, the controller will include a parameter which can be tuned to maximize a suitable measure such as oil recovery before gas breakthrough. Such controllers may be found using the Lyapunov method .
Definition 1. A Lyapunov function is a scalar function V ( t ) defined on a region D that is continuous, positive
definite V ( t ) > 0, for every t ≠ 0 , and has continuous first-order derivatives at every point of D . •
The existence of a Lyapunov function for which V ( x ) is negative definite on some region D containing the origin guarantees the asymptotic stability of the zero solution of the system. Exponential stability is a stricter form of asymptotic stability since it requires exponential decay towards the origin. More information on this method may be found in . Now consider the following Lyapunov function 1
1 V ( t ) = ∫ h 2 ( x , t ) dx 20
Using integration by parts and (1.8), the first derivative with respect to time t is give by • ⎛ ∂h ⎞ V ( t ) = u ( t ) h (1, t ) − ∫ h ( x, t ) ⎜ ( x, t ) ⎟ dx ⎝ ∂x ⎠ 0 2
Let’s denote the well radial as hw , then
min h ( x, t ) = hw
Substituting (1.14) to (1.13), yields
⎛ ∂h ⎞ V ( t ) ≤ u ( t ) h (1, t ) − min h ( x, t ) ∫ ⎜ ( x, t ) ⎟ dx x∈[0,1] ∂x ⎠ 0⎝
u ( t ) = −λ h (1, t )
λ > 0 . Using Poincare’s inequality , this yields 1 • 1 ⎛ hw ⎞ 2 ⎛h ⎞ V ( t ) ≤ − min ⎜ , λ ⎟ ∫ h ( x, t ) dx = − min ⎜ w , λ ⎟ V ( t ) 2 ⎝ 2 ⎠0 ⎝ 2 ⎠
which implies the Lyapunov function in bounded by
V (t ) = V (0) e
⎛h ⎞ − min ⎜ w , λ ⎟t ⎝ 2 ⎠
Hence, equilibrium h ( x, t ) ≡ 0 is globally exponentially stable. Remark that the rate controller (1.16) is in an explicit form and only requires measurement at the well heel. This is an advantage in terms of practical use. The height of the oil column at the well heel h (1, t ) may be estimated from seismic data and pressure measurement. The control gain
may be seen as a
tuning parameter which can be used to find an optimal solution to the optimization problem. In the numerical example below, the optimization problem is to maximize the oil recovery over a time horizon with different production constraints. There will be trade-off between the lengths of production time with the amount of oil obtained. Therefore, the term of optimal in this sense is to provide a high oil recovery in a short production time. Numerical Example The reservoir has a rectangular shape and consists of gas, oil, and water (Fig. 4). The oil layer located in the middle of the reservoir and has a thickness of 25 (m). The reservoir has the size of 2040 (m) x 1001 (m) x 225 (m) and was divided into 110 x 67 x 26 grid blocks (Fig. 5). Porosity is 0.25 and assumed homogeneous in all parts of the reservoir. The permeability on the oil and the gas layer is 300 (mD) - 1000 (mD), while in the water layer is 0.0001 (mD). Permeability in the water layer is assumed very low to avoid the effects of water coning which is not a topic of interest in this paper. The initial pressure was set to be 140 (Bara) at the water oil contact. The capillary pressure at the gas oil contact and the water oil contact are ignored. A horizontal well was placed in the middle of the reservoir. In order to capture the dynamics of the gas coning, the grid around the wells is made refiner using geometric progression rules. The pressure drop along the horizontal well is calculated using (1.9). The simulation runs include 120 days. In ECLIPSE simulation, RESV is used as the control mode for production, which means the production is controlled by the reservoir fluid volume target. Fig. 6 shows the depletion of gas oil contact from the beginning until the occurrence of gas breakthrough.
The proposed control rate is calculated using (1.16). The control gain λ is found using a simple search optimization method. The optimal control rate will then be compared with six different constant control rates ranging from the minimum to the maximum allowable rate, which is the usual strategy in operation. The maximum and minimum allowable rates are present due to operational limitation. For this simulation the minimum allowable limit is set to be 400 sm3/day while the maximum is set to be 2000 sm3/day. The simulation will be done by using four different constraints. The first constraint is the production time, where production will be stopped when the well began to produce the gas (gas breakthrough). The second, third, and fourth constraints are related to the total gas produced, where the gas processing capacity often has a certain limit. Fig. 7 shows gas oil ratio for different control strategies. The highest constant rate (2000 sm3/day) gives the fastest gas breakthrough while the lowest constant rate (400 sm3/day) gives the longest gas breakthrough, as expected. One may observes the relation between production rate and gas breakthrough time is nonlinear. A very small production rate (e.g. 1 sm3/day) will able to drain the reservoir in a very long period without the well experiences gas breakthrough. However, this is definitely not a good choice in an operational standpoint. Fig. 8 shows oil rates from different control strategies. The black color denotes the rate from (1.16). The optimal λ for this simulation is found to be 0.75. Table 1 shows the gas, oil, and water cummulative productions with gas breakthrough time as the production constraint. It can be observed that in a relatively short time, the optimal production rate (1.16) gives a higher oil recovery than using 800, 1000, 1500, and 2000 sm3/day constant rates. The optimal rate also gives the same amount oil recovery of 14.9 Ksm3 compared with constant rate of 600 sm3/day. However, it gives a lower oil recovery than using constant rate of 400 sm3/day. This is because the lowest constant production rate (400 sm3/day) would require a longer time period to reach the gas breakthrough, which causes longer production time as well. In this case, the optimal control rate requires only 18 days while the low constant rate requires 49 days. Table 2 shows the gas, oil, and water cummulative productions with the total gas production of 1000 Ksm3 as the production constraint. Here, the optimal production rate may be seen as the best option. Although the value is equal to the lowest constant production rate, it provides the largest oil recovery in a relatively short time. Moreover, the optimal control rate may be compared with the constant rate of 1000 sm3/day of which both of them requires the same production time. But it is clear that the optimal control rate gives 2.2 % more oil than the constant rate. Table 3 shows the gas, oil, and water cummulative productions with the total gas production of 2000 Ksm3 as the production constraint. Here, like in the previous case, the optimal production rate may be seen as the best options. It gives a higher oil recovery in a shorter production time. One may compare between the optimal production rate with the constant rate of 800 sm3/day since the have a similar production time. It can be observed that the optimal production rate is still better eventhough the difference is only 0.8 %. Table 4 shows the gas, oil, and water cummulative productions with the total gas production of 3000 Ksm3 as the production constraint. It can be observed, as the constraint increases, the difference between the optimal production rate and the constant rate becomes narrow. This happens because the production time had far exceeded the gas breakthrough time, while the rate control is made only for the subcritical phase. Therefore, the optimal production rate starts equaled by the constant rates.
Fluid Volume Rate (sm3/day) Constant ‐ 400 Constant ‐ 600 Constant ‐ 800 Constant ‐ 1000 Constant ‐ 1500 Constant ‐ 2000 Control Rate (1.16)
Production Time (days) 49 32 24 19 17 16 18
Gas Breakthrough Time (days) 49 32 24 19 17 16 18
Cummulative Production (Ksm3) Oil Water Gas 15.3 2.3 671 14.9 2.4 641 14.8 2.4 648 14.5 2.5 670 14.2 2.5 657 13.4 2.4 626 14.9 2.6 679
Table 1. Cummulative oil, gas, and water production until gas breakthrough.
Production Fluid Volume Rate Time (sm3/day) (days) Constant ‐ 400 63 Constant ‐ 600 42 Constant ‐ 800 32 Constant ‐ 1000 26 Constant ‐ 1500 22 Constant ‐ 2000 22 Control Rate (1.16) 26
Gas Breakthrough Time (days) 49 32 24 19 17 16 18
Cummulative Production (Ksm3) Oil Water Gas 18.8 2.7 1000 18.7 2.8 1000 18.8 2.9 1000 18.4 3.0 1000 18.2 3.0 1000 18.2 3.0 1000 18.8 3.0 1000
Table 2. Cummulative oil, gas, and water production until the total of gas production reach 1000 Ksm3 i.e. the total of gas production is as the production constraint.
Production Fluid Volume Rate Time (sm3/day) (days) Constant ‐ 400 94 Constant ‐ 600 63 Constant ‐ 800 47 Constant ‐ 1000 38 Constant ‐ 1500 30 Constant ‐ 2000 28 Control Rate (1.16) 46
Gas Breakthrough Time (days) 49 32 24 19 17 16 18
Cummulative Production (Ksm3) Oil Water Gas 24.1 3.2 2000 23.9 3.3 2000 23.7 3.5 2000 23.6 3.6 2000 23.3 3.6 2000 22.8 3.6 2000 23.9 3.6 2000
Table 3. Cummulative oil, gas, and water production until the total of gas production reach 2000 Ksm3 i.e. the total of gas production is as the production constraint.
Production Fluid Volume Rate Time (sm3/day) (days) Constant ‐ 400 121 Constant ‐ 600 81 Constant ‐ 800 61 Constant ‐ 1000 48 Constant ‐ 1500 37 Constant ‐ 2000 34 Control Rate (1.16) 66
Gas Breakthrough Time (days) 49 32 24 19 17 16 18
Cummulative Production (Ksm3) Oil Water Gas 27.6 3.5 3000 27.1 3.7 3000 27.0 3.8 3000 26.8 3.9 3000 26.5 4.0 3000 26.6 4.0 3000 27.2 3.9 3000
Table 4. Cummulative oil, gas, and water production until the total of gas production reach 3000 Ksm3 i.e. the total of gas production is as the production constraint.
Conclusions and Future Works This paper suggests an optimal rate controller for nonlinear PDEs associated with the gas coning flow inside the reservoir using the Lyapunov method. The rate controller is written explicitly and only requires measurement at the boundary of the reservoir. The controllers come with a control gain which can be used as a tuning parameter in an optimal control framework. For different production constraints, the simulation applying a fine-gridded ECIPLSE model shows that the proposed rate controller gives a better result in the total oil production compared with constant rates, especially in the subcritical phase. The optimal control rate can be use as a feedback controller in a closed loop system. The next challenge may be how to estimate the height of oil column at the well heel which is needed to calculate the control rate. Since seismic data is fairly expensive, the height needs to be estimated by combining estimation theory (filter method) and reservoir simulation.
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Figure 4. The reservoir consists of gas oil and water. Gas is located at the top while oil is located at the middle of the reservoir. Water layer is located at the bottom of the reservoir with very low permeability.
Figure 5. The reservoir was divided into 110 x 67 x 26.
(d) Figure 6. Simulation of gas coning using ECLIPSE. (a) shows the initial gas oil contact while (d) shows gas oil contact after 100 days. The snapshot was taken at the well heel (I=13).
Figure 7. Gas oil ratio for different control strategies. Red (constant 400 sm3/day), green (constant 600 sm3/day), blue (constant 800 sm3/day), cyan (constant 1000 sm3/day), purple (constant 1500 sm3/day), yellow (constant 2000 sm3/day), black (control rate (1.16)).
Figure 8. Oil rate for different control strategies. Red (constant 400 sm3/day), green (constant 600 sm3/day), blue (constant 800 sm3/day), cyan (constant 1000 sm3/day), purple (constant 1500 sm3/day), yellow (constant 2000 sm3/day), black (control rate (1.16)).