Exact Boundary Controller Design for a Kind of Enhanced Oil Recovery Models

and Applied Analysis 3 Assumption 2. For simplification, we assume that


Introduction
In recent years, the economy has developed rapidly over the years requiring a lot of energy sources in China, but it is impossible to largely import oil required.Many oil fields in China are developed by water flooding, but now, the recovery efficiency is low and water cut is over 80% because of the heterogeneity of reservoirs and high viscosity of oil.It is essential to increase the oil production of oil fields.As a result, enhanced oil recovery (EOR) has been a challenging field for different scientific disciplines.A mathematical model in [1] is developed to describe surfactant-enhanced solubilization of nonaqueous-phase liquids (NAPLs) in porous media.The goal in [2] is to find an optimal viscosity profile of the intermediate layer that almost eliminates the growth of the interfacial disturbances induced by mild perturbation of the permeability field.The mechanism of enhanced oil recovery using lipophobic and hydrophilic polysilicon (LHP) nanoparticles ranging in size from 10 to 500 nm for changing the wettability of porous media is analyzed theoretically in [3].It is shown in [4] that water-soluble hydrophobically associating polymers are reviewed with particular emphasis on their application in improved oil recovery (IOR).The solution properties of enhanced oil recovery are provided in [5,6].In order to enhance oil recovery and stabilize oil production, the study on EOR has been carried out for more than 20 years (also see [7][8][9][10][11]).
One of the strategies used in EOR is to use polymer flooding.Polymer flooding involves using a polymer additive to increase water viscosity, improve the water-oil mobility ratio, and enhance the displacement efficiency.Polymer flooding has been widely applied as an effective tertiary oilrecovery method in Daqing, Shengli, and other oilfields in China.
However, Different polymer flooding units have different static conditions and development status before polymer flooding.The production performance and behavior are also different.The quantitative characterization and prediction of polymer flooding performance have important guiding significance for polymer flooding scheme programming, performance evaluation, and adjustment.Hence, it is necessary to construct some mathematical models to illustrate the properties of polymer flooding.In [12][13][14][15], a 2 × 2 nonlinear 2 Abstract and Applied Analysis system model is presented to describe the polymer flooding of an oil recovery: where  = (, ) is the saturation of the aqueous phase (i.e., the solution of polymer and water, 0 ≤  ≤ 1),  = (, ) is the concentration of polymer in the water (0 ≤  ≤ 1). = (, ) [16] is the particle velocity of the aqueous phase. denotes the position in the reservoir and  denotes the time.In the polymer flooding, water thickened with polymer is injected into the reservoir.
Let  = (, ); system (1) can be written as where  ≜ ( 1 ,  2 ) = (, ), () = (, ) is a scalar function and usually referred to be the flow function.In this paper, we consider  to be rotationally invariant; namely, define that () = (‖‖) with ‖‖ = √ 2 1 +  2 2 .As a result, system (2) can be described as To the authors' knowledge, seldom researchers discussed the optimal control problem of system (3) (or (1)), but it is really interesting.Actually, in the last forty years, different optimal control schemes such as pinning control and impulsive control have been presented on all kinds of mathematical models of the engineering and physical application [17][18][19].It is worth noting that almost all of the discussed models in [17][18][19] are ordinary differential but system (3) is partial differential.A problem is arisen: how to discuss the control problem of the partial differential model (3)?By the constructive method, the authors in [20,21] discuss the global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws with linearly degenerate characteristics.Inspired by [20,21], we will discuss the exact boundary control problem of system (3) by using a constructed method.Hence, the main concern of this paper is to design an exact boundary controller for the EOR model (3).
The remainder of this paper is organized as follows.In Section 2, the exact boundary control problem and some Lemmas are presented.In Section 3, the main result is completed by a constructive method.Moreover, some important lemmas are also proposed in this section.In Section 4, an example is carried out to illustrate the effectiveness of the main result.Finally, conclusions are drawn in Section 5.
In order to solve the above boundary control problem, we need the following assumptions and Lemmas.
Remark 4. The discussed models in [20,21] are both linearly degenerate.However, model (10) in this paper is neither genuinely nonlinear nor linearly degenerate, which is more difficult and complicated to be discussed.
First, we need to discuss the lifespan of the Cauchy problem and Goursat problem.From the Cauchy problem ( 10) and ( 12), we have the following lemma.Lemma 8.If there exists  0 ∈ R such that  0 ()/| = 0 < 0, the Cauchy problem (10) and (12) must blow up in a finite time and the lifespan is dependent on the initial data.
Proof.For the first equation of system (10), the characteristic  2 () can be defined by One has / = 0; that is, ( 2 (), ) =  0 ().That is, the  0 norm of  is finite.Hence, we need to show that the first derivative of  must blow up in a finite time.
Remark 9. Note that, in the second equation of system (10), according to [22],   will always be bounded.Moreover,   will blow up if the following holds: Obviously, this does not hold since (⋅) is independent of the function .As a result,   will never blow up in a finite time.
For the Goursat problem (10) and ( 20), we have the following result.Lemma 10.If there exists  0 ∈ R such that R ( 0 ) > 0, the solution of the Goursat problem (10) and (20) must blow up in a finite time and the lifespan depends on the initial data.

Main Results
In this section, the boundary controllers will be designed.
Proof.One has the following.
Otherwise, if condition (1) is not satisfied, there exists a characteristic  = x2 () which passes through two points  = (0, 0) and  1 = ( 1 , ), 0 ≤  1 ≤ 1.According to the characteristic property, one has (, ) =  0 (0) =   ( 1 ), for ∀(, ) ∈ Ω and Ω is enclosed by the characteristics  =  1 (),  = x2 (), and the  axis.Note that the initial and terminal conditions are usually to be arbitrarily chosen.If one choose that  0 (0) ̸ =   ( 1 ), the system will not go from the given initial state to the desired terminal state no matter what control inputs are given.As a result, condition (1) should be satisfied.Also, with the similar analysis, condition (2) should be satisfied.
Step 3. Let Ω 3 be the domain enclosed by the characteristic , the characteristic , the straight line , and the straight line , where  is denoted by where  = (  −   )/(  −   ) is the slope / of the straight line .Consider the following system on the domain Ω 3 : with initial conditions For points  and , it is required that Along the characteristic  :  = Similarly, along the characteristic  :  =  2 (), one has According to Proposition 2.1 of [21], one has the fact that when  >  0 = max{−1/ max , 1/ min }.Then, from Assumption 2 and Remark 6, one has R 0 (  ) > 0, R 0 (  ) < 0. With Lemma 8, the Cauchy problem (37) and (38) with prescribed data on  3 () must blow up in a finite time.Here, we have interchanged the role of  and  variables.Hence, the time means the -axis.
Remark 17.In [10], an optimal control model of distributed parameter systems (DPSs) is presented to discuss the polymer injection strategies.Compared with [10], the differences of our paper are (1) the considered model is a hyperbolic system and the maximum principle does not hold here; (2) the desired outputs can be achieved by controlling the boundary inputs.

An Example
In this section, an example is presented to demonstrate the effectiveness of our results.

Remark 19.
Compared with [20,21], the difference in this paper is the fact that one cannot avoid the phenomenon of blowup.Hence, one important goal for EORs is to find the controllers ℎ 1 () and ℎ 2 () to make sure that the blowup points of EORs are beyond the domain we fixed.As a result, there exist more complications in the controlling process as described in Section 3. The example we give here is to illustrate that the controllers are achievable and it can be applied in real problems.

Conclusions
In this paper, we have discussed the exact boundary controllability of a class of enhanced oil recovery models.By using a constructed method, it has been shown that the enhanced oil recovery systems with nonlinear boundary conditions is exactly boundary controllable.Moreover, an interval of the control time has also been presented to be optimal.Finally, an example has been provided to illustrate the effectiveness of the obtained criterion.no.1301025B, the Open Fund of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation under Grant no.PLN1130, and the University Research Funds under Grants nos.2012XJZ029, 2012XJZT005, and 2013XJZT004.

Remark 11 .
According to Lemmas 8 and 10 and Remark 9, one knows that the blowup of the Goursat problem and the Cauchy problem only occurs in the solution .