Optimal Control of Polymer Flooding Based on Maximum Principle

Abstract

Polymer flooding is one of the most important technologies for enhanced oil recovery (EOR). In this paper, an optimal control model of distributed parameter systems (DPSs) for polymer injection strategies is established, which involves the performance index as maximum of the profit, the governing equations as the fluid flow equations of polymer flooding, and the inequality constraint as the polymer concentration limitation. To cope with the optimal control problem (OCP) of this DPS, the necessary conditions for optimality are obtained through application of the calculus of variations and Pontryagin’s weak maximum principle. A gradient method is proposed for the computation of optimal injection strategies. The numerical results of an example illustrate the effectiveness of the proposed method.

1. Introduction

It is of increasing necessity to produce oil fields more efficiently and economically because of the ever-increasing demand for petroleum worldwide. Since most of the significant oil fields are mature fields and the number of new discoveries per year is decreasing, the use of EOR processes is becoming more and more imperative. At present, polymer flooding technology is the best method for chemically EOR [1]. It could reduce the water-oil mobility ratio and improve sweep efficiency [2–5].

Because of the high cost of chemicals, it is essential to optimize polymer injection strategies to provide the greatest oil recovery at the lowest cost. The optimization procedure involves maximizing the objective function (cumulative oil production or profit) from a polymer flooding reservoir by adjusting the injection concentration. One way of solving this problem is direct optimization with the reservoir simulator. Numerical models are used to evaluate the complex interactions of variables affecting development decisions, such as reservoir and fluid properties and economic factors. Even with these models, the current practice is still the conventional trial and error approach. In each trial, the polymer concentration of an injection well is selected based on the intuition of the reservoir engineer. This one-well-at-a-time approach may lead to suboptimal decisions because engineering and geologic variables affecting reservoir performance are often nonlinearly correlated. And the problem definitely compounds when multiple producers and injectors are involved in a field development case. The use of the optimal control method offers a way out.

The optimal control method has been researched in EOR techniques in recent years. Ramirez et al. [6] firstly applied the theory of optimal control to determine the best possible injection strategies for EOR processes. Their study was motivated by the high operation costs associated with EOR projects. The objective of their study was to develop an optimization method to minimize injection costs while maximizing the amount of oil recovered. The performance of their algorithm was subsequently examined for surfactant injection as an EOR process in a one-dimensional core flooding problem [7]. The control for the process was the surfactant concentration of the injected fluid. They observed a significant improvement in the ratio of the value of the oil recovered to the cost of the surfactant injected from 1.5 to about 3.4. Optimal control was also applied to steam flooding by Liu et al. [8]. They developed an approach using optimal control theory to determine operating strategies to maximize the economic attractiveness of steam flooding process. Their objective was to maximize a performance index which is defined as the difference between oil revenue and the cost of injected steam. Their optimization method also obtained significant improvement under optimal operation. Ye et al. [9] were involved in the study of optimal control of gas-cycling in condensate reservoirs. It was shown that both the oil recovery and the total profit of a condensate reservoir can be enhanced obviously through optimization of gas production rate, gas injection rate, and the mole fractions of each component in injection gas. Daripa et al. [10–15] researched the basic physical mechanisms that contribute to poor oil recovery by EOR technologies and how to individually control each of these physical mechanisms. Brouwer and Jansen [16] and Sarma et al. [17] used the optimal control theory as an optimization algorithm for adjusting the valve setting in smart wells of water flooding. The water flooding scheme that maximized the profit was numerically obtained by combining reservoir simulation with control theory practices of implicit differentiation. They were able to achieve improved sweep efficiency and delayed water breakthrough by dynamic control of the valve setting.

For the previous work on optimal control of polymer flooding, Guo et al. [18] applied the iterative dynamic programming algorithm to solve the OCP of a one-dimensional core polymer flooding. However, the optimal control model used in their study is so simple that it is not adapted for practical oilfield development. As a result of the complicated nature of reservoir models with nonlinear constraints, it is very tedious and troublesome to cope with a large number of grid points for the state variables and control variables. To avoid these difficulties, Li et al. [19] and Lei et al. [20] used the genetic algorithms to determine the optimal injection strategies of polymer flooding and the reservoir model equations were treated as a “black box.” The genetic algorithms are capable of finding the global optimum on theoretical sense, but, as Sarma et al. [17] point out, they require tens or hundreds of thousand reservoir simulation runs of very large model and are not able to guarantee monotonic maximization of the objective function.

In this paper, an optimal control model of DPS for polymer flooding is established which maximizes the profit by adjusting the injection concentration. Then the determination of polymer injection strategies turns to solve this OCP of DPS. Necessary conditions for optimality are obtained by Pontryagin’s weak maximum principle. A gradient numerical method is presented for solving the OCP. Finally, an example of polymer flooding project involving a heterogeneous reservoir case is investigated and the results show the efficiency of the proposed method.

2. Mathematical Formulation of Optimal Control

2.1. Performance Index

Let Ω∈𝑅² denote the domain of reservoir with boundary 𝜕Ω, 𝑛 be the unit outward normal on 𝜕Ω, and (𝑥,𝑦)∈Ω be the coordinate of a point in the reservoir. Given a fixed final time 𝑡_𝑓, we set Ψ=Ω×[0,𝑡_𝑓], Γ=𝜕Ω×[0,𝑡_𝑓], and suppose that there exist 𝑁_𝑤 injection wells and 𝑁_𝑜 production wells in the oilfield. The injection and production wells are located at 𝐿_𝑤={(𝑥_𝑤𝑖,𝑦_𝑤𝑖)∣𝑖=1,2,…,𝑁_𝑤} and 𝐿_𝑜={(𝑥_𝑜𝑗,𝑦_𝑜𝑗)∣𝑗=1,2,…,𝑁_𝑜}, respectively. This descriptive statement of the cost functional must be translated into a mathematical form to use quantitative optimization techniques. The oil value can be formulated as ^𝑡𝑓₀_Ω𝜉_𝑜1−𝑓_𝑤𝑞_out𝑑𝜎𝑑𝑡,(2.1) where 𝜉_𝑜 is the cost of oil per unit mass (10⁴$/m³), 𝑓_𝑤(𝑥,𝑦,𝑡) is the fractional flow of water, and 𝑞_out(𝑥,𝑦,𝑡) is the flow velocity of production fluid (m/day). We define 𝑞_out(𝑥,𝑦,𝑡)≥0 at (𝑥,𝑦)∈𝐿_𝑜 and 𝑞_out(𝑥,𝑦,𝑡)≡0 at (𝑥,𝑦)∉𝐿_𝑜.

The polymer cost is expressed mathematically as ^𝑡𝑓₀_Ω𝜉_𝑝𝑞_in𝑐_pin𝑑𝜎𝑑𝑡,(2.2) where 𝜉_𝑝 is the cost of oil per unit volume (10⁴$/m³), 𝑐_pin(𝑥,𝑦,𝑡) is the polymer concentration of the injection fluid (g/L), and 𝑞_in(𝑥,𝑦,𝑡) is the flow velocity of injection fluid (m/day). We define 𝑞_in(𝑥,𝑦,𝑡)≥0 at (𝑥,𝑦)∈𝐿_𝑤 and 𝑞_in(𝑥,𝑦,𝑡)≡0 at (𝑥,𝑦)∉𝐿_𝑤.

The objective functional is, therefore, max𝐽=^𝑡𝑓₀_Ω𝜉_𝑜1−𝑓_𝑤𝑞_out−𝜉_𝑝𝑞_in𝑐_pin𝑑𝜎𝑑𝑡.(2.3)

2.2. Governing Equations

The maximization of the cost functional 𝐽 given by (2.3) is not totally free but is constrained by the system process dynamics. The governing equations of the polymer flooding process must therefore be developed to describe the flow of both the aqueous and oil phases through the porous media of a reservoir formation. The equations used in this paper allow for the adsorption of polymer onto the solid matrix in addition to the convective and dispersive mechanisms of mass transfer. Let 𝑝(𝑥,𝑦,𝑡), 𝑆_𝑤(𝑥,𝑦,𝑡) and 𝑐_𝑝(𝑥,𝑦,𝑡) denote the pressure, water saturation, and polymer concentration of the oil reservoir, respectively, at a point (𝑥,𝑦)∈Ω and a time 𝑡∈[0,𝑡_𝑓], then 𝑝(𝑥,𝑦,𝑡), 𝑆_𝑤(𝑥,𝑦,𝑡), and 𝑐_𝑝(𝑥,𝑦,𝑡) satisfy the following partial differential equations (PDEs).(i)The flow equation for oil phase 𝜕𝜕𝑥𝑘^𝑝𝑟_𝑜𝜕𝑝𝜕𝑥+𝜕𝜕𝑦𝑘^𝑝𝑟_𝑜𝜕𝑝𝜕𝑦−1−𝑓^𝑤𝑞_out=ℎ𝜕𝑎^𝑜𝜕𝑡,(𝑥,𝑦,𝑡)∈Ψ.(2.4)(ii)The flow equation for water phase 𝜕𝜕𝑥𝑘^𝑝𝑟_𝑤𝜕𝑝𝜕𝑥+𝜕𝜕𝑦𝑘^𝑝𝑟_𝑤𝜕𝑝𝜕𝑦+𝑞ⁱⁿ−𝑓_𝑤𝑞_out=ℎ𝜕𝑎^𝑤𝜕𝑡,(𝑥,𝑦,𝑡)∈Ψ.(2.5)(iii)The flow equation for polymer component 𝜕𝜕𝑥𝑘^𝑑𝑟_𝑑𝜕𝑐_𝑝𝜕𝑥+𝜕𝜕𝑥𝑘^𝑝𝑟_𝑐𝜕𝑝𝜕𝑥+𝜕𝜕𝑦𝑘^𝑑𝑟_𝑑𝜕𝑐_𝑝𝜕𝑦+𝜕𝜕𝑦𝑘^𝑝𝑟_𝑐𝜕𝑝𝜕𝑦+𝑞ⁱⁿ𝑐_pin−𝑓_𝑤𝑞_out𝑐_𝑝=ℎ𝜕𝑎^𝑐𝜕𝑡,(𝑥,𝑦,𝑡)∈Ψ.(2.6)(iv)The boundary conditions and initial conditions 𝜕𝑝𝜕𝑛|||_𝜕Ω=0,𝜕𝑆^𝑤𝜕𝑛||||_𝜕Ω=0,𝜕𝑐^𝑝𝜕𝑛||||_𝜕Ω𝑝(𝑥,𝑦,0)=𝑝₀=0,(𝑥,𝑦,𝑡)∈Γ,(2.7)𝑐(𝑥,𝑦),𝑆_𝑤(𝑥,𝑦,0)=𝑆^0𝑤(𝑥,𝑦),_𝑝(𝑥,𝑦,0)=𝑐^0𝑝(𝑥,𝑦),(𝑥,𝑦)∈Ω,(2.8)where the corresponding parameters are defined as 𝑟𝑘_𝑝=𝐾ℎ,𝑘_𝑑=𝐷ℎ,(2.9)_𝑜=𝑘^𝑟𝑜𝐵_𝑜𝜇_𝑜,𝑟_𝑤=𝑘^𝑟𝑤𝐵_𝑤𝑅_𝑘𝜇_𝑤,𝑟_𝑐=𝑘^𝑟𝑤𝑐_𝑝𝐵_𝑤𝑅_𝑘𝜇_𝑝,𝑟_𝑑=𝜙^𝑝𝑆_𝑤𝐵_𝑤𝑎,(2.10)_𝑜=𝜙1−𝑆^𝑤𝐵_𝑜,𝑎_𝑤=𝜙𝑆^𝑤𝐵_𝑤,𝑎_𝑐=𝜙^𝑝𝑆_𝑤𝑐_𝑝𝐵_𝑤+𝜌_𝑟(1−𝜙)𝐶_𝑟𝑝.(2.11) The constant coefficient 𝐾(𝑥,𝑦) is the absolute permeability (𝜇m²), ℎ is the thickness of the reservoir bed (m), 𝐷 is the diffusion coefficient of polymer (m²/s), 𝜌_𝑟 (kg/m³) is the rock density, and 𝜇_𝑜 (mPa·s) is the oil viscosity.

The oil volume factor 𝐵_𝑜, the water volume factor 𝐵_𝑤, the rock porosity 𝜙, and the effective porosity to polymer 𝜙_𝑝 are expressed as functions of the reservoir pressure 𝑝: 𝐵_𝑜=𝐵^𝑜𝑟𝐵1+𝐶_𝑜𝑝−𝑝_𝑟_𝑤,=𝐵^𝑤𝑟𝜙=𝜙_𝑟1+𝐶_𝑤𝑝−𝑝_𝑟,𝜙1+𝐶_𝑅𝑝−𝑝_𝑟_𝑝,=𝑓_𝑎𝜙,(2.12) where 𝑝_𝑟 is the reference pressure (MPa), 𝜙_𝑟, 𝐵_𝑜𝑟, and 𝐵_𝑤𝑟 denote the porosity, the oil, and water volume factor under the condition of the reference pressure, respectively, 𝑓_𝑎 is the effective pore volume coefficient, 𝐶_𝑜, 𝐶_𝑤, and 𝐶_𝑅 denote the compressibility factors of oil, water, and rock, respectively.

Functions relating values of the oil and water relative permeabilities 𝑘_𝑟𝑜 and 𝑘_𝑟𝑤 to the water saturation 𝑆_𝑤 are 𝑘_𝑟𝑤=𝑘_{𝑟𝑤𝑟𝑜}𝑆^𝑤−𝑆_𝑤𝑐1−𝑆_𝑤𝑐−𝑆_𝑜𝑟^𝑛𝑤𝑘_𝑟𝑜,=𝑘_{𝑟𝑜𝑐𝑤}1−𝑆^𝑤−𝑆_𝑜𝑟1−𝑆_𝑤𝑐−𝑆_𝑜𝑟^𝑛𝑜,(2.13) where 𝑘_{𝑟𝑤𝑟𝑜} is the oil relative permeability at the irreducible water saturation 𝑆_𝑤𝑐,𝑘_{𝑟𝑤𝑐𝑤} is the water relative permeability at the residual oil saturation and 𝑆_𝑜𝑟,𝑛_𝑤, and 𝑛_𝑜 are constant coefficients.

The polymer solution viscosity 𝜇_𝑝 (mPa·s), the permeability reduction factor 𝑅_𝑘, and the amount adsorbed per unit mass of the rock 𝐶_𝑟𝑝 (mg/g) which depend on the polymer concentration 𝑐_𝑝 are given by 𝜇_𝑝𝑅=𝜇_𝑤1+𝑎𝑝₁𝑐_𝑝+𝑎𝑝₂𝑐^2𝑝+𝑎𝑝^3𝑝_𝑘,=1+𝑅^𝑘max−1⋅𝑏_𝑟𝑘⋅𝑐_𝑝1+𝑏_𝑟𝑘⋅𝑐_𝑝𝐶_𝑟𝑝,=𝑎𝑐^𝑝1+𝑏𝑐_𝑝,(2.14) where 𝜇_𝑤 is the viscosity of the aqueous phase with no polymer (mPa·s), 𝑎𝑝₁,𝑎𝑝₂,𝑎𝑝₃, 𝑅_𝑘max,𝑏_𝑟𝑘, and 𝑎 (cm³/g) and 𝑏 (g/L) are constant coefficients.

The fractional flow of water 𝑓_𝑤 is given by, 𝑓_𝑤=𝑟^𝑤𝑟_𝑜+𝑟_𝑤.(2.15)

2.3. Constraint

Since the negative and overhigh injection polymer concentrations are not allowed, the constraint in polymer flooding is expressed mathematically as 0≤𝑐_pin≤𝑐_max,(2.16) where 𝑐_max is the maximum injection polymer concentration.

2.4. Optimal Control Formulation

The reservoir pressure 𝑝, the water saturation 𝑆_𝑤 and the polymer concentration 𝑐_𝑝 are the three state variables for the problem as formulated. The system state vector is denoted by 𝐮(𝑥,𝑦,𝑡)=𝑝,𝑆_𝑤,𝑐_𝑝^𝑇.(2.17)

The control for the process is the polymer concentration of injected fluid 𝑣(𝑥,𝑦,𝑡)=𝑐_pin,(𝑥,𝑦)∈𝐿_𝑤.(2.18) Then the OCP of DPS for polymer flooding has the general form, max_𝑣𝐽=^𝑡𝑓₀_Ωs.t.𝐟𝐮̇,𝐮,𝐮_𝑥𝐹(𝐮,𝑣)𝑑𝜎𝑑𝑡,(2.19)𝐠𝐮,𝐮_𝑥,𝐮_𝑦,𝐮_𝑥𝑥,𝐮_𝑦𝑦,𝑣=0,(𝑥,𝑦,𝑡)∈Ψ,(2.20)𝐮(𝑥,,𝑦𝐮,_𝑦0,)𝐮=_𝑥𝑥𝐮,𝐮_𝑦𝑦0=0,(𝑥,𝑦,𝑡)∈Γ,(2.21)0≤𝑣(𝑥≤,𝑦𝑣),_max(𝑥,𝑦)∈Ω,(2.22),(2.23) where 𝐮̇=𝜕𝐮/𝜕𝑡,𝐮_𝑙=𝜕𝐮/𝜕𝑙,𝐮_𝑙𝑙=𝜕²𝐮/𝜕𝑙²,𝑙=𝑥,𝑦, (2.19) denotes the performance index (2.3), (2.20) expresses the governing equations (2.4)–(2.6), (2.21) and (2.22) denote the boundary and initial conditions, respectively, and (2.23) denotes the injection polymer concentration constraint (2.16).

3. Necessary Conditions of Optimal Control

3.1. Maximum Principle of DPS

We desire to find a set of necessary conditions for the state vector, 𝐮, and the control, 𝑣, to be extremals of the functional 𝐽 (2.19) subject to the PDEs (2.20)~(2.22) and the constraint (2.23). A convenient way to cope with such an OCP of DPS (2.19)~(2.23) is through the use of distributed adjoint variables. The first step is to form an augmented functional by adjoining the governing equations to the performance index 𝐽. We define the Hamiltonian as 𝐻=𝐹+𝜆^𝑇𝐟,(3.1) where 𝝀(𝑥,𝑦,𝑡) is the adjoint vector. Then the argument functional is given by, 𝐽_𝐴=𝐽+^𝑡𝑓₀_Ω𝝀^𝑇𝐟𝐮̇,𝐮,𝐮_𝑥,𝐮_𝑦,𝐮_𝑥𝑥,𝐮_𝑦𝑦,𝑣𝑑𝜎𝑑𝑡=^𝑡𝑓₀_Ω𝐻𝐮̇,𝐮,𝐮_𝑥,𝐮_𝑦,𝐮_𝑥𝑥,𝐮_𝑦𝑦,𝑣𝑑𝜎𝑑𝑡.(3.2)

Following the standard procedure of the calculus of variables, the increment of 𝐽_𝐴, denoted by Δ𝐽_𝐴, is formed by introducing variations 𝛿𝐮, 𝛿𝐮_𝑥, 𝛿𝐮_𝑦, 𝛿𝐮_𝑥𝑥, 𝛿𝐮_𝑦𝑦, 𝛿𝐮̇, and 𝛿𝑣 giving Δ𝐽_𝐴=𝐽−_𝐴𝐽_𝐴𝐮+𝛿𝐮,𝐮_𝑥+𝛿𝐮_𝑥,𝐮_𝑦+𝛿𝐮_𝑦,𝐮_𝑥𝑥+𝛿𝐮_𝑥𝑥,𝐮_𝑦𝑦+𝛿𝐮_𝑦𝑦,𝐮̇+𝛿𝐮̇,𝑣+𝛿𝑣𝐮,𝐮_𝑥,𝐮_𝑦,𝐮_𝑥𝑥,𝐮_𝑦𝑦,𝐮̇,𝑣.(3.3) This formulation assumes that the final time, 𝑡_𝑓, is fixed.

Expanding (3.3) in a Taylor series and retaining only the linear terms give the variation of the functional, 𝛿𝐽_𝐴, 𝛿𝐽_𝐴=^𝑡𝑓₀_Ω𝜕𝜕𝐻𝐮^𝑇𝛿𝐮+𝜕𝐻𝜕𝐮_𝑥^𝑇𝛿𝐮_𝑥+𝜕𝐻𝜕𝐮_𝑥𝑥^𝑇𝛿𝐮_𝑥𝑥+𝜕𝐻𝜕𝐮_𝑦^𝑇𝛿𝐮_𝑦+𝜕𝐻𝜕𝐮_𝑦𝑦^𝑇𝛿𝐮_𝑦𝑦+𝜕𝐻𝜕𝐮̇^𝑇𝛿𝐮̇+𝜕𝐻𝜕𝑣𝛿𝑣𝑑𝜎𝑑𝑡.(3.4) Since the variations 𝛿𝐮, 𝛿𝐮_𝑙, 𝛿𝐮_𝑙𝑙 (𝑙=𝑥,𝑦), and 𝛿𝐮̇ are not independent can be expressed in terms of the variations 𝛿𝐮 by integrating the following three terms by parts: _Ω𝜕𝜕𝐻𝐮_𝑙^𝑇𝛿𝐮_𝑙𝑑𝜎=_Ω𝜕𝜕𝑙𝜕𝜕𝐻𝐮_𝑙^𝑇𝛿𝐮𝑑𝜎−_Ω𝜕𝜕𝑙𝜕𝜕𝐻𝐮_𝑙^𝑇𝛿𝐮𝑑𝜎,(3.5)_Ω𝜕𝜕𝐮𝐻_𝑙𝑙^𝑇𝛿𝐮_𝑙𝑙𝑑𝜎=_Ω𝜕𝜕𝑙²²𝜕𝜕𝐮𝐻_𝑙𝑙^𝑇+𝛿𝐮𝑑𝜎_Ω𝜕𝜕𝑙𝜕𝜕𝐮𝐻_𝑙𝑙^𝑇𝛿𝐮_𝑙−𝜕𝜕𝑙𝜕𝜕𝐮𝐻_𝑙𝑙^𝑇𝛿𝐮𝑑𝜎,(3.6)_𝑡𝑓0𝜕𝜕𝐻𝐮̇^𝑇𝛿𝐮̇=𝜕𝜕𝐻𝐮̇^𝑇𝛿𝐮||||^𝑡𝑓₀−^𝑡𝑓₀𝜕𝜕𝑡𝜕𝜕𝐻𝐮̇^𝑇𝛿𝐮𝑑𝑡.(3.7) Using the Green’s formula in (3.5) and (3.6), we obtain _Ω_{𝑙=𝑥,𝑦}𝜕𝜕𝐻𝐮_𝑙^𝑇𝛿u_𝑙+𝜕𝐻𝜕𝐮_𝑙𝑙^𝑇𝛿𝐮_𝑙𝑙=𝑑𝜎_Ω𝑙_=𝑥,𝑦−𝜕𝜕𝑙𝜕𝜕𝐻𝐮_𝑙+𝜕𝜕𝑙²²𝜕𝜕𝐮𝐻_𝑙𝑙^𝑇+𝛿𝐮𝑑𝜎_𝜕Ω𝜕𝜕𝐮𝐻_𝑥−𝜕𝜕𝑥𝜕𝜕𝐮𝐻_𝑥𝑥^𝑇𝛿𝐮+𝜕𝐻𝜕𝐮_𝑥𝑥^𝑇𝛿𝐮_𝑥−𝜕𝐻𝑑𝑦𝜕𝐮_𝑦−𝜕𝜕𝑦𝜕𝜕𝐮𝐻_𝑦𝑦^𝑇𝛿𝐮+𝜕𝐻𝜕𝐮_𝑦𝑦^𝑇𝛿𝐮_𝑦𝑑𝑥.(3.8) By substituting the above equations (3.7) and (3.8) into (3.4), the first variation 𝛿𝐽_𝐴 is expressed as 𝛿𝐽_𝐴=^𝑡𝑓_𝑡0_Ω𝜕𝜕𝐻𝐮−𝜕𝜕𝑥𝜕𝜕𝐮𝐻_𝑥−𝜕𝜕𝑦𝜕𝜕𝐻𝐮_𝑦+𝜕²𝜕𝑥²𝜕𝐻𝜕𝐮_𝑥𝑥+𝜕²𝜕𝑦²𝜕𝐻𝜕𝐮_𝑦𝑦−𝜕𝜕𝑡𝜕𝜕𝐻𝐮̇^𝑇+𝛿𝐮𝑑𝜎𝑑𝑡_{𝑡𝑓𝑡0}_𝜕Ω𝜕𝜕𝐮𝐻_𝑥−𝜕𝜕𝑥𝜕𝜕𝐮𝐻_𝑥𝑥^𝑇𝛿𝐮+𝜕𝐻𝜕𝐮_𝑥𝑥^𝑇𝛿𝐮_𝑥−𝜕𝐻𝑑𝑦𝜕𝐮_𝑦−𝜕𝜕𝑦𝜕𝜕𝐮𝐻_𝑦𝑦^𝑇𝛿𝐮+𝜕𝐻𝜕𝐮_𝑦𝑦^𝑇𝛿𝐮_𝑦+𝑑𝑥𝑑𝑡_Ω𝜕𝜕𝐻𝐮̇^𝑇𝛿𝐮||||^𝑡𝑓₀𝑑𝜎+^𝑡𝑓_𝑡0_Ω𝜕𝜕𝐻𝑣𝛿𝑣𝑑𝜎𝑑𝑡.(3.9)

When the state and control regions are not bounded, the variation of the functional must vanish at an extremal (the fundamental theorem of the calculus of variations). When the control region is constrained by a boundary, then the necessary condition for optimality is to maximize the performance index 𝐽_𝐴 with respect to the control 𝑣. This means that the variation 𝛿𝐽_𝐴 is 𝛿𝐽_𝐴𝑣^∗,𝛿𝑣≥0,(3.10) where 𝑣^∗ denotes the optimal control. Equation (3.10) is the weak minimum principle of Pontryagin. The necessary conditions for these two cases are the same except for the term involving the variation of the control, 𝛿𝑣. For polymer flooding problem there are higher and lower bounds on the control variable 𝑣 given as (2.16).

The following necessary conditions for optimality are obtained when we apply Pontryagin’s maximum principle.

(1) Adjoint Equations
Since the variation 𝛿𝐮 is free and not zero, the coefficient terms involving the 𝛿𝐮 variation in the first term of (3.9) are set to zero. This results in the adjoint equations as given by 𝜕𝐻𝜕𝐮−_{𝑙=𝑥,𝑦}𝜕𝜕𝑙𝜕𝜕𝐻𝐮_𝑙+𝜕²𝜕𝑙²𝜕𝐻𝜕𝐮_𝑙𝑙−𝜕𝜕𝑡𝜕𝜕𝐻𝐮̇=0.(3.11) Substitute the Hamiltonian (3.1) into (3.11) and the adjoint equations become 𝜕𝐹𝜕𝐮+𝜕𝜕𝐮𝐟−𝜕𝜕𝑡𝜕𝜕𝐮𝐟̇^𝑇+_{𝑙=𝑥,𝑦}𝝀𝜕𝜕𝑙²²𝜕𝐟𝜕𝐮_𝑙𝑙−𝜕𝜕𝑙𝜕𝜕𝐮𝐟_𝑙^𝑇𝝀+2𝜕𝜕𝑙𝜕𝜕𝐮𝐟_𝑙𝑙−𝜕𝐟𝜕𝐮_𝑙^𝑇𝜕𝝀𝜕𝑙+𝜕𝜕𝐮𝐟_𝑙𝑙^𝑇𝜕²𝝀−𝜕𝐟𝜕𝑙²𝜕𝐮̇^𝑇𝜕𝝀𝜕𝑡=0.(3.12) Equation (3.12) is a set of PDEs with nonconstant coefficients.

(2) Transversality Boundary Conditions
The adjoint boundary conditions are obtained from the second term of (3.9): 𝜕𝜕𝐻𝐮_𝑙−𝜕𝜕𝑙𝜕𝜕𝐮𝐻_𝑙𝑙^𝑇𝛿𝐮+𝜕𝐻𝜕𝐮_𝑙𝑙^𝑇𝛿𝐮_𝑙|||||_𝜕Ω=0,𝑙=𝑥,𝑦.(3.13)

(3) Transversality Terminal Conditions
Since the initial state is specified, the variation 𝛿𝐮|_𝑡=0 of (3.9) is zero. However, the final state is not specified; therefore, the variation 𝛿𝐮|_{𝑡=𝑡𝑓} is free and nonzero. This means that the following relation must be zero: 𝜕𝐻𝜕𝐮̇=𝜕𝜕𝑓𝐮̇^𝑇𝝀=0,at𝑡=𝑡_𝑓.(3.14)

(4) Optimal Control
With all the previous terms vanishing, the variation of the functional 𝛿𝐽_𝐴 becomes 𝛿𝐽_𝐴=^𝑡𝑓₀_Ω𝜕𝜕𝐻𝑣𝛿𝑣𝑑𝜎𝑑𝑡.(3.15) This equation expresses the direct influence of variation 𝛿𝑣 on 𝛿𝐽_𝐴. A necessary condition for the optimality of 𝑣^∗ is that 𝛿𝐽_𝐴≥0 for all possible small variations, 𝛿𝑣. Since there are lower and higher bounds on the control 𝑣 (2.9), we use the weak maximum principle to assert the following necessary conditions for optimality: 𝜕𝐻𝜕𝑣=0for0≤𝑣^∗≤𝑣_max,(3.16) when the control vector is unconstrained. Because the variation 𝛿𝑣 can only be negative along the lower bound, we have 𝜕𝐻𝜕𝑣≤0for𝑣^∗=0.(3.17) And because the variation 𝛿𝑣 can only be positive along the higher bound, we have 𝜕𝐻𝜕𝑣≥0for𝑣^∗=𝑣_max.(3.18)

3.2. Necessary Conditions of OCP for Polymer Flooding

Let 𝝀(𝑥,𝑦,𝑡)=(𝜆₁,𝜆₂,𝜆₃)^𝑇 denote the adjoint vector of OCP for polymer flooding. Applying the theory developed in Section 3.1 and substituting the governing equations (2.4)–(2.6) into (3.12), the adjoint equations, given by (3.12), reduce for the polymer flooding problem under consideration as given in, _𝑙_=𝑥,𝑦𝜕𝜕𝑙𝑘^𝑝𝑟_𝑜𝜕𝜆₁𝜕𝑙+𝜕𝜕𝑙𝑘^𝑝𝑟_𝑤𝜕𝜆₂𝜕𝑙+𝜕𝜕𝑙𝑘^𝑝𝑟_𝑐𝜕𝜆₃−𝑘𝜕𝑙_𝑝𝜕𝑟_𝑜𝜕𝑝𝜕𝜕𝑝𝑙𝜕𝜕𝜆𝑙¹+𝑘^𝑑𝜕𝑟_𝑤𝜕𝑝𝜕𝜕𝑝𝑙𝜕𝜕𝜆𝑙²+𝑘^𝑝𝜕𝑟_𝑐𝜕𝑝𝜕𝜕𝑝𝑙+𝑘^𝑑𝜕𝑟_𝑑−𝑞_out𝜕𝑝𝜕𝜕𝑐𝑙^𝑝𝜕𝜕𝜆𝑙³𝜉^𝑜𝜕𝑓_𝑤𝜕𝑝−𝜕𝜕𝑓𝑝^𝑤𝜆¹+𝜕𝑓^𝑤𝜕𝑝𝜆²+𝑐_𝑝𝜕𝑓_𝑤+𝜕𝑎_𝑜𝜕𝑝𝜆³𝜕𝑝𝜕𝜕𝜆𝑡¹+𝜕𝜕𝑎𝑝^𝑤𝜕𝜕𝜆𝑡²+𝜕𝜕𝑎𝑝^𝑐𝜕𝜕𝜆𝑡³=0,(𝑥,𝑦,𝑡)∈Ψ,_{𝑙=𝑥,𝑦}−𝑘^𝑝𝜕𝑝𝜕𝑙𝜕𝜕𝑆𝑟^𝑜_𝑤𝜕𝜆₁𝜕𝑙+𝜕𝜕𝑆𝑟^𝑤_𝑤𝜕𝜆₂𝜕𝑙+𝜕𝜕𝑆𝑟^𝑐_𝑤𝜕𝜆₃𝜕𝑙−𝑘^𝑑𝜕𝑟_𝑑𝜕𝑆_𝑤𝜕𝑐_𝑝−𝑞_out𝜕𝑙𝜕𝜕𝜆𝑙³𝜉^𝑜𝜕𝑓_𝑤𝜕𝑆_𝑤−𝜕𝑓^𝑤𝜕𝑆_𝑤𝜆₁+𝜕𝑓^𝑤𝜕𝑆_𝑤𝜆₂+𝑐_𝑝𝜕𝑓_𝑤𝜕𝑆_𝑤𝜆₃+𝜕𝜕𝑆𝑎^𝑜_𝑤𝜕𝜆₁𝜕𝑡+𝜕𝜕𝑆𝑎^𝑤_𝑤𝜕𝜆₂+𝜕𝑎_𝑐𝜕𝑡𝜕𝑆_𝑤𝜕𝜆₃𝜕𝑡=0,(𝑥,𝑦,𝑡)∈Ψ,_{𝑙=𝑥,𝑦}𝜕𝜕𝑙𝑘^𝑑𝑟_𝑑𝜕𝜆₃𝜕𝑙−𝑘^𝑝𝜕𝑝𝜕𝑙𝜕𝜕𝑟𝑐^𝑤_𝑝𝜕𝜆₂𝜕𝑙+𝜕𝜕𝑐𝑟^𝑐_𝑝𝜕𝜆₃−𝑞_out𝜕𝑙𝜉^𝑜𝜕𝑓_𝑤𝜕𝑐_𝑝−𝜕𝑓^𝑤𝜕𝑐_𝑝𝜆₁+𝜕𝑓^𝑤𝜕𝑐_𝑝𝜆₂+𝑐_𝑝𝜕𝑓_𝑤𝜕𝑐_𝑝+𝑓_𝑤+𝜕𝑎_𝑐𝜆³𝜕𝑐_𝑝𝜕𝜆₃𝜕𝑡=0,(𝑥,𝑦,𝑡)∈Ψ.(3.19)

The boundary conditions (2.7) of the DPS result in (𝜕𝐮/𝜕𝑙)|_𝜕Ω=0 and 𝛿𝐮_𝑙|_𝜕Ω=0,𝑙=𝑥,𝑦, in (3.13). The coefficients of the arbitrary variation 𝛿𝐮_𝑙|_𝜕Ω terms must be zero and yield the boundary conditions for the adjoint equations as given by 𝜕𝐻𝜕𝐮_𝑙−𝜕𝜕𝑙𝜕𝜕𝐮𝐻_𝑙𝑙=0,𝑙=𝑥,𝑦.(3.20) By substituting the governing equations (2.4)–(2.6) into (3.20), the boundary conditions of adjoint equations for the polymer flooding OCP are expressed as 𝑟^𝑜𝜕𝜆₁𝜕𝑙+𝑟^𝑤𝜕𝜆₂𝜕𝑙||||_𝜕Ω=0,𝜕𝜆³𝜕𝑙||||_𝜕Ω=0,𝑙=𝑥,𝑦,(𝑥,𝑦,𝑡)∈Γ.(3.21)

The following transversality terminal conditions at 𝑡=𝑡_𝑓 are obtained by substituting the governing equations (2.4)–(2.6) into (3.14): −𝜕𝑎^𝑜𝜕𝑝𝜆¹−𝜕𝑎^𝑤𝜕𝑝𝜆²−𝜕𝑎^𝑐𝜕𝑝𝜆³−𝜕𝑎_𝑜=0,𝜕𝑆_𝑤𝜆₁−𝜕𝑎^𝑤𝜕𝑆_𝑤𝜆₂−𝜕𝑎^𝑐𝜕𝑆_𝑤𝜆₃=0,−𝜕𝑎^𝑐𝜕𝑐_𝑝𝜆₃=0.(3.22) Since the coefficient terms involving the adjoint variables in (3.22) are not zero, the terminal conditions of adjoint equations in the OCP of polymer flooding can be simplified to 𝜆₁𝑥,𝑦,𝑡_𝑓=0,𝜆₂𝑥,𝑦,𝑡_𝑓=0,𝜆₃𝑥,𝑦,𝑡_𝑓=0,(𝑥,𝑦)∈Ω.(3.23) Equation (3.23) shows that the adjoint variables are known at the final time 𝑡_𝑓. Since the state variables are known at the initial time and the adjoint variables are known at the final time, the OCP is a split two-point boundary-value problem.

The variation of the performance index, 𝐽, reduces to the following simplified functional of the control variation: 𝛿𝐽_𝐴=^𝑡𝑓₀_Ω𝑞_in𝜉_𝑝+𝜆₃𝛿𝑣𝑑𝜎𝑑𝑡.(3.24) From the results of (3.16)–(3.18), the necessary condition for optimality of polymer flooding problem is 𝑞_in𝜉_𝑝+𝜆₃⎧⎪⎨⎪⎩=0,for0≤𝑣^∗≤0,for𝑣_∗≤𝑣_max,≥0,for𝑣_∗=0,=𝑣_max.(3.25)

4. Numerical Solution

We propose an iterative numerical technique for determining the optimal injection strategies of polymer flooding. The computational procedure is based on adjusting estimates of control function 𝑣 to improve the value of the objective functional. For a control to be optimal, the necessary condition given by (3.25) must be satisfied. If the control 𝑣 is not optimal, then a correction 𝛿𝑣 is determined so that the functional is made lager, that is, 𝛿𝐽_𝐴>0. If 𝛿𝑣 is selected as 𝛿𝑣=𝑤𝑞_in𝜉_𝑝+𝜆₃,(4.1) where 𝑤 is an arbitrary positive weighting factor, the functional variation becomes 𝛿𝐽_𝐴=^𝑡𝑓₀_Ω𝑤𝑞_in(𝜉_𝑝+𝜆₃)²𝑑𝜎𝑑𝑡≥0.(4.2) Thus, choosing 𝛿𝑣 in the gradient direction ensures a local improvement in the objective functional, 𝐽_𝐴. At the higher and lower bounds on 𝑣, we must make the appropriate weighting terms equal to zero to avoid leaving the allowable region.

The computational algorithm of control iteration based on gradient direction is as follows.

(1) Initialization
Make an initial guess for the control function, 𝑣(𝑥,𝑦,𝑡),(𝑥,𝑦)∈𝐿_𝑤,𝑡∈[0,𝑡_𝑓].

(2) Resolution of the State Equations
Using stored current value of 𝑣(𝑥,𝑦,𝑡),(𝑥,𝑦)∈𝐿_𝑤, integrate the governing equations forward in time with known initial state conditions. We use the finite difference method of a full implicit scheme for the PDEs as discussed in [21, 22]. The profit functional is evaluated, and the coefficients involved in the adjoint equations which are function of the state solution are computed and stored.

(3) Resolution of the Adjoint Equations
Using the stored coefficients, integrate the adjoint equations numerically backward in time with known final time adjoint conditions by (3.23). Compute and store 𝛿𝑣 as defined by (4.1).

(4) Computation of the New Control
Using the evaluated 𝛿𝑣, an improved function is computed as 𝑣(𝑥,𝑦,𝑡)^new=𝑣(𝑥,𝑦,𝑡)^old+𝛿𝑣(𝑥,𝑦,𝑡),(𝑥,𝑦)∈𝐿_𝑤,(4.3) where 0≤𝑣^new≤𝑣_max. A single variable search strategy can be used to find the value of the positive weighting factor 𝑤 which maximizes the improvement in the performance functional using (4.3).

(5) Termination
The optimization algorithm is stopped when the variation 𝛿𝑣 is too small to effectively change the performance measure, that is, when ||𝐽^new−𝐽^old||<𝜀,(4.4) where 𝜀 is a small positive number.

5. Case Study

In this section we present a numerical example of optimal control for polymer flooding done with the proposed iterative gradient method.

The two-phase flow of oil and water in a heterogeneous two-dimensional reservoir is considered. The reservoir covers an area of 421.02×443.8 m² and has a thickness of 5 m and is discretized into 90(9×10×1) grid blocks. The production model is a five-spot pattern, with one production well P1 located at the center of the reservoir (5,6) and four injection wells W1~W4 placed at the four corners (1,10), (9,10), (1,1), and (9,1) as shown in the permeability distribution map of Figure 1. Polymer is injected when the fractional flow of water for the production well comes to 97% after water flooding. The time domain of polymer injection is 0~1440 days and the polymer flooding project life is 𝑡_𝑓=5500 (days). Figures 2 and 3 show the contour maps of the initial water saturation 𝑆^0𝑤 and the initial reservoir pressure 𝑝⁰, respectively. The initial polymer concentration is 𝑐^0𝑝=0 (g/L). In the performance index calculation, we use the price of oil 𝜉_𝑜=0.0503(10⁴$/m³) (80($/bbl)), and the cost of polymer 𝜉_𝑝=2.5×10⁻⁴(10⁴$/kg). The fluid velocity of production well is 𝑞_out=7.225×10⁻³ (m/day), and the fluid velocity of every injection well is 𝑞_in=2.89×10⁻² (m/day). The PDEs are solved by full implicit finite difference method with step size 10 days. For the constraint (29), the maximum injection polymer concentration is 𝑐_max=2.2 (g/L). The parameters of the reservoir description and the fluid data are shown in Tables 1 and 2, respectively.

Table 1 

Parameters of reservoir description used in the example.

Table 2 

Fluid data used in the example.

Figure 1 

Permeability (𝜇m2) distribution.

Figure 2 

Initial water saturation contour map.

Figure 3 

Initial reservoir pressure (MPa) contour map.

The polymer injection strategies obtained by the conventional engineering judgment method (trial and error) are the same 1.8 (g/L) for all injection wells. The performance index is 𝐽=$1.592×10⁷ with oil production 32429 m³ and polymer injection 155520 kg. For comparison, the results obtained by engineering judgment method are considered as the initial control strategies of the proposed iterative gradient method. A backtracking search strategy [23] is used to find the appropriate weighting term 𝑤 and the stopping criterion is chosen as 𝜀=1×10⁻⁵. By using the proposed algorithm, we obtain a cumulative oil of 33045 m³ and a cumulative polymer of 151618 kg yielding the profit of 𝐽^∗=$1.624×10⁷ over the polymer flooding project life of the reservoir. The results show an increase in performance index of $3.2×10⁵. Figures 8 and 9 show the fractional flow of water in production well and the cumulative oil production curves of the two methods, respectively. It is obvious that the fractional flow of water obtained by iterative gradient method is lower than that by engineering judgment. Therefore, with the less cumulative polymer injection, the proposed method gets more oil production and higher recovery ratio. Figure 4 to Figure 7 show the optimal control policies of the injection wells W1~W4. As a result, the optimal injection polymer concentration profiles of W1 and W2 are significantly different from those of W3, W4. It is mainly due to the differences of the well positions and the distance to the production well, as well as the reservoir heterogeneity and the uniform initial water saturation distribution.

Figure 4 

Injection polymer concentration of well W1.

Figure 5 

Injection polymer concentration of well W2.

Figure 6 

Injection polymer concentration of well W3.

Figure 7 

Injection polymer concentration of well W4.

Figure 8 

Fractional flow of water for the production well P1.

Figure 9 

Cumulative oil production.

6. Conclusion

In this work, a new optimal control model of DPS is established for the dynamic injection strategies making of polymer flooding. Necessary conditions of this OCP are obtained by using the calculus of variations and Pontryagin’s weak maximum principle. An iterative computational algorithm is proposed for the determination of optimal injection strategies. The optimal control model of polymer flooding and the proposed method are used for a reservoir example and the optimum injection concentration profiles for each well are offered. The results show that the profit is enhanced by the proposed method. Meanwhile, more oil production and higher recovery ratio are obtained. And the injection strategies chosen by engineering judgment are same for all the wells, whereas the optimal control policies by the proposed method are different from each other as a result of the reservoir heterogeneity and the uniform initial conditions.

In conclusion, given the properties of an oil reservoir and the properties of a polymer solution, optimal polymer flooding injection strategies to maximize profit can be designed by using distributed-parameter control theory. The approach used is a powerful tool that can aid significantly in the development of operational strategies for EOR processes.

Acknowledgments

This work was supported by the Natural Science Foundation of China under Grant 60974039, the Natural Science Foundation of Shandong Province of China under Grant ZR2011FM002, the Fundamental Research Funds for the Central Universities under Grant 27R1105018A, and the Postgraduate Innovation Funds of China University of Petroleum.