Research ArticleEnhanced Oil Recovery for ASP Flooding Based on BiorthogonalSpatial-Temporal Wiener Modeling and IterativeDynamic Programming

Shurong Li ,1 Yulei Ge ,2 and Yuhuan Shi 3

1Automation School, Beijing University of Posts and Telecommunications, Beijing 100876, China2College of Information and Control Engineering, China University of Petroleum (East China), Qingdao 266580, China3CNPC EastChina Design Institute Co. Ltd., Qingdao 266071, China

Correspondence should be addressed to Shurong Li; lishurong@bupt.edu.cn

Received 29 January 2018; Accepted 30 July 2018; Published 17 October 2018

Academic Editor: Dan Selişteanu

Copyright © 2018 Shurong Li et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Because of the mechanism complexity, coupling, and time-space characteristic of alkali-surfactant-polymer (ASP) flooding,common methods are very hard to be implemented directly. In this paper, an iterative dynamic programming (IDP) based on abiorthogonal spatial-temporal Wiener modeling method is developed to solve the enhanced oil recovery for ASP flooding. Atfirst, a comprehensive mechanism model for the enhanced oil recovery of ASP flooding is introduced. Then the biorthogonalspatial-temporal Wiener model is presented to build the relation between inputs and states, in which the Wiener model isexpanded on a set of spatial basis functions and temporal basis functions. After inferring the necessary condition of solutions,these basis functions are determined by the snapshot method. Furthermore, a theorem is proved to identify parameters in theWiener model. Combined with the least square estimation (LSE), all unknown parameters are determined. In addition, theARMA model is applied to build the model between states and outputs, whose parameters are identified by recursive leastsquares (RLS). Thus, the whole modeling process for ASP flooding is finished. At last, the IDP algorithm is applied to solve theenhanced oil recovery problem for ASP flooding based on the identification model to obtain the optimal injection strategy.Simulations on the ASP flooding with four injection wells and nine production wells show the accuracy and effectiveness of theproposed method.

1. Introduction

With the old oil fields entering the later period of develop-ment, the moisture content of a reservoir is increasing, andthe oil production is reducing [1]. How to update thetechnique to ensure oil recovery is an important measure tostabilize oil production. ASP flooding is an important tertiaryoil recovery technique which is widely studied [2]. It canenhance the oil production evidently by use of the interactionamong alkali, surfactant, and polymer which can improve thephysicochemical property of a reservoir. The mechanismmodel of ASP flooding is a complex distributed parametersystem. It is hard to get the optimal injection strategy bya common method, because of the features of infinite dimen-sions, spatial-temporal coupling, and complex nonlinear

behavior. In addition, there is a lack of a uniform mathemat-ical model description because of the uncertainty of alkalireaction. It is important to build an accurate and easily tobe applied model in the research of ASP flooding.

In an enhanced oil recovery problem of ASP flooding, theinputs are the injection concentrations of alkali, surfactant,and polymer; the output is the moisture content of produc-tion wells; the state variables contain water saturation, pres-sure, and grid concentrations; and the performance index isusually the net present value (NPV). In applications, manydifferent kinds of methods are used to search the optimalinjection concentration. One kind is to solve the mechanismequations [3]: fluid equations, seepage equations, and so on.However, this method usually involves a lot of math opera-tions. The process of mathematic treatment always needs a

HindawiComplexityVolume 2018, Article ID 9248161, 19 pageshttps://doi.org/10.1155/2018/9248161

huge amount of calculation. Though many scholars havedone researches on modeling of ASP flooding, nearly all theworks are about improving or enriching the primary model.The main work they have done is to consider more influenc-ing factors or inductive research; the model becomes morecomplex and difficult. So the identification method isconsidered to approximate the ASP flooding system so thatthe mathematical operation process can be simplified. It willbe helpful to improve the computational efficiency andreduce the computational complexity of the algorithm [4].

In the current research, the block-oriented nonlinearmodels have been widely used because of their simplestructures and abilities to approximate nonlinear systemsand considering the dynamic characteristics. This kind ofmethod consists of two parts: the static nonlinear block andthe dynamical linear block. The Hammerstein model andWiener model are two common structures. The only differ-ence between them is the sort order of two blocks. For theWiener model, the dynamical linear block is in the front,while for the Hammerstein model, it is in the back [5]. In[6], an optimal two-stage identification algorithm was pre-sented for Hammerstein-Wiener systems where two staticnonlinear elements surrounded a linear block. Reference [7]introduced one Wiener modeling method, in which theoutput was expanded on a set of spatial basis functions andtemporal coefficients. After identifying the temporal coeffi-cients, the final model is built. At present, most researchesabout the block-oriented models focus on a lumped parame-ter system mainly. Very few of them involve the DPS.Though the above methods can get a relatively good model,their essence is a lumped parameter method. In the case ofmodel matching and modeling accuracy and the spatial-temporal characteristics, unmodeled dynamics need befurther studied.

Spatial-temporal decomposition technology [8, 9], whichcomes from Fourier series expansion, is a very useful model-ing method for DPS. A spatial-temporal variable can beexpanded into an infinite number of spatial basis functionsand corresponding temporal coefficients. Generally speaking,the first few primary spatial basis functions can reflect themaximal inner information of the system, which provides agood approximation because of their separation properties[10]. In this way, the spatial-temporal method can obtain afinite-dimensional model. In spite of these advances, a con-ventional spatial-temporal modeling method only considersthe spatial information, while the temporal informationwhich is important for analyzing system performance isignored. The biorthogonal decomposition [11], which haswide applications in image processing field, can expand thedistributed parameter on the spatial basis function and tem-poral basis function. It can reflect both space informationand time information.

An important condition of modeling is to guarantee theaccuracy which is highly dependent on the choice of spatialbasis functions. In particular, Karhunen-Loeve (KL) decom-position, which is called proper orthogonal decompositionand principal component analysis [12, 13], is a popularspatial-temporal decomposition approach to find principalspatial structures and reduce the dimension. Among all

linear expansions, KL expansion is the most efficient and,for a given approximation error, the number of KL basesrequired is minimal. As a result, KL decomposition canhelp to reduce the model dimension and the number ofestimated parameters.

Dynamic programming (DP) is a very good optimizationprocedure [14]. The basic idea is Bellman’s principle ofoptimality. In DP, the decision-making process is dividedinto many stages, and optimization computations are donesequentially from the last stage to the first stage. To overcomethe shortcomings of requiring a very large number of gridpoints (the menace of expanding grid) to get reasonableaccuracy, Luus [15, 16] suggested the use of a small numberof grid points, and to get sufficient accuracy to use DP inan iterative way, where after every iteration, the grid pointsare brought closer together. The details of this method callediterative dynamic programming (IDP) and a computer pro-gram written in FORTRAN for a typical optimal controlproblem are given by Luus [17]. Here, IDP is adopted to solvethe enhanced oil recovery of ASP flooding.

From the above, a new IDP algorithm based on biortho-gonal spatial-temporal modeling is developed to solve theenhanced oil recovery of ASP flooding in this paper. At first,the mechanism model for the enhanced oil recovery of ASPflooding is given. Then the biorthogonal spatial-temporalmodeling is presented to build the relation between the input(injection concentration of displacing agents) and states(water saturation). After giving the necessary condition forthe existence of basis function, the spatial basis functionand temporal basis function are obtained by the snapshotmethod based on the state samples. To identify the parameterin theWiener model, the LSE is adopted. During this process,a theorem is proved to help the identification. In addition, therelation between states and output (moisture content) isestablished by ARMA which is identified by RLS. Further-more, combining the established model with the indexNPV, the approximation optimization problem for ASPflooding is solved by IDP. At last, the proposed method isapplied to solve the enhanced oil recovery of ASP floodingwith four injection wells and nine production wells. Thesimulation result shows the accuracy and generalizationability of the proposed method.

This article is organized as follows: The mechanismmodel description of ASP flooding is described in Section 2.Section 3 presents the biorthogonal spatial-temporal Wienermodeling method based on biorthogonal decomposition. InSection 4, the optimization procedure with IDP is given. InSection 5, the simulation process and result with ourspatial-temporal modeling method and IDP are shown.Finally, a few conclusions are summarized in Section 6.

2. Mechanism Model Description for theEnhanced Oil Recovery of ASP Flooding

The following are the basic assumptions for ASP flooding:

(1) The stratum satisfies heterogeneity, the wholereservoir is isothermal, and the adsorption

2 Complexity

process complies with the Langmuir isothermaladsorption equation.

(2) The oil displacement system is composed of alkali,surfactant, and polymer. All displacing agents existin the water phase, while only the surfactant existsin the oil phase.

(3) The Darcy law is fit for the flow of the oil phase andwater phase.

(4) The phase equilibrium is set up instantly for allkinds of adsorptions which satisfy the generalizedFick’s law.

For a reservoir with the region x, y, z ∈Ω,Ω ∈ R3, ASPflooding consists of five main components: oil, water,polymer, surfactant, and alkali. Based on [18, 19], the belowmathematical model of ASP flooding can be drawn:

The seepage continuity equation for the oil phase is

∇ ⋅KkroBoμo

∇ po − ρogh + qo =∂∂t

ϕ 1 − SwBo

1

The seepage continuity equation for the water phase is

∇ ⋅Kkrw

BwRkμw∇ pw − ρwgh + qw = ∂

∂tϕSwBw

2

The adsorption diffusion equation for the polymer is

∇ ⋅KkrwcpBwRkμw

∇ pw − ρwgh + ∇ ⋅ Dw +Dwp

ϕpSwBw

∇cp

+ qc =∂∂t

ρr 1 − ϕ Crp +ϕpSwcpBw

3

The adsorption diffusion equation for the surfactant is

∇ ⋅KkrwcwsBwRkμw

∇ pw − ρwgh + ∇ ⋅KkrocosBoRkμo

∇ po − ρogh

+ ∇ ⋅ Do +DosϕsSoBo

∇cos + ∇ ⋅ Dw +DwsϕsSwBw

∇cws

+ qd =∂∂t

ϕsSocosBo

+ ϕsSwcwsBw

+ ρr 1 − ϕ Crs

4

The adsorption diffusion equation for the alkali is

∇ ⋅ DOHϕSwBw

∇cOH + ∇ ⋅KkrwcOHBwRkμw

∇ pw − ρwgh

+ ROH + qe =∂∂t

Ka

1 + KbcOH2 1 − ϕ SwcOH + ϕSwcOH

Bw

5

Furthermore, the dosage limit of displacing agents isdefined as

〠P

i=1∭ΩqincΘindσ ⋅ Ti ≤MΘ max 6

The slug injection strategy is usually used in ASP flood-ing, in which the slug only refers to the stage with displacingagents injected. Suppose that there are P slugs; then

cΘin t =cΘini, ti−1 ≤ t ≤ ti, i = 1, 2,… , P,0, ti−1 ≤ t ≤ t f , i = P + 1

7

P = 3 denotes the three-slug injection which is usuallyadopted in industrial production. Figure 1 gives the details.

The injection concentration limit and slug size limitsatisfy below conditions:

0 ≤ cΘin ≤ cΘ max,

〠P

i=1Ti = tP ,

8

where cΘ max denotes the maximum injection concentrationof displacing agents and Ti denotes the time length of eachslug which has the integer restriction, Ti = ti − ti−1. Note thatP + 1 th stage denotes the water flooding, in which cΘin ≡ 0.

The initial conditions and the terminal constraint ofwater content are

p x, y, z, t ∣t=0 = p0,Sw x, y, z, t ∣t=0 = S0w,cΘ x, y, z, t ∣t=0 = c0Θ,  x, y, z ∈Ω

9

0 500 1000 1500

The i

njec

tion

conc

entr

atio

n of

disp

laci

ng ag

ents

2000 2500 3000

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (T/d)

Figure 1: Three-slug injection scheme. The slug size is theinjection concentration, and the slug length is the injection timeof displacing agents.

3Complexity

The boundary conditions are selected as

∂p∂n

∣∂Ω = 0,

∂Sw∂n

∣∂Ω = 0,

∂cΘ∂n

∣∂Ω = 0

10

Among all the above equations, K is the absolute perme-ability of rock; kro and krw are the relative permeability of oiland water, respectively; po and pw are the pressure of oil andwater, respectively; So and Sw are the oil saturation and watersaturation, respectively; cΘ,Θ = p, s, OH are the concentra-tions of polymer, surfactant, and alkali, respectively; cij is themass concentration of component i in solution j; μo and μware the viscosity of oil and water, respectively; ϕ, ϕp, and ϕsare the rock porosity, reachable porosity of polymer, andreachable porosity of surfactant, respectively; ϕp,s = ϕf a,s,f a, f s are the reachable porosity factors; Rk is descendingcoefficient of relative permeability; Ka is the speed factorof the ion exchange and adsorption capacity; Kb is theadsorption constant of surfactant; vw is the seepage veloc-ity; ROH is the alkali consumption; Crp and Crs are theadsorption quality of unit mass of rock of polymer andsurfactant, respectively; qo and qw are the flow rate of oiland water in the standard state; qc, qd , and qe are the trans-port velocity of shaft flooding agents; Di, i ∈ w, o, OH isthe diffusion coefficient; Dij, i ∈ w, o , j ∈ s, p is thediffusion coefficient of component j in solution i; MΘpdenotes the maximum usage of displacing agents Θ; and∇ is the Hamilton operator, which can be written as ∇ = ∂/∂x i + ∂/∂y j + ∂/∂z k in a rectangular coordinate system.

The flow coefficients of the oil phase and waterphase are ro = Kkro/Boμo and rw = Kkrw/BwRkμw ,respectively. The moisture content f w x, y, z, t is thefunction of all spatial states p, Sw, cp, cs, and cOH andcan be defined as

f w = rwrw + ro

11

The concentration of surfactant is defined as

cs =qwcws + qocos

qw + qo12

Define qin x, y, z, t ≥ 0 and qout x, y, z, t ≥ 0 as theinjection term and production term, respectively, at thewell location x, y, z . In view of all the above descrip-tions, qo, qw, qc, qin, qout, and the control variablescΘin only work at the location of injection wells andproduction wells.

Suppose that the location sets of injection wells andproduction wells are ψw = x, y, z j,k, j ∈ κw, k ∈ ϑwj andψp = x, y, z j,k, j ∈ κp, k ∈ ϑp . The definitions of flow terms

(qo, qw, qc, qin, and qout) in (1), (2), (3), (4), and (5) can bedescribed as

qo =− 1 − f w qout, x, y, z ∈ ψp,0, x, y, z ∉ ψp,

qw =

−f wqout, x, y, z ∈ ψp,qin, x, y, z ∈ ψw,

0, x, y, z ∉ ψp ⋃ ψw,

qc =

qwcp, x, y, z ∈ ψp,qwcpin, x, y, z ∈ ψw,

0, x, y, z ∉ ψp ⋃ ψw,

qd =

qwcws + qocos, x, y, z ∈ ψp,qwcsin, x, y, z ∈ ψw,

0, x, y, z ∉ ψp ⋃ ψw,

qe =

qwcOH, x, y, z ∈ ψp,qwcOHin, x, y, z ∈ ψw,

0, x, y, z ∉ ψp ⋃ ψw

13

In the optimization model of enhanced oil recovery forASP flooding, the performance index is functional. In thispaper, the maximum of NPV is adopted as the performanceindex, which means the profit maximization [20].

max  JNPV =t f

01 + χ −t/ta ∭Ω Poil 1 − f w qout t

−〠ΘPΘqincΘin t dσ − Pcost dt,

14

where t f denotes the terminal flooding time, χ denotes thediscount rate, ta denotes the total discounting period, Poildenotes the price of oil, PΘ denotes the price of three displa-cing agents, and Pcost is the production cost per day.

The other physicochemical algebraic equations arepresented in Appendix.

3. Biorthogonal Spatial-Temporal WienerModeling Method

The ASP flooding system is a complex distributed parametersystem which is very difficult to be modeled with a generalmethod. In this paper, a biorthogonal spatial-temporalWiener modeling method based on Karhunen-Loeve decom-position is presented, which is shown in Figure 2. The spatialinformation and temporal information of samples arereflected on the spatial basis function and temporal basisfunction. During the modeling process for ASP flooding,

4 Complexity

the main problem is to identify an appropriate spatial-temporal model according to the input (injection concentra-tion of displacing agents cΘin,Θ = OH, s, p ), output(moisture content f w), and system state (pressure P, gridconcentration cΘ, and water saturation Sw). For easydescription, a set υ is introduced to represent all statevariables. The whole model for ASP flooding can be dividedinto three stages:

(1) Determine the spatial basis function and temporalbasis function according to system state υ x, y, z, t .

(2) Identify the biorthogonal spatial-temporal Wienersystem.

(3) Model for the moisture content f w t of productionwells and corresponding system states υ x, y, z, t .

3.1. Spatial-Temporal Wiener System. The structure of tradi-tional Wiener model is presented in [21]. According to thespecialty of ASP flooding, the entire system is transformedinto the DPS and the output identification is added. Thespecific structure is shown in Figure 3. It mainly consists oftwo parts: the distributed parameter identification for statesand the lumped parameter identification for moisture con-tent. For simplicity, only a one-dimensional method is shownin this paper. The expression of the three-dimensionalmethod is similarly to the one-dimensional situation.

The distributed parameter identification contains a lineardynamical block followed by a static nonlinear block N · :ℝ⟶ℝ, the description is as follows.

υ x, t =N v x, t =N 〠t

τ=0 Ωh x, z, τ cΘin z, t − τ dz + d x, t ,

15

where h x, z, q 1 × 1 is the transfer function of dynamicalblock, τ and t are the time variables, x and z denote spatialvariables defined on the domain Ω, while q is the time for-ward shift operator, v x, t denotes the system intermediatevariable, cΘin z, t ∈ℝ and υ x, t ∈ℝ stand for the inputand state output of the system at time t separately, andd x, t ∈ℝ includes the unmodeled dynamics and thestochastic disturbance. In addition, the integral operator isused for spatial operation and sum operator for temporal

operation [22, 23]. Here only the single-input single-outputsystem is considered for simplicity. It is obvious that thismethod is also applicable to the multi-input multi-outputsystem [24]. This part can realize the modeling betweeninputs and system states.

The output identification, which is also the lumpedparameter identification, contains a static block F · : ℝ⟶ℝ. Its aim is to build the relation between system states andtarget outputs. The formula description is shown in (16).

f w t = F υ x, t + d′ x, t , 16

where f w t ∈ℝ denotes the target output of the system andd′ x, t ∈ℝ stands for the unmodeled dynamics and thestochastic disturbance.

From the above, the whole model for ASP flooding isgiven. Then the problem is to identify N and h according tothe input-state data and F on the basis of state-output data.

3.2. Basis Function Determination Method

3.2.1. Biorthogonal Spatial-Temporal Decomposition. KLdecomposition can acquire a lower dimension model thanother spatial-temporal methods which are expanded by lin-ear basis functions [4]. That is to say, these basis functionsobtained by KL decomposition can reflect the system infor-mation at the most extent when the number of basis functionis constant. Here the KL decomposition is adopted to find thesystem dominant characteristics from sampled data.

For a general system, suppose that the system outputy xi, t N ,L

i=1,t=1 is uniformly distributed on the time andspace domain, the orthogonal spatial basis function isφi x

∞i=1, the temporal basis function is ψi t

∞i=1, and

the coefficient is αi∞i=1. Then the output can be expressed

as follows [25–27]:

y x, t = 〠∞

i=1αiφi x ψi t 17

The basis function is orthogonal; that is,

φi x , φj x =Ωφi x φj x dx =

0, i ≠ j,1, i = j,

ψi t , ψj t =Tψi t ψj t dt =

0, i ≠ j,1, i = j,

18

where ·, · denotes the inner product.In practical application, only a few dominant bases which

can reflect most system information are chosen. Then

yn x, t = 〠n

i=1αiφi x ψi t ,

αi = φi x ψi t , y x, t19

Nonlinear DPS

Basis functiondetermination

Spatial-temporalWiener system

Outputidentification

v(x,y,z,t)ˆ

c𝛩in(t) fw(t)

fw(t)ˆ

v(x,y,z,t)

Figure 2: The schematic diagram for biorthogonal spatial-temporalWiener modeling.

5Complexity

To obtain the dominant bases, we can get the belowminimization problem:

minφi x ,ψi t

 1N1L T Ω

y x, t − yn x, t 2dxdt,

s t   φi x , φi x = 1, φi x ∈ L2 Ω , i = 1,… , n,ψi t , ψi t = 1, ψi t ∈ L2 T , i = 1,… , n

20

Bring in a Lagrangian multiplier and convert (20) into anunconstrained optimization problem:

J = 1N1L T Ω

y x, t − 〠n

i=1αiφi x ψi t

2

dxdt

+ 〠n

i=1λi φi x , φi x − 1 + 〠

n

i=1ηi ψi t , ψi t − 1

= 1N1L T Ω

y2 x, t − 2y x, t 〠n

i=1αiφi x ψi t +ΛΛT dxdt

+ 〠n

i=1λi

Ωφ2i x dx − 1 + 〠

n

i=1ηi

Tψ2i t dt − 1 ,

21 a

Λ =

α1

α2

⋮

αn

T φ1 x 0 ⋯ 00 φ2 x 0 ⋮

⋮ 0 ⋱ 00 ⋯ 0 φn x

ψ1 t

ψ2 t

⋮

ψn t

,

21 b

ΛΛT = 〠n

i=1α2i 21 c

Take the variation of (21-a) with respect to φi x andψi t ; then

δJ = 1N1L T Ω

−2y x, t 〠n

i=1αiψi t δφi x

+ −2y x, t 〠n

i=1αiφi x δψi t dxdt

+ 〠n

i=12λi

Ωφi x δφi x dx + 〠

n

i=12ηi

Tψi t δψi t dt

22

Equation (22) can be rewritten as

δJ =Ω

1N1L T

−2y x, t 〠n

i=1αiψi t dt + 〠

n

i=12λiφi x δφi x dx

+T

1N1L Ω

−2y x, t 〠n

i=1αiφi x dx + 〠

n

i=12ηiψi t δψi t dt

23

The necessary condition of the extreme value for thisfunctional problem is δJ = 0. Since φi x and ψi t can bean arbitrary function, the below equation can be developed:

1N1L T

y x, t 〠n

i=1αiψi t dt = 〠

n

i=1λiφi x ,

1N1L Ω

y x, t 〠n

i=1αiφi x dx = 〠

n

i=1ηiψi t

24

For all sample points in the time domain and spacedomain, T f x, t dt =∑L

t=1 f x, t and Ω f x, t dx =∑Nz=1 f

x, t can be drawn approximately. Here, z = 1, 2,… ,Ncorrespond to x1, x2,… , xN . Then (24) can be written as

1N1L〠L

t=1y x, t 〠

n

i=1αiψi t = 〠

n

i=1λiφi x , 25 a

1N1L〠N

z=1y x, t 〠

n

i=1αiφi x = 〠

n

i=1ηiψi t 25 b

In (25-a), αiψi t can be regarded as the projection ofy x, t on φi x ; that is, αiψi t = y x, t , φi x . Thus,

1N1L〠L

t=1〠n

i=1y x, t

Ωφi ξ y ξ, t dξ = 〠

n

i=1λiφi x 26

Let R x, ξ = 1/N 1/L ∑Lt=1y x, t y ξ, t ; then

〠n

i=1 ΩR x, ξ φi ξ dξ = 〠

n

i=1λiφi x 27

Considering the mutual independence for i = 1,… , n,(27) is true if and only if the below condition is satisfied:

ΩR x, ξ φi ξ dξ = λiφi x 28

N(g)h(x,z,q)

d(x,t)

v(x,t) v(x,t)

d′(x,t)

fw(t)F(g)

c𝛩in(z,t) ˆ

Figure 3: Spatial-temporal Wiener system.

6 Complexity

Similarly, set R t, τ = 1/N 1/L ∑Nz=1y x, t y x, τ ,

and (29) can be obtained according to (25-b).

TR t, τ ψi τ dτ = ηiψi t 29

From the above, the necessary condition for theoptimization problem in (20) is

ΩR x, ξ φi ξ dξ = λiφi x ,

TR t, τ ψi τ dτ = ηiψi t ,

  i = 1,… , n

30

Therefore, the problem of solving (20) is transformedinto the problem to find the solution of necessary condi-tion (30). The snapshot method is introduced to obtainthe basis function. Since the sample data always distrib-utes discretely, the numerical solution method is usuallyadopted to solve (30). An eigenvalue problem of thematrix can be obtained by discretizing (30).

3.2.2. Snapshot Method. The snapshot method [28] can getthe solution of (30) effectively. Assume that the spatial basisfunction φi x and temporal basis function ψi t are thelinear combination of a series of snapshots:

φi x = 〠L

t=1γity x, t ,

ψi t = 〠N

z=1ωizy x, t

31

Substitute (31) into (30); then

1N1L Ω

〠L

t=1y x, t y ξ, t 〠

L

k=1γiky ξ, k dξ = λi 〠

L

t=1γity x, t ,

1N1L T

〠N

z=1y x, t y x, τ 〠

N

z=1ωizy z, τ dτ = ηi 〠

N

z=1ωizy x, t

32

Define the correlation function of two points as follows:

Ctk =1NL Ω

〠L

k=1y ξ, t y ξ, k dξ,

Cxz =1NL T

〠N

z=1y x, τ y z, τ dτ

33

Then the optimization problem in (30) is converted intothe below matrix eigenvalue problem:

Ctkγi = λiγi,Cxzϖi = ηiϖi,

34

where γi = γi1,… , γiLT and ϖi = ϖi1,… , ωiN

T denote theith eigenvectors.

According to (34), a series of γi and ϖi can be obtained.Then substitute them into (31); the spatial basis functionand temporal basis function can be determined. Since matrixC is a symmetric positive semidefinite, the basis functionscorresponding to γi and ϖi are orthogonal. After standardiza-tion treatment, we can get the final spatial basis function φix and temporal basis function ψi t .

3.2.3. Dimension Selection. In the above problem, supposethat the maximal number of the nonzero eigenvalue is K ,which satisfies K ≤min N , L . Sort these eigenvalues in adescending order, then λ1 ≥ λ2 ≥⋯ ≥ λK correspond to thespatial basis functions φ1 x , φ2 x ,… , φK x , and η1 ≥ η2≥⋯≥ ηK correspond to the spatial basis functions ψ1 t ,ψ2 t ,… , ψK t .

For the determination of basis function, the more num-ber the dominant basis function is, the better accuracy thesystem has. But this will lead to the complexity of modeling.So only a few domain bases which can describe most energyare selected here. Define that the total system energy is equalto the sum of all eigenvalues. Then the bigger the value is, themore energy the basis function reflects. For every basisfunction ϕi · with eigenvalue μi, which is obtained bybiorthogonal spatial-temporal decomposition, the propor-tion of energy is

Ei =μi

∑Kj=1μ j

35

Generally speaking, when the proportion is more than99%, it can be concluded that these bases can reflect the mostenergy; the corresponding n is the dimension finally.

Experience shows that for most spatial-temporal systems,the front several basis functions can represent nearly all theenergy. For an arbitrary basis function θi · n

i=1, the belowequation can be obtained [29]:

〠n

i=1y x, t , ϕi · 2 = 〠

n

i=1μi ≥ 〠

n

i=1y x, t , θi · 2 36

This shows that the KL method can get a lower orderthan other methods in modeling.

3.3. Biorthogonal Spatial-Temporal Model. For ASP flooding,the spatial basis function φi x

ni=1 and temporal basis

function ψi tli=1 can be obtained by decomposing the state

sample information with the method in Section 3.2. And theinput basis function ϕi x

mi=1 can be obtained by decom-

posing the injection concentration of displacing agents cΘin

7Complexity

combined with ψi tli=1. Then the expanded biorthogonal

spatial-temporal model is presented as follows:The input cΘin can be expressed as

cΘin z, t = 〠m

i=1ciϕi z ψi t , 37

where ci = cΘin z, t , ϕi z ψi t denotes the coefficient.The unmodeled dynamics and error d x, t can be

expressed as

dn x, t = 〠n

i=1diφi x ψi t , 38

where di t = d x, t , φi x ψi t denotes the coefficient.Suppose that the block h x, z, τ in (15) is absolutely

integrable on time domain 0,∞ at any spatial point x andz. That is to say, the dynamical block is stable. Then themathematical description of h x, z, τ is

hn,l x, z, τ = 〠n

i=1〠m

j=1〠l

k=1αi,j,kφi x ϕj z ψk τ , 39

where αi,j,k ∈ℝ i = 1,… , n, j = 1,… ,m, k = 1,… , l denotesthe constant coefficients corresponding to the basis functionsφi x , ϕj z , and ψk τ , respectively.

Since the unmodeled dynamics and error are addedbetween the two blocks of the Wiener model, we cannot getthe relation between the input and the state directly. Themethod of solving the inverse function is usually used. Toillustrate the mathematical description more precisely, atheorem is presented in this section.

Lemma 1 (see [30]). Assume that y = f u , u ∈D , and u =φ x , x ∈Q, u ∈ Z,D ∩ Z ≠∅ . If both y = f u and u = φ xexist inverse functions and they are u = f −1 y and x = φ−1 u ,respectively, the composite function y = f φ x must exist aninverse function; the function is x = φ−1 f −1 y .

Theorem 1. For a biorthogonal spatial-temporal Wienersystem in Figure 3, suppose that ξi x is the spatial basis func-tion, ςi t is the temporal basis function, ρi is the coefficients,and P · : ℝ⟶ℝ and P−1 · : ℝ⟶ℝ are the inverse func-tions of each other. If υ x, t =N v x, t =∑n

i=1ρiξi x ςi t Pv x, t , for an arbitrary given set of spatial basis functionϖi x and temporal basis function wi t , there must be a setof coefficients μi which can make v x, t =N−1 υ x, t =∑n

i=1μiϖi x wi t P−1 υ x, t true.

Proof. According to the property of the inverse function thatevery implicit function has an inverse function, there must bean inverse function for N · .

As υ x, t =∑ni=1ξi x ρi t P v x, t , P · and P−1 · are

the inverse functions of each other, according to Lemma 1:

v x, t = P−1 1∑n

i=1ρiξi x ςi tυ x, t 40

Take the derivative of (40) with respect to y x, t ; then

dv x, tdυ x, t = dP−1 u

du⋅

dudυ x, t = dP−1 u

du⋅

1∑n

i=1ρiξi x ςi t,

41

where u = 1/ ∑ni=1ρiξi x ςi t υ x, t .

For v x, t =N−1 υ x, t , take the derivative with respectto υ x, t ; then

dv x, tdυ x, t = 〠

n

i=1μiϖi x wi t ⋅

dP−1 udu

⋅du

dυ x, t

= 〠n

i=1μiϖi x wi t ⋅

dP−1 udu

,42

where u = υ x, t .To make sure that v x, t =N−1 υ x, t and υ x, t =

N v x, t are the inverse function of each other, accord-ing to the property of derivatives of inverse functions,(41) has to be equal to (42). Then μiϖi x wi t satisfiesthe below relation:

〠n

i=1μiϖi x wi t = 1

∑ni=1ρiξi x ςi t

43

From our experience, there are infinite pairs of wi t ,ϖi x and μi. Let χ = 1/n∑n

i=1ρiξi x ςi t ; then

〠n

i=1μiϖi x wi t = 1

nχ= 1n2χ

+ 1n2χ

+⋯ + 1n2χ

n

44

If μiϖi x wi t ≡ 1/n2χ, then for an arbitrary given setof wi t and ϖi x , there is the corresponding set of μi =1/ n2χwi t ϖi x , which can make (41) true. Therefore,Theorem 1 is proved.

In practical application, it is no necessary that (43) is trueprecisely. The appropriate values of wi t , ϖi x , and μi canbe selected approximately under a given error. Among allthe combinations, one group of wi t , ϖi x , and μi by theKL theory can be found.

Assume that N · is invertible; then v x, t can beexpressed as (45) according to Theorem 1.

v x, t = 〠n

i=1P−1 υ x, t Biφi x ψi t , 45

where Bi is the identification parameter.

8 Complexity

In addition, v x, t = h x, z, q cΘin z, t + d x, t . That is,

v x, t = 〠t

τ=0 Ω〠n

i=1〠m

j=1〠l

k=1αi, j,kφi x ϕj z ψk t − τ 〠

m

r=1ϕr z ψr τ crdz

+ d x, t46

Uniting (38), (45), and (46), then

〠n

i=1P−1 υ x, t Biφi x ψi t

= 〠t

τ=0 Ω〠n

i=1〠m

j=1〠l

k=1αi,j,kφi x ϕj z ψk t − τ 〠

m

r=1ϕr z ψr τ crdz

+ d x, t = 〠n

i=1φi x 〠

m

j=1〠l

k=1αi,j,k 〠

m

r=1 Ωϕj z ϕr z dz〠

t

τ=0ψk

t − τ ψr τ cr + 〠n

i=1diφi x ψi t

47

Let ϕj,r = Ωϕj z ϕr z dz and Lk,r t =∑tτ=0ψk t − τ cr ;

then

〠n

i=1P−1 υ x, t Biφi x ψi t

= 〠n

i=1φi x 〠

m

j=1〠l

k=1αi,j,k 〠

m

r=1ψr τ ϕj,rLk,r t + 〠

n

i=1diφi x ψi t

48

Project (48) on the temporal basis function ψs t andspatial basis function φh x with the Galerkin method.Considering the orthogonal property of temporal basisfunction, (48) can be inferred.

〠n

i=1 Ωφh x φi x dxP−1 υ x, t Bi

= 〠n

i=1 Ωφh x φi x dx〠

m

j=1〠l

k=1αi,j,k 〠

m

r=1ϕj,rLk,r t

+ 〠n

i=1 Ωφh x φi x dxdi

49

Since the spatial basis function is orthogonal, n equationscan be drawn from (49).

P−1 υ x, t Bi = 〠m

j=1〠l

k=1αi,j,k 〠

m

r=1ϕj,rLk,r t + di, i = 1,… , n

50

Set Lj,k t =∑mr=1ϕj,rLk,r t ∈ℝ, B = B1,… , Bn

T ∈ℝn,

d = d1,… , dn T ∈ℝn, and α j,k = α1,j,k,… , αn,j,k T ∈ℝn; then

P−1 υ x, t B = 〠m

j=1〠l

k=1α j,kLj,k t + d 51

Without the loss of generality, take α1,1 = I. Equation (51)can be rewritten as

d = B −ATL t , 52

where L t = L1,2 t ,… , L1,l t , L2,1 t ,… , Lm,l t , −P−1 υ

x, t T ∈ℝml and A = α1,2,… , α1,l, α2,1,… , αm,l, B ∈ℝn×ml.The only unknown parameter in (52) is A which can be

obtained by a quadratic criterion on the prediction errors[31] as follows:

A = arg minA1L〠L

t=1−L1,1 t −ATL t

2 53

Parameter A can be identified by using the LSEmethod [32]:

A = 1L〠L

t=1L t LT t

−1

−1L〠L

t=1L t L1,1T t 54

So far the value of A has been determined, that is, αand B. Parameter ci is decomposed from the input vari-able; it is the optimization variable actually. Thus, all theunknown parameters have been determined. Accordingto this method, the relation between the input (cΘin) andthe state (υ) can be built.

3.4. Output Identification. To acquire the mathematicaldescription between system state υ and output f w t , themultivariable ARMA identification [31] is introduced whichis identified by the recursive least squares (RLS) [33].Suppose that there are ns states in ASP flooding. The ARMAmodel is

f w t =U0 q−1 f w t +U1 q−1 υ1 x, t +U2 q−1 υ2 x, t+⋯ +Uns

q−1 υns x, t ,55

where U0 q−1 =∑N0i=1U0,iq

−i, U1 q−1 =∑N1i=1U1,iq

−i, U2 q−1

=∑N2i=1U2,iq

−i, and Unsq−1 =∑

Nnsi=1Uns ,iq

−i; q−1 is the back-ward shift operator which means backing up one samplingperiod; U0,U1,… ,Uns

are the matrix polynomial of q−1;N0 denote the time lag for output; and N1,… ,Nns

denotethe time lag for all states.

9Complexity

Convert the above problem into the linear regressionmodel,

f w t = θ ⋅ F x, t , 56

where

F x, t =

f w t − 1 T ,… , f w t −N0T T

υ1 x, t − 1 T ,… , υ1 x, t −N1T T

⋮

υns x, t − 1 T ,… , υns x, t −NnsT T

,

θ = U0,1,… ,U0,N0,U1,1,… ,U1,N1

,U2,1,… ,Uns ,1,… ,Uns ,Nns

57

Identify parameter θ with RLS in (58).

θ t = θ t − 1 + K t f w t − F x, t Tθ t − 1 ,

K t = P t − 1 F x, t F x, t TP t − 1 F x, t + μ−1,

P t = 1μ

I − K t F x, t T P t − 1 ,

58

where K t denotes the weight matrix, 0 < μ < 1 is theforgetting factor, and P t denotes the positive definitecovariance matrix.

Then the modeling for output identification is finished.In conclusion, the whole model for ASP flooding is deter-mined, which is composed of (15) and (55). Adding the netpresent value as the performance index, an optimizationproblem for ASP flooding can be constructed. Then theiterative dynamic programming (IDP) algorithm is appliedto solve the optimal injection strategy.

4. Optimization Procedures

The biorthogonal spatial-temporal Wiener modeling methodin Section 3 is also applicable to a three-dimensional space.For a practical reservoir given in Section 2, which is athree-dimensional system, since the moisture content f w ofproduction wells is related to the water saturation Sw ofcorresponding wells directly, only the relation between Swand f w for the same production well is considered here. Thenbuild the model for ASP flooding with proposed method.Equations (15) and (55) can be transformed into (59-b) and

(59-c). According to the discussions above, the below optimi-zation problem can be drawn:

max  JNPV =t f

01 + χ −t/ta ∭Ω Poil 1 − f w qout t

−〠ΘPΘqincΘin t dσ − Pcost dt,

59 a

Sw x, y, z, t =N v x, y, z, t

=N 〠t

τ=0 Ωh x, y, z, ξ, ζ, ς, τ cΘin ξ, ζ, ς, t − τ dz

+ d x, y, z, t ,

59 b

f w t = F f w t − 1 ,… , f w t −N0 , Sw x, y, z, t − 1 ,… , Sw

x, y, z, t −N1 ,

59 c

where x, y, z is the location of the production well, N0,N1denote the time lag of f w, Sw, and ξ, ζ, ς denotes thecoordinate of an arbitrary point in the three-dimensionalspace. cΘin,Θ = OH, s, p is the injection concentration forthree displacing agents, which is also the optimizationvariable in (59-a)–(59-c).

The next work is to get the optimal injection strategy forthe enhanced oil recovery of ASP flooding based on theapproximation optimization problem.

In this section, the IDP [34] is adopted to solve thisproblem based on the biorthogonal spatial-temporal Wienermodel. For simplicity, only the injection concentration isselected as the control variable. The specific solution proce-dure for ASP flooding is presented as follows:

(1) Divide the whole process of ASP flooding into P + 1stages. The optimization variable is cΘin k , k = 1,… ,P. Note that cΘin P + 1 ≡ 0.

(2) In initialization, give the slug length and select theinitial control feasible region rΘin and the initial con-

trol c 0Θin. The number of the initial discrete state grid

is M. Select R values for every grid. Set contractionfactor γ, maximal iteration lmax, and convergenceaccuracy ε. Start from generation l = 1.

(3) Set the current control domain r lΘ = rΘin.

(4) In the current feasible region, generate M − 1injection concentrations with a uniform strategy

[35] in the region c∗ l−1Θin k − r l−1

Θ k , c∗ l−1Θin k +

r l−1Θ k in which c∗ l−1

Θin is the optimal control strat-egy obtained from the previous generation. Specially,

10 Complexity

l = 1 corresponds to the initial condition. Calculatethe system as (59-b), and get M state trajectories.M state values Sw k − 1 , k = 1,… , P + 1 at thebeginning of each stage constitute M state gridsof the corresponding stage.

(5) Start from P + 1 th stage, and calculate the stateand performance index as (59-a)–(59-c). Recordthe result.

(6) Start from the Pth stage, and generate R injectionconcentration for every state grid Sw k − 2 as (60).

c lθin P = c∗ l−1

θin P + ζr l−1θ P , 60

where ζ denotes a diagonal matrix in which allelements on diagonal are random numbers belonging

to −1, 1 and c∗ l−1θin P denotes the optimal injection

concentration of the previous iteration. When gener-ating injection concentrations which not conform to(8), we deal with the control variables as follows:

c lΘin P =

0, c lΘin P < 0,

cΘ max, c lΘin P > cΘ max

61

Calculate the system state from tP−1 to tP in (8). Then

select the optimal injection concentration c∗ lΘin P + 1

which is closest to the value of current tP and calcu-late the system state from tP to tP+1. Compute thetotal performance index from tP−1 to tP+1. Compare

and record the injection concentration c∗ lΘin P with

the maximal NPV for every grid at stage P.

(7) Shift the time step forward, and repeat step (6) untilthe first slug. Compute the optimal performanceindex of the whole time domain as (59-a), and save

the corresponding injection concentration c∗ lΘin k

with the maximal NPV.

(8) Shrink the feasible domain of decision variables:

r l+1Θ k = γr l

Θ k , k = 1, 2,… , P + 162

Take the optimal injection concentration obtainedfrom step (7) as the center of the feasible domain inthe next iteration.

(9) Add the iteration time l = l + 1. If Jnew − Jold ≥ ε,go to step (4), and continue computing anditerating. If Jnew − Jold < ε, then save the valuesof optimal injection concentration, and finish theoptimization process.

As for the physicochemical equation constraints, thepenalty function method [36] is used to deal with theseconstraints. The other process keeps the same.

5. Solutions of Enhanced Oil Recovery for ASPFlooding Based on Spatial-TemporalModeling and IDP

5.1. Reservoir Description. Suppose that the reservoir of ASPflooding consists of four injection wells and nine productionwells. All wells distribute uniformly; there is one injectionwell at the center of every four production wells. The distri-bution of wells is shown in Figure 4.

5.1.1. Reservoir Parameters. The length is 630m, the width is630m, and the thickness is 19.990m. There are 7 layers in all,the thickness of each layer is 2.857m, and the net thickness is1.4286m. The depth of the upper surface is 2420m, theporosity of every layer is 0.3, and the pore volume is1.1097× 106m3. The initial grid concentration of ASP is0 g/L. The initial permeability, initial pressure, and initialwater saturation are shown in Figures 5–7. The grids of thereservoir are divided into three directions x, y, and z. Thegrids in x and y are divided into 21 units, and the grids in zare divided into 7 units. The total number is 21× 21× 7.

The water injection rate of each injection well is 83m3/d.The recovery rate of production wells is defined in Table 1.Using reservoir simulation software CMG 2010, the wholeoil production process can be obtained by simulationbased on the mechanism model of ASP flooding in Sec-tion 2. The whole production time lasts for 96 months,and the sampled output can be obtained by the snapshotmethod. The injecting time of ASP flooding is served as theinitial time, so the total number of time nodes is L = 97and the total number of space nodes is N = 21 × 21 × 7.The other reservoir parameters for ASP flooding keepthe same with [37].

5.2. Modeling and Verification for ASP Flooding. Given theASP flooding above, we build the approximate model withthe method proposed in this paper. In order to motivate thesystem sufficiently, simulations are executed 50 times on

ºS1-3

ºS2-3ºS2-2

ºS3-3ºS3-2ºS3-1

ºS2-1

ºS1-1 ºS1-2

ºL1 ºL2

ºL3 ºL4

Figure 4: The distribution diagram of well position.

11Complexity

1.00

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

S1-3

S2-3

S3-3

S1-2

S2-2

S1-1

S2-1

S3-1

S3-2

L2

L4

L1

L3

Figure 7: The distribution diagram of initial water saturation.

Table 1: The liquid volume quantity of production wells.

Wells S1-1 S1-2 S1-3 S2-1 S2-2 S2-3 S3-1 S3-2 S3-3

m3/day 20.75 41.5 20.75 83 41.5 41.5 20.75 41.5 20.75

S1-3 1505

1364

1222

1080

939

797

655

514

372

231

89

S2-3

S3-3

S1-2

S2-2

S1-1

S2-1

S3-1

S3-2

L2

L4

L1

L3

Figure 6: The distribution diagram of initial pressure. The unit is MPa.

Table 2: RMSE for different numbers of basis function.

n 1 2 3 4 5

RMSE 0.0751 0.0502 0.0296 0.0209 0.0197

19,332

17,414

15,496

13,576

S1-3

S2-3

S3-3

S1-2

S2-2

S1-1

S2-1

S3-1

S3-2

L2

L4

L1

L2

11,857

9738

7819

5900

3981

2062

143

Figure 5: The distribution diagram of initial permeability.

12 Complexity

CMG 2010. The software randomly generates differentinjection strategies for every month, and the process ofinjection totally lasts for 48 months. After obtaining thesamples, the modeling process with the proposed method iscarried out on MATLAB R2016b.

In order to improve the generalization of the identifi-cation model, the below basis function selection methodis used here. (a) Apply the biorthogonal spatial-temporaldecomposition method to get the spatial basis function,the temporal basis function, and coefficients of the 50 setsof grid water saturation samples. (b) Use every group ofbasis function to reconstitute the system states with other49 groups of coefficients, and calculate the average meansquare errors of all sampling points for these 49 times.(c) After executing this process for 50 times, select thegroup of basis function with the smallest mean squareerror to serve as the final basis functions for modeling.Then build the spatial-temporal model for ASP floodingwith the proposed method. The specific description isgiven in (59-a)–(59-c).

Regarding the determination of the number of the spatialbasis functions, the error index is defined as follows:

RMSE = ∑e x, y, z, t 2dxdydzdxdydz∑Δt , 63

where e x, y, z, t = Sw x, y, z, t − Sw x, y, z, t .Generally speaking, the more number the basis function

is, the more information of the system model reflects, andthe higher accuracy the model has. But at the same time,the model will be sensitive to the external disturbance andthe generalization ability will decrease. Besides, the dimen-sion of the model increases. On the other hand, the lessquantity the basis function has, the model will reflect lessinformation and the error may be very big. It is importantto choose the proper amounts of the spatial basis functionand temporal basis function. In order to test the influenceon the accuracy of modeling, we choose the differentnumbers of basis function to reconstruct the system with

AlkaliSurfactantPolymer

1009080706050Time (month)

The i

njec

tion

mas

s con

cent

ratio

n (k

g/m

3 )

4030201000

0.5

1

1.5

2

2.5

3

3.5

4

(a) The injection mass concentration

25201510

200

500

Time (month) Grid x

Wat

er sa

tura

tion

1000.40.50.60.70.80.9

1

(b) The output of real system

25201510200

500

Time (month) Grid x

Wat

er sa

tura

tion

1000.40.50.60.70.80.9

1

(c) The output of modeling

Figure 8: Comparison diagram of water saturation.

13Complexity

the coefficients and calculate the average RMSE which isshown in Table 2.

Note that the number of spatial basis function is equal tothat of temporal basis function. This is decided by the matrixoperations. According to Table 2, the accuracy of the modelincreases with the number of basis function. However, whenthe number is over 4, the RMSE of the model increasesslightly. So the number of both spatial basis function andtemporal basis function adopted in this paper is 4. Besidesthat, the time lag of moisture content is N0 = 5; the time lagof water saturation is N1 = 6.

With regard to the whole modeling process for ASPflooding, the errors are as follows:

The mean RMSE for grid water saturation is 0.0285, andthe mean RMSE for the moisture content of production wellsis 1.1378%. This demonstrates the good modeling ability.

In order to verify the generalization ability, the injectionconcentrations of ASP flooding are given randomly:

cpin = 2 7, 1 4, 1 1 kg/m3,

cOHin = 3 7, 2 8, 1 3 kg/m3,csin = 2 9, 1 3, 0 6 kg/m3

64

The displacing agent injection totally lasts for 48 monthswhich are divided into 3 slugs uniformly, and the rest time iswater flooding. The specific injection can be found inFigure 8(a). Because the whole system is five-dimensionalconsidering the time and value of water saturation, it cannotbe plotted in the picture directly. We choose the 21 gridsranging from 1, 11, 3 to 21, 11, 3 at the third layer; thenthe result and error of modeling are shown in Figures 8 and 9.

To analyze the generalization ability further, we computethe error with (63). The mean absolute errors for themoisture content of production wells and for the grid watersaturation are 1.2970% and 0.0318%, respectively. Figure 10compares the moisture content results. The dotted linedenotes the moisture content of numerical reservoirsimulation software CMG 2010. It represents the real result,which is called the original curve in Figure 10. And thecontinuous line is the moisture content of the proposedmodeling method. It can be known that the errors of nine

production wells are very small, which verifies that the wholemodel has better generalization ability.

From the above, the biorthogonal spatial-temporalWiener modeling method can model for ASP flooding well.It has good modeling accuracy and generalization ability.

5.3. Solutions of Enhanced Oil Recovery for ASP Flooding withIDP. For now, the specific spatial-temporal identificationmodel for ASP flooding has been acquired. Then apply theIDP algorithm according to Section 4 to optimize the optimalinjection strategy for alkali, surfactant, and polymer.

5.3.1. Parameter Setting. P = 3, Pp = 6 85$/kg, Ps = 8 31$/kg,Pa = 2 86$/kg, the production cost per day is Pcost = 500$/day, the discount rate χ = 1 5 × 10−3, and the oil price isreferenced to the WTI price [18], that is, 55$/barrel, which

25201510200

500

Time (month) Grid x

Erro

r of w

ater

satu

ratio

n

100−0.15

−0.1

−0.05

0

0.05

0.1

Figure 9: Modeling error of water saturation.

50

60

70

80

90

100

0 10 20 30 40 50

Time (month)

Com

paris

on o

f moi

sture

cont

ent (

%)

60 70 80 90 100

Original curveModeling curve

Figure 10: Comparison for the moisture content of nineproduction wells.

00

0.5

1

1.5

2

2.5

3

3.5

4

10 20 30 40 50 60Time (months)

The i

njec

tion

mas

s con

cent

ratio

n (k

g/m

3 )

70 80 90 100

MM: alkaliMM: surfactantMM: polymer

PM: alkaliPM: surfactantPM: polymer

Figure 11: Comparison of the optimal injection concentration.

14 Complexity

is equal to 0.346$/L. r = 0 85, cΘ max = 4, R = 5,M = 3, ε = 1 ×10−4, η = 0 7,MOH = 1300 t,Ms = 1000 t, andMp = 700 t. Theinitial control strategy is cOHin = 2 9 kg/m3, cpin = 2 1 kg/m3,and csin = 1 8 kg/m3.

In order to illustrate the accuracy of the proposedmethod, the mechanism model is also solved with the IDPalgorithm in Section 4 as a comparison. Before computation,the mechanism model is processed by the finite differencemethod. The results are presented in Figures 11–15. Forsimplicity, we use “PM” to denote the proposed method inthis paper and “MM” to denote the optimization based onthe mechanism model. After finishing the optimization, theoptimal injection strategy of the proposed method is

cOHin = 3 5, 2 59, 1 8 kg/m3,cpin = 2 26, 1 17, 0 69 kg/m3,

csin = 2 42, 1 5, 0 49 kg/m3

65

In Figure 11, the green line and black line denote thecontrol strategies of MM and PM corresponding to themechanism model and identification model, respectively. Itis clear that the result of PM is very close to that of MM,which can explain the effectiveness of PM partly.

The moisture content and oil production are shown inFigures 12 and 13. The full line denotes the moisture of nineproduction wells, and the dotted line denotes the oilproduction of nine production wells. In order to show theoil flooding effect clearly, the average moisture content andaverage oil production for nine production wells arecomputed and showed in Figures 14 and 15. From thecomparative analysis, the moisture content of the proposedmethod is very close to that of the mechanismmodel for bothmoisture content and oil production. For the result of theproposed method, the moisture content descends obviouslyat the early injection stage. This is because the more produc-tion is gained at the early stage, the more NPV is discountedto the present time. While the injection of displacing agentscan improve the oil production obviously, the injection

0

100

80

60

40

10 20 30 40 50Time (months)

The m

oistu

re co

nten

t of p

rodu

ctio

nw

ells fw

for P

M (%

)

Oil

prod

uctio

n of

nin

e pro

duct

ion

well

s (t)

60 70 80 90 100

S1-1S1-2S1-3

S2-1S2-2S2-3

S3-1S3-2S3-3

500

0

1000

1500

Figure 12: The moisture content and oil production of production wells for PM.

0

100

80

60

40

10 20 30 40 50 Oil

prod

uctio

n of

nin

e pro

duct

ion

well

s (t)

Time (months)

The m

oistu

re co

nten

t of p

rodu

ctio

nw

ells fw

for M

M (%

)

60 70 80 90 100

500

0

1000

1500

Figure 13: The moisture content and oil production of production wells for MM.

15Complexity

concentration at the early stage has a larger value. But after aperiod of development time, since the oil reserves aredecreasing and oil flooding efficiency is attenuating, themoisture content is rising gradually.

To illustrate the optimization effect, some essentialindexes are calculated by the MM and proposed method.The specific result is shown in Table 3.

By comparison, the results of the PM and MM are veryclose for nearly all the indexes. Although the productionand NPV have little differences, these can be ignored com-pared with their big cardinal number, since only a tiny differ-ence on injection strategy can accumulate a certain influenceon NPV. After being substituted by the identification model

in this paper, the oil production is decreased by 311t, andthe NPV is decreased by 0.08 million $ than the optimizationresult of the mechanism model. The relative error of oilproduction is only 0.22%, and that of NPV is only 0.11%.

In a word, the proposed method is effective for themodeling and optimization. It has good accuracy and gener-alization and can be used to solve the enhanced oil recoveryproblem of ASP flooding in oil exploitation.

6. Conclusion

In this paper, an iterative dynamic programming (IDP)method based on biorthogonal spatial-temporal Wienermodeling is developed to solve the enhanced oil recoveryfor alkali-surfactant-polymer (ASP) flooding. The maincontributions are the following: (1) a biorthogonal spatial-temporal Wiener modeling method is developed; (2) in orderto identify the static nonlinear part, a theorem is given andproved; (3) combined with IDP, the mechanism model isreplaced by identification which is built by the proposedmethod in solving the optimization problem of ASP flooding.The simulation result shows the accuracy and generalizationability of the proposed method. This paper presents aneffective method to solve the enhanced oil recovery of ASP

0

4

4.5×106

3.5

3

2.5

2

1.5

1

0.5

010 20 30 40 50

Time (months)

The a

vera

ge o

il pr

oduc

tion

of 9

pro

duct

ion

well

s (%

)

60 70 80 90 100

MMPM

Figure 15: The average oil production comparison of 9production wells.

Table 3: Parameter comparison.

ParametersMechanism model

(MM)Proposed method

(PM)

Consumption ofpolymer (t)

659.312 656.563

Consumption ofsurfactant (t)

701.184 702.777

Consumption of alkali (t) 1258.944 1257.350

Oil production (t) 143,371 143,060

NPV (million $) 47.05 46.97

0

100

95

90

85

80

75

70

65

60

5510 20 30 40 50

Time (months)MMPM

The a

vera

ge m

oistu

re co

nten

t com

paris

onof

pro

duct

ion

wells

(%)

60 70 80 90 100

Figure 14: The average moisture content comparison of production wells.

16 Complexity

flooding which can avoid solving the complex mechanismequations. It is also suitable for solving other complexoptimization problems in engineering.

Appendix

The relevant physicochemical algebraic equations areshown below.

A. Alkali Consumption

Many kinds of conditions can influence the alkali consump-tion, while only the adsorption consumption, acidoidconsumption, and heavy metal ion consumption are consid-ered in this paper to describe the mechanism well and simply.The alkali consumption is given through the chemicalreaction term ri, which is defined as follows:

ROH = −ϕSw∂∂t

r1 + r2 + r3 , A 1

where ri is the alkali consumption per unit volume ofdifferent effects; the specific forms are as follows [20]:

(a) The rapid alkali consumption r1 is caused by the ionexchange between Na+ in solution and H+ on therock surface. This effect can be expressed as theLangmuir isothermal adsorption equation:

r1 = r10 a1cOH1 + a1cOH

, A 2

where r1 is the maximal alkali consumption of thiseffect and a1 is the coefficient; they are set byexperimental data.

(b) The alkali consumption r2 caused by acidoid is

r2 = r2 HAw, cOH A 3

The curve of r2 under different acidity and alkalinitycan be obtained through an experiment.

(c) The alkali consumption r3 caused by Ca2+, Mg2+, andacidoid in oil is

r3 = r3 KCa2+ , KMg2+ , cOH , A 4

where KCa2+ and KMg2+ denote the solubility product.

B. Surface Tension

Surface tension varies with the concentration of alkali,surfactant, and polymer. The form is [20]

σ = σ cOH, cp, cs B 1

C. Capillary Pressure

The capillary pressure of the compound system can bedescribed as the function of oil/water capillary pressure andsurface tension; that is [20],

pcow Sw = CPC ⋅ϕ

Ka⋅ 1 − Sn

NPC , C 1

where CPC and NPC denote the constant and Sn is the wetphase saturation.

D. Surfactant Adsorption

Crs = C0rs

ascs1 + ascs

1 − bspH‐7

pHmax‐7, D 1

where C0rs and as are related to the ion concentration,

pHmax is the pH value of the injection fluid, and bs isthe coefficient [37].

E. Polymer Adsorption

Considering that the relation of polymer adsorption isreversible with salinity and is irreversible with concentration,the adsorption can be described as follows according to theLangmuir isothermal adsorption law [36]:

Crp = Cmaxrp

a1cp1 + b1cp

, E 1

where Cmaxrp is the maximal adsorption capacity to the rock

surface of the polymer under different salinity and a1 andb1 are the adsorption equilibrium constants.

F. Relative Permeability

The relative permeability of oil phase kro and water phase krwis usually given by the interpolation method in the process ofmodel solution. Here, the formula proposed by Ge et al. in[38] is adopted.

Krw,ro = A ⋅ 1 − SwB ⋅ Sw − C D, F 1

where A, B, C, andD denote the coefficients which areidentified by the data of kro and krw.

Since the addition of the displacing agent, the permeabil-ity reduction factor has to be considered.

Rk = 1 + Rmaxk − 1 ⋅ qθqmaxθ

, F 2

where qθ and Rk denote the adsorption capacity of displacingagent and permeability reduction factor of the water phaseunder different salinity, respectively.

17Complexity

G. Liquid Viscosity

For the displacing liquid of ASP flooding, the relationbetween viscosity μ and shear rate γ can be expressed bythe Meter equation as [38]

μASP = μw +μ0p − μw

1 + γ/γ1/2 Pa−1, G 1

where γ1/2 is the shear rate when viscosity μ is equal to themean value of μ0p and μw and Pa is the economic coefficient.

H. Residual Saturation

The residual saturation is relevant to the capillary numberNc; it can be obtained through the experiments [37].

Nc =∑ jμjV j

σ,

Srj = Srj Nc ,H 1

where Srj denotes the residual saturation of phase j.

Data Availability

The detailed mechanism model and model parameters ofASP flooding are given in the manuscript. The results arecomputed on the MATLAB software with the model andgiven parameters, while the relevant results are also givenin the manuscript.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural ScienceFoundation (Grant nos. 60974039 and 61573378), theNatural Science Foundation of Shandong Province (Grantno. ZR2011FM002), and the Fundamental Research Fundsfor the Central Universities (Grant no. 15CX06064A).

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