Abstract Thermal recovery can entail considerably highercosts than conventional oil recovery, so the use of compu-tational optimization techniques in designing and operatingthese processes may be beneficial. Optimization, however,requires many simulations, which results in substantial com-putational cost. Here, we implement a model-order reduc-tion technique that aims at large reductions in computationalrequirements. The technique considered, trajectory piece-wise linearization (TPWL), entails the representation ofnew solutions in terms of linearizations around previouslysimulated (and saved) training solutions. The linearized rep-resentation is projected into a low-dimensional space, withthe projection matrix constructed through proper orthog-onal decomposition of solution “snapshots” generated inthe training step. Two idealized problems are consideredhere: primary production of oil driven by downhole heatersand a simplified model for steam-assisted gravity drainage,where water and steam are treated as a single “effective”phase. The strong temperature dependence of oil viscosity
M. A. H. Rousset (�) · L. J. DurlofskyDepartment of Energy Resources Engineering,Stanford University,Stanford, CA 94305-2220, USAe-mail: [email protected]
is included in both cases. TPWL results for these systemsdemonstrate that the method can provide accurate predic-tions relative to full-order reference solutions. Observedruntime speedups are very substantial, over 2 orders of mag-nitude for the cases considered. The overhead associatedwith TPWL model construction is equivalent to the com-putation time for several full-order simulations (the preciseoverhead depends on the number of training runs), so themethod is only applicable if many simulations are to beperformed.
The production of unconventional oil often requires ther-mal recovery methods. Steam injection has been used forheavy-oil recovery for decades, and in recent years, steam-assisted gravity drainage (SAGD) has been applied for insitu production in oil sands. In some cases, the goal is tointroduce heat into the formation to facilitate chemical reac-tions. Examples include the in situ upgrading of bitumen(in the case of oil sands) or the in situ conversion of kero-gen (in the case of oil shales). Higher temperatures thanthose attainable with steam are required in such cases, andthese can be achieved through the use of downhole electricalheaters.
Thermal recovery processes are often expensive to imple-ment and in some cases can have significant environmentalimpact. The use of formal optimization procedures could
substantially benefit the economics and mitigate the risksassociated with thermal operations. This may be chal-lenging in practice, however, since simulations of thermaloperations are often computationally demanding, and opti-mization procedures typically require a large number ofsimulations.
Various approaches have been applied in related appli-cation areas for accelerating the many simulations that areneeded for computational optimization. Of particular inter-est here is the use of reduced-order models (ROMs), wherethe full-order (high-dimensional) representation is projectedinto a low-dimensional subspace in which most of the rele-vant dynamics can be captured. Such approaches have beenapplied for some subsurface flow problems, as discussedbelow, but they do not appear to have yet been studiedfor thermal problems. Our intent here is to extend ROMprocedures to prototype thermal models. This represents akey first step in the development of ROM-based surrogatemodels that can be used for the optimization of thermalproblems.
Many previous ROM approaches have applied properorthogonal decomposition (POD) to define a projectionmatrix that relates high-dimensional states to low-ordervariables [2, 16]. The POD approach requires first con-structing the so-called “snapshot matrix” which contains, asits columns, the state vectors from one or more full-ordertraining simulations (in an oil–water problem, these statevectors are simply the pressure and saturation fields at eachtime step). A singular value decomposition (SVD) of thesnapshot matrix is then performed, and the singular vec-tors associated with the largest singular values constitute theprojection matrix.
POD-based ROM approaches for subsurface flow havebeen presented by a number of investigators [8, 10, 15,20, 22]. For nonlinear problems, however, these techniquesachieved relatively modest runtime speedups—a factor of10 at best. Much larger runtime speedups (∼O(100 −1,000)) can be achieved, for nonlinear problems, throughthe use of the more approximate trajectory piecewise lin-earization (TPWL) procedure, which combines the PODrepresentation with linearization around saved states. TPWLwas originally developed by Rewienski and White [18] andhas since been used in a number of different applicationareas [4, 5, 21]. Initial applications for subsurface flow mod-eling were focused on two-phase oil–water systems [7, 12].Recent work has targeted the application of TPWL to morecomplex compositional problems [11].
In this paper, we apply TPWL to idealized thermal mod-els. We consider two such systems—primary productionof oil driven by downhole heaters and a simplified modelfor SAGD, in which we treat water and steam as a single“effective” phase. These models lack much of the complex-ity of actual thermal problems, but they do retain some of
the important nonlinear character, such as the strong depen-dence of oil viscosity on temperature, associated with ther-mal recovery processes. Therefore, this work can be viewedas an initial exploration of the application of reduced-ordermodeling to complex thermal problems.
This paper proceeds as follows. We first introduce thegoverning equations for the two idealized thermal mod-els. The TPWL formulation for these equations, along withthe underlying POD procedure, is then described. Next, wepresent results for examples of heater-driven primary pro-duction in three-dimensional systems and idealized SAGDfor cross-sectional models. We conclude with some generalobservations and suggestions for future work.
2 Governing equations for thermal processes
In this section, we present the equations governing pri-mary production driven by downhole heaters and our ide-alized model of SAGD. In both cases, the governing equa-tions entail statements of mass conservation combined withDarcy’s law, along with an energy conservation equation.
2.1 Primary production driven by downhole heating
In the case of primary production driven by downhole heat-ing, the fluid (oil) is modeled as a single component in asingle phase. Mass conservation is expressed as
∂
∂t(φρo) = ∇ · [ρoλok(∇p − ρog∇D)] − ρoq
wo , (1)
where t is time, φ designates porosity, ρo is the oil density,λo = 1/μo is the oil mobility (μo is the oil viscosity), kis the absolute permeability, assumed to be a diagonal ten-sor, p is pressure, g is gravitational acceleration, D is depth,and qw
o represents the oil source/sink term (the superscriptw denotes well and tilde means per unit volume).
Energy conservation is described as follows:
∂
∂t[φρoUo + (1 − φ)Ur] = ∇ · (ρoHouo) + ∇ · (κ∇T )
−qH − ρoHoqwo . (2)
Here, Uj designates the internal energy of the oil (j = o) orthe rock (j = r), where Uj = Cj(T − T 0), Cr is the volu-metric heat capacity of rock, Co is the specific heat capacityof oil, T is temperature, and T 0 is a reference temperature.In addition, Ho is the specific enthalpy of oil and uo is theDarcy velocity, given by uo = −λok(∇p − ρog∇D). The∇ · (ρoHouo) term represents heat transfer through con-vection, while the ∇ · (κ∇T ) term represents conductiveheat transfer, with κ the overall thermal conductivity (hereκ is taken to be a scalar). The ρoHoq
wo term represents the
Comput Geosci
energy transported by the fluid into production wells, andthe term qH designates the energy introduced through thedownhole heaters. We note that the model defined by Eqs. 1and 2 contains some of the key effects that arise in in situupgrading and in situ conversion processes. This model can-not be considered to be an accurate representation of eitherprocess, however, since many important effects, such asthe chemical reactions describing kerogen and/or heavy-oildecomposition, are not included.
In this work, heat injection is modeled by specifying theheater temperature. Numerically, this is achieved by fixingthe grid block temperature to be the heater temperature andthen setting the heat capacity of the rock in this block toa very large number (here, we use Cr = 108 Btu/ft3/◦R).This simple approach is accurate when the size of the heaterwell block is comparable to the size of the actual heater.When the block is much larger than the heater, this treat-ment overestimates the amount of heat injected (though theerror can be mitigated by specifying the block temperatureto be somewhat lower than the heater temperature). We notethat this approach was also used by Fan et al. [9] for the sim-ulation of in situ conversion. A more accurate heater welltreatment, which could also be applied here, was recentlydeveloped by Aouizerate et al. [1].
Boundary conditions at the domain boundaries (here andfor the idealized SAGD problem) are no flux of heat andmass. Details on the finite-volume numerical treatments thatare used to solve the full-order systems associated with theseequations are given in [19].
2.2 Idealized SAGD
Our simplified model of the SAGD process entails a two-phase formulation where we treat water and steam as asingle “effective” phase. Conservation of mass for thissystem is expressed as
∂
∂t(φρjSj) = ∇ · [ρjλjk(∇p − ρjg∇D)] − ρjq
wj , (3)
where j = o, w refers to the oil or water phase. Here, ρj
is the phase density and Sj represents the phase saturation.Phase mobility is given by λj = krj/μj, where krj is thephase relative permeability and μj is the phase viscosity.The source/sink term is designated qw
j .Energy conservation for this case can be stated as fol-
where subscripts o, w, and r designate oil, water, and rock,respectively. The definitions of Uw and uw are analogous tothose for Uo and uo given above. The left-hand side of Eq. 4
represents the accumulation of internal energy (U) in thefluids and rock. Terms representing convective heat transfer,conductive heat transfer, and sources/sinks of heat (throughthe production and injection of fluids) appear on the right-hand side of the equation.
A more realistic SAGD model would also include a vaporphase, and additional terms associated with the heat ofvaporization/condensation of steam would appear in Eq. 4.This full representation would, however, require us to treatphase disappearance and reappearance, and the associatedswitching of primary variables, which is required whenso-called “natural variables” are used in the numerical for-mulation (as is the case here). Variable switching introducescomplications within the context of TPWL because it canlead to incompatibilities between training runs and subse-quent prediction (test) runs. This issue was addressed inrecent work on the application of TPWL for compositionalsystems [11], and it was found that a better approach isto employ molar variables rather than natural variables, asthe use of molar variables eliminates the need for variableswitching. Given that our goal in this work is to perform aninitial investigation into the use of reduced-order treatmentsfor thermal processes, we believe that our idealized modelis sufficient for current purposes. However, if reduced-ordermodels are developed for practical SAGD applications, afull implementation based on the molar formulation (alongthe lines of that presented in [11]) might be required.
3 TPWL representation of thermal models
In this section, we describe the components of the TPWLformulation, including the linearized simulation model, theuse of POD to represent reduced states, and the final TPWLexpression. Key implementation issues are also discussed.Detailed developments of the general TPWL procedure foroil–water systems are provided in [7] and [12], and an oil–gas compositional formulation is presented in [11].
3.1 Linearized simulation equations
For both the primary production (Eqs. 1 and 2) and ideal-ized SAGD models (Eqs. 3 and 4), the full-order systemsare solved numerically using a standard finite-volume dis-cretization. For both cases, the resulting discrete system ofequations can be expressed as
g(xn+1, xn, un+1) = F(xn+1) + A(xn+1, xn)
+Q(xn+1, un+1) = 0, (5)
where x is the state vector containing the primary vari-ables for all grid blocks, u contains the well control values(bottom-hole pressures or BHPs, well rates, heater-welltemperatures), g is the residual vector we seek to drive
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to zero, F represents the flux terms, A the accumulationterms, and Q the source/sink terms. The superscripts n andn + 1 indicate time level. In the primary production prob-lem, x contains the pressure and temperature associatedwith all grid blocks, while in the idealized SAGD formula-tion, the state variables include the pressure, saturation, andtemperature in all grid blocks.
The system defined in Eq. 5 is nonlinear and is solvediteratively using Newton’s method:
J(ν)�x(ν+1) = −g(ν), (6)
where J(ν) is the Jacobian matrix, �x(ν+1) = x(ν+1)−x(ν) isthe update of the unknown vector over the Newton iteration,and g(ν) is the residual vector at iteration ν. The Jacobianmatrix corresponding to the converged solution at time leveln + 1 is given by
Jn+1 = ∂gn+1
∂xn+1. (7)
In order to apply TPWL, we must first simulate one ormore full-order models with the controls u specified to vary(in time) over the range of interest. These simulations arereferred to as “training” runs. The states xi , (converged)Jacobian matrices Ji , and derivatives with respect to the con-trols are all saved at each time step in the training runs.Then, results for new cases involving different time-varyingcontrols can be represented by linearizing Eq. 5 aroundthese previously computed solutions.
This linearization can be written as
gn+1 = gi+1 + ∂gi+1
∂xi+1(xn+1 − xi+1) + ∂gi+1
∂xi(xn − xi )
+ ∂gi+1
∂ui+1(un+1 − ui+1), (8)
where superscripts i and i + 1 designate information thathas been saved from the training runs and superscripts n andn+1 refer to time levels in the new simulation (informationat time level n is known). Equation 8 can be simplified bynoting that gn+1 = g(xn+1, xn, un+1) = 0 upon solutionand gi+1 = g(xi+1, xi , ui+1) = 0 because it is the residualof a training simulation. Introducing Eqs. 5 and 7, Eq. 8 thusreduces to
Ji+1(xn+1 − xi+1) = −∂Ai+1
∂xi(xn − xi )
−∂Qi+1
∂ui+1(un+1 − ui+1). (9)
Equation 9 can be used to solve for xn+1, since all otherterms are known. Because this is a linear equation, no iter-ation is required. All terms in Eq. 9 are, however, in thehigh-dimensional (full-order) space, so this equation wouldstill be time consuming to solve. To address this issue, thenext step in TPWL is to project the terms in Eq. 9 into low-dimensional space. This is accomplished through the use
of POD, applied in this context by, e.g., Van Doren et al.[20] and Cardoso and Durlofsky [7]. We now describe theconstruction of the POD basis matrix.
3.2 POD basis construction
Our description here of the application of POD is for thecase of primary production driven by downhole heaters(we will indicate later the additional treatments requiredfor the idealized SAGD model). For the primary pro-duction problem, there are two unknowns in each gridblock, pressure and temperature. During the training runs,the pressure and temperature states at each time step ofeach run are saved. These states, referred to as “snap-shots”, are assembled into snapshot matrices. Designatingthe pressure snapshots as x1
p, x2p, etc. and the temperature
snapshots as x1T , x2
T , etc., the snapshot matrices are givenby
Xp = [x1p x2
p . . . xLp ], (10)
XT = [x1T x2
T . . . xLT ], (11)
where L is the total number of snapshots. The matrices Xp
and XT are each of dimensions Nc×L, where Nc is the totalnumber of grid blocks in the full-order problem.
We then construct the basis matrices �p and �T byassembling the left singular vectors resulting from theSVD of Xp and XT . Specifically, for Xp, SVD allowsus to write Xp = U�VT , where U is a unitary matrixcontaining the left singular vectors of Xp, � is a diag-onal matrix containing the singular values of Xp (theentries in � are designated σi), and V is a unitary matrixcontaining the right singular vectors of Xp. The basismatrix �p is comprised of some number of the columnsof U.
The number of columns of U that is retained in �p,which we designate lp, can be based on an “energy” crite-rion, as discussed in, e.g., [7, 19]. Under this approach, wefirst compute the total amount of energy in the L nonzerosingular values, Et = ∑L
i=1 σ 2i . The value of lp is then
determined such that Elp = ∑lpi=1 σ 2
i represents a speci-fied fraction of Et . An analogous procedure is applied todetermine the number of columns (lT ) in �T .
An alternate approach for determining the number ofcolumns in the basis matrix, which entails the applica-tion of a basis optimization procedure, was shown to pro-vide improved numerical stability in oil–water problems[12]. It was also shown in that work that TPWL modelaccuracy does not increase monotonically with increas-ing numbers of columns in the basis matrices and that“overfitting” may occur when a large number of columnsis used. This suggests that some amount of numerical
Comput Geosci
experimentation will often be required to determine theappropriate number of columns to include in the basismatrices. This experimentation could entail assessment ofTPWL model performance, for various values of lp andlT (in our case), for new (test) simulations. Such anassessment can be performed very efficiently, as discussedin [12].
After the �p and �T matrices are constructed, they arecombined into a single overall basis matrix �, which allowsus to write
x =[
xp
xT
]
≈[
�p 00 �T
] [zp
zT
]
= �z, (12)
where zp and zT are the reduced variables. Because �
is orthonormal, we also have z = �T x. Defining l =lp + lT , the dimension of � is thus 2Nc × l. Becausel � 2Nc (� is a tall, skinny matrix), we are ableto represent high-dimensional states with relatively fewvariables.
Grid blocks containing heater wells are excluded fromthe POD projection, i.e., they are isolated and representedin full-order space. This is achieved by applying a high-resolution procedure to these grid blocks, as described byHe et al. [12]. Full details on the treatment of heater wellsare provided in [19].
For the idealized SAGD model, there are three unknownvariables per grid block (p, Sw, T), so an additional basismatrix, �S , is required. This matrix is constructed throughapplication of SVD to the saturation snapshot matrix, XS .The representation of the full-order states in terms ofreduced variables is then given by
x =⎡
⎣xp
xS
xT
⎤
⎦ ≈⎡
⎣�p 0 00 �S 00 0 �T
⎤
⎦
⎡
⎣zp
zS
zT
⎤
⎦ = �z. (13)
Fig. 1 Log-permeability field for primary production example. Heaterwells are shown in red and production wells in black
Fig. 2 Variation of oil density with T and p for primary productionexample
3.3 Final POD-TPWL representation
Now, once � has been constructed, the approximation x ≈�z is introduced into Eq. 9, which gives
Ji+1�(zn+1 − zi+1) = −∂Ai+1
∂xi�(zn − zi )
−∂Qi+1
∂ui+1(un+1 − ui+1). (14)
Fig. 3 Oil viscosity for primary production example
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0 200 400 600 800 1000
900
950
1000
1050
1100
1150
Hea
ter
Tem
pera
ture
s (R
)
Time (day)
Training 1,4,5 Training 2 Training 3
0 200 400 600 800 1000100
150
200
250
300
350
BH
P (
psi)
Time (day)
Training 1,2,3 Training 4 Training 5
Fig. 4 Control schedules for the five training simulations
This system is, however, overdetermined because there are2Nc equations (for the primary production problem) andonly l unknowns. We thus reduce the number of equationsby applying constraint reduction, i.e., by premultiplyingEq. 14 by a matrix �T (where � is of the same dimensionsas �):
�T Ji+1�(zn+1 − zi+1) = −�T ∂Ai+1
∂xi�(zn − zi )
−�T ∂Qi+1
∂ui+1(un+1 − ui+1)
(15)
which provides a (reduced) system of l equations in lunknowns.
In this work, we take � = �, which is referred to asa Galerkin projection. This approach is straightforward andprovides reasonable accuracy, though it is strictly optimalonly in certain cases. Another choice for �, which is basedon a least-square formulation and has been found to possessbetter numerical stability properties than Galerkin projec-tion, is applied in [11]. This treatment should be consideredin future work for the thermal systems addressed here.
Applying � = � in Eq. 15 and defining the reducedmatrices,
Ji+1r = �T Ji+1�, (16)
(∂Ai+1
∂xi
)
r= �T ∂Ai+1
∂xi�, (17)
(∂Qi+1
∂ui+1
)
r= �T ∂Qi+1
∂ui+1, (18)
we obtain the final (linear) TPWL representation:
zn+1 = zi+1 − (Ji+1r )−1
[(∂Ai+1
∂xi
)
r(zn − zi )
+(
∂Qi+1
∂ui+1
)
r(un+1 − ui+1)
]
. (19)
Because all terms in Eq. 19 are in the low-dimensional sub-space, the new reduced state vector zn+1 can be computedvery efficiently. Moreover, all of the time consuming oper-ations, such as the construction of the reduced matrices(Eqs. 16, 17, and 18) and the inversion of the reduced Jaco-bian matrix, are computed in the preprocessing stage. Oncea TPWL solution (zn+1) is found, state variables and otherquantities of interest can be reconstructed as required, on the
Fig. 5 Oil production rates forwells P1–P4 for three full-ordertraining simulations for primaryproduction driven by downholeheaters
0 200 400 600 800 10000
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ate
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/d)
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Pro
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ate
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ate
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Training 1Training 2Training 3
Training 1Training 2Training 3
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0 200 400 600 800 1000
900
950
1000
1050
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1150
Hea
ter
Tem
pera
ture
s (R
)
Time (day)
Test 1 Test 2
0 200 400 600 800 1000100
150
200
250
300
350
BH
P (
psi)
Time (day)
Test 1 Test 2
Fig. 6 Control schedules for the two test cases
entire domain or at selected locations such as well blocks,using the POD representation, i.e., by applying x ≈ �z.
3.4 Implementation issues
The definition of the time-varying controls u used inthe training runs can affect the accuracy of the result-ing TPWL model. In this work, these controls are spec-ified heuristically, with the goal of providing states thatare representative of those encountered in subsequent
test runs. When TPWL is used for optimization, it maybe difficult to define appropriate controls for the train-ing runs since the goal of the optimization is to deter-mine these controls. In such cases, retraining strategiescan be applied, in which the TPWL model is periodi-cally reconstructed (based on additional full-order trainingruns) during the course of the optimization. See [12] fordetails.
The selection of the low-dimension training solutions zi
and zi+1, around which the linearization is performed, isalso an important issue in TPWL methods. In the formu-lation used here, this is achieved by finding, at each timestep, the zi that is “closest” to the current TPWL solu-tion zn in terms of the Euclidean norm. The reduced stateat the next time step (from the same training run) defineszi+1.
The runtime (inline) TPWL computations are extremelyfast, though the preprocessing (offline) computationsneeded to construct the model can be significant. Specif-ically, the offline processing entails, in addition to per-forming the full-order training simulations, the construc-tion of the basis matrix � and the generation of thereduced states and matrices from their full-order coun-terparts. Typically, these computations require about thesame amount of computation as the training runs, e.g., iftwo training runs are used, the total overhead—includingtraining simulations and TPWL computations—would beequivalent to approximately four full-order simulations.Thus, it is not appropriate to construct a TPWL modelunless it can be used for a significant number of simu-lations. Because large numbers of simulations are indeedrequired for many applications, including computational
Fig. 7 Comparison of TPWLresults to the referencefull-order solution for primaryproduction driven by downholeheaters (first test case)
P1
Pro
d. R
ate
(stb
/d)
Time (day)
0
20
40
60
80
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Pro
d. R
ate
(stb
/d)
Time (day)
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ate
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Pro
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ate
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/d)
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0
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40
50
60
0
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40
60
80
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Fig. 8 Comparison of TPWLresults to the referencefull-order solution for primaryproduction driven by downholeheaters (second test case)
0 200 400 600 800 10000
20
40
60
80
100
P1
Pro
d. R
ate
(stb
/d)
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optimization, TPWL should be useful in a variety ofsettings.
4 Results for TPWL modeling of thermal recoveryprocesses
We now apply the TPWL procedure, as described above, tomodel primary production driven by downhole heaters andidealized SAGD processes.
4.1 Primary production driven by downhole heaters
The reservoir model for the primary production simulations,shown in Fig. 1, was generated geostatistically using theSGeMS [17] software package. The number of grid blocksin each direction is nx = 50, ny = 50, and nz = 30, fora total of 75,000 cells. Permeability is taken to be isotropic(kx = ky = kz) and follows a log-normal distribution witha mean value of 94 mD and a standard deviation σlogk of 2.Porosity is taken to be constant and equal to 0.3.
Each grid block is of dimensions 5 ft × 5 ft in the hor-izontal plane and 3 ft in the vertical direction. Thus, theoverall model is fairly small, of dimensions 250 ft × 250 ft× 90 ft. The model has four vertical production wells (des-ignated P1–P4), completed in layers 20–26, and 16 verticalheater wells, completed in layers 11–30. The well arrange-ment is shown in Fig. 1. The well spacing and well typesused in these simulations are motivated by those used inin situ conversion projects—see [9] for further discussion.Heater wells are represented in the full-order simulationby fixing the temperature in the grid blocks in which theyare located, as explained above. We use Stanford’s GeneralPurpose Research Simulator (GPRS) for the full-order sim-ulations [6, 14] for this case. GPRS was modified to outputthe vectors and matrices required by the TPWL procedure.
The rock heat capacity, Cr, is set to 35 Btu/ft3/◦R, the oilheat capacity Co to 0.5 Btu/lb/◦R (the units differ betweenCr and Co because Cr is a volumetric heat capacity), and thethermal conductivity, κ , to 25 Btu/ft/day/◦R. These valueswere taken from [3]. Note that temperatures are expressedin degrees Rankine (◦R), where ◦R =◦F + 459.67. These
Fig. 9 Correlations for water viscosity and density
Comput Geosci
Fig. 10 Correlations for oil viscosity and density
quantities are taken to be constant over the entire reservoir,and they are assumed not to vary with temperature. Spatialheterogeneity and temperature variation in these parame-ters could be readily incorporated if the necessary data areavailable.
The fluid is modeled as a single oil component in a singlephase. The oil is compressible, with density varying withpressure and temperature as follows:
ρo(p, T ) = ρ0 exp(cp
(p − p0
)− cT
(T − T 0
)), (20)
where ρ0 = 45 lb/ft3 is the reference density of oil at ref-erence pressure p0 = 14.7 psi and reference temperatureT 0 = 500 ◦R. The coefficient cp is the oil compressibilitywith pressure, taken to be constant and equal to 10−3 psi−1,and the coefficient cT is the oil compressibility with tem-perature, taken to be constant and equal to 10−3 ◦R−1. Thevariation of density with pressure and temperature is shownin Fig. 2.
Oil viscosity varies with temperature according to
μo(T ) = a exp
(b
T − T 0
)
, (21)
where μo is in units of centipoise, a = 0.5 and b = 800 aretwo empirical coefficients, and T 0 is as defined above. Thevariation of viscosity with temperature is shown in Fig. 3.
Note that, over the range T = 540 ◦R to T = 1,400 ◦R, μo
varies by a factor of about 1,000.Using this reservoir model, we run five training simula-
tions with different time-varying BHPs for production wellsand temperature schedules for heater wells. The controlsused in the training simulations are displayed in Fig. 4. Bysetting the controls in the training simulations to cover abroad range of pressures and temperatures, our intent is toconstruct a TPWL model that can capture a wide range ofbehavior.
The oil production rates obtained for the first three train-ing simulations are displayed in Fig. 5. These runs alluse the same BHP controls, though the heater tempera-ture schedules differ from run to run. Note that for thesecond training simulation, in which we apply the mostheat (highest heater temperatures), the oil peak occurs ear-lier and peak production is the largest. The fourth andfifth training simulations entail the same heater tempera-ture schedules as in the first training run, but they havedifferent BHP controls. The oil production rates for theseruns are quite similar to those for the first training run,which indicates that oil production is more sensitive tothe temperature schedule than to BHP schedule. For thesake of clarity, these results are not displayed in Fig. 5.In all cases, of the four production wells, P4 produces
Fig. 11 Oil and water production rates and water injection rate for the three full-order training simulations
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Fig. 12 Comparison of TPWL test-case results to the reference (full-order) solution for oil and water production rates and water injection rate,with �p = 60 psi
the least amount of oil. This is likely due to the factthat the average permeability of the grid blocks intersectedby this well is the lowest of that for the four productionwells.
The training simulations are performed using a max-imum time step of 5 days. This results in 1,132 savedstates for the five training runs (about 226 time stepsper run, on average). The basis matrix �, obtained byapplying POD, contains a total of 146 columns, of which90 correspond to �p and 56 to �T . This basis matrixcaptures 99.99 % of the system energy for pressure and99.997 % of the system energy for temperature (withenergy quantified as explained above). Note that the full-order model contains 2Nc = 150,000 unknowns, incontrast to the 146 unknowns associated with the TPWLmodel.
We now assess the accuracy of TPWL for new cases,with control schedules that differ (for both producer BHPand heater temperature) from those used in the training runs.These runs are referred to as test cases, and their respec-tive BHPs and heater temperatures are displayed in Fig. 6.The TPWL results for the test cases will be compared tothe results from full-order reference solutions provided byGPRS.
Results for oil production from the four producers, gen-erated using TPWL (blue circles) and GPRS (green lines)for the first test case, are displayed in Fig. 7. We see thatthe TPWL model reproduces the reference solution accu-rately, though a slight discrepancy is visible around thepeak in the results for producer P1. Specifically, the errorin the TPWL model in oil rate for P1 at 850 days is about10 %.
In the second test case, we use a different set of BHPand temperature controls, depicted by the dotted linesin Fig. 6. The oil production results for this case, forthe full-order GPRS and TPWL models, are displayedin Fig. 8. We see that TPWL is again able to repro-duce the reference solution accurately. Oil production peaks
earlier in this case than in the first test case, consis-tent with the higher heater temperatures used for theseruns.
Additional results for a variety of test cases, along withthe detailed quantification of error in oil production rates,are provided in [19]. These results illustrate the generalapplicability of TPWL for this problem, though they alsohighlight the fact that TPWL model accuracy may degradeas the controls for the test cases become “further” from thecontrols used in training runs. This motivates the need formodels of TPWL error. Results in [19] also demonstratethe impact of the number of training runs on TPWL modelperformance. There, it was shown that TPWL solution accu-racy increased considerably for some cases when multipletraining runs were used instead of a single training run.
For this case, one full-order simulation using GPRSrequires about 1.7 h. The total TPWL overhead computa-tion is equivalent to about ten full-order simulations (sincefive training runs were used). TPWL solutions requireonly about 12 s, which represents a runtime speedup ofabout 500. Because our TPWL model is implemented in
Fig. 13 Oil saturation maps reconstructed from the full-order refer-ence solution (left) and TPWL (right), with �p = 60 psi
Comput Geosci
(a) Δ p = 40 psi (b) Δ p = 35 psi (c) Δ p = 25 psi
Fig. 14 Comparison of TPWL test-case results to reference (full-order) solutions for oil production rate for various �p values
MATLAB, we expect that this speedup could be furtherimproved.
4.2 Idealized SAGD (homogeneous case)
We now apply the TPWL method for the idealized SAGDmodel. The first reservoir model considered is a homoge-neous two-dimensional y-z system, with ny = 121 andnz = 31. Each grid block has dimensions 2 ft×2 ft, and themodel extends 300 ft in the x-direction. Permeability is setto 5 D (5,000 mD) in the y-direction, and in the z-direction,it is set to 2.5 D (2,500 mD). One injection well is completedin grid block (61, 21) and one production well is completedbelow, in grid block (61, 29), which is the third layer fromthe bottom of the model. The two wells are thus spaced16 ft apart. Results are presented in terms of the pressuredifference between the two wells, �p.
Porosity and thermal conductivity are taken to be con-stant and are set to 0.25 and 24 Btu/ft/day/◦R, respectively.We additionally specify Cr = 34.5 Btu/ft3/◦R, Co =0.17 Btu /lb/◦R, and Cw = 0.423 Btu/lb/◦R. As shownin Fig. 9, water density is taken to be much lower than thatof actual liquid water, with the goal of capturing, to someextent, the behavior of steam. Water viscosity, shown inFig. 9, is also lowered relative to actual values for liquidwater.
Oil viscosity as a function of temperature is displayedin Fig. 10. This variation, which is again very significant,is representative of a fairly heavy oil (very heavy oils candisplay even more variation). In our idealized SAGD sim-ulation runs, we do not model the initial steam circulationphase aimed at lowering oil viscosity prior to steam injec-tion. We may therefore consider the initial oil viscosityin our model as already lowered from an earlier heatingphase. Oil density, shown in Fig. 10, also varies with pres-sure and temperature. Relative permeability functions foroil and water are specified as kro = 0.4(So − 0.2)1.5 and
krw = 0.4(Sw − 0.2)1.5, where the minimum saturation foreither phase is 0.2.
With this model, we run three full-order training sim-ulations using ConocoPhillips’ thermal-compositional sim-ulator (the simulator was modified to output the datarequired by TPWL). For the three training simulations,the pressure differences between the injection and produc-tion wells are set to 30, 45, and 50 psi. The oil produc-tion, water production, and water injection flow rates forthese training runs are presented in Fig. 11. Significantoil production does not occur until the “steam chamber”develops and the oil reaches a sufficiently high temper-ature. The second (small) peak in oil production corre-sponds to the chamber reaching the lateral boundaries of thereservoir.
We save snapshots from these training simulations andperform singular value decomposition of the three snapshotmatrices, as explained above. The resulting � matrix has385 columns; 42 of these columns correspond to �p, 309 to�S , and 34 to �T . As in the previous case, we use energycriteria to determine lp, lS , and lT .
The first set of test results for this case, shown in Fig. 12,are for �p = 60 psi (note that this �p value is outside therange of the training runs). TPWL results are depicted withblue circles and the full-order solution with green circles.We see that the two solutions match closely, which indicates
Fig. 15 Permeability field (log scale) for the heterogeneous modelused for idealized SAGD simulation
Comput Geosci
Fig. 16 Oil and water production rates and water injection rate for the five full-order training simulations (heterogeneous case)
that TPWL is able to capture key effects in this complicatedproblem. For this case, we also reconstruct the full oil sat-uration field, at various times, using the TPWL solution.These saturation fields are shown in Fig. 13 along with thefull-order (reference) results. For this homogeneous perme-ability example, the saturation map obtained from TPWL isvery close to that of the reference solution.
We complete the testing of the TPWL model by com-puting results for three additional �p values: 40, 35, and25 psi. These values all differ from the �p values used inthe training simulations. Results for oil production rates forthe TPWL and reference solutions, shown in Fig. 14, indi-cate that the TPWL model retains a high level of accuracyover this range of �p.
4.3 Idealized SAGD (heterogeneous case)
We now consider a heterogeneous permeability distribution.The system is again two-dimensional, though it now con-tains 121 × 50 grid blocks, each of dimensions 2 ft × 2 ft.The model is 300 ft thick. The permeability field is shownin Fig. 15. The water and oil properties are the same as in theprevious (homogeneous) case. Wells are located 16 ft apart,
in grid blocks (61, 40) for the injector and (61, 48) for theproducer.
For this example, we run five full-order training sim-ulations, from which we save the snapshots and Jacobianinformation at every time step. Oil and water productionrates and water injection rates from these runs are displayedin Fig. 16. The pressure differences between the injectionand production wells for these cases were specified as 50,32, 22, 17, and 15 psi, which cover a wide range of operat-ing conditions. For this case, the overall � matrix contains453 columns (96 columns correspond to �p, 316 to �S , and41 to �T ).
For the first test case with this heterogeneous model, weset �p = 60 psi (which is again above the maximum �p
used in the training runs). TPWL results along with thefull-order reference solution are shown in Fig. 17. Oil pro-duction rates from TPWL display close agreement with thereference solution until about 1,500 days, after which weobserve some mismatch. The error in oil rate at 2,600 days(the local maximum in the TPWL solution) is about 20 %.Water production and injection rates from the TPWL modelare in very close agreement with those from the referencesimulations throughout the entire simulation.
Fig. 17 Comparison of TPWL test-case results to the reference (full-order) solution for oil and water production rates and water injection rate,with �p = 60 psi (heterogeneous case)
Comput Geosci
Fig. 18 Oil saturation maps reconstructed from the full-order reference solution (left) and TPWL (right), with �p = 60 psi (heterogeneous case)
We again reconstruct the full oil saturation field (from theTPWL solution) at various times to enable comparison toreference results. These comparisons are shown in Fig. 18.Although the TPWL solution captures the general behavior,there is clearly more deviation between the two solutions forthis heterogeneous case than was observed for the homo-geneous example (see Fig. 13). We describe below someways in which TPWL accuracy could be improved for thisexample.
Finally, we test the TPWL model for �p values of 40, 35,and 25 psi. Oil production rates for these cases, for both theTPWL and reference solutions, are plotted in Fig. 19. Somedeviation between the two solutions is observed after about1,000 days for �p = 40 psi (e.g., 20 % error in the TPWLresult at 2,250 days), though otherwise the agreement is veryclose.
The full-order simulations of the idealized SAGD pro-cess require about 20 min of computation. The (MATLAB)TPWL simulations require around 5.5 s, which means theTPWL procedure provides a runtime speedup of about 200.The smaller speedup in this case than that in the primaryproduction example (where we observed a speedup of 500)
is likely due to the use of different simulators for the full-order runs and to the fact that we include more columnsin the � matrix in the idealized SAGD cases. The over-head required to construct the TPWL model for this caseis equivalent to about seven full-order simulations for thehomogeneous case (where we used three training runs) andten full-order simulations for the heterogeneous case (fivetraining runs).
5 Concluding remarks
In this paper, we applied a reduced-order modeling proce-dure, namely TPWL, to nonlinear thermal reservoir simu-lation problems. TPWL entails the representation of newsystem states as linearizations around nearby saved states(generated during one or more training runs), along withprojections into a subspace of low dimension. The basismatrix required for this projection is determined by applyingproper orthogonal decomposition to the states (snapshots)computed during training runs. The models considered hereare much simpler than those used in realistic thermal simu-
(a) Δ p = 40 psi (b) Δ p = 35 psi (c) Δ p = 25 psi
Fig. 19 Comparison of TPWL test-case results to the reference (full-order) solutions for oil production rate for various �p values (heterogeneouscase)
Comput Geosci
lations, though they do include some key nonlinearities suchas the strong dependence of viscosity on temperature. Thetwo systems that were considered are primary production ofsingle-phase oil driven by downhole heaters and a simpli-fied model for steam-assisted gravity drainage, where waterand steam are treated as a single effective phase.
TPWL was shown to provide results of reasonableaccuracy, relative to full-order (high-fidelity) referencesolutions, for these two systems. For the most compli-cated example considered—simulation of idealized SAGDin a heterogeneous reservoir—some discrepancy betweenthe full-order and TPWL solutions was observed, thoughTPWL performance was still acceptable. The TPWL modelruns very quickly, and runtime speedups of 200–500 wereobserved for the cases considered. The degree of speedupdepends in part on the number of columns retained in thebasis matrices and also on the computational efficiencyof the simulator used for the full-order model. We expectthat the speedups achieved here could be improved throughoptimization of the TPWL code. The preprocessing com-putations required for TPWL are, however, computationallydemanding. Specifically, the construction of the TPWLmatrices entails about the same amount of computation asthe training runs. For this reason, it would not be appropri-ate to construct a TPWL model unless it can be used formany simulations. Because large numbers of simulationsare typically required for optimization, data assimilation,and uncertainty quantification, TPWL is indeed applicablein these areas (see [7, 12] for use of TPWL in optimizationand [13] for its use in data assimilation).
There are several ways in which the TPWL proceduresfor thermal modeling presented here can be extended andenhanced. For example, He et al. [12] introduced a proce-dure for optimizing the number of columns included in thebasis matrices for oil–water problems, and this was shownto improve numerical stability. They also presented a tech-nique for the high-fidelity resolution of selected grid blocks(such as well blocks) within the TPWL representation, andthis was shown to provide improved accuracy for key wellquantities. It has also been shown in [11] that enhancedaccuracy can be achieved through use of different constraintreduction schemes (i.e., by using a different choice for �,with � �= �, in Eq. 15). The performance of these proce-dures for thermal problems should be evaluated. It will alsobe necessary to incorporate more realistic physics into thethermal models considered. For example, the SAGD modelshould be extended to include oil, water, and steam phases.Extensions in some of these directions could yield practicalreduced-order models for thermal processes.
Acknowledgments We thank the industrial affiliates of the SmartFields Consortium at Stanford University and the Geosciences andReservoir Engineering Technology Organization at ConocoPhillips,for the partial funding of this work.
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