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MOX-Report No. 04/2019
Conservative multirate multiscale simulation ofmultiphase flow in heterogeneous porous media
Delpopolo Carciopolo, L.; Formaggia, L.; Scotti, A.; Hajibeygi,
H.
MOX, Dipartimento di Matematica Politecnico di Milano, Via Bonardi 9 - 20133 Milano (Italy)
[email protected] http://mox.polimi.it
Conservative multirate multiscale simulation of
multiphase flow in heterogeneous porous media
Ludovica Delpopolo Carciopoloa,∗, Luca Formaggiaa, Anna Scottia, HadiHajibeygib
aMOX, Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133Milano, Italy
bFaculty of Civil Engineering and Geosciences, Department of Geoscience andEngineering, Delft University of Technology, Stevinweg 1 , 2628CV Delft, Netherlands
Abstract
Accurate and efficient simulation of multiphase flow in heterogeneous porousmedia motivates the development of space-time multiscale strategies for thecoupled nonlinear flow (pressure) and saturation transport equations. Theflow equation entails heterogeneous high-resolution (fine-scale) coefficientsand is global (elliptic or parabolic). The time-dependent saturation profile,on the other hand, has local sharp gradients (fronts) where the accuracyof the solution demands for tight time-step sizes. Therefore, accurate flowsolvers need to resolve the spatial multiscale challenge, while advanced trans-port solvers need to also resolve the challenge related to time-step size. Inthis work, we develop the first integrated multirate multiscale method whichimplements a space-time conservative multiscale framework for sequentiallycoupled flow and transport equations. The method solves the pressure equa-tion with a multiscale finite volume method at the spatial coarse scale, thetransport equation is solved by taking different time-step sizes at differentlocations of the domain. At each time step, a coarse time step is taken, andthen based on an adaptive recursive strategy, the front region is sharpenedthrough a local-fine-scale time stepping strategy. The accuracy and effi-ciency of the method is investigated for a wide range of heterogeneous test
∗Corresponding author.Email addresses: [email protected] (Ludovica Delpopolo
Carciopolo), [email protected] (Luca Formaggia), [email protected](Anna Scotti), [email protected] (Hadi Hajibeygi)
Preprint submitted to Elsevier February 3, 2019
cases. The results demonstrate that the proposed method provides a promis-ing strategy to minimise the accuracy-efficiency tradeoff by developing anintegrated space-time multiscale simulation strategy.
Keywords: Conservative multirate methods, Iterative multiscale methods,Multiscale finite-volume method, Multiphase flow, Porous media
1. Introduction
Simulation of multiphase flow in highly heterogeneous porous media withclassical numerical methods typically becomes too expensive to be performedat the resolution the heterogeneous parameters are defined. This is due to thepresence of multiple scales of interest, both in space and in time. In space,parameter heterogeneity, in particular that of permeability, strongly affectsthe flow and needs to be properly resolved by the grid. In time, the presenceof different flow velocities, due to the different mobilities of the fluid phases,can impose a strict restriction on the time step size to achieve stability andaccuracy in the whole simulation domain.
To resolve the simulation complexity with respect to the space, multiscalefinite element [1, 2, 3] and finite-volume [4, 5] methods have been developedto construct a spatial coarse-scale systems for elliptic [6, 7, 8] and parabolic[9, 10] flow equations with fine-scale heterogeneous coefficients. This wasachieved by introducing locally-computed basis functions. After solving thecoarse scale system, the approximate fine-scale solution can be found by inter-polating the coarse-scale solution with the local basis functions. The methodalso allows for convergent error reduction towards the fine-scale referencesolution with the possibility of conservative velocity construction after anyiteration step [5]. The method has been successfully integrated with bothsequential [11] and fully implicit [12] coupling approaches. Among manyother extensions, recent advancements include extensions of the method tocompositional [13, 14, 15, 16, 17] and geo-thermal flows [18, 19] in fracturedheterogeneous porous media [20, 21, 22, 23, 24, 25]. An important feature isthat the entire multiscale procedure can be formulated algebraically, whichallows for convenient integration with existing commercial simulators for two-level [26, 27, 28] and dynamic multilevel [29] simulations.
Once the pressure solution is obtained, the saturation equations has to besolved. We point out that, even if an unconditionally stable implicit method
2
is used to integrate the equations in time, small time steps are necessary toachieve a desired accuracy of the strong gradients present in the solution.
The temporal aspect of the simulation challenge has been addressed bythe development of multirate methods [30]. Multirate methods avoid refiningthe entire temporal domain in order to achieve a certain required accuracy.Instead, they allow for flexible selection of time-step size within the domainbased on the local flow characteristics. Different than, yet complementaryto, Adaptive Implicit (AIM) [31, 32, 33, 34, 35, 36] and potential order-ing [37] methods, multirate methods do not aim at an increase of stabilityby combining explicit-implicit integration strategies; instead, they allow forflexible time-step size in the domain to achieve the desired accuracy. Moreprecisely, the system is subdivided into two subsystems, one containing theactive (fast) components that need small time steps, and the other composedby the latent (slow) components, where a larger time steps can be used.
In the early developments of the method the subsystems were character-ized and partitioned a priori, by exploiting a priori knowledge of the problemat hand [38]. Later extensions included a self-adjusting strategy to selectthe fast components automatically [39]. Note that most of the developmentswithin the multirate literature address only hyperbolic wave transport equa-tions [40, 41]. They also mainly implement nonconservative procedure whichcan lead to stability issues for coupled flow-transport systems in porous mediaapplications. Very recently, a conservative multirate method was introducedfor only hyperbolic transport equations [42, 43]. Of particular interest is thedevelopment of a conservative multirate methods for heterogeneous porousmedia applications, and their integration within nonlinear multiphase flowsimulations.
In this paper we present the first conservative multirate multiscale methodfor multiphase flow simulation in heterogeneous porous media. The pressure-saturation coupling is treated via a sequential implicit formulation [44]. First,the pressure is solved with the multiscale finite volume method. The iter-ative multiscale procedure is applied to guarantee the desired accuracy ofthe pressure solution. Once the good-approximate multiscale pressure solu-tion is obtained, a conservative velocity field is constructed by solving localpressure equations subject to Neumann flux from the multiscale pressure so-lution. This conservative velocity field is then used to solve the saturationequation using a multirate method with a given accuracy tolerance. Themultirate method employs, initially, a coarse-scale time step everywhere inthe domain to estimate the updated saturation field. Then, based on the
3
error estimate criterion, the location of high sensitivity (fast dynamics) isdetected and solved with a smaller time step. The integration of the refinedtime-step zone (fast dynamics) and the rest of the domain (slow dynamics) isdone via a flux-constrained formulation that guarantees local mass balance.This combination of a space-time multiscale for flow and transport allowsfor minimizing the tradeoff between accuracy and efficiency by coarseningand refining both the space and time computational grids. Note that due tothe accuracy of the multiscale local basis functions as well as the iterativemultiscale procedure, the global stage of pressure solver is always performedat coarse-scale.
The accuracy and efficiency of the proposed method is investigated forseveral test cases. In particular, we count both the fraction of the fine-scale time steps as well as the nonlinear iterations performed to achieve themultiscale solutions. These are important quantities for the quantificationof a method performances with a research code where computing time istoo sensitive to the optimization of the implementation and the hardwarecharacteristics.
The paper is organized as follows. Governing equations for multiphaseflow in porous media at continuum (Darcy’s) scale are described in Section2. In Section 3, first the space and time discretization schemes adopted forboth flow and transport equations are presented. Then the multiscale multi-rate method for sequential implicit multiphase flow simulations is described.Section 4 contains several numerical tests that demonstrate the effectivenessof the approach, followed by a concluding Section where we outline possiblefurther developments.
2. Governing equations
Let Np be the number of phases present in the porous medium. Given ad-dimensional domain Ω ⊂ Rd, mass balance for an incompressible phase αreads
∂
∂t(φSα) +∇ · (uα) = qα, α ∈ 1, ..., Np, (1)
where φ is the porosity, Sα is the saturation, uα is the phase Darcy velocityand qα the source term.
Without gravitational nor capillary effects, the Darcy phase velocity isgiven by
uα = −λαK · ∇p in Ω, (2)
4
where p is the pressure and λα = krα/µα is the phase mobility, i.e., the ratioof the relative permeability krα (which is a function of the phase saturations)and the phase viscosity µα [45]. The permeability K is a heterogeneous ten-sor, though in this paper it is considered as a scalar heterogeneous quantity.The saturation of the phases must fulfill the constraint
∑Npi=1 Si = 1, i.e. the
pore space is fully filled with fluid phases.Under the incompressible fluid and porous solid assumption, the pres-
sure equation is obtained by summing all the phase balance equations [46],obtaining
−∇ · (λtK · ∇p) = qt in Ω, (3)
where λt =∑Np
i=1 λi is the total mobility and qt =∑Np
i=1 qi is the total source
term. Once the pressure equation is solved, the total velocity ut =∑Np
i=1 ui iscomputed and then the (Np−1) saturation balance Eqs. (1) are solved, wherethe fractional flow function fα = λα/λt is used to compute the respectivephase velocity, i.e., uα = fαut, leading to
φ∂Sα∂t
+∇ · (fαut) = fαqt, α ∈ 1, ..., Np−1. (4)
Note that the Np-th saturation can be found by using the constraint, i.e.,
SNp = 1−∑Np−1i=1 Si. Equations (3) and (4) are coupled by the total velocity
ut and the (nonlinear) phase relative permeabilities krα. These equationstogether with proper boundary and initial conditions form a well-posed cou-pled system for Np unknowns. Let the boundary ∂Ω of the computationaldomain Ω be decomposed into a non-overlapping Dirichlet ΓD and NeumannΓN parts, where ∂Ω = ΓD ∪ ΓN and ΓD ∩ ΓN = ∅. The boundary conditionsfor the pressure equations read
p = pD on ΓD
(λtK · ∇p) · n = ut · n = h on ΓN
The boundary conditions for the transport equations have to be defined onlyat the inflow (upwind) boundary Γ− = x ∈ ∂Ω : ut · n < 0, i.e.,
Sα = Sα on Γ− α ∈ 1, ..., Np−1. (5)
The phase saturations are defined at the initial time t = 0 by
Sα = S0α in Ω α ∈ 1, ..., Np−1, (6)
5
where S0α are the known initial distribution of phase saturation in the domain.
Note that a unique solution for pressure can be found if |ΓD| > 0, where |ΓD|is the measure of the Dirichlet boundary part.
3. Multirate multiscale approach
Sequential implicit simulation consists of separately solving for the mainunknowns of Eqs. (3) and (4) implicitly in time, at each time step ∆t fromcurrent time tn to the next simulation time tn+1 [47]. Having the currenttime step properties, the solutions at next time step (n + 1) are found byfirst solving for pressure, i.e.,
−∇ ·(λt(S
nα) K · ∇pn+1
)= qt in Ω. (7)
Then, the total velocity is computed as
utn+1 = −λtK · ∇pn+1, (8)
and, finally, the new saturation values are found by solving
φSn+1α − Snα
∆t+∇ ·
(fn+1α ut
n+1)− fn+1
α qt = 0 in Ω. (9)
The fine-scale (in space) discrete system is obtained by using the finite-volume-based two-point-flux-approximation scheme. Let Th be the fine scalemesh with rectangular or hexahedral control volumes K ∈ Th . The setof faces e of generic element K is denoted by EK . The flow equation isdiscretized as
−∑
eKL∈EK
FeKL = |K|qtK ∀K ∈ Th, (10)
where |K| stands for the measure (volume for three dimensional elements,area in two dimensional elements) of element K, eKL is the face of elementK shared with element L, and the numerical flux FeKL is given by
FeKL = τK|L(pL − pK) ∀eKL ∈ EK , (11)
where pK is the constant pressure approximation in the current cell K andpL is the pressure value in the neighboring cell. The interface trasmissibilityvalue τK|L is the harmonic average of the neighboring cell parameters, i.e.,
τK|L =|eKL|
dKσλtKKK
+ dσLλtLKL
, (12)
6
where |eKL| is the area of the interface (length in two-dimensional domains),dKσ is the distance between the center of the current cell K and the interfaceσ = eK|L and similarly, dσL is the distance between the interface and thecenter of the cell L.
The discrete saturation equation reads
φSn+1αK− SnαK
∆t+
1
|K|∑
eKL∈EK
F n+1eKL
(utKL − qtK ) = 0 ∀K ∈ Th. (13)
The discrete flux F n+1eKL
is calculated based on the first − order upwindmethod. If the upstream and downstream saturation values with respectto the total velocity are denoted by SU and SD, respectively, the interfaceflux reads
F n+1eKL
=
fα(Sn+1
αU) if uTKL > 0,
fα(Sn+1αD
) otherwise.(14)
where uTKL = ut · nKL, and nKL is the unit external normal to the faceeKL of K. The fractional flow is a nonlinear function of the saturation, sothe Newton linearization following an iterative strategy is used to solve thenonlinear transport equation. Note that the flux function is non-convex,therefore, the Newton method may diverge for big time steps, even if animplicit discretization scheme is used. To ensure the convergence of the non-linear method, the modified Newton approach proposed in [48] has been used.
For efficient and accurate solution of the flow and the transport equa-tions, i.e., Eqs. (10) and (13), respectively, we propose a multirate multiscalemethod. Figure 1 provides an algorithmic overview of the method for oneglobal time step. As illustrated, the multiscale finite volume method is usedto solve the pressure equation, and the multirate method for the transportequation. More details will be provided in the following sub-sections.
3.1. Multiscale method for flow
On the given fine-scale computational grid, the multiscale method im-poses two sets of coarse grids, namely, primal and dual coarse grids. Theprimal coarse grid defines the control volumes to solve the pressure equationat coarse scale. The dual grid, on the other hand, provides local supports forcomputation of the multiscale basis functions. Note that other local func-tions (block solvers), such as well-functions near fine-scale source terms, can
7
Multiscale for flow
Multirate for transport
Compute (or adaptively up-date) multiscale basis functions
1. Multiscale correction: δpν+1/2
2. Smoother correction: δpν+1
residual< tolp
Multiscale finite volume (conser-vative velocity at coarse-scale)
Compute locally conserva-tive velocity at fine-scale
no
yes
Asigne the global domainto the critical zone EA
Compute approximate fine-scale (in time) fluxes at
the coarse-fine boundaries
Solve nonlinear trasport for EA
loop 1
loop 2
Compute error estimators for EA
Error < tols
Update the sub-critical zone EARefine the time step forEA (finer-scale in time)
Are all cellsat the global
coarsetime-step?
Set thefinest-in-timezone as EA
END
no
yes
no
yes
Figure 1: Description of the multirate multiscale algorithm for a time step.
8
Figure 2: Illustration of the multiscale grids. Shown on the right and left are a coarse cellΩk and a dual-coarse cell Ωh, respectively.
be also computed on either of the coarse grids [28]. As shown in Figure 2,the dual-coarse grid is constructed by connecting the coarse nodes.
Let M be the number of coarse cells and N the dual coarse cells in thedomain. The multiscale method provides an approximation p′ of the fine-scale system solution pf using a superposition expression which reads
pf (x) ≈ p′(x) =M∑k=1
Φk(x)pk. (15)
Here, Φk is the basis function belonging to the coarse node k, which is foundby assembling the local basis functions corresponding to that node in eachdual coarse cell, i.e., Φk =
∑Nh=1 Φh
k(x). Also, pk is the coarse-scale solutionat the coarse node k. The basis functions are obtained by solving
−∇ ·(λtK · ∇Φh
k
)= 0 in Ωh
−∇|| ·(λtK · ∇||Φh
k
)= 0 on ∂Ωh
Φhk(xi) = δik ∀xi ∈ 1, ...M
(16)
on dual-coarse cells. These basis functions can be found algebraically withsome operations on the fine-scale system matrix, as explained in [28]. Note,reduced dimensional conditions are imposed on the local domain boundaries∂Ωh. The basis functions will be collected in the columns of the prolongationmatrix P . A restriction operator R is also obtained to construct a finite-volume coarse scale system. Entries of R are 1 and 0 as explained in [28].
9
More precisely, the entry rij is 1 only if the fine-scale i belongs to the coarsecell j.
For a given pressure equation at fine scale Ap = b, where A and b are thesystem matrix and right-hand-side vector, respectively, the iterative multi-scale procedure reads as follows.
1. Multiscale stage: δpν+12 = P(RAP)−1R rν = M−1
MSFV rν
2. Smoothing stage: pν+1 = M−1ILU(0) r
ν+ 12
Here, the residual vector is updated with the last available pressure, e.g.,rν = b−Apν . Steps 1 and 2 are repeated iteratively until the residual norm isbelow the desired threshold. Note that the ILU(0) factorization is taken forthe fine-scale smoother stage, as it has been found the most computationallyefficient choice [28]. Also, for fine-scale source terms, before entering thisiterative procedure, the initial guess pν is improved by block-solvers with thesupport of the dual-coarse cells embedding the fine-scale source term. Thiscan be sees as a form of well function [49].
Once the desired pressure is obtained, an additional MSFV stage is em-ployed to obtain coarse-scale conservative velocity field. Then local Neumannpressure equations over the primal coarse cells are solved, subject to the ve-locity from the multiscale approximate pressure field. This stage results infine-scale locally conservative velocity which can be used readily to updatethe saturation transport equations [5].
3.2. Conservative multirate method for transport equation
This section presents the details of the conservative multirate method forefficient solution of the saturation Eq. (9). Note that Eq. (9) is a nonlinearhyperbolic equation whose solution may exhibit strong localized variationsand fronts. The multirate method takes different time step sizes in differentparts of the spatial domain, in order to increase the computational efficiencywhile preserving the accuracy. The presented method is based on a fluxpartitioning strategy, to preserve the local mass conservation.
The multirate method procedure can be summerized as following: firstthe approximate solution at time tn+1 for all components will be computed.This means that, with coarse-scale time step ∆t all cells will be assigned tothe set of the critical zone EA, i.e.,
Sn+1αK
= SnαK −1
φ|K|∑
eKL∈EK
un+1tKL
F n+1eKL
. (17)
10
Here,
F n+1eKL
=
∆tfα(Sn+1
αU) if un+1
TKL> 0,
∆tfα(Sn+1αD
) otherwise.(18)
The value of the numerical fluxes at all cell interfaces is then checked,using an appropriate error estimator. If a flux is rejected on the basis of theerror estimator, all the cells which contribute to its definition are collectedin the set of new sub-critical zone EA. Accordingly, we can define the set ofactive fluxes (the quality of the flux is not satisfy) and the set of the boundaryfluxes between a fine-coarse region as
EKA = set of faces of the element K : ηKL ≤ tol ∀eKL ∈ EKEKL = set of faces of the element K : ηKL > tol ∀eKL ∈ EK
where ηKL is the error estimator value for the flux at the face eKL.The new sub-critical zone will be solved (locally) with the smaller time
step, subject to an approximate flux boundary condition. The approximationis made by scaling the coarse-scale fluxes by the ratio of the new to the coarsetime step. This approximation guarantees mass conservation at the globalcoarse time step. The new saturation value between a fine-coarse region iscomputed as
Sn+ 1
2αK = SnαK −
1
φ|K|∑
eKL∈EKL
1
2un+1tKL
F n+1eKL− 1
φ|K|∑
eKL∈EKA
un+ 1
2tKL
Fn+ 1
2eKL . (19)
If the new time step is such that all fluxes are accepted, the solution at timetn+1 reads
Sn+1αK
= SnαK −1
φ|K|∑
eKL∈EKL
1
2un+1tKL
F n+1eKL− 1
φ|K|∑
eKL∈EKA
un+1tKL
F n+1eKL
. (20)
The above description is an example of a simplified case where only 1 re-finement took place. In practice, however, the method allows for an arbitrarylevel of refinements until the necessary flux quality is reached. More precisely,a self-adjusting strategy is used, where the sub-critical zones are iterativelydefined until all fluxes satisfy the error threshold value. The method, there-fore, entails two main loops, as shown in Figure 1. Loop 1 detects and
11
x
tStep 1
x
tStep 2
x
tStep 3
x
tStep 4
x
tStep 2
Loop 1 Loop 1
(a)
(b)
x
tStep 0
Loop 2 Loop 2
Figure 3: Schematic illustration of the integration in time with the multirate strategy fora global time step. In row (a) the Loop 1 has been applied to locally refine in time untilthe flux quality is satisfied. In row (b) Loop 2 is employed to advance the sub-criticalzones in time until the global coarse time synchronization is reached. Note that Loop 1and Loop 2 are fully integrated, meaning that for each step of Loop 2, Loop 1 will becalled to maintain the flux quality.
integrates the sub-critical regions until the flux quality check is satisfied. Onthe other hand, Loop 2 advances the sub-critical zones in time until the globaltime-step synchronization takes place. Figure 3 illustrates a schematic exam-ple of how the two loops perform. The thick lines represent the sub-criticalzones EA. The highlighted sub-critical zones in this figure illustrate the cellsfor which the transport equation has been solved either for refinement (Loop1) or advancing (Loop 2) in time.
As the flux error estimator, an a posteriori error estimator is used tocheck convergence [50], based on the difference of the fluxes at the currentand previous time step:
η(i)tm,K ' |F (i)
eKL− F (i−1)
eKL|. (21)
12
Here, superscript (i) denotes the i-th step of the refinement inside the given
global time step while F(i−1)eKL represents the final approximate flux at the
previously accepted time step.
4. Numerical Results
To test the performance of the multirate multiscale method we considerthe numerical test cases presented in [44], In particular, the top and thebottom layers of the SPE10 test case are used as the permeability field dis-tribution [51].
The accuracy of the multirate multiscale solutions are quantified by com-paring them with the fine scale solution both in space and in time (referredto as “reference solution”, denoted with the sub-index “ref”). The pressureand saturation errors are calculated as
EP =|p− pref ||pref |
, (22)
andES = |S − Sref |, (23)
respectively.In the following test cases, the less viscous fluid (gas) is injected at the
top-left cell (1, 1) with a non-dimensional rate of 10 and the more viscous fluid(oil) is produced at the bottom-right cell (220, 60) at zero non-dimensionalpressure. No-flow boundary conditions are applied at the boundaries for alltest cases. There are 220× 60 fine-scale grid cells, over which the multiscalemethod imposes 20 × 12 coarse grid cells. Quadratic relative permeabilityfunctions are employed as kg(Sg) = S2
g and ko(Sg) = (1−Sg)2. The viscosityratio µo/µg is set equal to 10.
4.1. Case 1
The top-layer permeability field of the SPE10 test case, as shown in Figure4, is considered. The simulation starts at time t = 0 and ends at time t = 600.Global (coarse-scale) time steps are performed with the fixed size of 7.25. Theflux quality tolerance is set to 10−4, and the Newton iterative convergencethreshold is 10−8.
Figure 5 shows the solution of the multirate method in time and fine-scalein space at t = 600.
13
50 100 150 200
10
20
30
40
50
60-5
0
5
Figure 4: Natural logarithm of the SPE10 top-layer permeability distribution.
0 50 100 150 200
0
20
40
60
5
10
15
0 50 100 150 200
0
20
40
60 0
0.2
0.4
0.6
0.8
Figure 5: Solution at final time t = 600 for the global pressure (left) and gas saturation(right) compute with the multirate method on the fine scale grid in space.
As shown in Fig. 6, only the cells near the front are solved at the fine-scale in time. These cells are associated with a fast process and fine-scaletime steps guarantee the required accuracy.
Figure 7 shows the pressure and saturation errors of the simulation withcoarse time steps (top plots) and multirate (MR). Both of them have usedfine-scale grid in space. It is clear from this plot that the multirate techniqueimproves the solution taking only a small fraction of the cells at the fine-scaletime step. Note that, since pressure and saturation equations are coupled,improved solutions with multirate method are obtained for both pressure andsaturation.
The l2-norm of the pressure and saturation errors can be defined as
E2P (t) =||p(t)− pref (t)||2||pref (t)||2
(24)
andE2S(t) = ||S(t)− Sref (t)||2, (25)
respectively.Figure 8 shows the l2-norm of the error of the solutions obtained with
multirate (in time) and fine-scale in space.Figure 9 shows that, compared with the single-rate fine-scale in time
solver, the proposed multirate method applies more Newton iterations to
14
0 50 100 150 200
0
20
40
60 0
0.2
0.4
0.6
0.8
0 50 100 150 200
0
50
0 50 100 150 200
0
20
40
60 0
0.2
0.4
0.6
0.8
0 50 100 150 200
0
50
0 50 100 150 200
0
20
40
60 0
0.2
0.4
0.6
0.8
0 50 100 150 200
0
50
Figure 6: Gas saturation solution and active components at times t = 222.094, 435.062,and 599.094.
EP
(fine-space, coarse-time)
0 50 100 150 200
0
502
4
6
8
10-3 E
S (fine-space, coarse-time)
0 50 100 150 200
0
50
0
0.02
0.04
0.06
0.08
EP
(fine-space, MR-time)
0 50 100 150 200
0
502
4
6
8
10-3 E
S (fine-space, MR-time)
0 50 100 150 200
0
50
0
0.02
0.04
0.06
0.08
Figure 7: Errors for the pressure (left column) and errors for the gas saturation (rightcolumn) at final time t = 600for the fine space grid, coarse time steps solution (top row)and for the fine space grid, multirate solution (bottom row).
15
0 20 40 60
Time step
0
0.005
0.01
0.015
0.02
0.025
EP(2
nd
norm
)
fine-space MR-time
fine-space coarse-time
0 20 40 60
Time step
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
ES(2
nd
norm
)
fine-space MR-time
fine-space coarse-time
Figure 8: Pressure relative error l2-norm (left) and saturation error l2-norm (right) duringthe simulation time for the multirate approach with a fine space grid and the coarse timesteps approach with a fine space grid.
converge to the nonlinear saturation solutions at each global (coarse-scale)time step. This motivates the definition of an indicator to estimate theoverall computational cost as the cumulative sum (in time) of the number ofNewton iterations multiplied by the number of active components. This costfunction is indeed much lower for the multirate solver compared to that of thesingle-rate, fine-scale in time simulator. This is due to the fact that the stepsperformed with multirate method by taking large ∆t require more Newtoniterations, as expected, but this number drops quickly when refinements areperformed and the sub-critical zones are solved locally at smaller time stepsizes. For this test case the computational cost of the multirate solution isless than one third that of the fine-scale reference solver (the precise ratio is0.29).
Figure 10 shows the CFL number associated with the adaptive time stepsfor multirate simulations, computed using the maximum value of the ana-lytical flux derivative with respect to the saturation. Large portions of thedomain (far from the front) exhibit large CFL numbers, while a smallerfraction (near front) advance with smaller CFL numbers. This illustratesthe effectiveness of the proposed multirate method. In the simulations, therefinement in time is stopped once it leads to CFL = 0.8.
Now that the multirate method in time is fully investigated, we combine itwith the multiscale method in space. Figure 11 reports the solution errors ofthe multiscale in space and coarse-scale in time (top) as well as the multiscalein space and multirate in time (bottom). There are no virtual differences
16
0
1
2
3N
ew
ton
ite
r x a
ctive
c10
5 fine-space, MR-time
0 100 200 300 400 500 600
1.8249e+07
0
1
2
Ne
wto
n ite
r x a
ctive
c
105 fine-space, fine-time
0 100 200 300 400 500 600
6.03768e+07
Figure 9: Number of active cells multiplied by the number of Newton iterations at eachtime step for the multirate (MR) and fine-scale in time solvers.
0 500 1000 15000
5
10
15
20
25
Figure 10: CFL numbers (based on maximum analytical fractional flow derivative value)at each time step of the multirate method.
17
EP
(MS-space, MR-time)
0 50 100 150 200
0
20
40
60
0.02
0.04
0.06
0.08
0.1
0.12
ES
(MS-space, MR-time)
0 50 100 150 200
0
20
40
60 0
0.1
0.2
EP
(MS-space, coarse-time)
0 50 100 150 200
0
20
40
60
0.02
0.04
0.06
0.08
0.1
0.12
ES
(MS-space, coarse-time)
0 50 100 150 200
0
20
40
60 0
0.1
0.2
Figure 11: Errors for the pressure (left column) and errors for the gas saturation (rightcolumn) at final time for fine space grid, coarse time steps approach (top row) and for themultirate multiscale method (bottom row).
between the two sets of solutions, because in both cases the errors introducedby the multiscale in space dominate the overall errors.
Figure 12 shows the simulation errors, similarly to Figure 11, but with theiterative multiscale solver in space. Here, the two-stage multiscale solver isemployed until the l2-norm of the pressure residual is equal to 10−3 (top) andequal to 10−5 (bottom). For both the simulations the number of smoothingper iteration step was set to ns = 5.
Figure 13 presents the multirate multiscale errors compared with thoseof multiscale in space single-rate coarse-step in time. It is clear that themultirate method improves the solution, by applying fine-scale time stepssizes only close to the front. This is further elaborated in Figure 14, whichshows the l2-norm of the pressure and saturation errors at each global timestep.
Finally, the number of Newton iterations multiplied by the number ofactive cells throughout the simulation time is shown in Figure 15. As thefine scale in space and time (reference) solution is expected to employ theleast number of Newton loops, the proposed multirate multiscale method iscompared with the reference solution. It is clear that even if the pressuresolution obtained with the multiscale method is less accurate than the fine-scale ones with respect to the fine grid solution, the number Newton iterationrequired to solve the transport equation is comparable with that required bythe multirate method with a fine-grid pressure solution.
18
EP
(MS-space, MR-time) tol (iMS) = 1e-3
0 50 100 150 200
0
20
40
60
2
4
6
10-3 E
S (MS-space, MR-time) tol (iMS) = 1e-3
0 50 100 150 200
0
20
40
60 0
0.1
0.2
EP
(MS-space, MR-time) tol (iMS) = 1e-5
0 50 100 150 200
0
20
40
60
2
4
6
10-3 E
S (MS-space, MR-time) tol (iMS) = 1e-5
0 50 100 150 200
0
20
40
60 0
0.05
0.1
Figure 12: Errors for the pressure (left column) and errors for the gas saturation (rightcolumn) at final time for the multirate iterative multiscale approach with tolerance equalto 10−3(top row) and equal to 10−5 (bottom row).
EP
(MS-space, coarse-time)
0 50 100 150 200
0
20
40
60
2
4
6
8
1010
-3 ES
(MS-space, coarse-time)
0 50 100 150 200
0
20
40
60 0
0.02
0.04
0.06
0.08
EP
(MS-space, MR-time)
0 50 100 150 200
0
20
40
60
2
4
6
8
1010
-3 ES
(MS-space, MR-time)
0 50 100 150 200
0
20
40
60 0
0.02
0.04
0.06
0.08
Figure 13: Errors for the pressure (left column) and for the gas saturation (right column)at final time for the coarse time steps, iterative multiscale approach with tolerance equalto 10−5 (top row) and for the multirate method combined with the iterative multiscaleapproach with tolerance equal to 10−5 (bottom row).
19
0 20 40 60
Time step
0
0.005
0.01
0.015
0.02
0.025
EP(2
nd
norm
)
MS-space MR-time
MS-space coarse-time
0 20 40 60
Time step
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
ES(2
nd
norm
)
MS-space MR-time
MS-space coarse-time
Figure 14: Pressure relative error l2-norm (left) and saturation error l2-norm (right) dur-ing the simulation time for the multirate iterative multiscale approach and the iterativemultiscale, coarse time steps approach. The iterative multiscale tolerance is 10−5.
0
1
2
3
Ne
wto
n ite
r x a
ctive
c
105 MS-space, MR-time
0 100 200 300 400 500 600
1.79411e+07
0
1
2
Ne
wto
n ite
r x a
ctive
c
105 fine-space, fine-time
0 100 200 300 400 500 600
6.03768e+07
Figure 15: Number of active cells multiplied by the number of Newton iterations at eachtime step for the multirate multiscale and reference solvers. The iterative multiscaletolerance is 10−5.
20
50 100 150 200
10
20
30
40
50
60-5
0
5
Figure 16: Bottom-layer logarithmic permeability distribution.
0 50 100 150 200
0
20
40
60 0
0.1
0.2
0.3
0.4
0 50 100 150 200
0
20
40
60 0
0.2
0.4
0.6
0.8
Figure 17: Solution of pressure (left) and gas saturation (right) at final time t = 600computed with the multirate method.
4.2. Case 2
In the second test case, the permeability field of the SPE 10 bottomlayer, which has higher contrasts and more a channelized distribution, isused (Figure 16).
As in the previous case, the simulation starts at time t = 0 and ends attime t = 600. The global time step is equal to 5.5. Figure 17 shows thesolution at final time for the multirate in time, fine-scale in space approach.The flux tolerance is equal to 10−3 and the Newton convergence tolerance is10−8.
Figure 18 illustrates the active cells for which the saturation transportequation is solved at fine-scale in time, for three different times.
As in the previous test case, first the multirate approach is employed onfine-grid in space. Figure 19 shows the errors for single-rate coarse time steps(top) and the multirate method (bottom). Both the pressure error (left) andthe saturation error (right) are improved by the multirate technique. Notethat for this challenging case, the coarse-scale in time solutions do not capturethe right velocity values at the front region. The multirate method, on theother hand, leads to significantly improved solutions with only employingfine-scale time steps in the vicinity of the front.
Figure 20 shows the pressure and saturation relative errors at each globaltime step for the multirate and the coarse time step approaches.
21
0 50 100 150 200
0
20
40
60 0
0.2
0.4
0.6
0.8
0 50 100 150 200
0
50
0 50 100 150 200
0
20
40
60 0
0.2
0.4
0.6
0.8
0 50 100 150 200
0
50
0 50 100 150 200
0
20
40
60 0
0.2
0.4
0.6
0.8
0 50 100 150 200
0
50
Figure 18: Gas saturation solution and active components at times t = 103.176, 298.443,and 599.493.
EP
(fine-space, coarse-time)
0 50 100 150 200
0
502
4
6
8
10-4 E
S (fine-space, coarse-time)
0 50 100 150 200
0
50
0
0.01
0.02
EP
(fine-space, MR-time)
0 50 100 150 200
0
502
4
6
8
10-4 E
S (fine-space, MR-time)
0 50 100 150 200
0
50
0
0.01
0.02
Figure 19: Errors for the pressure (left column) and for the gas saturation (right column)at final time for the coarse time steps, fine space grid approach (top row) and for themultirate method with a fine space grid (bottom row).
22
0 50 100
Time step
0
0.01
0.02
0.03
0.04
0.05
0.06
EP(2
nd
norm
)
fine-space, MR-time
fine-space, coarse-time
0 50 100
Time step
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
ES(2
nd
norm
)
fine-space, MR-time
fine-space, coarse-time
Figure 20: Pressure relative error l2-norm (left) and saturation error l2-norm (right) duringthe simulation time for the multirate approach with a fine space grid and the coarse timesteps approach with a fine space grid.
Figure 21 presents the computational complexity of the methods, i.e. thenumber of Newton iterations multiplied by the number of active cells, formultirate (top), fine-scale in time (center), and the coarse-scale in time (bot-tom) simulation approaches. The computational complexity of the multiratemethod is 0.24 of the reference (fine-scale in space and time) simulation ap-proach.
The CFL numbers for multirate method are presented in Figure 22. Notethat, as in the previous test case, the CFL number is calculated based onthe maximum analytical fractional flow derivative value. Thus, clearly, thecoarse-scale (global) time steps have very high CFL values.
Finally, the multirate multiscale method is investigated. The iterativemultiscale solver tolerance is set to 1e-6 with ns = 30 ILU(0) smoothingsteps per iterations. Note that no iterative accelerator is used in this work[28]. We used the same size for the global time steps of the multirate testcase.
Figure 23 illustrates the pressure and saturation errors for the solutions ofthe iterative multiscale in space and coarse-scale in time (top) and multiratein time (bottom) methods. Figure 24 shows the relative errors at the globaltime steps for the coarse-scale and multirate in time approaches, where bothemploy the iterative multiscale in space simulation approach. Finally, Figure25 shows the number of Newton iterations multiplied by the active (criticalsub-domain) cells for the three different approaches.
23
0
1
2
3
New
ton ite
r x a
ctive c
105 fine-space, MR-time
0 100 200 300 400 500 600
2.76699e+07
0
1
2
New
ton ite
r x a
ctive c
105 fine-space, fine-time
0 100 200 300 400 500 600
1.14418e+08
0
1
2
New
ton ite
r x a
ctive c
105 fine-space, coarse-time
0 100 200 300 400 500 600
1.7886e+07
Figure 21: Complexity of the problem: number of Newton iteration per active componentsrequired at each time step for the multirate approach (top), fine time steps approach(center) and coarse time steps approach (bottom).
24
0 1000 2000 30000
20
40
60
80
Figure 22: CFL numbers at each time steps for the multirate in time (fine-scale in space)approach.
EP
(MS-space, coarse-time)
0 50 100 150 200
0
502
4
6
8
10-4 E
S (MS-space, coarse-time)
0 50 100 150 200
0
50
0
0.01
0.02
EP
(MS-space, MR-time)
0 50 100 150 200
0
502
4
6
8
10-4 E
S (MS-space, MR-time)
0 50 100 150 200
0
50
0
0.01
0.02
Figure 23: Errors for the pressure (left column) and for the gas saturation (right column)at the final time for the iterative multiscale coarse time steps approach (top row) and forthe multirate iterative multiscale approach (bottom row).
25
0 50 100
Time step
0
0.01
0.02
0.03
0.04
0.05
0.06
EP(2
nd
norm
)
MS-space, MR-time
MS-space, coarse-time
0 50 100
Time step
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
ES(2
nd
norm
)
MS-space, MR-time
MS-space, coarse-time
Figure 24: Pressure relative error l2-norm (left) and saturation error l2-norm (right) dur-ing the simulation time for the multirate iterative multiscale approach and the iterativemultiscale coarse time steps approach.
5. Conclusions
In this paper we developed the first multirate multiscale method for cou-pled flow-transport equation in heterogeneous porous media. In order tocontrol the errors in each multiscale dimension, i.e. time and space, an iter-ative multiscale strategy was used to preserve the spatial accuracy while aself-adaptive flux-based refinement strategy was used for temporal accuracy.The proposed approach has been applied to the implicit pressure implicit sat-uration approximation of multiphase flow, to have the benefit of large timesteps and CFL numbers. At the same time, the flux approximation as wellas the iterative multiscale procedure guarantees the local conservation massboth in space and in time.
Proof-of-concept, numerical tests show that pressure and saturation so-lutions improve compared with those obtained from the coarse-scale in timesimulations, with only a small fraction of the cells being solved at the fine-scale in time. The investigations include systematical comparisons of bothsolution error history and computational complexity. Overall, the proposedmethod allows for advancing the simulation in both space and time account-ing for the intrinsic multiscale nature of the problem. As such it develops apromising simulation approach for large-scale multiphase simulations.
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1
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New
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ctive c
105 MS-space, MR-time
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