Higher order, locally conservative temporal integration methods for
modeling multiphase ow in porous media
C.T. Miller a
,M.W. Farthing ay
, C.E. Kees az
,and C.T. Kelley bx
a
Center for the Advanced Study of the Environment
Department of Environmental Sciences and Engineering, Box 7400
University ofNorth Carolina
Chapel Hill, NC27566-7400, USA
b
Center for Research inScientic Computation
Department of Mathematics, Box 8205
North CarolinaState University
Raleigh,NC 27695-8205, USA
Traditionalapproaches for solving single-and multiphase uid ow in porous medium
systems rely on low-order methods in both space and time. Temporal approximations
often use either xed time-step methods or empirically based adaptive strategies that
adjust the time step based on the number of iterationsrequired for the nonlinear solver
to converge. Over the last several years, several single- and multiphase ow problems
havebeen solved using higherorder temporalintegration methods based onformalerror
estimation and control. This approach has led to robust and eÆcient solvers. In this
work, we summarize the recent advances in applying higher order temporal integration
methodsto a range of multiphase ow problems. We also extend this work to derive an
approximate solutionto Richards' equationusing mixed nite element methods inspace
and to formulate asolution approachfor three-phase ow.
1. INTRODUCTION
Overthe lastseveral years, advancementsinmodelingmultiphaseowandtransportin
porousmediumsystemshaveledtoincreasedeÆciency androbustnessinsimulatingsuch
systems; improved spatial and temporal discretizationschemes and advances in iterative
algebraicsolver approaches [1,2] have been largely responsible.
We have seen steady progress in temporal integration schemes. Usinglow-order
xed-time-step solution approaches for diÆcult multiphase ow problems is known to be
in-eÆcient [3]. Moreover, while heuristic time-adaption schemes are more eÆcient than
xed-time-step methods, they have given way to more sophisticated schemes: temporal
SupportedbyNIEHS grant2P42 ES05948,NSFgrantDMS-0112653,andNCSC
y
SupportedbyaDOEComputationalSciencesFellowship,NSFgrantDMS-0112653,andNCSC
z
UndertheauspicesofUCAR,VisitingScientistProgram,P.O.Box3000,Boulder,CO80307-3000,USA
integration methods estimateand controlthe temporal error through adaptingboth the
temporalstep size and the order of the temporal approximation. They are an especially
appealingclassofmethods[4],andconsiderableworkhasbeendoneonusingthem,along
with standardnite dierenceapproaches, toapproximate Richards'equation(RE)[3{8]
and, more recently, two-phase ow[9].
Since many subsurface systems are irregularly shaped, the ideal spatial discretization
approach would capture such features eÆciently. Because of this trait, nite element
approaches have received considerable attention in the literature, but standard
Bubnov-Galerkin methodsdonot conservemass locally. Mixednite element methods(MFEMs)
do not suer from this restriction, but only recently have these methods been combined
with adaptive higher order temporal integration schemes for saturated ow [10]. To
the best of our knowledge, the literature does not yet include treatments of multiphase
ow problems approximated using higher order adaptive step-size temporal integration
combined with MFEM approaches in space. We believe such approaches are promising.
Further, we know of no published report of a higher order temporalintegration
approx-imation for three-phase ow in porous media, althoughthis too seems like a potentially
useful advance.
Theoverallgoalofthisbriefworkistoadvancehigherordertemporalintegration
meth-ods to solve multiphase ow problems. Specically, our objectives are (1) to summarize
aneective and eÆcient approachfor adaptive, higherorder temporalintegration; (2)to
formulate ahigherorder MFEM modelforRE;(3)toillustratethe adaption of temporal
approximation order and step size for a eld-scale application; and (4) to formulate an
adaptivehigher order temporalapproximationfor three-phase ow inporous media.
2. TEMPORAL INTEGRATION
Low-order temporal integration approaches have been the standard for solving the
partial dierential equations (PDE's) that arise from water resources problems. As an
alternative, the method of lines (MOL) approach applies anapproximate method to the
spatial derivatives present in the original PDE and then approximates the resultant set
of ordinary dierential equations (ODE's) using the methods available for solving such
systems. Thesetwomethodsareequivalentwhenthesamelow-ordermethodsinspaceand
timeare usedtosolvethe originalPDE. However, decouplingthe solutionapproachusing
the MOL allows for the straightforward use of sophisticated time integration strategies
that have some attractive properties: excellent stability and accuracy, a range of orders
of approximation, adjustable step size, and built-in error estimation and control and to
meet user-specied criteria. MOL approaches clearly provide a robust and eÆcient way
to solve many water resources problems, althoughmany unanswered questions remain.
Choosing an integration strategy is one such question. We have investigated a range
of temporal integration approaches and advocate using a dierential algebraic equation
(DAE) approachbased on higher order,xed-leading-coeÆcientbackward dierence
for-mulas (FLCBDF's)[4,11]. We consider DAE's of the form
f[t;y(t);y 0
(t)]=0 (1)
variables with respect totime, y 0
. Applying a FLCBDF approximation toEquation (1)
yieldsa system of potentiallynonlinear algebraicequationsof the generalform
f(t;y; y+)=0 (2)
where isa constantthat depends onthe step size andorder of the approximation,and
isa constantrelated tothe predicted solution atthe unknown time step, y p
l +1
, wherea
predictor-corrector approachisused [4]. Equation(2) istypicallysolved usinganinexact
Newton method of the form
J m
Æ m
= f (3)
where J is the Jacobian of the nonlinear equations described by Equation (2), m is an
iteration index, and Æ is a Newton correction. The linear system solves needed to
com-pute Æ can be performed using direct methods for simple one-dimensional problems or
preconditioned Krylov-subspace methodsfor multidimensionalapplications [9].
The orderof the approximation and the step size can be chosen tomeet user-specied
absolute and relative error tolerances. Using a weighted norm approximation of the
dierence between the corrector and predictor solutionof the generalform
l =K
l kÆ
l k
W
<1 (4)
where istheerrorestimate,K
l
isacoeÆcientdependentontheFLCBDFapproximation,
and the subscript W denotes a weighted norm that depends on the user-specied error
tolerances. Complete details of this approachare available inthe literature [9,12].
We have found DAE approaches more eÆcient than the more traditional MOL
ap-proaches. The latter require the system of ODE's to be written in explicit form [3].
Moreover, DAE approachesbenetfromobject-oriented solutionmethods[9],and MOL/
DAEmethodscanbeused toderivemassconservativeapproximationsofmultiphase ow
by specifying explicit algebraicconstraints[9].
3. RICHARDS' EQUATION
We have shown that anMOL/DAE approachcan yielda robust and eÆcient
approxi-mationofRE[3]. Wehavealsoshownhowthistimeintegrationapproachcanbeextended
toderive a locallyconservativemixed hybridnite element(MHFEM) approximation to
single-phase ow in three spatial dimensions; such eorts also result in an eÆcient
solu-tion[10]. Here, weextendourpreviousworkbyderivinganadaptivetemporalintegration
approach forRE, which may be appliedto irregulardomainswhile maintaininga locally
conservative solution.
Although using a MHFEM discretization with a MOL/DAE approach showed several
benetsoverstandard approachesfor owinaconned, heterogeneousaquifer,anumber
of issuesshould be consideredbeforetranslatingits advantages for single-phaseow over
to RE.As mightbeexpected, these issues arise fromthe need to account forthe
nonlin-earity inherent in multiphase ow. Specically, the nonlinear nature of the constitutive
and spatial discretizations used as well as the techniques required to solve the resulting
nonlinear and linear systems.
Inadditiontothe standard MFEM [13], both the MHFEM[14] andthe enhanced
cell-centered dierence method (ECDM) [15,16] have been applied to RE. The ECDM can
alsobeseenasanonstandardnitedierencetechnique obtainedfromastandardMFEM
approachby usingcertainnumericalquadrature rules[17]. Likeothermorecommon
spa-tialapproximations, the MFEM discretizationshavevaried intheir choice of formulation
for RE, p-S-k relations,linear and nonlinear solution strategies, and time discretization.
The temporal approximations, however, have all been low order [13,14,16] with either
xed time step or empirical adaption. The low-order time discretization is of particular
concern for applying the MHFEMto the pressure head formof RE (PRE), whichcauses
signicantmass balanceerrors for certainproblems [18].
The PREcan begiven by
@^
@ +^
@
@ !
@
@t
+ r(q)^ =0 in[0;T] (5)
where and [0;T] are the physical and temporal domains, is the pressure head, ^is
a normalized density, and is the volume fraction for the aqueous phase. We use the
commonextension of Darcy's lawto variably saturated ow [19]
q= k
r ( )K
s
(r d)^ (6)
Here K
s
is the saturated hydraulic conductivity, k
r
is the relative permeability, and d is
a vector accounting for the acceleration of gravity. To describe the interdependence of
uid pressure, saturation, and therelative permeabilitywe use the p-S-k relationsof van
Genuchten [20] and Mualem [21], while we assume slight compressibility for the aqueous
phase density [22].
The boundary and initialconditions are given by
= b
on
D
;t2[0;T]
un=u b
on
N
;t 2[0;T]
= 0
in ;t=0 (7)
where u isthe mass ux.
TheECDM discretizationissuitablefor use with logicallyrectangularmeshes. Details
canbefoundin[17,15,23]. Briey,itformulatesanexpandedsystemforEquations(5){(7)
which includes anadjusted gradient
~
u = (r d)^ with
u = k^
r K
s ~
u (8)
It proceeds by identifying subdomains of over which the problem coeÆcients vary
smoothly. It forms local approximations of Equations (5){(8) over these subdomains
using the lowest order Raviart-Thomas space and certain numerical quadrature rules
[17]. Theselocalapproximationsarethen coupledby dening Lagrangemultipliersalong
We extended previous work by developing an a temporally adaptive ECDM solution
for a PRE. We used an available ECDM spatial discretization method [17], an
object-oriented FLCBDFDAE temporalintegration approach[4], and solved the resultant
sys-tem ofequationsusing apreconditionedgeneralized minimalresidual(GMRES) iterative
solver from the PETSc library [24]. Simulations were performed on an IBM RS/6000
SP supercomputer with 720 processors at the North Carolina Supercomputing Center
(www.ncsc.org).
As an example, we consider a typical unsatured ow problem for a hillslope domain.
The idealized hillslope geometry is dened in terms of a mapping F from a reference
domain ^
=[0;24][0;1] (meters) given by x
1 =x^
1 and
x
2 =
(
^ x
2
+0:1^x
1
sin(x^
1
=25) for x^
1 16
^ x
2
+1:477 otherwise
(9)
Giventhismapping,itisnaturaltoidentifytwosubdomainsof ^
forx^
1
<16andx^
1 >16.
The saturated conductivity and parameter values for the van Genuchten-Mualem p-S-k
relationswere then taken fromthe literature for a sand medium[25].
The initial value of was set to be at hydrostatic equilibrium with the water table,
locatedatx
2
=1=3. Noowconditionsweredened fortherightandbottomboundaries.
Homogeneous Neumann conditions were also used on the left boundary except for the
region given by x
2
1=3 where Dirichlet conditions were set to hydrostatic equilibrium
with the water table. The top boundary uxinto the domain wasgiven by
u b
= (
2sin(t) for5:33 x
2
13:33and t1=24
0 otherwise
(10)
A contour plot of at t = 0:1 on a 9060 mesh is shown in Figure 1. Figure 1
alsotraces the the choice of timestep and temporalapproximationorder overthe course
of the simulation for the DAE integrator with both full adaption and the integration
order restricted toone (BE-A).The data points shown are values averagedover every 10
time steps. The fully adaptive simulation required 336 seconds of total CPU time on 32
processors while the BE-A run required 1085.0 seconds. The BE-A run would be more
eÆcient than typical low-order heuristically based integration schemes and much more
eÆcient than xed-time-step low-order integration schemes. These results are consistent
with previous ndings documenting the advantages of the MOL/DAE approach used in
this work [3,10].
4. THREE-PHASE FLOW
Given a mass-conservative spatial discretization, we can obtain a mass conservative
temporal discretization for RE and two-phase ow models by employing a semi-explicit
index-oneformulation,aswe demonstrated in [22]. Here we briey outlinethis approach
for three-phase owmodels.
The semi-discretesystem of three-phase ow equationscan be writtenas
@
(
i;j
i;j )=O
di;j
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
2
4
6
8
10
12
14
16
18
20
22
0.5
1
1.5
2
x [m]
y [m]
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time [days]
average order DAE
average
∆
t
⋅
10
4
DAE
average
∆
t
⋅
10
4
BE−A
Figure1. left: Spatialdistributionofwaterphasevolumefractionforhill-slopesimulation.
right: Temporalintegration order and step size as afunction of simulation time for
hill-slope simulation.
where N is the number of discrete spatial nodes, O
d
is the approximate divergence of
uid phase mass ux at each node, and the subscripts w, n, and a are phase qualiers
correspondingtoawettingandanon-wettinguidphaseandair,respectively. Tosimplify
the presentation we have assumed that the form of the equations are the same on the
interior and boundary of the domain. We have also assumed that relations suÆcient to
obtainformalclosureareavailable;thatis,ateachspatialnodethereisathreecomponent
vector of unknowns, y
j
. Forexample, we could choose y
j =( w;j ; w;j ; g;j
) for common
closurerelations. Followingtheapproachgivenin[22],wederiveasemi-explicitindex-one
DAE fromeqn 11given by
@M i;j @t O di;j = 0 M i;j i;j (y i ) i;j (y i
) = 0
for i=w;n;a and j =1;:::;N (12)
where M represents the uid phase bulk density in the porous medium. Applying the
BDF time discretization aboveyields the nonlinear system
M j;i + i O dj;i = 0 M j;i j (y i ) j (y i
) = 0
for i=w;n;a and j =1;:::;N (13)
Thissystemcanbesolved eÆcientlyusingacombinationofstraightforwardalgebraic
ma-nipulationsandNewton'smethodasshownin[22]. Aslongaseqn13issolvedaccurately,
the numerical model is mass conservative in the sense that the discrete change in uid
5. CONCLUSIONS
The method of lines is a straightforward approach for applying sophisticated adaptive
temporalintegration methodstoPDE modelsofowand transportinporous media. We
havedemonstrated, in particular,that transient nonlinear models ofmultiphase owcan
besolvedeÆcientlyusingvariableorder,variablestepsize BDFmethodsinthecontextof
theMOL.Further,thesemethodscanbeappliedinconjunctionwithsophisticatedspatial
discretization techniques, such as MFEMs, toyield accurate, mass conserving, solutions
to complexmultiphase problems onirregulardomains.
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