Fast Stimulation of Polymer Injection

Fast Simulation of Polymer Injection inHeavy-Oil Reservoirs on the Basis ofTopological Sorting and Sequential

SplittingKnut-Andreas Lie, Halvor Møll Nilsen, Atgeirr Flø Rasmussen, and Xavier Raynaud, SINTEF ICT

Summary

We present a set of algorithms for sequential solution of flow andtransport that can be used for efficient simulation of polymerinjection modeled as a compressible two-phase system. Our for-mulation gives a set of nonlinear transport equations that can bediscretized with standard implicit upwind methods to conservemass and volume independent of the timestep. In the absence ofgravity and capillary forces, the resulting nonlinear system of dis-crete transport equations can be permuted to lower triangularform with a simple topological-sorting method from graph theory.This gives an optimal nonlinear solver that computes the solutioncell by cell with local iteration control. The single-cell systemscan be reduced to a nested set of nonlinear scalar equations thatcan be bracketed and solved with standard gradient or root-brack-eting methods. The resulting method gives orders-of-magnitudereduction in run times and increases the feasible timestep sizes. Infact, one can prove that the solver is unconditionally stable andwill produce a solution for arbitrarily large timesteps. For caseswith gravity, the same method can be applied as part of a nonlin-ear Gauss-Seidel method. Altogether, our results demonstrate thatsequential splitting combined with single-point upwind discretiza-tions can become a viable alternative to streamline methods forspeeding up simulation of advection-dominated systems.

Introduction

Heavy-oil resources are estimated to be more than twice the resour-ces of conventional crude oil. Reservoirs containing heavy oil arechallenging to produce, primarily because the oil has much higherviscosity than water, which will cause injected water to fingerthrough the reservoir and leave a large percentage of the hydrocar-bons behind as residual or nonrecoverable oil (Gao 2011).Enhanced oil recovery is therefore essential to increase each field’slifetime and ultimately the world’s oil-producible resources.

Polymer flooding is a widely used enhanced-oil-recovery(EOR) strategy (Wassmuth et al. 2009; Kang et al. 2011; Fletcheret al. 2012), in which a dilute solution of water-soluble polymermolecules is injected to increase the effective water viscosity andthereby establish more-favorable mobility ratios between theinjected and the displaced fluids. The addition of long-chain poly-mer molecules causes a reduction in the permeability and mayalso cause the water to behave like a non-Newtonian fluid (i.e., beresistant to flow at low velocities). This allows preferential fillingof highly permeable zones and increases the areal sweep effi-ciency. Altogether, this results in a highly nonlinear fluid behaviorand increased stiffness of the governing transport equations. Inparticular, because the water viscosity is strongly affected by thepolymer concentration, it is crucial to capture polymer frontssharply to resolve the nonlinear displacement mechanismscorrectly.

Polymer fronts are, unlike water fronts, not self-sharpening, andhigh numerical resolution is therefore required to limit the numeri-cal diffusion that would otherwise bias or deteriorate simulationresults. High numerical resolution can be achieved with a fine grid,by introducing a higher-order discretization, or through a combina-tion of these two approaches. Herein, we will focus on high grid re-solution, and consider methods for simulating polymer flooding onhigh-resolution geocellular grid models, which are currently out-side the reach of conventional solvers based on fully implicit dis-cretizations. To enable simulation of large grid models, we apply asequential solution method in which pressure and saturations/con-centrations are updated in separate steps [e.g., as is commonly usedin streamline simulators (Datta-Gupta and King 2007; Thiele et al.2010)]. For the particular models considered herein, the splittingconserves mass and volume and has no theoretical restriction onthe pressure and transport steps. This enables us to use efficientsolvers that are specially targeted at the equations found in each ofthe sequential steps. For the pressure step, we use a standard two-point discretization, combined with highly efficient, algebraic mul-tigrid, linear solvers. For the saturation/concentration equations,we apply an upstream mobility-weighted, finite-volume discretiza-tion in space and an implicit, first-order discretization in time. Thischoice may seem strange, given our goal of solving models with ahigh number of cells. However, our motivation is that geocellularmodels tend to have large differences in face areas and cell vol-umes, which effectively preclude the use of explicit timesteppingmethods that are common for high-order discretization methods.Instead, we will develop a set of nonlinear solvers that are robustover a large span of Courant-Friedrichs-Lewy (CFL) numbers andso fast that numerical diffusion can be reduced by the use of highgrid resolution and timesteps that are near the CFL limit of explicitschemes for the cells that contribute most to the numerical diffu-sion. The rationale is that first-order implicit and explicit schemesin practice will have equal numerical diffusion if the timestep of theimplicit method is approximately the same as the CFL restrictionfor explicit schemes in cells that have neither high flow nor smallvolumes. Moreover, localized methods such as the discontinuousGalerkin method can be applied if a higher approximation order isrequired for the spatial discretization (Natvig and Lie 2008a).

The key to developing highly efficient transport solvers is theobservation that the density contrast between the injected waterand the oil is typically (much) smaller in the recovery of heavy oilthan in traditional oilfield operations, which implies a looser cou-pling between viscous and gravity forces. We exploit this loosecoupling to introduce a splitting between advection and gravitysegregation in the transport step. In the absence of gravity, the ad-vective transport equations can be discretized by the first-order,implicit, upstream-mobility-weighted method to give a discretenonlinear system. By performing a topological sort on the fluxgraph, the nonlinear system can be permuted to triangular formand solved cell by cell with local control over the nonlinear itera-tions (Natvig and Lie 2008a; Lie et al. 2012a). This reorderingmethod is related to the potential ordering methods proposed byKwok and Tchelepi (2007) and Shahvali and Tchelepi (2013), inwhich equations are reordered on the basis of phase velocitiesrather than total velocity. When gravity is present, one can

Copyright VC 2014 Society of Petroleum Engineers

This paper (SPE 163599) was accepted for presentation at the SPE Reservoir SimulationSymposium, The Woodlands, Texas, USA, 18–20 February 2013, and revised forpublication. Original manuscript received for review 15 May 2013. Revised manuscriptreceived for review 18 October 2013. Paper peer approved 7 January 2014.

J163599 DOI: 10.2118/163599-PA Date: 12-December-14 Stage: Page: 991 Total Pages: 14

ID: jaganm Time: 15:32 I Path: S:/3B2/J###/Vol00000/140015/APPFile/SA-J###140015

December 2014 SPE Journal 991

formulate an effective nonlinear Gauss-Seidel method that reliesentirely on solving single-cell problems. Each single-cell problemcan be solved by use of the Newton-Raphson method, or it can bereduced to a nested set of scalar nonlinear equations that can bebracketed and thus is guaranteed to have a unique solution for ar-bitrary large timesteps. Altogether, this gives a very robust and ef-ficient method that is fully competitive with state-of-the-artstreamline methods (Alsofi and Blunt 2010; Thiele et al. 2010;Clemens et al. 2011), and exploits the same physical features ofthe governing equations. Our method, however, is solely based onstandard finite-volume discretizations and hence avoids potentialpitfalls, such as lack of mass conservation, allocation of fluxes tostreamlines, mapping between streamlines and physical grid, andtracing of streamlines in irregular geometries, that complicate theimplementation of streamline methods.

Mathematical Model and Discretization

In the following, we will consider a set of polymer models thathave many of the features that are typically supported by contem-porary commercial simulators. To this end, the polymer-floodingprocess will be described by an immiscible, two-phase, three-component flow model that may include the effects of adsorption,rock and fluid compressibilities, pressure-dependent transmissibil-ity, and gravity. Capillary effects are disregarded herein, but canbe included by operator splitting in the same way as for streamlinemethods. In summary, the model is essentially the same as in acommercial simulator (Eclipse 2009). The numerical methodspresented in the following can also apply to models that accountfor dead pore volume (PV)—wherein polymer cannot reach thesmallest pores so that the effective PV /ð1� SdpvÞ available forthe flowing polymer solution is smaller than the PV / of the rock.For brevity, this effect has been left out of the presentation.

Mathematical Model. To derive a model, we start from the con-servation equations for oil, water, and polymer:

@

@tðqa/SaÞ þ r � ðqa~vaÞ ¼ 0; a 2 fw; og; ð1aÞ

@

@tðqw/SwcÞ þ @

@t½qr;refð1� /refÞa� þ r � ðcqw~vwpÞ ¼ 0:

� � � � � � � � � � � � � � � � � � � ð1bÞ

Here, qa, ~va, and Sa denote density, velocity, and saturation ofphase a, respectively; the porosity is denoted by / and is assumedto be a function of pressure only; c denotes the polymer concen-tration; and ~vwp in the velocity of water containing diluted poly-mer. The function a models the amount of polymer that isadsorbed in the rock. The time scale of adsorption is much fasterthan that of mass transport and we will assume that adsorptiontakes place instantaneously so that a is a function of c only. Thereference rock density is qr;ref , and the reference porosity is /ref .Sources and sinks may be included in a manner equivalent toboundary conditions, and are left out of Eqs. 1a and 1b.

To get a solvable system, Eqs. 1a and 1b must be supplied by aset of closure relations. Simple pressure/volume/temperature be-havior is modeled through the formation-volume factors ba ¼baðpÞ, defined by qa ¼ baqS

a, where qSa is the surface density of

phase a. Inserting this into Eqs. 1 and 1b, the system can be simpli-fied by dividing each equation with the relevant surface density qS

a,

@

@tðba/SaÞ þ r � ðba~vaÞ ¼ 0; a 2 fw; og ð2aÞ

@

@tðbw/SwcÞ þ @

@t½ð1� /refÞa� þ r � ðbwc~vwpÞ ¼ 0; ð2bÞ

where for convenience we have introduced the shorthand a ¼aqr;ref=q

Sw.

Darcy’s law for the two phases can be written as ~va ¼�Kka½rp� baðpÞqS

agrz�, where K is rock permeability, ka ¼ka=la is the fluid mobility for phase a, g is the gravity constant,

and z is the coordinate in the vertical direction. We introduce thetotal mobility k ¼ ko þ kw and let fa ¼ ka=k denote the fractionalflow of phase a. As long as the relative permeabilities ka are non-negative, monotonic, and equal to zero for Sa ¼ 0, the fractionalflow functions faðSaÞwill be monotonic and have endpoint valuesfað0Þ ¼ 0 and fað1Þ ¼ 1. With this, the equation for the total ve-locity~v ¼~vw þ~vo reads

~v ¼ �Kk rp�X

a

baqSa fagrz

!: ð3Þ

Next, we describe a simple fluid model for polymer diluted inwater. The concentration of polymer is so small that the onlyphysical property affected is the viscosity. To model the viscositychange of the mixture, we will use the Todd-Longstaff model(Todd and Longstaff 1972). This model introduces a mixing pa-rameter x 2 ½0; 1� that takes into account the degree of mixing ofpolymer into water. Assuming that the viscosity lm of a fullymixed polymer solution is a function of the concentration, theeffective polymer viscosity is defined as

lp;eff ¼ lmðcÞxl1�xp ;lp ¼ lmðcmaxÞ: ð4Þ

The viscosity of the partially mixed water is given in a similarway by

lw;e ¼ lmðcÞxl1�xw : ð5Þ

The effective water viscosity lw;eff is defined by interpolatinglinearly between the inverse of the effective polymer viscosityand the partially mixed water viscosity:

1

lw;eff

¼ 1� c=cmax

lw;e

þ c=cmax

lp;eff

: ð6Þ

To take the incomplete mixing into account, we introduce thevelocity of water that contains polymer, which we denote~vwp. Forthis part of the water phase, the relative permeability is denotedby krwp and the viscosity is equal to lp;eff . We also consider thetotal water velocity, which we still denote ~vw and for which theviscosity is given by lw;eff . Darcy’s law then gives us

~vw ¼ �krw

lw;effRkðcÞKðrpw � qwgrzÞ;

~vwp ¼ �krwp

lp;effRkðcÞKðrpw � qwgrzÞ ¼ mðcÞ~vw; � � � � � ð7Þ

because we assume that the presence of polymer does not affectthe pressure and the density. We have also assumed that the rela-tive permeability does not depend on mixing, so that krwp ¼ krw.The function RkðcÞ denotes the actual resistance factor and is anondecreasing function that models the reduction of the perme-ability of the rock to the water phase because of the presence ofadsorbed polymer. Herein, we set RkðcÞ ¼ 1þ ðRrf � 1ÞCa

pðcÞ=Ca

pðcmaxÞ, where Rrf is the residual resistance factor defined as the

ratio of the initial-water mobility to the water-solution mobilityafter polymer flooding, Ca

p is the amount of polymer adsorbed,

and CapðcmaxÞ is the maximum-possible adsorbed polymer; all

three depend on the rock type and are thus spatially dependent. Auseful formula for m(c) can easily be derived with

mðcÞ ¼lw;eff

lp;eff

¼ 1� c

cmax

� �lp

lw

� �1�x

þ c

cmax

" #�1

: ð8Þ

To simplify the notation in the following, we write Eq. 7 as

~vw ¼ f ðS; cÞ~v þ~gw;~vwp ¼ mðcÞf ðS; cÞ~v þ~gwp: ð9Þ

Here, the terms ~gw and ~gwp denote gravity contributions; theirexact form will be given in the Gravity Splitting subsection andis not important in the preceding discussion. Fig. 1 shows

. . . . . . . . . .

. . . . . . . . . .

. . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . .

. . . . . . . .



992 December 2014 SPE Journal

representative plots of the functions f ðS; cÞ and mðcÞc, the nonli-nearity of which will play a prominent role in the following.

Discretization. To discretize Eqs. 2a and 2b, we first introduce agrid consisting of cells fCig that each have a bulk volume Vi, inte-grate over each cell in space, and apply a backward-Euler discreti-zation for the temporal derivative. This gives the discretetransport equations

ðba;i/iSa;iÞnþ1 � ðba;i/iSa;iÞn þDt

Vi

Xj

ðba;ijva;ijÞnþ1 ¼ 0;

� � � � � � � � � � � � � � � � � � � ð10aÞ

½bw;i/iSw;ici þ ð1� /ref;iÞai�nþ1� ½bw;i/iSw;ici þ ð1� /ref;iÞai�n

þ Dt

Vi

Xj

ðba;ijcijvwp;ijÞnþ1 ¼ 0 � � � � � � � � � ð10bÞ

Here, subscript i denotes quantities associated with cell Ci andsubscript ij denotes quantities associated with the interfacebetween cells Ci and Cj. Superscripts denote timesteps. To derivea discrete pressure equation, we sum the two phase equations,Eqs. 10a and 10b, and use the condition Sw þ So ¼ 1 to obtain

/nþ1i � /n

i

Xa

bna;i

bnþ1a;i

Sna;i

!

þDt

Vi

Xj

Xa

bnþ1a;ij

bnþ1a;i

ðfa;ijvij þ ga;ijÞnþ1 ¼ 0: � � � � � � � ð11Þ

Here, fa;ij and ga;ij denote upstream-weighted fractional-flow andgravity contributions of phase a, respectively, whereas vij denotesthe flux corresponding to the total velocity ~v, which can beexpressed as vij ¼ Tijðpi � pj þ gijÞ, where Tij denotes the transmis-sibility between cells Ci and Cj. The transmissibility is computed bythe industry-standard approach in which the fluid-mobility part isdiscretized with single-point, phase-based upstream weighting andthe fluid-independent part by a standard two-point flux approxima-

tion that involves harmonic averaging. The equation necessary tocompute discrete pressures and fluxes is now formed by combiningEq. 11 with the expression for the discrete flux.

Solution Strategy

Our overall system will consist of a pressure equation, Eq. 11, andtwo transport equations, Eq. 10a with a ¼ w for the water satura-tion and Eq. 10b for the polymer concentration. To solve thiscoupled system, we use a sequential implicit solution procedure,as outlined in Algorithm 1, which separates and solves the pressureand transport equations in consecutive steps. Herein, the pressureand transport equations are solved only once per timestep, but ifneeded each timestep can be iterated to reduce the (nonlinear)residuals to an arbitrarily small tolerance. In the pressure step,when solving Eq. 11, we use the values of saturation and concen-tration given at the previous timestep so that the superscript nþ 1in the equation refers only to the pressure values.

The dynamics of the transport problem are generally deter-mined by the balance between viscous and gravity forces. Forheavy-oil systems, however, gravity segregation is typically aweak effect compared with viscous flow because of large injectionrates and small differences in the densities of the oil and waterphases. This means that the transport will mainly be cocurrent,giving a unidirectional flow property that can be exploited to de-velop efficient transport solvers. If gravity effects are negligible,one can prove that this sequential implicit splitting method will beunconditionally stable in the sense that it is mass conservative,preserves consistency between pressures and masses, has no time-step restriction between pressure and transport steps, and has notimestep restrictions on the transport steps. (The proof will be pre-sented in a forthcoming paper.) In the next subsection, AdvectionStep: The Single-Cell Problem, we will also argue that each trans-port step can be solved by use of a finite number of operations, in-dependent of the time size of the splitting step Dt ¼ tnþ1 � tn.

Advection Step: The Single-Cell Problem. If gravity isneglected and a monotonic method such as the standard two-pointflux-approximation scheme is used to discretize the pressure

0 10

0.5

1

S

Fractional flow f (S, c) Fractional flow f(S, c) Coefficient m(c)

c = 0

c = 1/3c = 2/3

c = 1

ω = 0

0 10

0.5

1

S

c = 0c = 1/3 c = 2/3 c = 1

ω = 1

0 10

1

c/cmax

ω = 0ω = 0.2ω = 0.4ω = 0.6ω = 0.8ω = 1

ccmax

Fig. 1—Left: Plots of the fractional flow functions f (S,c), for x50 and x51. Here, c denotes ccmax

. We observe, as expected, that thefractional flow is a decreasing function of the polymer concentration, which explains the use of polymer to increase oil recovery.When there is no polymer (c50) or the solution is fully saturated with polymer (c51), the fractional flow functions are independentof x, but we notice that the decrease in fractional flow for intermediate values of the concentration is more important in the fullymixed case (x51); that is, the polymer is more effective in a fully mixed solution. Right: Plot of mðcÞc. This coefficient is equal tothe ratio of the polymer flux to the water flux. It can be seen as the analog of a relative permeability in a multiphase system, whereconcentration plays the role of saturation (Booth 2008). From the plot, we observe that x is directly related to the degree of nonli-nearity contained in the model, with the linear case corresponding to a fully mixed solution, x51.

Algorithm 1—Sequential Splitting Method for Pressure and Transport.




equation, the velocity field ~v has no loops. The discrete equiva-lence of this property is that the directed graph describing thefluxes is acyclic and hence can be topologically sorted. By use ofthe reordering induced by the topological sort, all cell neighbors ofcell Ci can be divided into two index sets: UðiÞ denotes all upwindneighbors and DðiÞ denotes all downwind neighbors. If we useupstream-cell evaluation for the fractional-flow function fij, thediscrete transport equations can be written as two residualequations:

Rw;iðSnþ1; cnþ1Þ ¼ ðbi/iSiÞnþ1 � ðbi/iSiÞn

þDt

Vi

Xj2UðiÞ½ f ðSj; cjÞbijvij�nþ1

þDt

Vif ðSiciÞnþ1

Xj2DðiÞðbijvijÞnþ1;

� � � � � � � � � � � � � � � � � � � ð12aÞ

Rc;iðSnþ1; cnþ1Þ ¼ ½bi/iSici þ aðciÞð1� /ref;iÞ�nþ1

�½bi/iSici þ aðciÞð1� /ref;iÞ�n

þ Dt

Vi

Xj2UðiÞ½mðcjÞcjf ðSj; cjÞbijvij�nþ1

þ½mðciÞcif ðSi; ciÞ�nþ1 Dt

Vi

Xj2DðiÞðbijvijÞnþ1:

� � � � � � � � � � � � � � � � � � � ð12bÞ

All upwind cells Cj for j 2 UðiÞ will appear before cell Ci in thereordered numbering. If the residual equations of Eqs. 12a and 12bare solved in the order given by a topological sort of the total flux,the saturations Snþ1

j and concentrations cnþ1j have already been

computed when we visit cell Ci. Hence, the only unknowns in Eqs.12a and 12b are the values of saturation and concentration in thecell Ci, Snþ1

i and cnþ1i . In other words, performing a topological sort

enables us to permute the nonlinear system of discrete transportequations, Eq. 12a and 12b, to a block-triangular form, as illustratedin Fig. 2. Because of the block-triangular form, the solution can becomputed one cell at a time, as outlined in Algorithm 2. Thismethod was first introduced by Natvig et al. (2006, 2007) for lineartransport equations and by Aarnes et al. (2006) for incompressibletwo-phase flow. The method was later extended to systems of trans-

port equations and higher-order methods by Natvig and Lie(2008a). In particular, Natvig and Lie (2008b) demonstrated two-orders-of-magnitude speedup for several examples of incompressi-ble and slightly compressible two-phase flow. One order of magni-tude was attributed to ordering of the equations, and one order ofmagnitude was attributed to localization of the nonlinear iterations.

In an implicit method, all cells are solved simultaneously,which means that the number of iterations is dictated by the cellthat converges slowest. Algorithm 2 gives us local control over theiteration process, meaning that the number of iteration stepsneeded per cell is no longer forced to be the maximum needed bythe worst cell. With local control, the average number of steps ismuch reduced, for example by exploiting that the residual will beidentical to zero ahead of the displacement fronts in many floodingscenarios. Hence, no iterations are required in the correspondingcells. This is demonstrated by Lie et al. (2012b). To achieve fur-ther gains in efficiency compared with a straightforward Newton-Raphson method, we must devise an efficient method for solvingeach of the single-cell problems defined in Eqs. 12a and 12b. Tosimplify the presentation, we write the residual equations in amore compact form. That is, we let S and c denote Snþ1

i and cnþ1i

and introduce seven constants, k0;…; k6, in which we collect termsinvolving cells Cj, j 2 UðiÞ that have already been computed,quantities that depend on the known pressure solution ðpnþ1

i ; vnþ1ij Þ

at time nþ 1, and terms from the previous timestep n. Then, wecan write Eqs. 12a and 12b as a 2�2 nonlinear system of the form

RwðS; cÞ ¼ k0 þ k1Sþ k2 f ðS; cÞ ¼ 0; ð13aÞ

RcðS; cÞ ¼ k3 þ k4Scþ k5aðcÞ þ k6mðcÞcf ðS; cÞ ¼ 0:

� � � � � � � � � � � � � � � � � � � ð13bÞ

Here, k0 and k3 are independent of Dt, whereas the remainingdepend linearly, which means that the nonlinear parts of the sys-tem increase linearly in Dt.

The straightforward way of solving this system is by applyinga gradient method such as the Newton-Raphson method. Whereassuch a method will be efficient in many cases, it is not guaranteedthat the method will converge, or even that Eqs. 13a and 13b haveunique solutions. Fortunately, Lie et al. (2012b) recently provedthat Eqs. 13a and 13b have unique solutions for a simpler flowmodel without polymer adsorption and resistance factor. The keyidea in the proof is to use Eq. 13a to formally eliminate S as a

. . . . . . . . . . . .

Sparsity pattern Topologically sorted Sparsity patternNatural order

1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17

18 19 20 21 22 23 24 25

26 27 28 29 30 31 32 33 34

35 36 37 38 39 40 41 42

43 44 45 46 47 48 49 50 51

52 53 54 55 56 57 58 59

0 10 20 30 40 50 60

0

10

20

30

40

50

60

nz = 148

1 2 6 11 18 33 35 36

3 4 5 10 17 32 34 37 38

7 8 9 16 31 39 40 41

12 13 14 15 30 45 46 47 48

19 20 21 29 44 51 52 53

22 23 26 28 43 50 55 56 57

24 25 27 42 49 54 58 59

0 10 20 30 40 50 60

0

10

20

30

40

50

60

nz = 148

Fig. 2—Illustration of the reordering idea for a quarter-five-spot simulation on a Voronoi grid with injection in the lower-left corner(Cell 1) and production in the upper right (Cell 59). The two plots to the left show the natural numbering of the cells together withthe resulting sparsity pattern of the discrete transport equations. The two plots to the right show the new ordering induced by thetopological sort and the resulting sparsity pattern.

Algorithm 2—Advection Step for a System that Can Be Completely Reordered.




function of c, and then insert this function SðcÞ into Eq. 13b toobtain a nonlinear scalar equation Rc½SðcÞ; c� ¼ 0. In general, SðcÞcannot be computed analytically, but must instead be computednumerically by solving Eq. 13a for each fixed c. The resultingalgorithm for the single-cell problem is hence a nested loop ofiterations over nonlinear scalar equations hðcÞ ¼ 0 that each canbe solved robustly using bracketing scalar root-finding algorithms.If hðaÞ differs in sign from hðbÞ, the function h crosses zero atleast once in the interval ½a; b�, which is thus guaranteed to containa root. Given a valid initial interval, bracketing algorithms cannotfail to find a root if the function h is well behaved and will termi-nate in a finite number of iterations for a given tolerance on the re-sidual. This means that the approximate solution can be computedin a finite number of iterations, which in practice will increasewith the local CFL number. However, the nested bracketing algo-rithm guarantees that there is an upper bound that is independentof the size of the timestep. The same can easily be proved for thesystem considered herein under the assumption of equal fluidcompressibilities. The proof can also be extended to models withdifferent fluid compressibilities, provided care is taken when eval-

uating ba;ij so that ba;ij ¼ baðpnþ1ij Þ if ð fovij þ go;ijÞ and pnþ1

ij �pnþ1

i have opposite signs, and ba ¼ baðpnþ1i Þ otherwise. Likewise,

the proof can be extended to models that account for dead PV(Eclipse 2009), provided that one makes certain modifications toeliminate the instability inherent in the model that allows for thecomputation of infinite polymer concentrations. Details will begiven in forthcoming papers.

There are several robust bracketing root-finding methods forsolving scalar equations, such as bisection or modified regula falsi,as well as hybrid methods, such as Brent’s method, which combinebisection, secant, and inverse quadratic interpolation. In our imple-mentation we have used a modified regula falsi method describedby Ford (1995), the Pegasus method, for both solves. This methodonly requires the evaluation of residuals, which are simple toimplement and typically inexpensive to compute compared withthe Jacobians required by gradient methods. Bracketing methodsmay not always have optimal convergence, but can still be used asnonlinear solvers in their own right or as robust fallback strategiesfor more-efficient, gradient-based methods when solving the sin-gle-cell problem, as demonstrated by Lie et al. (2012b).

Advection Step: Nonlinear Gauss-Seidel Iterations. In the gen-eral case, we are not guaranteed that the total velocity ~v does notcontain loops (i.e., blocks of mutually dependent cells that cannotbe solved one by one in order). This happens in particular if gravityeffects are included in the total velocity ~v or if a nonmonotonicmethod is used to discretize the pressure equation, Eq. 11. Fig. 3shows examples of loops occurring in discretized transportequations.

In (Natvig and Lie 2008a, b), the coupled multicell systemswere solved by use of a straightforward stabilized Newton method.However, numerical experiments indicate that it is more efficient

to use a simple, iterated, nonlinear Gauss-Seidel method in whichwe solve the multicell block one cell at a time, and repeat the pro-cess until an acceptable converged solution is obtained. Thismethod will be outlined in the following. By applying the reorder-ing, we ensure an optimal block structure of the nonlinear systemin which the number of the loops and the size of each loop are mini-mized. The loop segments are solved in the correct order so that,for a given loop, the equations are well posed and the onlyunknowns are the saturations and concentrations in the cells thatbelong to the loop. Let us now consider a loop of size N‘. We dropthe superscript nþ 1 and, with a slight abuse of notation, let S ¼½S1;…; SN‘

� and c ¼ ½c1;…; cN‘� denote the values of saturation

and concentration, respectively, in the cells that belong to the loop.The residuals given by Eqs. 12a and 12b take the following form,

RwðS; cÞ ¼ k0 þ k1

S1

..

.

SN‘

264

375þ k2

f ðS1; c1Þ...

f ðSN‘; cN‘Þ

264

375; ð14aÞ

RcðS; cÞ ¼ k3 þ k4

S1c1

..

.

SN‘cN‘

264

375þ k5

aðc1Þ...

aðcN‘Þ

264

375

þ k6

mðc1Þc1f ðS1; c1Þ...

mðcN‘ÞcN‘

f ðSN‘; cN‘Þ

264

375; ð14bÞ

where k0 and k3 are vectors and k1, k2, k4, k5, k6 are matrices thatdo not depend on S and c and are defined analogously as in Eqs.13a and 13b. In the nonlinear Gauss-Seidel method, we loopthrough all the cells that belong to the loop and in each cell Ci

solve the local 2�2 nonlinear single-cell problem to update Si andci, using values from the previous timestep or iteration step in allupwind cells. This process is repeated until the residuals arebelow a prescribed tolerance. Let superscript k denote the itera-tions of the Gauss-Seidel algorithm. Then, for k¼ 1, 2, …, wecompute sequentially Sk

i and cki for i ¼ 1;…;N‘ by solving the

single-cell problems

0¼Rkw;iðS;cÞ

¼Rw;iðSk1;…;Sk

i�1;S;Sk�1iþ1 ;…;Sk�1

N‘;ck

1;…;cki�1;c;c

k�1iþ1 ;…;ck�1

N‘Þ;

� � � � � � � � � � � � � � � � � � �ð15aÞ

0¼Rkc;iðS;cÞ

¼Rc;iðSk1;…;Sk

i�1;S;Sk�1iþ1 ;…;Sk�1

N‘;ck

1;…;cki�1;c;c

k�1iþ1 ;…;ck�1

N‘Þ:

� � � � � � � � � � � � � � � � � � � ð15bÞ

The single-cell problems of Eqs. 15a and 15b take the forms ofEqs. 13a and 13b and can be solved by a Newton-Raphson

. . . . .

. . . . . . . . .

Fig. 3—Illustration of loops occurring in discretized transport equations. The plots show the sparsity pattern of the nonlinear sys-tem for four different cases with increasing influence of loops, from left to right: an almost completely reorderable system contain-ing only two small loops; a minor fraction of the cells involved in seven small loops; the majority of cells involved in eight loops;and all cells connected in one big loop. When solving the nonlinear system, the local residual equations for the cells inside the redsquares are coupled and must be solved simultaneously.




method or in the same robust manner as described previously.The convergence of the method can be established for small time-steps, but the proof is omitted for brevity. The proof also showsthat the convergence speed increases when the dependence of theresiduals with respect to the components belonging to the cells

downward in the loop is small, as in@Rw;i

@Sj

��þ @Rw;i

@cj

�� 1 and

@Rc;i

@Sj

��þ @Rc;i

@cj

�� 1 for all i and j > i.

In the special case in which the cells C1;…;CN‘are reorder-

able, we have@Rw;i

@Sj¼ @Rw;i

@cj¼ @Rc;i

@Sj¼ @Rc;i

@cj¼ 0 for all i and

j > i, and the solution can then be found in exactly one iteration.The necessary steps for solving the advection problem are sum-marized in Algorithm 3.

Gravity Splitting. If gravity effects cannot be neglected, weneed to introduce an additional operator splitting for the transportequations, Eqs. 10a and 10b, to be able to use the reorderingmethods discussed in the two previous subsections. This operator-splitting method was first introduced within streamline simulation(Gmelig Meyling 1990, 1991; Bratvedt et al. 1996), but can alsooffer certain benefits for finite-volume methods, as discussed byLie et al. (2012a).

First, we solve the advective equations that account for viscouseffects (for brevity, subscript w has been dropped),

ðb/S�Þnþ1i � ðb/SÞni þ

Dt

Vi

Xj

½bf ðS�; c�Þv�nþ1ij ¼ 0; ð16aÞ

½b/c�S� þ ð1� /refÞaðc�Þ�nþ1i � ½b/cSþ ð1� /refÞaðcÞ�ni

þ Dt

Vi

Xj

½bmðc�Þc�f ðS�; c�Þv�nþ1ij ¼ 0; � � � � � � ð16bÞ

by use of the Gauss-Seidel method described in Algorithm 3. Thisgives us intermediate saturation and concentration distributionsS�;nþ1 and c�;nþ1, which are then used as initial conditions for a sec-ond set of segregation equations that account for gravity effects:

½b/ðS� S�Þ�nþ1i þ Dt

Vi

Xj

½bkof ðS; cÞðqw � qoÞgKrz�nþ1ij ¼ 0;

� � � � � � � � � � � � � � � � � � � ð17aÞ

fb/cðS� S�Þ þ ð1� /refÞ½aðcÞ � aðc�Þ�gnþ1i

þ Dt

Vi

Xj

½bmðcÞckof ðS; cÞðqw � qoÞgKrz�nþ1ij ¼ 0:

� � � � � � � � � � � � � � � � � � � ð17bÞ

In the segregation equation, all cells will in principle be coupledand must be solved for simultaneously by use of the Newton-Raphson method. However, stratigraphic models are often built sothat they have skewed pillars near faults and vertical pillars else-where. For models that only contain vertical cell columns, thesegregation equations will decouple between columns and can besolved one column at a time. Linearizing the segregation equation

inside a single column leads to a tridiagonal system that can besolved very efficiently by Gaussian elimination. Likewise, numer-ical experiments show that the use of a nonlinear Gauss-Seidelalgorithm, similar to the one outlined in the Advection Step: Non-linear Gauss-Seidel Iterations subsection, can also be very effi-cient for cases with small density differences. To this end, wevisit the cells inside the column in an alternating pattern, sweep-ing both from above and below. Solving for multiple columns isan operation that is straightforward to parallelize. In the generalcase, sets of cells that can be solved independently (in parallel)can be identified as connected components in the cell graphdefined such that cells are connected across a face if ~n �~g ¼ 0.Then, one can use a similar nonlinear Gauss-Seidel method to it-erate over all cells inside each connected component.

Finally, we note that by summing Eqs. 16a and 16b and Eqs.17 and 17b, we obtain

ðb/SÞnþ1i � ðb/SÞni þ

Dt

Vi

Xj

f½bf ðS�; c�Þv�nþ1ij

þ ½bkof ðS; cÞðqw � qoÞgKrz�nþ1ij g ¼ 0

� � � � � � � � � � � � � � � � � � � ð18aÞ

½b/cSþ ð1� /refÞaðcÞ�nþ1i � ½b/cSþ ð1� /refÞaðcÞ�ni

þ Dt

Vi

Xj

n½bmðc�Þc�f ðS�; c�Þv�nþ1

ij

þ½bmðcÞckof ðS; cÞðqw � qoÞgKrz�nþ1ij

o¼ 0; � � � ð18bÞ

which implies, after summing over all the cells, thatXi

ðb/SÞnþ1i ¼

Xi

ðb/SÞni ; andX

i

½b/cSþ ð1� /refÞaðcÞ�nþ1i

¼X

i

½b/cSþ ð1� /refÞaðcÞ�ni : � � � � � � ð19Þ

In other words, the total masses of water and polymer are con-served. The total mass of oil is in general not conserved and thisdiscrepancy can be seen as a splitting error of the pressure andtransport equations. In the special case of equal compressibilities,bo ¼ bw, the total mass conservation of oil is recovered, becauseit follows from the discrete pressure equation, Eq. 11, and thetotal mass conservation of water equation, Eq. 19.

Numerical Examples

In this section, we will present a few numerical experiments thatdemonstrate the utility of our numerical strategy and verify andvalidate the corresponding simulator that was implemented aspart of the Open Porous Media (OPM) initiative (OPM 2012).The simulator is available as free and open-source code and canbe downloaded from the OPM website under the GPLv3 license.For simplicity, all simulations reported in the following were per-formed on a single core. In the following examples, all 2�2 sin-gle-cell problems are solved either by the bracketed solverdiscussed previously or by a Newton-Raphson solver with thenested bracketing method as a fallback if Newton iterations fail.

A standard Newton-Raphson method for the full polymer sys-tem would consist of linearizing Eqs. 10a and 10b, solving the

. . .

Algorithm 3—Advection Step for a System that Can Only Be Partially Reordered.




resulting 2N � 2N system, and iterating. We do not make a directcomparison of the standard Newton-Raphson and the reorderedsolvers herein, but instead refer the reader to Natvig and Lie(2008b) for a comparison for the two-phase case without polymer.Here, one order of magnitude of speedup was observed by use ofreordering for the linearized Newton-Raphson equations, and anadditional order by use of reordering to localize the nonlinear iter-ations. Because the polymer equations represent a more complexnonlinear system, one can expect at least similar speed up.

The overall computational method involves several types oftimesteps that must be specified for the sequential splittings andimplicit discretizations: one for the sequential splitting of pressureand transport, one for the sequential splitting of the transport stepinto advection and segregation steps, one for the implicit advec-tion solver, one for the segregation solver, and one for the com-pressible pressure solver. These timesteps need not be equal, butmust have a certain internal consistency to ensure that the outertimesteps of the sequential splittings are multiples of the innertimesteps of the subequations. Moreover, to get an optimal algo-rithm, each step size should be chosen to reflect the time scales(or stiffness/nonlinearity) of the corresponding coupling or subeq-uation. In practice, one will always choose the timestep of thepressure/transport splitting and the pressure solver to be the same;for guidelines of how to choose this timestep, we refer the readerto the literature on streamline methods. For each pressure step,one may perform one or more transport steps that each may con-sist of one or more advection and segregation steps. This is dis-cussed in more detail by Lie et al. (2012a). Herein, all timestepsare assumed to be equal for simplicity, unless explicitly statedotherwise.

Example 1: Polymer Slug in a 1D Reservoir. In the first exam-ple, we will verify our simulator against a commercial simulator(Eclipse 2009) for three different polymer models of increasingcomplexity. To this end, we consider a 1D horizontal and homo-geneous, 1500-m-long reservoir that is initially filled with oil with5-cp viscosity. To produce the oil, we first inject water with 0.5-cp viscosity from the left side for 300 days, and then a polymerslug for 500 days, before continuing injection of pure water foranother 2,000 days. The polymer will increase the water viscosityby a factor of 20. The base-case model uses linear relative perme-

abilities as reported in Table 1: water viscosity 0.5 cp, oil viscos-ity 5 cp, no adsorption, no dead pore space, and a Todd-Longstaffmixing parameter x ¼ 1. In the second model, we include adsorp-tion as given in Table 1 and dead pore space equal to 0.15. In thethird model, the mixing parameter is set to x ¼ 0:5. The com-puted solutions plotted in Fig. 4 show excellent agreement for allthree cases, with only a slight deviation for the third model. Forthe last case, we also include solutions for a series of runs withlonger timesteps, which shows that one can increase the timestepso that the polymer front travels 1–2 cells per timestep withoutadversely affecting the resolution of the polymer slug. Notice thatthis corresponds to a CFL number that is significantly larger thanthe explicit limit for the whole system.

Example 2: Diagonal Flow in a 3D Cube. In the second exam-ple, we will investigate how our numerical methods scale with anincreasing number of grid cells. To this end, we consider an ideal-ized, synthetic test case consisting of a homogeneous reservoir inthe form of a cube described on a grid with n� n� n grid cells.The purpose of the example is to investigate how the transportsolver scales with the number of cells, and hence we disregardgravity and use a constant number of 10 timesteps for all grid res-olutions. (In a real physical case, the extent to which gravity canbe neglected would depend upon the strength of the injection raterelative to the gravity forces.) A total of 1 PV of water is injectedin the cell at the bottom southwest corner, and fluids are producedfrom the cell at the upper northeast corner. Polymer is injected inthe period between 0.2 and 0.5 PV injected. No-flow boundaryconditions are prescribed on all outer boundaries. The fluids aredescribed using the base-case model from Example 1.

Fig. 5 reports the central-processing-unit (CPU) times for thepressure and transport solvers. The pressure solver uses the alge-braic multigrid linear solver from dune:istl (Blatt and Bastian2007). As expected, the run time of the pressure solver does notscale linearly with the number of cells, but increases with an aver-age factor of 1.13. (This factor would most likely be higher forhighly heterogeneous petrophysical data.) The transport solver, onthe other hand, scales linearly because the fluxes do not containany loops; hence, the discrete transport equations can be permutedto a sequence of 2�2 single-cell problems.

Example 3: Gravity Convection. We have tested the perform-ance of the nonlinear Gauss-Seidel solver for the transport step fora worst-case scenario: a gravity convection cell in which all gridcells are mutually dependent. We have a 100�100-m domain inthe x–z-plane, with constant permeability equal to 0.1 darcy, unitporosity, and two phases with densities of 1000 kg/m3 and 600 kg/m3. Initially, the light and heavy fluids occupy the left and rightparts of the domain, respectively; gravity will then seek to redis-tribute the fluids so that the heavy fluid ends up at the bottom andthe light one on top, similar to a rotational motion. There are nosources or sinks, and no polymer. We have run all cases both withthe Gauss-Seidel-based transport solver (including segregation byoperator splitting), and a traditional Newton-Raphson solver.

In Fig. 6, the left plot shows the effect of increasing spatial re-solution. The case is run for 2,000 days in 10 uniform timesteps.There is an increase in computational cost per cell as we increaseresolution. The primary reason for this is the increased number ofiterations needed, by both the Gauss-Seidel and Newton algo-rithms, as the number of cells in the loop increase. The averagenumber of iterations for the Gauss-Seidel method is reported inTable 2. One may note that the increase in cost is not as large asthe increase in number of iterations. This is because a sparseupdating procedure is used for Gauss–Seidel, at each iterationmarking cells that have upstream saturation changes greater thana certain threshold (10�9 was used for this case). Only the markedcells are then updated at the next iteration, which rapidly reducesthe cost of successive iterations.

The right-hand plot of Fig. 6 shows the effect of increasing thenumber of timesteps. The case is run for 2,000 days with a varyingnumber of uniform timesteps on a 100�100-cell grid. As expected,

TABLE 1—FLUID DATA FOR EXAMPLE 1

Relative Permeabilities

Sw krw kro

0.2 0.0 1.0

0.7 0.7 0.0

1.0 1.0 0.0

Viscosity

c lmðcÞ=lw

0.0 1.0

7.0 20.0

Adsorption

c aðcÞ0.0 0.0000

2.0 0.0015

8.0 0.0025

Compressibility

Phase Value (bar�1)

Rock 3.00�10�6

Water 3.03�10�6

Oil 1.25�10�7




reducing the timestep significantly reduces the computational costper cell. This effect is so large that it may even take less time intotal to use more timesteps. With shorter timesteps, a smaller partof the total domain undergoes significant changes. Fewer itera-tions are then needed for convergence, as seen in Table 2.

For both comparisons, the Gauss-Seidel method works well: Itis stable and quite fast, more so than the Newton method, whichstarts to suffer convergence failures for the largest timesteps. TheNewton method implemented is very straightforward; although noattempt has been made to order the linearized equations, that

103 104 105 106 10710–1

100

101

102

103

Number of cells for n×n grid

CP

U ti

me

[sec

onds

]

Pressure

Transport

104 105 106

10–4

Number of cells for n×n grid

CP

U ti

me

per

cell

[sec

onds

]

Pressure

Transport

Fig. 5—CPU times for the n3n3n reservoir in Example 2. The left plot shows total CPU time as a function of the size of the model.The right plot shows the CPU time per cell as a function of the size of the model.

0 500 1000 1500 2000 2500

Polymer and water at position 2/3 − BASE Polymer and water at position 2/3 − ADS + DPS

Polymer and water at position 2/3 − ADS + DPS + MIX

days0 500 1000 1500 2000 2500

days

0 500 1000 1500 2000 2500days

0 500 1000 1500 2000 2500days

opm−polymerbaseline



Fig. 4—Comparison of 1D polymer slug computed by the OPM polymer solver (red lines) and a commercial simulator (black lines)for three different model configurations: base-case model, base-case model with adsorption and dead pore space, and with mixingparameter x50.5. All simulations used 5-day timesteps. The lower-right plot also shows solutions computed with timesteps of 10,20, 40, 80, and 160 days for the OPM solver.




would probably not have made a difference in this particular testcase, which is designed to be nonreorderable.

Example 4: Polymer-Front Resolution. The aim of our work isto make an efficient solver that can be used to reduce computa-tional costs, or alternatively, for the same computational time,increase the spatial resolution to decrease the error from numeri-cal diffusion. In this example, we highlight the latter with anidealized example consisting of a 1000-m, horizontal, 1D reser-voir with uniform permeability of 0.1 darcy and uniform porosityof 0.5, except in the interval between 50 and 100, in which the po-rosity is 0.005. The purpose of this region is to mimic the effectof small grid cells or fast flow near wells, which will alwaysappear in real reservoir simulations. The fluids have a viscositycontrast of 10, and polymer modifies the water viscosity by a fac-tor of 5 at maximum concentration.

Fig. 7 reports simulations with timesteps corresponding toCFL numbers � � 10, 1, and 0.1 in the high-porosity region. Asexpected, our implicit method is accurate for � 1 in the high-porosity region, despite that this corresponds to 100-times-largerCFL numbers and hence very large timesteps in the low-porosityregion. The reason that the solver works so well is that the maincontribution to the numerical diffusion will come from the high-porosity region where the fronts travel longest in time. To seethis, we can look at the worst-case scenario of a passivelyadvected wave, /ut þ aux ¼ 0, which would correspond to theadvection of the polymer concentration for x ¼ 1. First, we trans-form the equation to time-of-flight coordinates ðt; sÞ, wheres ¼ x/. Then, we observe that the effective equation for our

numerical discretization is given as @tuþ a@su ¼1

2ða2Dtþ

aDsÞ@2s u. This means that a discontinuity propagated by the

implicit advection scheme for a time t will be smeared to a width

O½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðaDtþ DsÞ � t

p�. In our case, we have two different regions:

a region of length ‘L with low porosity /L, and a region of length‘H with high porosity /H . Because the timestep and the grid sizesare constant for the whole domain, we have ðaDtþDx/LÞ < ðaDtþ Dx/HÞ. The residence time in each of theregions are tL ¼ sL=a ¼ ‘L/L=a and tH ¼ sH=a ¼ ‘H/H=a. Thismeans that the smearing will be dominated by the region with thelargest accumulated time of flight. For a front that travels fromx ¼ 0 to x ¼ 500, we have that sH ¼ 450 � 0:5 ¼ 900 � sL ¼50 � 0:005. In other words, the main contribution to the numericalsmearing will come from the high-porosity region in which theCFL number, and hence the numerical smearing, is low. Thisargument can easily be extended to more-realistic setups in threedimensions and suggests that an optimal solution strategy for animplicit discretization of the advection step is obtained by mini-mizing the CFL number and the grid size, measured in terms oftime of flight, in the regions that contribute most to the accumu-lated time of flight for a given traveling wave.

For explicit methods, the timestep is severely restricted byfast flow in small cells or fast-flowing regions, such as thosearound wells, even if these regions contribute little to the solu-tion in terms of time of flight. Regridding is often not an optionbecause the small grid cells are introduced to represent impor-tant flow connections in the system and the fast-flowing regionsneed to keep their resolution to minimize the error in the pres-sure solver. Compared with an explicit scheme, our implicit

0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 7—Comparison of 1D polymer-slug calculation using dif-ferent number of timesteps. The colored lines represent watersaturation (green) and polymer concentration (blue) computedwith 102 timesteps (dashed), 103 timesteps (dash-dotted), and104 timesteps on a grid with 1,000 cells. The black lines repre-sent a simulation with 104 timesteps and 200 cells. The totaltime is 5 years. The case has uniform porosity of 0.5, except inthe interval (50, 100), where the porosity is 0.005.

103 104

10−5

10−4

Gauss-SeidelNewton-Raphson

0 100 200 300 4000

1

2

3

4

5

6x 10–5

Gauss-SeidelNewton-Raphson

Fig. 6—CPU times per cell per timestep used by the transport solvers in Example 3. Left: logarithmic plot of run time in secondsvs. the number of cells. Right: logarithmic plot of run time in seconds vs. the length of the timestep.

TABLE 2—AVERAGE NUMBER OF ITERATIONS REQUIRED BY GAUSS-SEIDEL SOLVER IN EXAMPLE 3

Size 25�25 50�50 100�100 200�200

Average Iterations 15.5 28.7 65 111

Timesteps 5 8 10 16 20 40 80

Average Iterations 190 134 111 80 76 69 25




method can achieve less numerical diffusion with the perform-ance gained through the use of long timesteps in the fast-flowingregions to increase the resolution in slow-flowing regions thatdominate the numerical diffusion of a wave propagating throughthe reservoir. This stipulated gain in accuracy is, of course, casespecific, and will depend on the ratio of the largest CFL numbersto the CFL number in the regions corresponding to the largestaccumulation in time of flight. Compared with traditionalimplicit methods, our method can use the orders-of-magnitudeefficiency gain to reduce timestep and cell size in the sameregions. The need for higher grid resolution is clearly seen bycomparing the solid lines in Fig. 7 (black lines, 200 cells; blue/green lines, 1,000 cells), which both use very small timesteps.Another interesting feature is seen by comparing the dashed andthe black lines. For the black line (200 cells), the numerical dif-fusion is dominated by the grid cells, whereas for the dashedlines (1,000 cells), the error is dominated by the timestep and ismore sensitive to front speeds.

Example 5: The Norne Field. To validate our method, we con-sider a reservoir model representing Norne, a Norwegian Sea res-ervoir. The experiments presented in the following represent aplausible simulation case, but do not represent a real polymer-injection operation. The Norne model (NTNU 2012) is only usedto provide realistic reservoir geometry, petrophysical parameters,and well positions. Fluid properties and well schedules are syn-thetic, but are based on realistic heavy-oil cases from elsewhere inthe world. We emphasize that as far as the authors know, Statoiland the Norne license partners have no plans to inject polymerinto the field as discussed herein.

The Norne model shown in Fig. 8 consists of approximately45,000 cells and is a faulted corner-point grid with fairly complexand heterogeneous geometry: The ratio of cell volumes of the larg-est to the smallest cell is 3.6�105, whereas the ratio of the largestto the smallest interface area is 2.7�109. Most cells have sixneighbors, but some have as many as 21, because of faults. Thepermeability ranges from 0.3 md to 4 darcies. The model contains20 wells, of which six are injectors and 14 are producers. The res-ervoir is initially filled with oil and connate water, and simulatedallowing for compressible rock and fluids. For the fluid model, weuse the data described in Table 3 and Fig. 8; water and oil densitiesof 962 and 1080 kg/m3, respectively; water and oil viscosities of0.48 and 180 cp, respectively; and a Todd–Longstaff mixing pa-rameter x ¼ 1. In this model, the pillars of the corner-point gridare skewed, and the segregation equations are therefore globallycoupled. We have made the approximation when solving the seg-regation of treating the columns as uncoupled from each other. Webelieve this is reasonable because of small density differences, butthe accuracy of the assumption is not investigated in detail herein.

We compare a plain water-injection scenario lasting 4,000days to a polymer-injection scenario in which a polymer solutionis injected between Days 300 and 800. The 4,000 days have beencomputed over 73 steps with timesteps ranging from 1 day ini-tially to 100 days in the latter half. The timesteps were small ini-tially to establish the displacement fronts in the near-well regions.As the fronts move into the reservoir, the timestep is increased.Beyond this, the steps were chosen quite arbitrarily. In the leftpart of Fig. 9 we have plotted the total water cut of both scenarios,whereas the right part shows water-cut curves for the three pro-ducers with earliest breakthrough. The plots clearly demonstratethat the polymer has the effect of delaying water breakthroughand changing the shape of the curves.

The CPU times of the two simulations are reported in Table 4,where we clearly see that the computation time is dominated bythe pressure part, which takes five times longer than the transportsolver for polymer flooding and eight times longer for waterflood-ing. For the transport solver, a large part of the run time is used bythe gravity-segregation solver; running with no gravity reducesthe run time by a factor of 2–3 in both cases. The effect of gravity

TABLE 3—FLUID DATA FOR EXAMPLE 5

Viscosity

c lmðcÞ=lw

0 1.0

0.5 3.6

1.0 6.3

1.5 12.5

2.0 25.8

3.0 48.0

Adsorption

c aðcÞ c aðcÞ0.000 0.000000 1.250 0.000019

0.250 0.000010 1.500 0.000019

0.500 0.000012 1.750 0.000020

0.750 0.000016 2.000 0.000030

1.000 0.000018 3.000 0.000030

Compressibility

Phase Value (bar�1)

Rock 3.0�10�6

Water 5.0�10�5

Oil 5.0�10�5

Dead pore space¼0.05; residual resistivity factor¼ 1.3.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

krw

kro

Fig. 8—Left plot: The Norne model with injection wells shown in blue and production wells shown in red. Right plot: relative per-meability curves.




is small in this case because the density difference between thetwo phases is only 12%, and Fig. 10 shows that the water-cutcurves with and without gravity are almost indistinguishable.Neglecting gravity should therefore be a reasonable assumption.

The second effect that contributes to increase the run time iscompressibility. With incompressible rock and fluids, the pressureequation becomes linear and does not require any iterations. As aresult, the run time is reduced by a factor of 3.7 compared withthe nonlinear pressure solves in the compressible case. The water-cut curves are similar to the compressible case, but the waterbreakthrough comes significantly later. Allowing for rock com-pressibility with the porosity multiplier restricted to being a linear

function of pressure gives curves closer to the base case, whilekeeping the low computational cost of a linear pressure equation.

Example 6: A Heavy-Oil Field. As a last validation test, weconsider a realistic case modeled after a real-field model of aheavy-oil reservoir. The model consists of 600,000 cells and hasapproximately 40 wells. Petrophysical data, fluid properties, andwell positions are from the real model, but the injection strategyand well controls are purely synthetic. To produce oil, polymer isinjected along with water from all injection wells at a concentra-tion that increases the water viscosity by a factor of 50. Polymerand water are assumed to be perfectly mixed (i.e., we use x ¼ 1).

TABLE 4—CPU TIMES IN SECONDS FOR THE NORNE MODEL MEASURED BY USE OF A SINGLE CORE ON

2.5-GHz INTEL i7 PROCESSOR

Case

Compressible Model Incompressible Model

Polymer Flooding Water Injection Polymer Flooding Water Injection

Pressure solver 100 seconds 99 seconds 27 seconds 27 seconds

Transport solver 19 seconds 12 seconds 19 seconds 11 seconds

Without gravity 7.8 seconds 5.7 seconds 3.8 seconds 2.4 seconds

0 500 1000 1500 2000 2500 3000 3500 40000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Days since start of injection

Wat

er c

ut a

ll w

ells

Water cut with polymer

comp + grav, water + polymercomp + nograv, water + polymerincomp + grav, water + polymerincomp + nograv, water + polymer

0 500 1000 1500 2000 2500 3000 3500 40000

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Wat

er c

ut a

ll w

ells

Water cut water only

comp + grav, watercomp + nograv, waterincomp + grav, waterincomp + nograv, water

Fig. 10—Comparison of water-cut curves for the base-case model and models without gravity and/or compressibility for polymerflooding (left) and water injection (right).

0 500 1000 1500 2000 2500 3000 3500 40000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


Wat

er c

ut

Pol

ymer

inje

ctio

n pe

riod

water

water + polymer

0 500 1000 1500 2000 2500 3000 3500 40000

0.2

0.4

0.6

0.8


Wel

l 1

0 500 1000 1500 2000 2500 3000 3500 40000

0.5

1


Wel

l 2

0 500 1000 1500 2000 2500 3000 3500 40000

0.5

1


Wel

l 3

waterwater + polymer

Fig. 9—Comparison of water-cut curves for the Norne test case with water injection and polymer flooding.




Fig. 11 shows oil saturation, polymer concentration (>0.1), andthe number of iterations used (>0) in the last timestep, with the outersurfaces of the reservoir superimposed. To compute the solution, weuse 10 timesteps of 100 days and the bracketing solver for the single-cell problems. Fig. 11 reports the corresponding number of iterationsused in the last timestep for all methods. A key observation from Fig.11 is that because the injection fronts only contact a relatively smallnumber of cells, the nonlinear iterations will be effectively localizedto regions around wells and near the oil/water contact, where eitherthe water saturation or the polymer concentration changes. Thisshould lead to significant reduction in computational cost comparedwith a global Newton-Raphson solve.

For the pressure step, we use AGMG (Notay 2010; AGMG 2012)as linear solver because it turned out to be slightly more efficient thandune:istl for this particular case. Each pressure solve converges withinfour nonlinear iterations and takes approximately 10 seconds on aLinux workstation with a 2.6-GHz Intel i7 processor. Motivated bythe previous example (and preliminary numerical experiments), weneglect the influence of gravity. With this assumption, one transportstep takes approximately 0.5 seconds. This means that we obtain 20transport steps for the cost of a single pressure step. Because the cou-pling of the pressure and transport is not very tight, it can be advanta-geous to compute multiple timesteps in the advective solver for eachpressure step to increase the resolution of polymer fronts without sig-nificantly affecting the total run time. These results are encouraging,but further research is needed to find an optimal timestepping strategyfor achieving the best accuracy for the given computational cost.

Concluding Remarks

We have presented a highly efficient simulation method for realis-tic models of polymer flooding in heavy-oil reservoirs. The new

simulation method is implemented as part of the OPM initiativeand is freely available from the website (OPM 2012) under theGPLv3 license. The current software implements a polymermodel that includes dead pore space, adsorption, and permeabilityreduction. The simulator is particularly efficient for systems inwhich rock compressibility dominates fluid compressibilities andgravity only governs slow processes; such systems are highly rele-vant for EOR of heavy oil.

The key points to the efficiency of our new method are toexploit the loose coupling between flow and transport to enablespecial solvers for the respective equations; to exploit the weakinfluence of gravity and split the transport calculation into an ad-vective and a segregation part; and to localize the nonlinear solvesof the advective part of transport step to significantly reduce com-putational costs. Initial testing indicates that the method workswell for a wide range of timesteps and is robust with respect tolarge differences in PV and face areas that are typically present inrealistic high-resolution geocellular models. In particular, if nestedbracketing is used for solving the localized single-cell problems,either as a full solver or as a fallback strategy, we have an uncondi-tionally stable method that avoids timestep restrictions induced bystability problems. However, further research is needed to under-stand what the feasible timestep ranges are for practical reservoirsimulation in terms of accuracy vs. efficiency.

Research is also needed to develop efficient localization strat-egies for the nonlinear iterations for flow fields that contain largeloops that may be caused by gravity effects. As we have demon-strated previously, neglecting gravity effects will in certain casesgive a significant speedup without an adverse effect on the overall ac-curacy of the simulation. However, we are not aware of any criterionthat can accurately and robustly determine whether gravity effectscan be neglected. Currently, this would typically be decided by

Oil saturation Polymer concentration

Histogram of iterations (logarithmic) Nonlinear Iterations

0 5 10 15 20100

101

102

103

104

105

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

2 4 6 8 10 12 14 16 18 20 22 24

0.9 0.5 1 1.5 21

Fig. 11—Simulation of polymer injection for a real-field model. All plots show quantities from the last step in a simulation of 1,000 days.




considering density ratios or by simulating spatial or temporal subsetsof the model. We refer the interested reader to a recent paper (Lieet al. 2012a) for a preliminary discussion of how gravity affects the ef-ficiency of the type of operator-splitting methods presented herein.

The reordering method discussed herein can be very fast whenrun on a single core. However, because the implicit updating pro-cedure needs to visit cells sequentially in a predetermined order,the resulting methods are predominantly serial and not necessarilystraightforward to implement efficiently on a parallel computer.The most obvious approach to this end would be to exploit thedelineation of the reservoir volume that is inherent in the reorder-ing procedure. The reordering method can easily be extended tolabel cells with the injection wells (or inflow boundaries) they areinfluenced by (Shahvali et al. 2012), and with these labels, onecan identify independent regions that can be computed concur-rently. This way, each concurrent thread can start in a unique welland march the solution outward until one encounters cells that arealso influenced by one of the other threads. Although this obvi-ously would not provide perfect scaling across a large number ofcores, it could potentially provide sufficient speedup on worksta-tion computers with tens of cores, provided that the model con-tains a sufficient number of injection wells.

Nomenclature

a ¼ normalized adsorption polymer concentrationa ¼ adsorption polymer concentration

ba ¼ formation volume factor of phase ac ¼ polymer concentration

cmax ¼ maximum polymer concentration

kra ¼ relative permeability of phase akrwp ¼ relative permeability of polymer

K ¼ permeability tensorp ¼ pressure

pa ¼ pressure of the phase aRa ¼ residual value for the conservation equation of the phase aRc ¼ residual value for the conservation equation of the polymerRk ¼ actual resistance factorRrf ¼ residual resistance factorSa ¼ saturation of the phase a~va ¼ flux of the phase a~vwp ¼ polymer flux

Vi ¼ volume of the cell iDt ¼ timesteplm ¼ viscosity of the fully mixed polymer solutionla ¼ viscosity of the phase a

lp, eff ¼ effective polymer viscositylw, e ¼ viscosity of the partially mixed water

lw, eff ¼ effective water viscosityqa ¼ density of the phase aqS

a ¼ density of the phase a at surface conditionqr, ref ¼ reference rock density

/ ¼ porosity/ref ¼ reference porosity

x ¼ mixing parameter

Subscripts

a ¼ phase (a = w for water, a = o for oil)i ¼ cell number

i, j ¼ face number (interface between cells i and j)

Superscript

n ¼ timestep

Acknowledgments

The authors would like to thank Statoil Petroleum A/S for fundingand access to data sets for simulator testing. The authors wouldalso like to thank Statoil (operator of the Norne field) and itslicense partners Eni and Petoro for the release of the Norne data.

Further, the authors acknowledge the Center for Integrated Opera-tions at the Norwegian University of Science and Technology(NTNU) for cooperation and coordination of the Norne cases.Finally, the authors thank Ove Sævareid for valuable input andEclipse simulations, and Vegard Kippe, Alf Birger Rustad, SteinKrogstad, Bard Skaflestad, and Kristin M. Flornes for fruitful dis-cussions and feedback on our work.

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Knut-Andreas Lie is chief scientist and research manager atSINTEF Information and Communication Technology (ICT) inOslo, Norway, and professor at the University of Bergen, Nor-way. His research interests include gridding and discretizationof complex reservoir models, fast-simulation tools, upscalingand multiscale methods, modeling of subsurface carbon diox-ide (CO2) storage, and open-source software. Lie holds a PhDdegree in applied mathematics from NTNU, Trondheim, Nor-way. He is an associate editor for SPE Journal.

Halvor M. Nilsen is a senior scientist at SINTEF ICT. He hasworked on discretization methods, streamline and front-track-ing methods, linear and nonlinear solvers, upscaling methods,modeling of CO2 storage, and open-source software. Lieholds a PhD degree in physics from the University of Bergen.

Xavier Raynaud is a research scientist at SINTEF ICT. He hasworked on polymer models and upscaling, fully implicit andsplitting methods, higher-order methods, adjoint methods,and optimization. Raynaud holds a PhD degree in appliedmathematics from NTNU.

Atgeirr F. Rasmussen is a research scientist at SINTEF ICT. Hisresearch includes upscaling methods, streamline tracing, non-linear solvers, finite-volume and discontinuous Galerkin meth-ods, fast-flow diagnostics tools, open-source software,domain-specific languages. and automatic differentiation.Rasmussen holds a PhD degree in applied mathematics fromthe University of Oslo, Norway.




http://dx.doi.org/10.1016/j.advwatres.2007.05.015

http://dx.doi.org/10.1016/j.advwatres.2007.05.015

http://www.ipt.ntnu.no/~norne/wiki/doku.php

http://www.ipt.ntnu.no/~norne/wiki/doku.php

http://www.opm-project.org/

http://www.opm-project.org/






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