Study on Mechanism of Viscoelastic Polymer Transient Flow in...

Research ArticleStudy on Mechanism of Viscoelastic Polymer Transient Flow inPorous Media

Huiying Zhong1 Weidong Zhang1 Hongjun Yin1 and Haoyang Liu2

1Key Laboratory for EnhancedOil ampGas Recovery of theMinistry of Education Northeast PetroleumUniversity Daqing 163318 China2PetroChina Research Institute of Petroleum Exploration amp Development Beijing 100083 China

Correspondence should be addressed to Huiying Zhong zhhy987126com

Received 2 July 2017 Revised 16 September 2017 Accepted 16 October 2017 Published 2 November 2017

Academic Editor Yingfang Zhou

Copyright copy 2017 Huiying Zhong et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Oil recovery including conventional and viscous oil can be improved significantly by flooding with polymer solutions Thischemical flooding method can increase oil production and it can improve the macrodisplacement efficiency and microsweepefficiencies In this study we establish physical models that include the dead-end and complex models based on the pore-networkpattern etched into glass using the snappyHexMesh solver in OpenFOAM These models capture the complexity and topologyof porous media geometry We establish a mathematical model for transient flows of viscoelastic polymers using computationalfluid dynamics simulations and we study the distributions of pressure and velocity for different elasticity scenarios and differentflooding process The results demonstrate that the pressure difference increases as the relaxation time decreases before the flowreaches its steady state For a steady flow elasticity can give rise to an additional pressure difference which increases with increasingelasticity Thus the characteristics of pressure difference vary before and after the flow becomes steady this phenomenon is veryimportant Velocity contours become more widely spaced with elasticity increase This suggests that elasticity of the polymersolutions contributes to the microsweep efficiency The results of the study provide the necessary theoretical foundation forlaboratory experiments and development of methods for polymer flooding and can be helpful for the design and selection ofpolymers for polymer flooding

1 Introduction

With continuing development of energy infrastructure thesupply and consumption of energy have come to play animportant role in the industrialization and urbanizationof world countries Oil plays the most important rolebecause as of today it cannot be fully replaced by otherenergy sources To ensure the sufficient production of oilmany measures have been adopted including exploration fordiscovering new oil reserves creating national oil strategicreserve systems and avoiding import risk based on themarket fluctuations of oil futures Enhanced oil recovery fromdeveloped reservoirs is an important strategy to face theabove-described challenges [1 2] Among various chemicalflooding techniques for enhanced oil recovery polymerflooding is especially promising Of late polymer floodingresearch has matured enough for commercial applicationsTaking the Daqing oilfield as an example 54 industrial blockshave been displaced by polymer flooding Compared with

water flooding polymer flooding increased oil recovery by13 Polymer flooding-based recovery of second-type oillayers enhanced oil recovery by 10 compared with waterflooding [3] In addition polymer flooding has been usedin heavy-oil reservoirs [4] especially for offshore heavyoil Polymer flooding is suitable for operation on offshoreplatforms because it does not require complex and additionalsurface facilities given a relatively limited platform spaceBohai Bay is the most important offshore oilfield in Chinawhich produces heavy oil and polymer flooding has signifi-cantly increased the oil recovery there In three years in thepilot area of four injection wells and six production wellsincremental oil production increased to 76 times 104m3 [5 6]The main factor contributing to the success of this techniqueis that the viscosity of water can be increased by polymersresulting in a better macroscale sweep efficiency [7 8] Interms of microscopic mechanisms the polymer elasticity cancontribute to pulling out residual oil droplets Thus on the

HindawiGeofluidsVolume 2017 Article ID 8763951 8 pageshttpsdoiorg10115520178763951

2 Geofluids

microscale polymer elasticity in porous media is importantfor increasing the oil recovery [9ndash12]

To investigate the pore-scale flow mechanisms research-ers have employed microscale visual experiments and the-oretical computational fluid dynamics (CFD) methods andstudied the effects of viscosity and elasticity on flow charac-teristics A two-dimensional (2D) etched-silicon microscalemodel having the geometrical and topological characteristicsof stone has been prepared Experimental results for heavyoil displaced by water and polymer show that at zeroand low polymer concentrations relatively long and widefingers of injectant develop and slowly recover As polymerconcentration increases manymore relatively fine fingers areformed [13] Wang and Xia compared viscoelastic polymersand pure viscous glycerin solutions having the same viscosityto displace oil by conducting experiments with glass etchedsystem [14ndash16] By comparing images after flooding by twofluids polymer elasticity was shown to be responsible forpulling out residual oil droplets The oil recovery associatedwith polymer solutions was enhanced by 7ndash16 comparedwith glycerin having the same viscosity Vermolen et al [17]designed and performed core flooding experiments usingglycerol and polyacrylamide in which they validated thatpolymer elasticity can enhance the displacement efficiencyThe results of these experiments showed that for low viscositycrude oil extra oil was recovered when using polymers Thisconfirms that high-elasticity polymers can reduce the amountof residual oil compared with water flooding

The polymer elasticity can enhance the displacementeffect as has been validated by the above experimental stud-ies Theoretical investigations about the flow characteristicsof polymer solutions in the context of oil industry are lackingSome researchers have studied steady-flow polymer solutionsfor 4 1 contraction and dead-end pores using theoreticalmethod The finite difference method was used to study thestream function velocity and pressure contours for differentviscoelastic polymer flows in microscale pores The results ofthese studies show that with increasing polymer elasticity theflow area increases while the residual oil area decreases Thevelocity and pressure also increase [18ndash20] Static numericalsimulations (water-based fluid flowing around an immobileoil droplet) showed that additional forces are present in thecase of elastic fluids Dynamic simulations revealed strikingdifferences in the behavior of elastic and inelastic fluids oildroplets underwent deformation over time when inelasticfluids were injected but the droplet did not deform in thepresence of elastic fluids [21] Owing to the nonlinearityassociated with elasticity the numerical simulation of vis-coelastic flows remains challengingThe stabilized method offinite elements was used in some studies [22] This methodyielded high elasticity (Weissenberg number gt 2) and theresults suggest that viscoelasticity may play a significant rolein determining the behavior of polymer flows Earlier studiesof viscoelastic behavior focused on steady flows In 2011Jafari [23] developed a spectral element method to obtain anaccurate numerical solution for viscoelastic fluid flows Theobjective was to introduce a new algorithm for overcomingthe drawbacks associated with the direct reformulation ofthe classical constitutive equation as a logarithmic one The

research effort culminated in a comprehensive study on tem-poral growth of spuriousmodes Prasuna et al [24] solved theproblem of viscoelastic unsteady flow through two imperme-able parallel plates using Laplace transformation techniqueThe equations were solved for stage-steady and unsteadymotion respectively The researchers obtained expressionsfor the flow rate and shear stress on the walls In mosttheoretical studies simplified models such as the dead-endmodel contraction structures and expansion-contractionstructures were adopted [25] However the structure of areal porous medium is very complex and so is likely toaffect the flow characteristics Hence it is very importantto consider the complexity of pores in studies of polymerelasticity In this case the finite volume method [26ndash29] islikely to bemore stable Also the existing studiesmainly focuson viscoelastic steady flows and although some researchhas been done on unsteady flow those studies address thealgorithms rather than the flow characteristics of viscoelasticfluids in porous media Transient flow characteristics (beforea flow reaches a steady state) are important leading tosome potentially interesting phenomena that are ignored inexperiments In this study we investigated the complex poregeometries based on etched-in glasses model This 2D modelcan capture the geometrical and topological characteristicsof pores in a realistic manner Furthermore we studiedthe microscopic characteristics of transient flows using theviscoelastic flow solver in OpenFOAMThese results providetheoretical support and guidance to studies that aim toenhance oil recovery using polymer flooding

2 Physical Model

Towphysicalmodels that represent the pores have been estab-lished in this studyThefirst one is the dead-endmodel whichcaptures a simple pore The second one that models morecomplex pore structures is represented using the etched-inglass model The dead-end model is schematically shown inFigure 1 The residual oil in the dead end often cannot bedisplaced by water The real etched-in glass model is 40 times40mm in size and was used in microscale visual experimentswith pure viscous glycerin and elastic polymer flooding Itsporosity is 35 and its permeability was 2200 times 10minus3 120583mFigure 2(a) shows the saturated oil imaged using a high-resolution camera Figure 2(b) shows a region from Figure 1as a digitized model to demonstrate the computationalspeed and convergence and area under consideration is 1 times1mm The diameter of the smallest pore was sim002mm Inconclusion this model provides a sufficiently detailed pictureof a porous medium to allow studies of the polymer flowbehavior

First the image was digitized using Solidworks to obtaina black and white presentation and then the black pixelscorresponding to the pore boundary were extracted (Fig-ure 1(c)) The skeleton of the pores must be imported usingsnappyHexMesh a solver in OpenFOAM and is shown inFigure 1(c) The nonorthogonal grid was obtained using thesnappyHexMesh solver in OpenFOAM and it is shown inFigure 1(d) The number of mesh elements in the model wassim399220 The minimum radius was 002mm The mesh file

Geofluids 3

25 ＧＧ

05ＧＧ

10 ＧＧ

05 ＧＧ

Figure 1 The dead-end model cells

(a) Original saturated oil micromodel (b) The digitized model

(c) The stl file exported from Solidworks (d) Meshes exported from snappyHexMesh

Figure 2 The complex model and the mesh formation workflow

was designed so it could be accepted by any solver in Open-FOAM

3 Mathematical Model

The following simulation addressed a viscoelastic transientflow in an incompressible and isothermal 2D system Thegoverning equations included the continuity equation themomentum equation and the constitutive equation

The continuity equation isnabla sdot (U) = 0 (1)

The momentum equation is

120597 (120588U)120597119905 + nabla sdot (120588UU) = minusnabla119901 + nabla sdot 120591

(2)

where U is the velocity vector in ms 119880119909 is the velocity inthe 119909 direction and 119880119910 is the velocity in the 119910 directionOther parameters were as follows flow time 119905 in s polymerdensity 120588 in kgm3 stress tension 120591 in Pa which had threecomponents with 120591119909119909 and 120591119910119910 corresponding to the first andsecond normal stresses respectively and 120591119909119910 correspondingto the shear stress pressure 119901 in Pa

4 Geofluids

7734E minus 0055800E minus 0053867E minus 0051933E minus 0050000E + 000

(a) 01 s


(b) 04 s


(c) 07 s


(d) 1 s

Figure 3 Velocity contour with time (120582 = 006 s)

The polymer molecules with large relaxation time do nothave sufficient time to stretch recoil and adjust themselvesto the elongation and contraction of flow through poresand throats in the reservoir rock Therefore the resultantelastic strain produces high apparent viscosity Viscoelasticpolymer exhibits nonlinear rheological responses which arecaptured using the constitutive equation Strong experimen-tal evidence indicates that the first normal stress differenceis dominant and is larger than the second normal stressdifferenceTheOldroydB equation can describe this propertyof polymer solutions [30] as follows

120591 + 120582 nabla120591= 120578119901119863 (3)

where nabla120591= 120597120591120597119905+U sdotnabla120591minus120591 sdotnablaUminus[120591 sdotnablaU]119879 120578119901 is the polymerviscosity in Pasdots119863 is the deformation rate tensor in sminus1 120582 isthe relaxation time in s which captures the elasticity of thepolymer solutions It has been suggested that the inverse ofthe frequency at which 1198661015840 and 11986610158401015840 intersect corresponds tothe characteristic relaxation time of the polymer solution

To ensure the convergence of numerical solutions weused theDEVSS approach for solving the systemof equationsThe stress tensor was divided into two parts including thesolvent contribution part (viscous effect) and polymer contri-bution part (elastic effects) together these allow representingclearly the nonlinear viscoelasticity phenomena [31] Thecorresponding equation was

nabla sdot 120591 = nabla sdot 120591119904 + nabla sdot 120591119901 (4)

where nabla sdot 120591119904 is the solvent contribution to stress in Pa andnabla sdot 120591119901 is the polymeric contribution in Pa

120591119904 = 2120578119904119863 (5)

The parameter 120591119901 could be calculated from the Oldroyd Bconstitutive differential equation and the momentum equa-tion was rewritten as follows

120597 (120588U)120597119905 + nabla sdot (120588UU) minus nabla sdot 120591119901 + nabla sdot (120578119901nablaU)

minus (120578119904 + 120578119901) nabla sdot (nablaU) = minusnabla119901(6)

where 120578119904 is the solvent viscosity in Pasdots and 120578119901 is polymericviscosity in Pasdots

Equations (1) (2) and (6) constitute the governingequations for solving the problemThe finite volume methodwas used to solve the above equations The steps used forsolving the system are given below

(1) For a given initial velocity field pressure field andstress tensor field momentum equation is solved obtaininga new velocity field where the pressure gradient and stressdivergence are calculated by explicit method

(2) Using the new velocity field according to theRhieChow interpolation idea construct the Poisson equationfor pressure solve for the pressure field and correct thevelocity to satisfy the continuity equation For the transientproblem we use the PISO method and for the steady-stateproblem we use the SIMPLE method

(3) Using the velocity field that satisfies the continuityequation use the constitutive equation of the viscoelasticfluid to calculate the stress tensor

(4) Repeat steps (1)ndash(3)The equations were solved using the OpenFOAMrsquos vis-

coelasticFluidFoam solverIn these models the density of the polymer flow was

803 kgm3 and viscosity was 8mPasdots The relaxation timevalues were 006 s 01 s and 015 s

4 Flow Characteristics in the Dead-End Model

In these flow cases the flow velocity has components in boththe 119909 and 119910 directions The flow velocity was defined

119880 = radic1198802119909 + 1198802119910 (7)

Geofluids 5

= 006

= 010

= 015

2 4 6 80t (Ｍ)

20

40

60

80

100

Δp

(10minus4０)

Figure 4 Pressure difference between inlet and outlet with different elastic polymer flow

p000458

00001000200030004

(a) 01 s

p000458

00001000200030004

(b) 05 s

p000458

00001

0002

0003

0004

(c) 2 s

Figure 5 The pressure distribution diagram of the polymer solution injected at different times (120582 = 006 s)

Figure 3 shows the velocity contours at different timesIt can be seen that the maximal velocity decreases as timeincreases Because the process is unsteady the polymer flowsinto the main flow pore from the inlet This is manifested asan oscillation owing to the elasticity of the polymer solutionsThe flow becomes steady at 119905 = 1 s Thus the flow velocitybecomes constant at 119905 = 1 s The maximal velocity does notchange with time

The pressure differences between the inlet and the outletfor different relaxation times are shown in Figure 4 Forthe same relaxation time the pressure difference increaseswith the time and it becomes constant for a steady flow Atthe same time for transient flows the pressure differenceincreases with relaxation time decrease For steady flows thedifference increases with relaxation time increaseThe kineticenergy is transformed into elastic potential energy thusthe pressure difference is larger for smaller relaxation timesbefore the flow becomes steady Elastic potential energy istransformed into kinetic energy when the flow is steady thusin this regime the difference is larger for larger relaxationtimes

5 Flow Characteristics in the Complex Model

Based on the dead-end model we simulated the flow ofviscoelastic polymers in a complexmicroscalemodel Figures5 6 and 7 show the results of these simulations for differentsolutions of elastic polymers

Figures 5 6 and 7 demonstrate that in the porousmedia the inlet-outlet pressure difference increases withtime and for the same relaxation time the pressure is muchhigher in pores compared with throats The figures showthat the pressure difference increases as the relaxation timedecreases for the same injection time when the flow istransient This occurs because the kinetic energy is trans-formed into elastic energy the elasticity increasing and thekinetic energy decreasing and then the pressure differencereducing

Figure 8 shows the pressure difference between the inletand outlet of the polymer solution at different times Forthe same injection time before the flow becomes steady thefigure shows that the pressure difference increases with theelasticity decrease For steady flows the pressure difference

6 Geofluids

p00035

0

0001

0002

0003

(a) 01 s

p00035

0

0001

0002

0003

(b) 05 s

p00035

0

0001

0002

0003

(c) 2 s


p00028

0

0001

0002

(a) 01 s

p00028

0

0001

0002

(b) 05 s

p00028

0

0001

0002

(c) 2 s


= 006

= 010

= 015

1 2 3 4 5 6 70t (Ｍ)

20

40

60

80

100

120

140

Δp

(10minus3０)

Figure 8 Pressure difference versus flow time at different relaxationtime

increases as the relaxation time increases thus the elasticityof the polymer solution can produce an additional pressuredrop The greater the elasticity the stronger the pressuredrop This result is consistent with the flow in the dead-endmodel

Figure 9 shows the velocity contour with the flow timeat the same relaxation time (120582 = 15 s) For flow times under

1 s the flow velocity increases however for a steady flow thevelocity decreases and reaches a constant Figure 10 shows thevelocity contour with the different relaxation time for steadyflows As shown in Figure 10 the flow velocity increases withthe elasticity increasing Using the samemaximal velocity theobtained contour is shown in Figure 11

Figure 11 also shows that when the maximal and minimalvelocities are the same for different relaxation times thearea swept by the same contours become larger as the relax-ation time increases in three local areas This phenomenondemonstrates that the elasticity of a polymer solution canincrease the microsweep area and sweep efficiency in porousmedia

6 Conclusions

In this paper a mathematical model to describe the flowviscoelastic polymer solution in a porous medium wasestablished and solved using the OpenFOAM solver Themicroscale model based on the etched-glass can capture thecomplexity and topology of porous media

For both models the simulations of transient viscoelasticpolymer flows revealed that the flow pressure differenceincrease as the elasticity decrease before the flow becamesteady For a steady flow the pressure difference increasedas the elasticity increased The flow velocity increased withflow time and in the flow steady state the flow velocity wasconstant

The simulations of velocity contours in the micromodelshowed that the microsweep area increased as the relaxation

Geofluids 7


(a) 119905 = 01 s


(b) 119905 = 07 s


(c) 119905 = 20 s

Figure 9 Velocity contours with different relaxation time through complex micromodel (120582 = 015 s)


(a) 120582 = 006 s


(b) 120582 = 01 s


(c) 120582 = 015 s


4830E minus 004

3623E minus 004

2415E minus 004

1208E minus 004

0000E + 000

(a) 120582 = 006 s


(b) 120582 = 01 s


(c) 120582 = 015 s

Figure 11 Local velocity contours with different relaxation time through complex micromodel (119905 = 20 s)

time increased The microsweep efficiency increased as theelasticity increased

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work presented in this paper was financially supportedby the National Natural Science Funds for Young Scholars ofChina (Grant no 51604079) the Natural Science Foundation

of Heilongjiang Province (Grant no E2016015) and the Uni-versity Nursing Program for Young Scholars with CreativeTalents in Heilongjiang Province (Grant no UNPYSCT-2016126) which are gratefully acknowledged

References

[1] J Lu P J Liyanage S Solairaj et al ldquoNew surfactant develop-ments for chemical enhanced oil recoveryrdquo Journal of PetroleumScience and Engineering vol 120 pp 94ndash101 2014

[2] D Denney ldquoProgress and Effects of ASP Floodingrdquo Journal ofPetroleum Technology vol 65 no 01 pp 77ndash81 2015

8 Geofluids

[3] Y Zhu ldquoCurrent developments and remaining challenges ofchemical flooding EOR techniques in Chinardquo in Proceedingsof the SPE Enhanced Oil Recovery Conference pp 11ndash13 KualaLumpur Malaysia 2015

[4] SDoorwar andKKMohanty ldquoViscous Fingering duringNon-Thermal Heavy Oil Recoveryrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Denver Colorado USA

[5] W Zhou J Zhang G Feng et al ldquoKey technologies of polymerFlooding in offshore oilfield of Bohai Bayrdquo in Proceedings of theSPE Asia Pacific Oil and Gas Conference and Exhibition PerthAustralia 2008

[6] W Zhou J Zhang M Han W Xiang G Feng and WJiang ldquoApplication of hydrophobically associatingwater-solublepolymer for polymer flooding in china offshore heavy oilfieldrdquoin Proceedings of the International PetroleumTechnology Confer-ence Dubai UAE 2007

[7] H L Chang ldquoPolymer flooding technology - yesterday todayand tomorrowrdquo Journal of Petroleum Technology vol 30 no 8pp 1113ndash1128 1978

[8] J J Sheng B Leonhardt and N Azri ldquoStatus of polymer-flooding technologyrdquo Journal of Canadian Petroleum Technol-ogy vol 54 no 2 pp 116ndash126 2015

[9] J Y Yoo and Y Na ldquoA numerical study of the planar contractionflow of a viscoelastic fluid using the simpler algorithmrdquo Journalof Non-Newtonian Fluid Mechanics vol 39 pp 89ndash109 1991

[10] D Wang and G Wang ldquoThe influence of viscoelasticity on dis-placement efficiency-frommicro-tomacroscalerdquo in Proceedingsof the SPE annual technical conference and exhibition pp 11ndash14Anaheim California USA 2007

[11] D Wang and G Wang ldquoLarge scale high viscous-elastic fluidflooding in the field achieves high recoveriesrdquo in Proceedingsof the SPE Enhanced Oil Recovery Conference Kuala LumpurMalaysia 2011

[12] MDelshad D H KimO AMagbagbeola C Huh G A Popeand F Tarahhom ldquoMechanistic interpretation and utilizationof viscoelastic behavior of polymer solutions for improvedpolymer-flood efficiencyrdquo in Proceedings of the 16th SPEDOEImproved Oil Recovery Symposium pp 1051ndash1065 2008

[13] M Buchgraber T Clemens L M Castanier and A R KovscekldquoA microvisual study of the displacement of viscous oil bypolymer solutionsrdquo SPE Reservoir Evaluation and Engineeringvol 14 no 3 pp 269ndash280 2011

[14] D Wang J Cheng and Q Yang ldquoViscous-elastic polymercan increase micro-scale displacement efficiency in coresrdquo ActaPetrolei Sinica vol 21 no 5 pp 45ndash51 2000

[15] D Wang J Cheng and H Xia ldquoImprovement of displacementefficiency of cores by driving forces parallel to the oil-waterinterface of viscous-elastic fluidsrdquo Acta Petrolei Sinica vol 23no 5 pp 47ndash52 2002

[16] H Xia D Wang and Z Liu ldquoStudy on the mechanism of poly-mer solutionwith visco-elastic behavior increasingmicroscopicoil displacement efficiencyrdquo Acta Petrolei vol 22 no 4 pp 60ndash65 2001

[17] E Vermolen M Van Haasterecht and S Masalmeh ldquoA sys-tematic study of the polymer visco-elastic effect on residualoil saturation by core floodingrdquo in Proceedings of the SPE EORConference at Oil Gas West Asia Muscat Oman 2014

[18] X Yue L Zhang and Z Liu ldquoViscoelastic flow andmicroscopicdisplacement mechanism of polymer solution in reservoirporesrdquo Petroleum Geology and Recovery Efficiency vol 9 no 3pp 4ndash6 2002

[19] L Zhang X Yue and Z Liu ldquoFlow field of viscoelastic fluid inpores with dead endsrdquo Journal of Hydrodynamics vol 17 no 6pp 747ndash755 2002

[20] L Zhang X Yue and Z Liu ldquoFlow performance of viscoelasticpolymer solution in dead end of reservoir poresrdquo PetroleumExploration and Development vol 31 no 5 pp 105ndash108 2004

[21] A Afsharpoor and M T Balhoff ldquoStatic and dynamic CFDmodeling of viscoelastic polymer trapped oil displacement anddeformation at the pore-levelrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition New Orleans LouisianaUSA 2013

[22] MQi JWegner L Falco and L Ganzer ldquoPore-scale simulationof viscoelastic polymer flow using a stabilized finite elementmethodrdquo in Proceedings of the SPE Reservoir Characterisationand Simulation Conference and Exhibition pp 16ndash18 AbuDhabi UAE 2013

[23] A Jafari ldquoSimulation of time-dependent viscoelastic fluid flowsby spectral elementsrdquo EPFL 2011

[24] T G Prasuna R M V Murthy N C P Ramacharyulu and GV Rao ldquoUnsteady flow of a viscoelastic fluid through a porousmedia between two impermeable parallel platesrdquo Journal ofEmerging Trends in Engineering and Applied Sciences (JETEAS)vol 1 no 2 pp 225ndash229 2010

[25] J Cai W Wei X Hu and D A Wood ldquoElectrical conductivitymodels in saturated porous media a reviewrdquo Earth-ScienceReviews vol 171 pp 419ndash433 2017

[26] A M Afonso P J Oliveira F T Pinho and M A AlvesldquoDynamics of high-Deborah-number entry flows a numericalstudyrdquo Journal of Fluid Mechanics vol 677 pp 272ndash304 2011

[27] F Pimenta and M A Alves ldquoStabilization of an open-sourcefinite-volume solver for viscoelastic fluid flowsrdquo Journal of Non-Newtonian Fluid Mechanics vol 239 pp 85ndash104 2017

[28] S Xue and G W Barton ldquoAn unstructured finite volumemethod for viscoelastic flow simulations with highly truncateddomainsrdquo Journal of Non-Newtonian Fluid Mechanics vol 233pp 48ndash60 2016

[29] L Zhao J Ouyang W Zhou Y Xie and J Su ldquoThe simplerGMRES method combined with finite volume method forsimulating viscoelastic flows on triangular gridrdquo Advances inEngineering Software vol 87 pp 57ndash67 2015

[30] R B Bird W E Stewart and E N Lightfoot Transport phe-nomena John Wiley and Sons inc 2nd edition 2007

[31] J L Favero A R Secchi N S M Cardozo and H JasakldquoViscoelastic flow analysis using the software OpenFOAM anddifferential constitutive equationsrdquo Journal of Non-NewtonianFluid Mechanics vol 165 no 23-24 pp 1625ndash1636 2010

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of


EarthquakesJournal of


Mining


Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 201

International Journal of

OceanographyInternational Journal of


Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering


GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of


OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in


MineralogyInternational Journal of


MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Geological ResearchJournal of


Geology Advances in

2 Geofluids

microscale polymer elasticity in porous media is importantfor increasing the oil recovery [9ndash12]

To investigate the pore-scale flow mechanisms research-ers have employed microscale visual experiments and the-oretical computational fluid dynamics (CFD) methods andstudied the effects of viscosity and elasticity on flow charac-teristics A two-dimensional (2D) etched-silicon microscalemodel having the geometrical and topological characteristicsof stone has been prepared Experimental results for heavyoil displaced by water and polymer show that at zeroand low polymer concentrations relatively long and widefingers of injectant develop and slowly recover As polymerconcentration increases manymore relatively fine fingers areformed [13] Wang and Xia compared viscoelastic polymersand pure viscous glycerin solutions having the same viscosityto displace oil by conducting experiments with glass etchedsystem [14ndash16] By comparing images after flooding by twofluids polymer elasticity was shown to be responsible forpulling out residual oil droplets The oil recovery associatedwith polymer solutions was enhanced by 7ndash16 comparedwith glycerin having the same viscosity Vermolen et al [17]designed and performed core flooding experiments usingglycerol and polyacrylamide in which they validated thatpolymer elasticity can enhance the displacement efficiencyThe results of these experiments showed that for low viscositycrude oil extra oil was recovered when using polymers Thisconfirms that high-elasticity polymers can reduce the amountof residual oil compared with water flooding

The polymer elasticity can enhance the displacementeffect as has been validated by the above experimental stud-ies Theoretical investigations about the flow characteristicsof polymer solutions in the context of oil industry are lackingSome researchers have studied steady-flow polymer solutionsfor 4 1 contraction and dead-end pores using theoreticalmethod The finite difference method was used to study thestream function velocity and pressure contours for differentviscoelastic polymer flows in microscale pores The results ofthese studies show that with increasing polymer elasticity theflow area increases while the residual oil area decreases Thevelocity and pressure also increase [18ndash20] Static numericalsimulations (water-based fluid flowing around an immobileoil droplet) showed that additional forces are present in thecase of elastic fluids Dynamic simulations revealed strikingdifferences in the behavior of elastic and inelastic fluids oildroplets underwent deformation over time when inelasticfluids were injected but the droplet did not deform in thepresence of elastic fluids [21] Owing to the nonlinearityassociated with elasticity the numerical simulation of vis-coelastic flows remains challengingThe stabilized method offinite elements was used in some studies [22] This methodyielded high elasticity (Weissenberg number gt 2) and theresults suggest that viscoelasticity may play a significant rolein determining the behavior of polymer flows Earlier studiesof viscoelastic behavior focused on steady flows In 2011Jafari [23] developed a spectral element method to obtain anaccurate numerical solution for viscoelastic fluid flows Theobjective was to introduce a new algorithm for overcomingthe drawbacks associated with the direct reformulation ofthe classical constitutive equation as a logarithmic one The

research effort culminated in a comprehensive study on tem-poral growth of spuriousmodes Prasuna et al [24] solved theproblem of viscoelastic unsteady flow through two imperme-able parallel plates using Laplace transformation techniqueThe equations were solved for stage-steady and unsteadymotion respectively The researchers obtained expressionsfor the flow rate and shear stress on the walls In mosttheoretical studies simplified models such as the dead-endmodel contraction structures and expansion-contractionstructures were adopted [25] However the structure of areal porous medium is very complex and so is likely toaffect the flow characteristics Hence it is very importantto consider the complexity of pores in studies of polymerelasticity In this case the finite volume method [26ndash29] islikely to bemore stable Also the existing studiesmainly focuson viscoelastic steady flows and although some researchhas been done on unsteady flow those studies address thealgorithms rather than the flow characteristics of viscoelasticfluids in porous media Transient flow characteristics (beforea flow reaches a steady state) are important leading tosome potentially interesting phenomena that are ignored inexperiments In this study we investigated the complex poregeometries based on etched-in glasses model This 2D modelcan capture the geometrical and topological characteristicsof pores in a realistic manner Furthermore we studiedthe microscopic characteristics of transient flows using theviscoelastic flow solver in OpenFOAMThese results providetheoretical support and guidance to studies that aim toenhance oil recovery using polymer flooding

2 Physical Model

Towphysicalmodels that represent the pores have been estab-lished in this studyThefirst one is the dead-endmodel whichcaptures a simple pore The second one that models morecomplex pore structures is represented using the etched-inglass model The dead-end model is schematically shown inFigure 1 The residual oil in the dead end often cannot bedisplaced by water The real etched-in glass model is 40 times40mm in size and was used in microscale visual experimentswith pure viscous glycerin and elastic polymer flooding Itsporosity is 35 and its permeability was 2200 times 10minus3 120583mFigure 2(a) shows the saturated oil imaged using a high-resolution camera Figure 2(b) shows a region from Figure 1as a digitized model to demonstrate the computationalspeed and convergence and area under consideration is 1 times1mm The diameter of the smallest pore was sim002mm Inconclusion this model provides a sufficiently detailed pictureof a porous medium to allow studies of the polymer flowbehavior

First the image was digitized using Solidworks to obtaina black and white presentation and then the black pixelscorresponding to the pore boundary were extracted (Fig-ure 1(c)) The skeleton of the pores must be imported usingsnappyHexMesh a solver in OpenFOAM and is shown inFigure 1(c) The nonorthogonal grid was obtained using thesnappyHexMesh solver in OpenFOAM and it is shown inFigure 1(d) The number of mesh elements in the model wassim399220 The minimum radius was 002mm The mesh file

Geofluids 3

25 ＧＧ

05ＧＧ

10 ＧＧ

05 ＧＧ











(2)


4 Geofluids


(a) 01 s


(b) 04 s


(c) 07 s


(d) 1 s



120591 + 120582 nabla120591= 120578119901119863 (3)





120591119904 = 2120578119904119863 (5)














119880 = radic1198802119909 + 1198802119910 (7)

Geofluids 5

= 006

= 010

= 015

2 4 6 80t (Ｍ)

20

40

60

80

100

Δp

(10minus4０)


p000458

00001000200030004

(a) 01 s

p000458

00001000200030004

(b) 05 s

p000458

00001

0002

0003

0004

(c) 2 s








6 Geofluids

p00035

0

0001

0002

0003

(a) 01 s

p00035

0

0001

0002

0003

(b) 05 s

p00035

0

0001

0002

0003

(c) 2 s


p00028

0

0001

0002

(a) 01 s

p00028

0

0001

0002

(b) 05 s

p00028

0

0001

0002

(c) 2 s


= 006

= 010

= 015

1 2 3 4 5 6 70t (Ｍ)

20

40

60

80

100

120

140

Δp

(10minus3０)






6 Conclusions




Geofluids 7


(a) 119905 = 01 s


(b) 119905 = 07 s


(c) 119905 = 20 s



(a) 120582 = 006 s


(b) 120582 = 01 s


(c) 120582 = 015 s


4830E minus 004

3623E minus 004

2415E minus 004

1208E minus 004

0000E + 000

(a) 120582 = 006 s


(b) 120582 = 01 s


(c) 120582 = 015 s





Acknowledgments



References



8 Geofluids





































Mining


Journal of









Journal of




Advances in











Geology Advances in

Geofluids 3

25 ＧＧ

05ＧＧ

10 ＧＧ

05 ＧＧ











(2)


4 Geofluids


(a) 01 s


(b) 04 s


(c) 07 s


(d) 1 s



120591 + 120582 nabla120591= 120578119901119863 (3)





120591119904 = 2120578119904119863 (5)














119880 = radic1198802119909 + 1198802119910 (7)

Geofluids 5

= 006

= 010

= 015

2 4 6 80t (Ｍ)

20

40

60

80

100

Δp

(10minus4０)


p000458

00001000200030004

(a) 01 s

p000458

00001000200030004

(b) 05 s

p000458

00001

0002

0003

0004

(c) 2 s








6 Geofluids

p00035

0

0001

0002

0003

(a) 01 s

p00035

0

0001

0002

0003

(b) 05 s

p00035

0

0001

0002

0003

(c) 2 s


p00028

0

0001

0002

(a) 01 s

p00028

0

0001

0002

(b) 05 s

p00028

0

0001

0002

(c) 2 s


= 006

= 010

= 015

1 2 3 4 5 6 70t (Ｍ)

20

40

60

80

100

120

140

Δp

(10minus3０)






6 Conclusions




Geofluids 7


(a) 119905 = 01 s


(b) 119905 = 07 s


(c) 119905 = 20 s



(a) 120582 = 006 s


(b) 120582 = 01 s


(c) 120582 = 015 s


4830E minus 004

3623E minus 004

2415E minus 004

1208E minus 004

0000E + 000

(a) 120582 = 006 s


(b) 120582 = 01 s


(c) 120582 = 015 s





Acknowledgments



References



8 Geofluids





































Mining


Journal of









Journal of




Advances in











Geology Advances in

4 Geofluids


(a) 01 s


(b) 04 s


(c) 07 s


(d) 1 s



120591 + 120582 nabla120591= 120578119901119863 (3)





120591119904 = 2120578119904119863 (5)














119880 = radic1198802119909 + 1198802119910 (7)

Geofluids 5

= 006

= 010

= 015

2 4 6 80t (Ｍ)

20

40

60

80

100

Δp

(10minus4０)


p000458

00001000200030004

(a) 01 s

p000458

00001000200030004

(b) 05 s

p000458

00001

0002

0003

0004

(c) 2 s








6 Geofluids

p00035

0

0001

0002

0003

(a) 01 s

p00035

0

0001

0002

0003

(b) 05 s

p00035

0

0001

0002

0003

(c) 2 s


p00028

0

0001

0002

(a) 01 s

p00028

0

0001

0002

(b) 05 s

p00028

0

0001

0002

(c) 2 s


= 006

= 010

= 015

1 2 3 4 5 6 70t (Ｍ)

20

40

60

80

100

120

140

Δp

(10minus3０)






6 Conclusions




Geofluids 7


(a) 119905 = 01 s


(b) 119905 = 07 s


(c) 119905 = 20 s



(a) 120582 = 006 s


(b) 120582 = 01 s


(c) 120582 = 015 s


4830E minus 004

3623E minus 004

2415E minus 004

1208E minus 004

0000E + 000

(a) 120582 = 006 s


(b) 120582 = 01 s


(c) 120582 = 015 s





Acknowledgments



References



8 Geofluids





































Mining


Journal of









Journal of




Advances in











Geology Advances in

Geofluids 5

= 006

= 010

= 015

2 4 6 80t (Ｍ)

20

40

60

80

100

Δp

(10minus4０)


p000458

00001000200030004

(a) 01 s

p000458

00001000200030004

(b) 05 s

p000458

00001

0002

0003

0004

(c) 2 s








6 Geofluids

p00035

0

0001

0002

0003

(a) 01 s

p00035

0

0001

0002

0003

(b) 05 s

p00035

0

0001

0002

0003

(c) 2 s


p00028

0

0001

0002

(a) 01 s

p00028

0

0001

0002

(b) 05 s

p00028

0

0001

0002

(c) 2 s


= 006

= 010

= 015

1 2 3 4 5 6 70t (Ｍ)

20

40

60

80

100

120

140

Δp

(10minus3０)






6 Conclusions




Geofluids 7


(a) 119905 = 01 s


(b) 119905 = 07 s


(c) 119905 = 20 s



(a) 120582 = 006 s


(b) 120582 = 01 s


(c) 120582 = 015 s


4830E minus 004

3623E minus 004

2415E minus 004

1208E minus 004

0000E + 000

(a) 120582 = 006 s


(b) 120582 = 01 s


(c) 120582 = 015 s





Acknowledgments



References



8 Geofluids





































Mining


Journal of









Journal of




Advances in











Geology Advances in

6 Geofluids

p00035

0

0001

0002

0003

(a) 01 s

p00035

0

0001

0002

0003

(b) 05 s

p00035

0

0001

0002

0003

(c) 2 s


p00028

0

0001

0002

(a) 01 s

p00028

0

0001

0002

(b) 05 s

p00028

0

0001

0002

(c) 2 s


= 006

= 010

= 015

1 2 3 4 5 6 70t (Ｍ)

20

40

60

80

100

120

140

Δp

(10minus3０)






6 Conclusions




Geofluids 7


(a) 119905 = 01 s


(b) 119905 = 07 s


(c) 119905 = 20 s



(a) 120582 = 006 s


(b) 120582 = 01 s


(c) 120582 = 015 s


4830E minus 004

3623E minus 004

2415E minus 004

1208E minus 004

0000E + 000

(a) 120582 = 006 s


(b) 120582 = 01 s


(c) 120582 = 015 s





Acknowledgments



References



8 Geofluids





































Mining


Journal of









Journal of




Advances in











Geology Advances in

Geofluids 7


(a) 119905 = 01 s


(b) 119905 = 07 s


(c) 119905 = 20 s



(a) 120582 = 006 s


(b) 120582 = 01 s


(c) 120582 = 015 s


4830E minus 004

3623E minus 004

2415E minus 004

1208E minus 004

0000E + 000

(a) 120582 = 006 s


(b) 120582 = 01 s


(c) 120582 = 015 s





Acknowledgments



References



8 Geofluids





































Mining


Journal of









Journal of




Advances in











Geology Advances in

8 Geofluids





































Mining


Journal of









Journal of




Advances in











Geology Advances in








Mining


Journal of









Journal of




Advances in











Geology Advances in

Study on Mechanism of Viscoelastic Polymer Transient Flow in...

Documents

Transcript of Study on Mechanism of Viscoelastic Polymer Transient Flow in...