Automatic correction and simplification of geological maps and cross-sections for numerical simulations

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Automatic correction and simplification of geological

maps and cross-sections for numerical simulations

Pierre Anquez, Jeanne Pellerin, Modeste Irakarama, Paul Cupillard, Bruno

Lévy, Guillaume Caumon

To cite this version:

Pierre Anquez, Jeanne Pellerin, Modeste Irakarama, Paul Cupillard, Bruno Lévy, et al.. Automatic

correction and simplification of geological maps and cross-sections for numerical simulations. Comptes

Rendus Géoscience, Elsevier Masson, 2019, 351 (1), pp.48-58. �10.1016/j.crte.2018.12.001�.

�hal-02136697�

Tectonics,

Tectonophysics

Automatic

correction

and

simplification

of

geological

maps

and

cross-sections

for

numerical

simulations

Pierre

Anquez

*

,

Jeanne

Pellerin

,

Modeste

Irakarama

,

Paul

Cupillard

,

Bruno

Le´vy

,

Guillaume

Caumon

a

Universite´ deLorraine,CNRS,GeoRessources,54000Nancy,France

b

Universite´ catholiquedeLouvain,MEMA,1348Louvain-la-Neuve,Belgium

c

INRIA,ProjectAlice,54600Villers-le`s-Nancy,France

1. Introduction

Geologicalmodelshonoringsubsurfacedataarecentral objectsinvolvedinawidespectrumofapplications(Ringrose andBentley,2015).Amodel,whetheritbethreedimensional ortwodimensional,is generallydedicatedtoanswerone questionatagivenscale,e.g.,contaminanttransport(e.g.,

HuysmansandDassargues,2009),theimpactofsubsurface structuresongroundmotion(e.g.,BenitesandOlsen,2005; Chaljub et al., 2010), the geomechanical behavior of a reservoir(e.g., Seguraetal.,2011;Verdonetal.,2013).In mostcases,thechoiceofthisscaleandoftheadaptedspatial resolution is complicated because very small geological features can have an important impact on the physical process.Forinstance,thinshaledrapesinalluvialdeposits mayhavealargeimpactonfluidcirculations(e.g.,Issautier etal.,2013;Jacksonetal.,2000,2009;Massartetal.,2016). Tosimulatethephysicalprocessesinnaturalgeological settings,mostmethodsrelyonaspatialdiscretizationofthe model,usingameshoragrid(weusethetermmeshforan unstructuredspatial discretization).Themeshrepresents themodelatagivenlevelofdetailandisusedtosimulatea

ARTICLE INFO Articlehistory:

Received8October2018

Acceptedafterrevision5December2018 Availableonline23January2019 HandledbyIsabelleManighetti Keywords:

Geoinformatics Cross-sections Repairandsimplification Graphformalization Wavepropagation

ABSTRACT

Incorporating prior geologicalknowledgein geophysicalprocessmodels often meets practicalmeshingchallengesandraisesthequestionofhowmuchdetailistobeincluded inthegeometricmodel.Weintroduceastrategyto automaticallyrepairandsimplify geologicalmaps,geologicalcross-sectionsandtheassociatedmesheswhilepreserving elementary consistency rules. Toidentify featuresbreaking validity and/or the thin featurespotentiallyproblematicwhengeneratingamesh,weassociateanexclusionzone witheachmodelfeature(horizon,fault).Whenthesezonesoverlap,boththeconnectivity andthegeometryofthegeologicallayersareautomaticallymodified.Theoutputmodel enforces specificpracticalquality criteria on themodeltopology and geometry that facilitatesthegenerationofameshwithlowerboundsonminimumanglesandminimum localentitysizes.Ourstrategyisdemonstratedonaninvalidgeologicalcross-sectionfrom areal-casestudyintheLorrainecoalbasin.Wefurtherexploretheimpactsofthemodel modificationsonwave propagationsimulation.We showthatthedifferencesonthe seismograms due to model simplifications are relatively small if the magnitude of simplificationsisadaptedtothephysicalproblemparameters.

C _{2018PublishedbyElsevierMassonSASonbehalfofAcade´miedessciences.Thisisan}

openaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/ by-nc-nd/4.0/).

* Correspondingauthor.GeoRessources,UL/CNRS/CREGU,ENSG,2,rue duDoyen-Marcel-Roubault,54518Vandœuvre-le`s-Nancycedex,France.

E-mailaddresses:Pierre.Anquez@univ-lorraine.fr(P.Anquez),

Jeanne.Pellerin@uclouvain.be(J.Pellerin),

Modeste.Irakarama@univ-lorraine.fr(M.Irakarama),

Paul.Cupillard@univ-lorraine.fr(P.Cupillard),Bruno.Levy@inria.fr

(B.Le´vy),Guillaume.Caumon@univ-lorraine.fr(G.Caumon).

ContentslistsavailableatScienceDirect

Comptes

Rendus

Geoscience

ww w . sci e nc e di r e ct . com

https://doi.org/10.1016/j.crte.2018.12.001

1631-0713/C _{2018PublishedbyElsevierMassonSASonbehalfofAcade´miedessciences.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense}

givenphysicalprocess.Itshouldthereforecaptureallthe geological features having a significant impact on that physical process and meetthe requirements of the PDE discretizationmethodintermsofmeshquality.Thesetwo objectives may oftenbeconflicting because geometrical complexity is inherentin geologicalmodels(thin layers, stratigraphic unconformities, small fault displacements) (GrafandTherrien,2008;Karimi-FardandDurlofsky,2016; Merlandetal.,2014;MustaphaandDimitrakopoulos,2011; Pellerinetal.,2015).Onesolutionistosolvethedifferential equationsonameshthatdoesnotnecessarilyaccountsfor allgeologicaldiscontinuities,e.g.,usingtheextended finite-elementmethod(XFEM)(Moe¨setal.,1999).Otheroptions aretodecreasetheexpectedmeshqualitycriteria,ortofirst simplifythegeometryoftheinputmodel.

In practice, most models are manually modified by skilled practitioners who are able to ensure that the generatedmeshwillfulfilltheprojectrequirements.Afew automatic strategies exist tosimplify the model before generatingitsmesh,butnoneensureseitherthevalidityof the outputmodel orits geologicalconsistency ( Karimi-FardandDurlofsky,2016;MustaphaandDimitrakopoulos, 2011;Pellerinetal.,2014).

Another practical challengealsoarises,since geomo-delingmethodsdonotalwaysgenerateasealedboundary representation. In this case, repairing (or sealing) is required toprepare the model for meshing.This repair process (e.g., Barequet and Kumar, 1997; Elsheikh and Elsheikh,2012; Euleretal.,1998)aims atremovingthe smallgaps andintersectionsbetween modelfeaturesto maketheirmeshesconformable.

Inthispaper,weintroduceanautomaticmethodtorepair and simplify two-dimensional (2D) geological models (Fig. 1): geological maps and cross-sections. The output model enforces specific practical quality criteria on the modeltopologyandgeometrytofacilitatethesuccessofthe generation of a good mesh, e.g., minimum angles and minimumlocalentitysizes.Todetecttheinvalidfeaturesof the inputmodel,wedefineexclusion zonesaroundinput model features (e.g., horizons, faults). The connections betweenmodelfeaturesandtheinvalidconfigurationsare modeledbyagraph,whichcapturesessentialinformationfor modelediting(Section3).Ourmethodoperatesonthegraph andaimsatremovingalltheinvalidfeatures. Eachgraph operationcorrespondstoeitheramodificationofthemodel connectivity,amodificationofthemodelgeometry,orboth. Toremovefeaturesbreakingvalidityandtolocallysimplify complexgeometricalstructures,wemodifytheconnectivity

and the geometry of the lines representing geological interfaces (Section 4). We apply the presented editing method to a geometrically complex invalid geological cross-sectionanddiscusstheimpactofmodelsimplifications onseismicwavepropagationsimulations(Section5). 2. Definitions

2.1. 2Dgeologicalmodels

Inthispaper,werefertogeologicalmapsandgeological cross-sectionsbythegenerictermof2Dgeologicalmodels (Fig. 1a). 2D geological models are defined by Surfaces representingcross-sectionsingeologicallayers.Theyare delimitedbytheLinesthatrepresentgeologicalinterfaces (e.g.,horizons,faults,unconformities)andbyCornerpoints representing interface contacts (e.g., fault–fault, fault– horizon)andfaulttips.ModelSurfaces,LinesandCorners are called model entities. This definition follows the RINGMesh library (Pellerin et al., 2017) on which our implementationisbased.

The geometry of the model is determined by the geometry of all its entities (Fig. 1b). The incidence relationshipsbetween theseentitiesdefinethetopology (orconnectivity)ofthemodel(Fig.1c).

2.2. 2Dmodelvalidity

Toensurethecorrectnessofmostoperationsmadeona 2D model, this model must be valid. The validity of a geological model is defined by a set of criteria on its topology, its geometry, and the consistency between topologyandgeometry.Thissectionsummarizestherules weusetoverifythatageologicalmodelisvalid.Sincethe model Surfaces are defined and controlled by their boundary Lines and Corners, validity conditions are formulatedascriteriaonthemodelLinesandCorners.

Topological validity. Topological validity conditions are strongvaliditycriteria ensuring modelintegrity(Ma¨ntyla¨, 1988;Requicha,1980;Sakkalisetal.,2000).The relations-hipsbetweenmodelentitiesareconstrainedtoensureavalid topologyforageologicalmodel.Inourcase,thetopological validityconditionson 2Dmodelsare:(1)a Cornershould haveatleastoneincidentLine,and(2)aLineshouldhave exactlyone(closedLine)ortwoboundaryCorners.

Dependingon themodeled geologicalelement,these constraintscanbestronger.ACornerhasonlyoneincident LineifandonlyiftheLinerepresentsafault(Caumonetal.,

Fig.1.(a)Geological2Dmodelsaredefinedbyageometryandatopology.(b)Thegeometryissetbythegeometricaldiscretizations(meshes)ofthemodel entities:theSurfaces,theLinesandtheCorners.(c)Thetopologycorrespondstotheboundary/incidencerelationshipsbetweenmodelentities.

2003).SuchCornersarecalledfreeCornersandcorrespond tofaultandfracturetips.

Consistencybetweentopologyandgeometry.Themodel geometryshouldbeconsistentwiththemodeltopology. Theintersectionsbetweentwoentitiesmustbeasubsetof theirboundaries(Requicha,1980).Asaconsequence,two neighboring Linesmust share thesame Cornersat their common extremities (Fig. 2a), and there can be no intersection between two Lines representing conformal horizons(Fig.2b).

Geometricalvalidity.Wedefinecriteriaontheshapesof modelentities(Fig.3a).Thesecriteriarelyonthedefinition ofminimalentitylocalsizes(i.e.Linelengths,Surfacelocal widths that are equivalent to the distance between surface-boundaryLines)andtheminimalanglesbetween twoadjacentLines.Thesevaliditycriteriadependonthe requirementsofnumericalmethodsrunonthemodel.As Surfacemeshesareconstrainedby themodel interfaces, thesecriteriaon model entitiesaimat improvingmesh qualitymetricssuchasminimalangleorminimaledgesize (e.g.,Shewchuk,2002),andatimprovingtheaccuracyof thecomputationsaswellasthesimulationcomputation time.

We usealsovalidity criteria on the discretizationof modelentities(Fig.3b).Themeshedentitiesshouldhavea valid mesh, i.e. composed of non-empty triangles with

edgesthatintersectproperly(seeFreyandGeorge,2000, forextendeddefinitions).

Entityexclusionzonestodefinegeometricalvalidity.To detect geometrical invalid features, we define for each modelCornerandLineeianexclusionzoneZðeiÞR2from

agivenminimumdistanceandminimumangleatCorner (Fig. 4). Exclusion zones are defined implicitly using distancestotheentityeitodetermineifapointbelongsto

theexclusionzoneornot.Exclusionzonesrepresentareas thatcannotoverlapotherentityexclusionzonesandwhich shouldnotinclude,evenpartially,anyothermodelentity. More formally, we can write the geometrical validity conditionbetweentwomodelentitiesas:

ei;ej2M2ejei6¼ej)ZðeiÞ\Zej¼? ; (1)

whereMeisthesetofmodelentitiesandZ(ei)isthe

exclusion zone ofthe modelentity ei. Theintersections

between exclusion zones define invalid features of the inputmodel(Fig.4,right).

Input minimal entity sizes and minimal angles may varyspatiallyandfromoneentitytoanother,depending onavailabledata,physicalproperties,orpriorknowledge ontheimportanceofeachmodelfeatureforthe applica-tion. As a consequence, the thickness of the exclusion zones, which is deduced from these input data, is not necessarilyconstant.NotethatanexclusionzoneZ(ei)can

bereducedtotheentityeiitself,locallyoreverywhere(no

conditionaboutanglesandlocalsizes).

3. Proposedapproach

Ourmainobjectiveistogeneratea2Dgeologicalmodel that respects validity conditions from an invalid input model.Inadditiontotheinputmodel,adesiredminimal mesh size and minimal angle are given as input parameters. These geometrical parameters define an exclusionzoneforeachmodelCornerandforeachmodel Line.Priorgeologicalknowledge,e.g.,thetypeofcontact

Fig. 2.Typical invalid configurations breaking consistency between modeltopologyandgeometryare(a)non-conformitybetweenadjacent Surfacesand(b)intersectionsbetweenLines.

Fig.3.Geometricalinvalid features.(a)Entityinvalidconfigurations. (b)Meshinvalidconfigurations.

Fig. 4.Detection of invalid model zones from the intersections of exclusionzonesdefinedaroundthemodelentities.Noticethatthetwo criteriaarecombined:theanglecriterionis combinedonthetipsof exclusionzonesin(a).

betweenlayers(conformableornot),mayadditionallybe usedtoconstrainthemodel-editingstrategy.

3.1. Elementarymodeleditingoperations

To remove the identified invalid model features, we performelementaryoperationsthatareonlygeometrical operationsorbothtopologicalandgeometricaloperations (Fig.5).

Thetwogeometricaloperations,modelfeature reshap-ingandmeshediting,donotchangethemodeltopology (Fig.5a);theyaimatthickeningorenlargingsmallmodel features.Thethreetopologicaloperations,entitymerging, entitysplitting,andentityremoval,modifythenumberof modelentitiesandtheirconnectivity(Fig.5b);theyaimat deleting thin model features by collapsing them. These operations reduce the topological model complexity (Pellerinetal.,2015).

3.2. Overview

We introduce a graph, named invalidity graph G,to supportourmethod.ThegraphGstoresateachstepthe necessary information needed to solve the editing problem.Theinputgeologicalmodelsareeditedinthree steps(Fig.6).

1.Analysis of the input model and detection of the topologicalandthegeometricalinvalidfeatures(Section 4.1).TheyarestoredinthegraphG.

2.Determination of the editing operations to perform (Section4.2).ThesecorrectionsaremadeonthegraphG andformulatedbyelementarygraphoperations. 3.Reconstruction of the output geological model. The

editedgraphGisinterpretedingeometricalterms:the topologyoftheedited modelisfirst determined,and thenthegeometricalembeddingcorrespondingtothis topologyisfound(Section4.3).

Ourstrategydisconnectsgrapheditingfromgeological modelediting.Theadvantageisthatthedecision-making about corrections, performed into the graph G, is (i) performedonlyafterhavingidentifiedtheinvalidfeatures, and(ii)decoupledfromeffectivemodelmodifications.It allows applying the modifications only once after all correctingdecisionshavebeenmade.Theinputmodelis thus kept as a reference model to avoid a too large differenceaftermodifications.Moreover,itavoidsmixing topological and geometrical changes that often lead to instabilities or inconsistencies (e.g., Euler et al., 1998; Pellerinetal.,2014).

3.3. Problemformulationusingagraph

TheinvaliditygraphGisakeypointofourmethod:(i)it is a simple abstract representation storing the invalid configurations as graph edges, (ii) it supports the formulation of elementary model editing operations as successive elementary graph operations, and (iii) it disconnects decision making about model editing from theimplementationofthemodifications:thedecisionsare made on the graph before being performed on the geologicalmodel.

Graphdefinition.WedefinetheinvaliditygraphG=(N,E) whereNdenotesthesetofgraphnodesandEdenotesthe setofundirectedgraphedges(Fig.7).Amodelentitytype (eitherCornerorLine)isassignedtoeachgraphnodeNi2

N.Inaddition,weassigntoagraphnodeNithesetofinput

model parts thenode represents, denotedby p(Ni). For

example, a graph node colored as a Corner node, can representtwoinputmodelCornersandonepartofaninput modelLine.

TherearetwokindsofedgesinE.First,theedgesinthe subset Ec _{represent incidence–boundary relationships}

betweennodes(blueedgesinFig.7). Second,theedges inthesubsetEd_{representinvalidconfigurationsbetween}

nodes(rededgesinFig.7).AgraphedgeEd_(A,B)between

nodesAandBisalsoassociatedwithaproperty,denoted p(Ed_{(A,B)),whichstorestheinputmodelentitypartsofA}

Fig. 5.The most common geometrical and topological operations performedtofixgeologicalcross-sections.

and B where their exclusion zones intersect. Note that p(Ed_(A,B))

(p(A)[p(B)).

Problemreformulation.Invalidconfigurationsare repre-sentedbythesetofedgesEd.Correctingthemodelthus consistsin theremoval of all theinvalid edgesEd_.The

graphissaidvalidwhenthesetEd_{isempty.Removingthe}

invalidedgesEdimpliesmodificationsonthegraphGthat arethen translatedintomodifications on thegeological model.

4. 2Dmodelcorrectionandsimplificationmethod 4.1. Analysisoftheinputmodel

Inthisfirststep,theinputmodelisexaminedtodetect the configurations breaking its validity and thus to initialize the graph G. The graph is initialized by first creatingonenodeforeachmodelCornerandeachmodel Line.Atinitialization,eachnoderepresentingamodelLine correspondstothisLinefromitsfirstboundaryCornertoits last boundary Corner.The connectivityedges Ec _{are set}

with respect to the input model boundary–incidence relationships.

EdgesEd_{areaddedintothegraphG(Fig.7)whenever}

an invalid configuration is detected. Topological issues, suchashorizonfreeborders,aredetectedbyinspectingthe graph of entity connectivities. A geometrical search is performedtofindthenearestmodelentityEnearesttowhich

the free Corner Cfree should be connected. The nearest

model entity is determined by using the anisotropic distance function induced by the free Corner exclusion zone(i.e.theconvexgaugefunction,seeAppendixAfor definitions).AnedgeEd_(C

free,Enearest)isaddedbetweenthe

noderepresentingCfreeandtheoneforEnearest.

Forintersectionsbetweentheexclusion zonesoftwo model entities ei and ej, an edge Ed(Nei,Nej) is added

betweentwonodesNe_iandNe_j(Fig.7).Thepropertyonthis

edge,p(Ed(Nei,Nej)),correspondstothesetofpointsofthe

entitieswhereZ(ei)andZ(ej)intersect.

Inourimplementation,theexclusionzoneofaLineis equaltotheunionofthediskscenteredonallthepointsof theLine.Thisdefinitionisequivalenttothedefinitionof the dilation of the Line by a disk in mathematical

Fig.6. Workflowofourgeologicalmodel-editingapproachusinganinvaliditygraph.

Fig.7.Inputgeologicalmodelentityexclusionzonesareusedtodetectissues.Theseissuesarethenrepresentedbyinvalidityedges(inred)inthegraphG. Then,thegraphGiseditedtoremovetherededges,whichprovidesinformationforoutputreconstruction.

morphology (e.g.,Serra, 1986). The exclusion zone of a Cornerisgivenbycornerdilationbyachosenconvexshape (e.g.,disks,ellipses,rectangles).Inthisprocess,theuseof anellipseorarectanglewhosemainaxisisalignedonthe LineincidenttotheCornermakesitpossibletodetectfree Corners that are close to another Line. Intersections betweenexclusionzonesareeasilydetectedbycomputing convexshapeintersections,e.g.,usingtheGJKalgorithm (GilbertandFoo,1990).Weidentifyalltheentitypoints (not only its vertices) whose corresponding disks (or convex shapes) compose the intersections between exclusionszones.Theseidentifiedpointsarerecordedin thepropertyofthecorrespondinginvaliditygraphedge. Thedetectionofintersectionscanbeimplementedusing exactpredicates(e.g.,Le´vy,2015)forincreasedrobustness. 4.2. Eliminationofgraphinvalidedges

The goalof this stepis todetermine how tocorrect invalid configurations, i.e. how to edit G to remove all edges of Ed_{. We perform successive graph elementary}

operationstoremovealltheinvalidedges(Fig.8).These graphoperationscorrespondtooperationsonthe geologi-calmodel.

1.Edgedeletion.Theedgebetween Aand Bis removed without affecting A and B (Fig. 8a). This operation corresponds to a geometrical editing of the model withoutmodificationofitstopology.

2.Edgecontraction.TheedgebetweenAandBisremoved by mergingA and B(Fig.8b). This operation is only performedbetweennodesofthesameentitytypeand corresponds to the merger of two geological model entities.

3.Node removal. One node is removed, leading to a removaloftheincidentedges(Fig.8c).Thisoperation correspondstotheremovalofageologicalmodelentity considered as unimportant or a duplicated model feature.Priorgeologicalknowledge,givenas input,is requiredtodeterminewhichnodeisremoved. 4.Nodesplitting.Onenodeis splitintotwo newnodes

(Fig.8c).Thisoperationcanonlybeperformedonanode representingaLineandcorrespondstothesubdivision ofamodelLineintwoconnectedparts.Thisleadstothe removalorredistributionoftheincidentedges.

Thedifferentoptionsareconstrainedby(i)theentity typeofthenodes(seeAppendixC),(ii)theselectedediting strategygivenasinput(e.g.,seekingeitherpreservationor simplificationofmodelconnectivity), (iii)thegeological types of the entities, e.g., edge deletion cannot solve horizonfree-borderinvalidconfigurations.

WefirstprocesstheedgesbetweenaLinenodeanda Corner node and then the other edges. This choice is motivatedbythefactthatLine-Corneristhemostfrequent configurationforhorizonfree-borderinvalidedge.Starting withthisconfigurationallowstacklingfirstthepotential non-conformityrepairproblem,andthenthe simplifica-tionone.BecausethecollisionsofLineexclusionzonesmay notconcernthewholeLines,operationsonLine-Lineinvalid edgeslikeedgecontractionornoderemovalmayneed,asa pre-process,nodesplitting(seeAppendixB).

4.3. Reconstructionofthegeologicalmodel

The graph G contains all the information needed to generatetheoutputmodel. ThenodesN andthesetof incidence edges Ec _{directly define the output model}

topology. Each graph node is converted into an output model entity and each graph connectivity edge is translatedintoanentityincidence–boundaryrelationship. Fordeterminingthegeometricalembeddingofthemodel entities,thenodepropertiesp(Ni),8i2jNjcontainthelists

ofinputmodelentitypartsthatshouldbeusedasinput. Inourapproach,Cornersarefirstgeometricallysetand then Lines are resampled between their bounding Corners.ThepositionoftheoutputCornersisdetermined depending on the input model entities listed in the correspondingnode property.Ifa nodepropertyis only composedofinputCorners,thegeometricpositionofthe new Corner is set at the barycenter of input Corners.However, ifa nodepropertyincludesalsoinput Line parts, the given barycenter of input Corners is projected on each input Line part and finally the new Cornerissetatthebarycenteroftheseprojectedpoints. Thisavoids,forinstance,thegenerationofkinksonfault traces when connecting a horizon free extremity. Once CornersandLineshave beenresampled,itis possibleto compute thenumber of surfaces composing theoutput modelandtoretrievetheboundaryrelationshipsbetween Lines and Surfaces. The Surfaces can be meshed using various tools, e.g., Triangle (Shewchuk, 1996), Gmsh (GeuzaineandRemacle,2009)ormmg2d(https://www. mmgtools.org/).

Fig.8.Thefourelementarygraphoperationsperformedforremoving invalidedges(Ed

5. Application

Apracticaldifficultyintheapplicationoftheproposed method is to choose the appropriate level of model simplification to preserve the accuracy of physical computations. To investigate this problem further, we proposeapreliminarysensitivitystudy:westartfroman invalid model and perform a model topological repair (Section5.1)withoutgeometricalsimplification,thenwe performmodelsimplificationsandanalyzetheimpacton wavepropagationsimulations(Section5.2).

5.1. Sealingofa2Dgeologicalmodel

In this first application, we repair a non-watertight verticalcross-sectionfroma3Dgeologicalsurface-based modelbuiltbyCollonetal.(2015);thismodelrepresentsa restricted area of theLorraine coal basin (northeastern France). Theoriginal 3D modeldescribes thegeological regionalstructures(13faultsand2horizons)and71 sub-verticalcoalbeds(Fig.9)modeledbytriangularsurfaces.A north–south cross-section almost perpendicular to the Saint-Nicolasfaultisextractedfromthis3Dmodel.The2D cross-sectionis composed of246 Cornersand 126Lines representingtheSaint-Nicolasfault,thetopography,the Permo-Triassic base horizon, and the 71 coal beds (Fig. 10a). The cross-section in Fig. 10a shows that the modelisnotwatertight;therearesmallgapsandoverlaps at contacts between coal bed Linesand faultLines and boundaryLines.Theseissuesareduetotheremeshingof coalbedsurfacesduringthebuildingofthe3Dgeological model(Collonetal.,2015).

To seal the 2D model, we apply our graph-based approach. Minimal local entity size and minimal angle valuesaresettozerobecausetheobjectiveisonlytorepair this model without simplification. Moreover, we use additional geological knowledge: the coal veins are conformableonetoanother.Thisconstrainsourmethod by preventing mutual branching of vein Lines. As a consequence,veinfreeborderscanonlybeconnectedto the fault, to the Permo-Triassic base horizon, or to a boundaryLine.Thegeneratedoutput2Dmodel(Fig.10b)is watertight.Ascorrectingthesetopologicalissuesinvolves

line splitting operations, the number of model lines evolvedfrom126to367.Theexecutiontimeforrepairing this cross-sectionis0.17sona laptopwitha 2.50GHz processorIntel1

CoreTM_i7-6500U.

5.2. Impactof2Dmodelsimplificationonwavepropagation Theobjectiveofthissecondapplicationistoshowthat we are able to slightly simplify a model without significantlychangingtheresultsofseismicwave propa-gation simulations. This is possible by directly using physical parameters from seismic wave simulations as inputofourmethod.

Definition of a velocity model. From the geometrical model,weextractasub-domainonwhichacousticwave propagations are run. These simulations are performed usingavelocity-stressformulationfinite-difference meth-odonCartesiangrid(Virieux,1984)in2D.IntheMerlebach area,coalbedsthicknessesvaryfromafewcentimetersto 5 m (Collon et al., 2015); for simplicity, model lines representingcoalbedsarerasterizedbysettingaconstant veinthicknessof1m.Wesetthegridsizeequalto0.33m; thuseachcoalbedisrepresentedbyobliquerowsofthree cells. The griddimensions are1800 cells by1800 cells, coveringthe600-m-by-600-msub-domainrepresentedby thedashedlineinFig.10b.Wedefinethreerockdomains (Fig.11a):thePermo-Triassicconglomeratesatthetop (P-wavevelocity=3000ms1_{,inblueinFig.11a),thedeeper}

Westphalian and Stephanianclaystones,sandstonesand conglomerates (P-wavevelocity=4 000 ms1_{,in redin}

Fig. 11a), and intercalated coal beds (P-wave velocity=2400ms1_{,ingreeninFig.11a).Notethatwe}

ignoretheP-wavevelocityvariationwithdepthandthat weusetheconstantdensityapproximationforsimplicity. We set a seismic source (Ricker wavelet) with a maximum wave frequency fmax=300Hz. As a

conse-quence,theminimumwavelengthis8m,whichissampled by 24 grid points.The grid samplingis considered fine enoughtoreducenumericaldispersion (atleast10grid points,Virieux,1984).Theseismicresolutionrisgivenby the quarter-wavelength criterion (e.g., Yilmaz, 2001, chapter11): r

l

y

where

l

v

minimumP-wavevelocity.

Simplifiedmodel.Weapplyourgraph-basedmethodto simplify the generated sealed model (Section 5.1). The objectiveistoremovethesmallmodelfeatures,e.g.,the smalllinescausedbyveryclosefault–horizoncontacts.To doso,theminimallocalentitysizevalueissettothevalue oftheseismicresolution,i.e.2m(seeEq.(2)).Asadditional geologicalknowledge,wepreventmergersbetweencoal veins precisely identified in Collon et al. (2015). As a consequence,modelmodificationsareexclusivelylocated atcoalveinboundaries,i.e.alongtheSaint-Nicolasfault, thePermo-Triassicbasehorizon,andthemodelboundary (Fig.11c). Theexecution timeforsimplifyingthis cross-sectionis 0.25son a laptopwitha 2.50GHzprocessor

Fig.9.3D geologicalmodelaroundtheMerlebachmine (North-East France)built by Collon etal.(2015) andcomposed ofsurfaces for horizons,faultsandcoalbeds.Thegreynorth–southsurfaceisthevertical planesupportingthe2Dcross-section.

Intel1

CoreTM _{i7-6500U. The difference between the}

velocitymodelmincorrespondingtotheinitialgeometry

(Fig.11a)andthevelocitymodelmoutcorrespondingtothe

simplified geometry is defined by

dm

(Fig.11b).

Simulations. The velocity-stress formulation finite-difference method(Virieux, 1984) used for simulations is definedwithsecond-orderaccurateoperatorsin time and eight-order accurateoperators in space. We apply Perfectly Matched Layers (PML) absorbing boundary conditions(CollinoandTsogka,2001)onthetwolateral sidesandthebottomofthesub-domaingrid.Weapplythe rigid-surface condition (also called zero-displacement conditionorhomogeneousDirichletcondition);see,e.g., Virieux,1984,atthetopofthedomain.Wavesarethus reflected on the top of the domain, simulating the reflections on thesurface:thefree surface is simplified

toahorizontallineabovetheregionofinterest.Weplace onereceiverbygridpointatthetopofthedomain.The acoustic sourceis located on thefault, rightunder the locationoftheprincipalmodelmodifications(Fig.11).By doingso,weexpecttocapturethehighestimpactofmodel modificationsontherecordedseismograms.Simulations are run for t=0.5 s after the shock with a time discretization of 104 _{s. Two simulations are run: one}

using the velocity model min corresponding to the 2D

modelrepairedinSection5.1,consideredasthereference, and the other one using the velocity model mout

correspondingtothesimplifiedmodel.

Results. We compare the difference

du

simulatedinmout:wedefine

du

Cartesian grids is here verypractical for computingthe difference

du

Fig.10.(a)Inputnon-watertightcross-section(notethegapsandtheoverlapsondetailedviews).(b)Sealedoutputcross-section.Gapsandoverlapshave beenremoved,resolvingnon-watertightnessfeatures.Thedottedlinesrepresentsthesub-domainonwhichwerunwavepropagationsimulations.

Fig.11.(a)Velocitymodelcorrespondingtothesimplifiedmodelmout.(b)Differencebetweentheinputvelocitymodelandthesimplifiedvelocitymodel

RMS(rootmeansquare)waveformdifferenceforagiven receiver using the definition by Geller and Takeuchi (1995):

d

The mean waveform difference, for all receivers, is 2.18%withamaximaldifferenceof3.28%forthereceiver positioned at x=481 m (Fig. 12). This difference is reasonably small because the modifications on the geologicalmodelhavemagnitudeslowerthantheseismic resolutionr.

To check the impact of the same simplifications on higher-frequency wavefields, we ran additional simula-tions:wekeptthesameparameters,butdoubledthewave frequency (fmax=600 Hz). Using this frequency, the

minimumwavelengthis4m,whichissampledby12grid points,andtheseismicresolutionr(Eq.(2))isnowaround 1.00 m. Using fmax=600 Hz, the difference on the full

waveformsisnow12.45%,withamaximaldifferenceof 19.44%forthereceiverpositionedatx=438m(Fig.13). Thewavepropagationdifferencesaresignificantlyhigher atthisfrequency,asexpected,sincethemagnitudeofthe simplificationsonthegeologicalmodel(i.e.2m)islarger thantheseismicresolution(i.e.1m).

6. Discussion

Geological model simplifications impact physical simulations,but wehave shownthat theeffectscanbe reduced to an acceptable amount (below 4% in our exampleforfmax=300Hz)byconstrainingthemagnitude

of geological model modifications below the spatial resolutionimposedbythesimulatedphysicalphenomena. Ifthemagnitudeofmodificationsisbiggerthantheseismic resolution,thedifferenceissignificantlybigger(morethan 10% in our examplefor fmax=600 Hz).In theproposed

method,thismagnitudeisdirectlycontrolledbythesizeof theentityexclusionzonessetasinputofthemethod.Asa consequence,itispossibletousepriorknowledgeabout thewavefieldfrequencycontenttochoosethe simplifica-tion parameters that have a negligible impact on the simulationresults.

The presented applicationshows simulations perfor-medusingaFiniteDifferenceschemeonCartesiangrids. This is intentionally a very simple application, using a commonFDschemeandsimplevelocitymodels.Wehave shownthatitispossibletouseasinputofourmethodthe physicalparametersofthesimulationtolimittheimpact ofsimplificationsonsimulationresults.

However, because the two grids have the same dimensions, wecannot measure on this applicationthe benefitsofsimplificationsonsimulationruntime.Togo further, theFD scheme canbeimproved usingmethods accounting more for discontinuitiesof themodels(e.g., Kristeketal.,2016;Moczoetal.,2002).Anotherpossibility istoexplicitlyadaptthespatialdiscretizationtothemodel discontinuitiesusing2Dunstructuredmeshes.Bydoingso, P-SV wave propagation simulations can be run on 2D unstructured meshes using Discontinuous Galerkin scheme (e.g.,Ka¨serand Dumbser,2006; Peyrusse etal., 2014),ortriangularSpectralElements(e.g.,Merceratetal., 2006).Thiswouldallowcomparingboththesimplification effects onsimulation accuracy,butalsoon computation time directly on the generated unstructured meshes, withoutanyrasterizationstep.

7. Conclusion

Theproposedframeworktoedit2Dgeologicalboundary modelseasesmodeleditingbyautomaticallydetectingthe invalid configurationsand correcting them.Thismethod canbeusedtocorrecttheissuesbreakingtheunphysical watertightnessof2Dgeologicalmodels.Itis alsoableto simplify model interfacegeometries and connectivity to improvethequalityofthegeneratedmeshes.

We have shown that, for the problem of wave propagation, our simplification method has a limited impactonthesimulationresults,providedthattheamount ofsimplificationissmallwithregardtothesensitivityofthe

Fig.12.(a)RMSwaveformdifferenceasafunctionoftheabscissaxfora maximumwavefrequencyequalto300Hz.(b)Seismogramsrecordedby thereceiverx=481m(maximalRMSwaveformdifference).Thetrace fromthesimplifiedmodel(red)isalmostidenticaltothetracefromthe referencemodel(black).

Fig.13.(a)RMSwaveformdifferenceasafunctionoftheabscissaxfora maximumwavefrequencyequalto600Hz.(b)Seismogramsrecordedby thereceiverx=438m(maximalRMSwaveformdifference).Thetrace fromthesimplifiedmodel(red)hasmoredifferencefromthetracefrom thereferencemodel(black)thanfora300Hzsourcewavelet.

physical simulations. This condition can be checked in seismic applications for which minimal velocity and maximumfrequency areknown.However, thechoice of the simplificationsizecannotbeeasily expressedforall physicalprocessessuchasfluidflowandtransport.Inthat case, our method makes it possible to experiment numericallydifferentmagnitudesofsimplificationandto measuretheirimpactontheconsideredphysicalprocess. Moreover,ourmethodiscanbecustomizedtoinclude priorgeologicalknowledgebyadaptingthesimplification strategiestopreservethegeologicalmodelfeaturesdeemed essential.Thisopensinterestingperspectivestoadaptthe simplificationstrategiesbothintermsofpriorknowledge andintermsofsensitivityofthephysicalproblems.

However, thepresent version of themethod cannot detectunsolvableconfigurationsgiveninputparameters onminimumanglesandminimalfeaturesizes(e.g.,whena stack of horizon lines are much closer than the input minimumfeaturesize).Insomecases,invalid configura-tions may still be present after the application of the method.Insomepathologicalcases,thewholeprocedure should, therefore, be executed several times, and the exclusion zones may need to be reduced to obtain an outputmodelthatconformstothegivenvaliditycriteria. Our ultimate objective is to extend the proposed method to repair and simplify 3D geological models. Extensionto3Dofanygeometricaloperationsisalways extremely difficult. Model editing is highly non-trivial because many 3D geometrical operations are difficult, whilethecorresponding2Doperationsaresimpleandwell understood,e.g.,meshgeneration.However,wedobelieve thatthedissociationbetweentopologicaloperationsand geometrical operations that we propose is key for a successful3Dsimplificationmethod.Themainchallenge will be to fix the 3D geometrical embedding once the topologyisfixed.

Acknowledgments

This work was performed in the frame of the RING projectatthe Universite´ deLorraine.Wethankfortheir supporttheindustrialandacademicsponsorsoftheRING– GOCAD Consortium managed by ASGA (http://www. ring-team.org/consortium).J.Pellerinissupportedbythe European Research Council, grant agreement ERC-2015-AdG-694020. Software corresponding to this paper is availabletosponsorsintheRINGsoftwarepackagesSCAR (model editing) and SIGMA (wave propagation simula-tions).WeacknowledgeE.Chaljub,M.Karimi-Fard,andN. Glinskyfor theirconstructive remarksthathelped usto improvethequalityofthispaper.Theauthorswouldalso liketothankArnaudBotellafor fruitfuldiscussions.The authorsacknowledgeINRIAfortheGeogramlibraryusedin RINGMeshandParadigmfortheSKUA–GOCADsoftware. AppendixA. Supplementarydata

Supplementarydataassociatedwiththisarticlecanbe found,intheonlineversion,athttps://doi.org/10.1016/j.crte. 2018.12.001.

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