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英文原文ACatastropheModelofCoalandGasDelayOutburstLIUBaoxian,WANGZeyun,LIULi(DepartmentofCivilConstruction,XihuaUniversity,Chengdu610039SichuanChina)Abstract:Catastrophetheorycaneffectivelysolvecatastropheproblemthatcanpossibleappearincontinuumsystem.Themechanismofcoalandgasdelayoutburstisstudiedwithcatastrophetheory.Itisconcludedthatthedelaytimeofcoalandgasdelayoutburstisproducedbythecombinationofgeostressesgasinsidecoal,physicsmechanicscharactersofcoalandexternalaction(explosion).Accordingtotheaboveresearch,thewayofpreventingcoalandgasdelayoutburstispresentedKeywords:coalandgasdelayoutburst;catastrophe;prevention0IntroductionCoalandgasdelayoutburstreferstotheoutburstnotoccurredduringtheexplosiveextractionattheexplosionarea,butoccurredwithinacertainperiodaftertheexplosion.DelayoutburstoccurredfrequentlyinChinaandcausedsubstantiallossesinlifeandproperties,suchasDonglinCoalMineandYutianbaoCoalMineofNantongMiningBureau,ShuijiangCoalMineofFulingarea,SanbaoCoalMineofBeipiaoMiningBureau,TanjiaCoalMineandLiwang-miaoCoalMineofBaishaMiningBureau,HonglingCoalMineofShenyangMiningBureau,LiuzhiMiningBureauandsoon.TheissueofcoalandgasdelayoutburstwasputforwardbythescholaroftheformerSovietUnioninthe1950s,anditalsowasstudiedinChina.Thereareseveralmainideasaboutthemechanismofcoalandgasdelayoutburst.(1)Thecauseofdelayoutburstisthatthereisapreparationstagefortheoutburst-releasingofpotentialenergyandcrushingofthecoalmass.(2)Thedecisivefactorofdelayoutburstmightbethathardcoalwithrelaivehighcarryingcapacityexistinnewcoal-facewhichcontinuetoaccumulatetheelasticenergy.(3)Delayoutburstresultsfromtheplasticdeformationundertheimpactofexternalforce.(4)Theoccurrenceofdelayoutburstisresultedfromthecombinedfunctionsofthestoragesituationandvariousactions(internalandexternalfactors)duringthecourseofextraction.(5)Delayoutburstisassociatedwithcoalmasspressurereliefregion.Delayoutburstoccurs,asittakessometimeforoutburstwiththerelativelylongpressurereliefregion.(6)Delayoutburstoccursbecausethereissomethickofsoftcoalseam.(7)Delayoutburstoccurswhenthecreepinggas-containingcoalproressedfromthecreepingstageII(stabilizationstage)totheaccelerateddestructionstageIII(destabilizationcreepingstage)undertheimpactofstressraiseaftercoalmassenteredintothecreepingstage.Inthebeginningof1970s,CatastropheTheorywasfoundedbyThom,aFrenchmathematician.CatastropheTheoryisanewbranchofstudyingincontinuousphenomenon.Itcaneffectivelysolvecatastropheproblemthatcanpossiblyappearincontinuumsystem.Themainmethodisthatallkindsofphenomenonaresummeduptodifferentkindoftopologicalstructureanddiscussincontinuousstatecharacterneartoallkindsofcriticalpoints.CatastropheTheoryisespeciallyapplicabletostudythesystemwhichinternalactionsarenotknown.Inrecentyears,manyscholarsathomeandbroadusedittostudyearthquake,rock-burst,rockfailureandinstabilityandsoon,inordertoexplainandsolveincontinuousphenomenonofgeoscience.Becauseofcomplexityofcoalandgasdelayoutburst,thepaperwilltrytostudyitusingCatastropheTheory.1CatastropheModelofCoalandGasDelayOutburstSincecoalandgasdelayoutburstisakindofout.burst,itshouldalsobefailureandinstabilityofcoalseamfallingandgasoverflowingcausedbykineticenergywhichgasinternalenergyanddeformationstrainenergyofcoalmasstranslateinto.Thekindsofdelayoutburstreleasingenergyhastwolikeinstantaneousoutburst,oneiselasticstainenergycausedbythegeostressesactingoncoalskeleton,theotherisgasinternalenergywhichiscontainedbycoalmassandpore.Soasimplifiedmechanicsmodel(Fig.1)isadoptedtostudydelayoutburstinthepaper.Asectionistakeinthedirectionofworkingface(fig.1(a)),andthesectioninthedirectionofLissimplifiedtoafreelysupportedbeam(fig.1(b))whichlengthisL,verticalwidthish,horizontalwidthisb(Lh,Lb),elasticitycoefficientisE,p(t)isgeostresses,uniformstressq(t)isthestresssyntheticallyreflectinggasinternalenergyandsoon.Becausesuchasgeostressesandgasinternalenergyvarywithtimewhenworkingfaceisexcavatedsuddenly,p(t)andq(t)arethefunctionsoftime.Theaxisflexibilitycurveofthebeamcanbeexpressedasequation(1)usingFourierseries.Fig.1ThesimplifiedmechanicsmodelofcoalmassWiththesecondseriesbeingapproximated,canbedenotedasequation(2).(2)whereisthelengthofcurve,isflexibility,Yisflexibilityoftheaxismidpoint.Becausethetotalpotentialenergyofanystructuresystemiscomposedbystrainenergyandtheworkthatloadsdowithcorrespondingdisplacement,thepotentialfunctionofthebeaminthemechanicsmodeliswrittenasequation(3).(3)whereUisthestainenergyofthebeam,W1istheworkofthegeostressesp(t)doing,W2istheworkoftheuniformstressq(t)doing.Supposethedeformationobeyingconstantsectionhypothesis,thestrainεdistributingobeyequation(4).(4)whereRissemidiameterofcurvatureoftheneutercurve,yisthedistancebetweenthepointofsectionandthemeutercurve.Thenthestressofthesectioncanbewrittenasequation(5).(5)ThemomentMofthesectioncanbeexpressedasequation(6).(6)whereistheinertialmomentofthebeam.(7)hence(8)Thenthestainenergyofthebeamcanbewrittenasequation(9).(9)Usingtheequation(6),theeqation(9)canberewrittenasequation(10).(10)whereKiscurvatureofthebeam.(11)usingtheequation(11),theequation(10)canbeexpressedasequation(12).(12)Thecompressvalueofthebeamcanbewrittenasequation(13)undergeostressesacting.(13)Sotheworkofp(t)canbeexpressedasequation(14).(14)Theworkofuniformstressq(t)doingcanbeexpressedasequation(15).(15)Usingtheequation(12),equation(14)and(15),thepotentialfunctionofthebeaminequation(3)becomesequation(16).(16)Thepotentialfunctioncanbeexpressedasequation(17)throughtheintegralfunctionoftheequation(16)isoutspreadusingTaylorseries.(17)Usingequation(17)canbeexpressedasequation(18).(18)SotheequationofbalancedsurfaceVcanbeexpressedasequation(19).(19)Theequationofbifurcationaggregatecanbeexpressedasequation(20).(20)or(21)Basedonthereferencepoint(p(t),q(t))thatisdeterminedbygeostressesp(t)anduniformstressq(t),thebalancestateofCoalmassisdiscussed:(1)PointAisthecriticalpointofsystemstabilizationandbalance(Fig.2),andobeysequation=0,thatistosaythatitobeysequationq(t)=0,p(t)=.Becauseitispossibletospanthebifurcationaggregateonlywhenequationa≤0isobtained,thenecessaryconditionofsystemsuddenjump(coalandgasoutburst)is(23)Theequation(22)alsocanbewrittenas(24)Hence,instabilityofcoalmasscontaininggas(coalandgasoutburst)isrelatedtocoalmassstructureandphysicsmechanicscharactersofcoalmasscontaininggas.Fig.2Catastrophemodelofcoalandgassystem(2)Whenp(t)andq(t)obeyequation<0,thereferencepoint(p(t),q(t))isinsidethebifurcationaggregate;whenp(t)andq(t)obey>0,thereferencepoint(p(t),q(t))isoutsidethebifurcationaggregate.Solongasthereferencepoint(p(t),q(t))doesnotspanthebifurcationaggregate,thesuddenjumpofthesystemdoesn&thappen,thatistosaythatcoalandgasoutburstdoesn&thappen.Afterworkingfacebeingexcavatedsuddenly,if(p(t),q(t))spansbifurcationaggregatesuddenlyundertheactingofstress-raiseinworkingface,thesuddenjumpbreakofcoalandgassystemhappens,thatistosaythatcoalandgasinstantaneousoutbursthappens;Iftheactingofstress-raiseinworkingfacecannotmake(p(t),q(t))spanthebifurcationaggregatesuddenly,butthechangeofp(t)andq(t)canmake(p(t),q(t))spanbifur-cationaggregatewith.inacertainperiodsafterworkingfacebeingexcavatedsuddenly,thesuddenjumpofcoalandgassystemhappens,thatistosaythatcoalandgasdelayout.bursthappens,andthetimebetweenworkingfacebeingexcavatedsuddenlyandthechangeofp(t)andq(t)making(p(t),q(t))spanbifurcationaggregateisthedelaytimeofdelayoutburst.Ifthechangeofp(t)andq(t)doesnotmake(p(t),q(t))spanbifurcationaggregateallthetime,coalandgasoutburstordelayoutburstdoesnothappen.(3)Theuniformstressq(t)isrelatedtogeostressesp(t),gasinternalenergy,coalphysicsmechanicscharacters(suchascharactersofcoaladsorptionandseepage),andalsorelatedtoexternalaction(suchasexplosion),Sowethinkthatthedelaytimeofcoalandgasdelayoutburstisproducedbytheombinationofgeostresses,gasinsidecoal,physicsmechanicscharactersofcoalandexternalaction(explosion).(4)Althoughtheabovecatastrophemodelofcoalandgasdelayoutburstshouldbestillfartherstudied,theconclusionisconsistentwiththatofdocument[6]thatthefristnecessaryconditionofcoalandgasoutburstishighgeostressesandthesecondnecessaryconditionofoutburstissuddenchangeofstressstate.Thisprovesthemodeliscorrect.2PredictandPreventionofCoalandGasDelayOutburstPredictandpreventionofcoalandgasdelayout..burstarestudiedverylittleathomeandbroad.Traditionalmethodsofpredictandpreventionarestilladoptedandhavelittleeffect.Accordingtotheaboveresearch,itisconcludedthatthedelaytimeofcoalandgasdelayoutburstisproducedbythecombinationofgeostresses,gasinsidecoal,physics-mechanicscharactersofcoalandexternalaction(explosion).Equation(23)isthenecessaryconditionofdelayout..burstandequation(21)isplenarycondition.Itisnecessarytoobserveandcontrolthechangeofgeostresses,gasandphysics-mechanicscharactersintimetopreventthereferencepointspanningbifurcationaggregateinordertoensurecoalsafelyexploitation[7].3ConclusionAcatastrophemodelofcoalandgasdelayoutburstisobtainedinthepaperusingCatastrophetheor.Themodelcanexplainthemechanismofthedelaytime,andofferanewwaytofartherstudycoalandgasdelayoutburst.Itisconcludedthatthedelaytimeisproducedbythecombinationofgeostresses,gasinsidecoal,physics-mechanicscharactersofcoalandexternalaction(explosion),andifthechangeofp(t)andq(t)makethereferencepoint(p(t),q(t))spanbifurcationaggregate,coalandgasoutburstordelayoutburstwillhappen.Itispossibletopreventcoalandgasdelayoutburstinordertoensurecoalsafetyexploitationthroughobservingandcontrollingthechangeofgeostresses,gasandphysicsmechanicscharactersintimetopreventthereferencepointspanningbifurcationaggregate.References[1]SaundersDT,AnIntroductiontoCatastrophe[M],NewYork,Springer,1980.[2]YinGuangzhi,Researchonnonlinearthoryinrockmechanicsandpredictionofrockburst,[Ph.D.Dissertation][M].Chongqing:ChongqingUniversity,1999.[3]XuZenghe,XuXiaohe.Cuspcatastropheofoccurrenceconditionsandhysteresisofrockburstsinpillarworkings[J],TheChineseJouralofNonferrousMetals,1997,7(2),17~23.[4]Henley,S.CatastropheTheoryModelsinGeology[J],MathematicalGeology,1976,8(6),649~655.[5]ZhangWohua,JinYi,ChenYunmin.Mechanismofenergyreleaseduringcoal/gasoutbursts[J],ChineseJournalofRockMechanicsandEngineering，2000,19(SUPPL)，829~835.[6]YuBufanandBaiFanandLiuMing.Thepredictandpreventionofcoalminegas[J].Beijing:EconomyPublishingCompanyofChina,1987.[7]LiuBaoxian,Researchonphysics-mechanicscharacteristicsofthedelayoutburstcoalanditsdelayoutburstmchanism[D].Chongqing:ChongqingUniversity,2000.

中文譯文煤與瓦斯延期突出的突變理論研究劉保縣,王澤云,劉立(西華大學(xué)建筑與土木工程系,四川成都610039)摘要:突變理論能比較有效地解決光滑系統(tǒng)中可能出現(xiàn)的突變問題。本文利用突變理論對(duì)煤與瓦斯延期突出進(jìn)行了研究,認(rèn)為延期突出的滯后時(shí)間是由地應(yīng)力、煤體中的瓦斯、煤的物理力學(xué)特性及外作用(爆破等)等的共同作用所產(chǎn)生的,并依此提出了延期突出的預(yù)測(cè)預(yù)報(bào)方法。關(guān)鍵詞:煤與瓦斯延期突出；突變；預(yù)測(cè)0引言煤與瓦斯延期突出是指，突出并不是在爆炸時(shí)的爆炸區(qū)域發(fā)生的，而是在爆炸發(fā)生一段時(shí)間以后才發(fā)生。煤與瓦斯延期突出是中國(guó)煤礦一種常見的自然災(zāi)害，引起人員和財(cái)產(chǎn)的巨大損失。在我國(guó)的許多煤礦，比如南桐礦務(wù)局的東林煤礦、魚田堡煤礦、涪陵地區(qū)的水江煤礦、北票礦務(wù)局的三寶煤礦、白沙礦務(wù)局的坦家煤礦、里王廟煤礦、沈陽(yáng)礦務(wù)局的紅菱礦和六枝礦務(wù)局等，都曾多次發(fā)生過延期突出。早在20世紀(jì)的50年代，原蘇聯(lián)的學(xué)者就提出了延期突出的問題，國(guó)內(nèi)同時(shí)也對(duì)此相繼進(jìn)行了研究，關(guān)于煤與瓦斯延期突出有幾個(gè)主要的研究觀點(diǎn)，（1）煤與瓦斯延期突出發(fā)生的原因是突出有一個(gè)準(zhǔn)備階段—潛能釋放與煤體被破碎；（2）突出的決定性因素可能是承載能力高的硬煤在新的煤層不斷積累彈性能量；（3）延期突出是煤體受外力作用后的結(jié)果；（4）延期突出的發(fā)生是煤層賦存狀況與采掘過程中的各種作業(yè)（內(nèi)，外因）綜合共同作用的結(jié)果（5）延期突出與煤體卸壓區(qū)有關(guān)，有些觀點(diǎn)認(rèn)為，突出由于卸壓區(qū)相對(duì)較長(zhǎng)，沖破這層阻礙要一定的時(shí)間，所以發(fā)生延期突出的現(xiàn)象；（6）延期突出的發(fā)生是由于工作面前方有一定厚度的軟媒體存在，（7）延期突出是煤體進(jìn)入蠕變階段以后，在應(yīng)力集中的作用下，蠕變的含有瓦斯的煤體從蠕變第Ⅱ階段（穩(wěn)態(tài)階段）進(jìn)入加速破壞的第Ⅲ階段（失穩(wěn)蠕變階段）時(shí)發(fā)生的。在20世紀(jì)70年代，突變理論是由法國(guó)數(shù)學(xué)家托姆創(chuàng)立的。突變理論成為用來研究不連續(xù)現(xiàn)象的一個(gè)新興分支。它可以有效的解決突變中存在的難題，并有可能在一個(gè)連續(xù)的系統(tǒng)中反映出來。其主要研究方法是將各種現(xiàn)象歸納到不同類別的拓?fù)浣Y(jié)構(gòu)中，并且討論各類臨界點(diǎn)附近的非連續(xù)性態(tài)特征，從而能夠有效地解決光滑系統(tǒng)中可能出現(xiàn)的突變問題，突變理論特別適用于內(nèi)部作用尚屬未知系統(tǒng)的研究。近些年來，國(guó)內(nèi)外的許多學(xué)者把該理論應(yīng)用帶地震、沖擊地壓、巖石破裂失穩(wěn)等方面，試圖解釋和解決地球科學(xué)中的不連續(xù)現(xiàn)象和問題。由于煤與瓦斯延期突出的復(fù)雜性，本文擬利用突變理論對(duì)煤與瓦斯延期突出機(jī)理進(jìn)行研究。1煤與瓦斯突出的突變理論模型煤與瓦斯延期突出既然是一種突出，也應(yīng)該是一種由于儲(chǔ)存于煤層中的瓦斯內(nèi)能、變形應(yīng)變能轉(zhuǎn)化為動(dòng)能而發(fā)生的煤層崩塌、瓦斯溢出的失穩(wěn)破壞。延期突出的過程與瞬時(shí)突出的過程一樣，釋放的能量只有兩種：一種是煤體骨架受地應(yīng)力作用而產(chǎn)生的彈性應(yīng)變能；另一種是貯存在煤體及孔隙內(nèi)的瓦斯內(nèi)能。所以，本文采用圖1所示的簡(jiǎn)化力學(xué)模型進(jìn)行研究，在圖1所示的工作面上沿L方向取一截面，并且將L方向的截面簡(jiǎn)化為一簡(jiǎn)支平直梁（圖1（b）），其中梁長(zhǎng)為L(zhǎng)，垂直寬度為h，水平寬度為b（L》h，L》b），彈性模量為E，p（t）為地應(yīng)力，而均布力q（t）是瓦斯內(nèi)能等的綜合反映。因?yàn)楫?dāng)工作面瞬間進(jìn)尺后，地應(yīng)力及瓦斯內(nèi)能等都隨時(shí)間發(fā)生變化，因此p（t）和q（t）應(yīng)是時(shí)間的函數(shù)。梁結(jié)構(gòu)的軸線撓度曲線可用傅里葉級(jí)數(shù)表示：圖1煤體的簡(jiǎn)化力學(xué)模型在二級(jí)近似的條件下，可用傅里葉級(jí)數(shù)表示成方程（2）：(2)式中：為撓度，Y為軸線中點(diǎn)的撓度。由于任何結(jié)構(gòu)體系的總勢(shì)能應(yīng)由應(yīng)變能和載荷在相應(yīng)位移上所做的功組成，所以，本力學(xué)模型中梁結(jié)構(gòu)的勢(shì)函數(shù)可寫為方程（3）：(3)式中：U為梁的應(yīng)變能，W1為地應(yīng)力p(t)所做的功，W2為均布力q(t)所做的功。假設(shè)變形服從平截面假設(shè)，則應(yīng)變?chǔ)欧植挤墓剑?）：(4)式中：R為中性線的曲率半徑，y為截面上的點(diǎn)距中性線的距離。所以，截面上的應(yīng)力可表示為方程（5）：(5)截面上的力矩M可表示為方程（6）：(6)這里令為梁的慣性力矩：(7)因此可得，(8)所以梁的應(yīng)變能可表示為方程（9）：(9)將方程（6）帶入，則方程（9）可表示為方程（10）(10)式中：K為梁的曲率，表達(dá)式為：(11)將方程（11）帶入，方程（10）可表示為方程（12）：(12)在地應(yīng)力的作用下，梁的壓縮量可表示為方程（13）：(13)所以，p(t)所做的功可表示為方程（14）：(14)水平均布力q（t）所做的功為：(15)把方程（12）、（14）、（15）帶入方程（3），得梁的勢(shì)函數(shù)，表示為方程（16）：(16)對(duì)式（16）被積函數(shù)按泰勒級(jí)數(shù)展開，經(jīng)整理可近似得勢(shì)函數(shù)（17）的表達(dá)式為：(17)令則方程（17）可表示為（18）：(18)所以平衡曲面M的方程為表示為（19）：(19)分歧點(diǎn)集所滿足的方程可表示為（20）：(20)即(21)現(xiàn)根據(jù)地應(yīng)力p(t)和水平應(yīng)力q(t)所確定的控制點(diǎn)(p(t),q(t))，討論煤體受力平衡狀態(tài)。(1)A點(diǎn)是系統(tǒng)穩(wěn)定平衡的臨界點(diǎn)，見圖2，滿足方程=0，也就是滿足方程q(t)=0，p(t)=。因?yàn)橹挥挟?dāng)a≤0時(shí)，才出現(xiàn)跨越分歧集的可能，所以可以得到系統(tǒng)發(fā)