考慮任意壓力分布的裂縫誘導(dǎo)應(yīng)力場(chǎng)計(jì)算模型及其應(yīng)用

彭 瑀,李勇明,趙金洲

(西南石油大學(xué)油氣藏地質(zhì)及開(kāi)發(fā)工程國(guó)家重點(diǎn)實(shí)驗(yàn)室,四川成都610500)

考慮任意壓力分布的裂縫誘導(dǎo)應(yīng)力場(chǎng)計(jì)算模型及其應(yīng)用

彭 瑀,李勇明,趙金洲

(西南石油大學(xué)油氣藏地質(zhì)及開(kāi)發(fā)工程國(guó)家重點(diǎn)實(shí)驗(yàn)室,四川成都610500)

通過(guò)復(fù)分析和斷裂力學(xué)Westergaard應(yīng)力函數(shù)方法建立局部受均勻內(nèi)壓力作用下的誘導(dǎo)應(yīng)力分布模型,并以此為基本單元結(jié)合數(shù)值積分復(fù)化矩形法的思路,疊加得到考慮任意壓力分布情況的誘導(dǎo)應(yīng)力計(jì)算模型。采用先計(jì)算再虛實(shí)分離的方法進(jìn)行編程,驗(yàn)證單元模型的精度。模擬縫內(nèi)壓降、裂縫不對(duì)稱(chēng)延伸和凈壓力周期性波動(dòng)對(duì)裂縫誘導(dǎo)應(yīng)力的影響。結(jié)果表明:經(jīng)典的裂縫誘導(dǎo)應(yīng)力模型僅是局部受均勻內(nèi)壓力模型的特解;考慮縫內(nèi)壓降后,遠(yuǎn)井位置形成縫網(wǎng)的可能性大幅度下降,天然裂縫的穩(wěn)定性顯著提高;裂縫不對(duì)稱(chēng)延伸使裂縫附近的壓應(yīng)力區(qū)向短半縫偏移,但遠(yuǎn)離裂縫區(qū)域的誘導(dǎo)應(yīng)力仍關(guān)于裂縫中垂線(xiàn)對(duì)稱(chēng);縫內(nèi)凈壓力周期性波動(dòng)對(duì)誘導(dǎo)應(yīng)力的影響范圍在五分之一半縫長(zhǎng)以?xún)?nèi),求解遠(yuǎn)離裂縫區(qū)域的應(yīng)力時(shí),可以將平均壓力直接帶入經(jīng)典公式計(jì)算。

壓裂;誘導(dǎo)應(yīng)力;凈壓力;不對(duì)稱(chēng)延伸;數(shù)學(xué)模型

水力裂縫周?chē)恼T導(dǎo)應(yīng)力分布對(duì)于分段壓裂縫間應(yīng)力干擾[1]、后續(xù)裂縫起裂[2-3]和原有天然裂縫(結(jié)構(gòu)弱面)的穩(wěn)定性[4-5]具有重要影響。國(guó)內(nèi)外學(xué)者針對(duì)該問(wèn)題展開(kāi)了大量研究,Atkinson等[6]通過(guò)解析和數(shù)值方法研究了近井彎曲裂縫的應(yīng)力和位移分布規(guī)律;Taghichian等[7]通過(guò)商業(yè)軟件模擬了水力裂縫應(yīng)力陰影區(qū)域的尺寸和特征;Cheng、Bunger和Dahi等[8-10]分別通過(guò)解析、有限元和擴(kuò)展有限元方法研究了在誘導(dǎo)應(yīng)力影響下水力裂縫和天然裂縫的相互作用機(jī)制。但目前多數(shù)文獻(xiàn)中都沒(méi)有考慮縫內(nèi)流體壓降對(duì)誘導(dǎo)應(yīng)力的影響,還缺少一種快速、準(zhǔn)確的可以計(jì)算任意縫內(nèi)壓力分布的誘導(dǎo)應(yīng)力模型。筆者在前人研究的基礎(chǔ)上,通過(guò)復(fù)分析和Westergaard應(yīng)力函數(shù)方法建立局部受均勻壓力作用下的誘導(dǎo)應(yīng)力分布模型,并驗(yàn)證其精度,借鑒數(shù)值積分的思路,疊加得到考慮任意壓力分布情況的誘導(dǎo)應(yīng)力計(jì)算模型,并模擬水力裂縫縫內(nèi)壓降、裂縫不對(duì)稱(chēng)延伸和凈壓力周期波動(dòng)對(duì)應(yīng)力分布的影響。

1 單元模型的建立與求解

根據(jù)Westergaard的理論[11-12],若有一解析函數(shù)Z能夠令形如式(1)的應(yīng)力函數(shù)滿(mǎn)足I型裂紋的等應(yīng)力邊界條件,則根據(jù)解析函數(shù)的性質(zhì)和柯西黎曼條件可以得到其應(yīng)力分布,

式中,Z為某一解析函數(shù),MPa;z為解析函數(shù)Z的自變量,m;x和y為直角坐標(biāo)參數(shù),m。

求解裂縫在任意局部均勻壓力作用下的誘導(dǎo)應(yīng)力分布模型,假設(shè)在無(wú)限大地層中有一條處在平面應(yīng)變狀態(tài)的裂縫,長(zhǎng)2a。以裂縫中心為原點(diǎn),縫長(zhǎng)方向?yàn)閤軸,縫面法向?yàn)閥軸建立直角坐標(biāo)系,如圖1所示。

Tada等[13]通過(guò)對(duì)縫面任意位置受集中應(yīng)力作用解的卷積得到了解析函數(shù)Z:

式中,a、b和c為圖1中對(duì)應(yīng)點(diǎn)橫坐標(biāo),m;pn為某一段所受均勻內(nèi)壓,MPa。

圖1 計(jì)算任意壓力作用下裂縫誘導(dǎo)應(yīng)力的基本單元Fig.1 Basic computation element of induced stress under arbitrary inner pressure

計(jì)算式(2)還需要解析函數(shù)Z的導(dǎo)數(shù),對(duì)式(3)求導(dǎo)可得:

通常在采用Westergaard應(yīng)力函數(shù)方法求解彈性力學(xué)問(wèn)題時(shí)[14],會(huì)要求對(duì)式(3)和式(4)進(jìn)行虛實(shí)分離,得到直觀(guān)的應(yīng)力分布表達(dá)式。但觀(guān)察式(3)和式(4)可以發(fā)現(xiàn),其虛實(shí)分離相當(dāng)困難,即便將反三角函數(shù)代換成對(duì)數(shù)函數(shù)[15],其結(jié)果仍然須按照[-a,0]、[-c,0]、[-b,0]、[0,0]、[b,0]、[c,0]、[a,0]7個(gè)點(diǎn)展開(kāi)復(fù)變函數(shù)[16],導(dǎo)致最終解分為8個(gè)區(qū)域,十分不利于計(jì)算,因此筆者在編程時(shí)采用了先計(jì)算再分離的思路,并不直接對(duì)式(3)和式(4)進(jìn)行處理,而是將a、b、c、z等參數(shù)賦值按照復(fù)變函數(shù)運(yùn)算規(guī)則計(jì)算得到復(fù)數(shù)解后,再對(duì)其進(jìn)行虛實(shí)分離帶入式(2)。

2 單元模型的驗(yàn)證

根據(jù)上述方法編程計(jì)算得到的縫面任意位置(此例取b=0.5a,c=0.6a)受均勻壓力作用下的y方向正應(yīng)力分布如圖2(a)所示,當(dāng)b=-a、c=a時(shí),計(jì)算結(jié)果如圖2(b)所示。

從圖2(a)可以看出,在縫內(nèi)局部區(qū)域受均勻內(nèi)壓作用時(shí),裂縫尖端仍然具有奇異性,應(yīng)力分布明顯不再關(guān)于中垂線(xiàn)對(duì)稱(chēng),圖2(b)模擬的結(jié)果與使用經(jīng)典解析模型計(jì)算的結(jié)果完全一致[17],證明式(2)～(4)構(gòu)成的縫內(nèi)局部區(qū)域受均勻內(nèi)壓作用的誘導(dǎo)應(yīng)力解正確,原有的經(jīng)典解析模型只是該模型的特例(b=-a、c=a)。進(jìn)一步根據(jù)數(shù)值積分復(fù)化矩形法的思路,在求解任意壓力作用下的誘導(dǎo)應(yīng)力分布時(shí),可以將其近似表示為n個(gè)局部均勻壓力導(dǎo)致的誘導(dǎo)應(yīng)力之和,顯然n越大,求解結(jié)果越精確。根據(jù)內(nèi)壓力變化,累次疊加后的解析函數(shù)Z及其導(dǎo)數(shù)可以表達(dá)為

式中,di為按縫長(zhǎng)方向進(jìn)行離散分段的第i個(gè)結(jié)點(diǎn);pi為對(duì)應(yīng)結(jié)點(diǎn)的壓力,MPa;n為總的離散段數(shù)。

3 壓降對(duì)誘導(dǎo)應(yīng)力的影響

Economides等[18]認(rèn)為發(fā)生完全層間滑移的水力裂縫可以在水平截面上應(yīng)用(廣義)平面應(yīng)變假設(shè),因此可以將擬三維模型模擬輸出的壓降曲線(xiàn)作為本模型的輸入?yún)?shù)。以縫口凈壓力為參考?jí)毫?裂縫半長(zhǎng)為參考長(zhǎng)度,可以由擬三維模型得到一組無(wú)因次凈壓力與無(wú)因次距離的關(guān)系[19-20](當(dāng)然這組關(guān)系并不是唯一確定的,這里只是作為后續(xù)模擬的輸入?yún)?shù)使用),如圖3所示。對(duì)曲線(xiàn)進(jìn)行如圖所示的處理,將光滑的凈壓力曲線(xiàn)離散為一系列不連續(xù)的矩形均勻應(yīng)力(紅色方框)。再將圖3中各參數(shù)映射到圖2的實(shí)際尺寸下,并帶入到式(2)～(4)進(jìn)行疊加處理,即可得到圖4。

比較圖2(b)和圖4可知,當(dāng)考慮水力裂縫內(nèi)的流動(dòng)壓降時(shí),誘導(dǎo)應(yīng)力的壓應(yīng)力區(qū)和拉應(yīng)力區(qū)都顯著減小(拉正壓負(fù))。對(duì)于縫間應(yīng)力干擾研究,如果考慮了縫內(nèi)壓降,轉(zhuǎn)向區(qū)域會(huì)明顯減小,在遠(yuǎn)井位置形成裂縫網(wǎng)絡(luò)的可能性會(huì)大幅度降低;而對(duì)于起裂壓力的計(jì)算,考慮縫內(nèi)流體壓降會(huì)使周?chē)渌c(diǎn)的起裂附加應(yīng)力減小,經(jīng)典模型對(duì)于起裂附加應(yīng)力的計(jì)算過(guò)于保守。

圖2 不同縫內(nèi)均勻壓力作用下的誘導(dǎo)應(yīng)力分布Fig.2 Induced stress distribution under different local uniform inner pressure

圖3 無(wú)因次凈壓力和無(wú)因次距離的關(guān)系Fig.3 Relationship between dimensionless net pressure and dimensionless distance

圖4 流體壓降對(duì)誘導(dǎo)應(yīng)力分布的影響Fig.4 Influence of pressure drop on induced stress distribution

從圖5差值圖版可以看出,與無(wú)壓降模型相比,考慮縫內(nèi)壓降對(duì)誘導(dǎo)應(yīng)力分布的影響主要集中在裂縫尖端。因此,實(shí)際水力壓裂施工中,天然裂縫的穩(wěn)定性大大高于文獻(xiàn)[4]的估計(jì);在分析天然裂縫的穩(wěn)定性時(shí),應(yīng)該考慮采用本文中推薦的誘導(dǎo)應(yīng)力計(jì)算公式。

圖5 不考慮縫內(nèi)壓降與考慮縫內(nèi)壓降的差值分布Fig.5 Stress distribution difference between no pressure drop and taking pressure drop into consideration

4 非對(duì)稱(chēng)延伸對(duì)誘導(dǎo)應(yīng)力的影響

常規(guī)壓裂施工中經(jīng)常會(huì)出現(xiàn)裂縫兩翼長(zhǎng)短不一的情況。在縫高擴(kuò)展分析時(shí),一般也認(rèn)為上縫高會(huì)略大于下縫高。因此,不論將平面應(yīng)變假設(shè)應(yīng)用在縫長(zhǎng)方向還是縫高方向都會(huì)面臨非對(duì)稱(chēng)延伸的問(wèn)題。經(jīng)典模型只能考慮縫內(nèi)壓力均勻分布的邊界條件,不能模擬裂縫非對(duì)稱(chēng)延伸的影響。建立的考慮任意壓力分布的模型則可以通過(guò)對(duì)邊界條件的調(diào)整模擬非對(duì)稱(chēng)延伸對(duì)裂縫誘導(dǎo)應(yīng)力分布的影響。

圖6給出了上下半縫高(左右半縫長(zhǎng))分別為14和6 m的誘導(dǎo)應(yīng)力分布云圖。其中x軸上的-4 m處為注液的入口,因此為最高壓應(yīng)力區(qū)。從裂縫誘導(dǎo)應(yīng)力的整體分布看,壓應(yīng)力區(qū)在靠近裂縫的位置不對(duì)稱(chēng)分布,短半縫周?chē)纬煽p網(wǎng)的可能性更大;但在遠(yuǎn)離裂縫的位置(距離裂縫二分之一半縫長(zhǎng)以外),壓應(yīng)力區(qū)逐漸轉(zhuǎn)變?yōu)閷?duì)稱(chēng)分布,由于平均壓力的下降,遠(yuǎn)離裂縫區(qū)域的應(yīng)力絕對(duì)值遠(yuǎn)小于圖2(b)中的對(duì)應(yīng)位置。長(zhǎng)半縫的拉應(yīng)力區(qū)明顯大于短半縫,更容易導(dǎo)致周?chē)奶烊涣芽p剪切滑移,這一區(qū)域會(huì)給微地震的檢波器傳遞更多的破裂信號(hào)[21]。

圖6 裂縫非對(duì)稱(chēng)延伸對(duì)誘導(dǎo)應(yīng)力的影響Fig.6 Influence of asymmetric propagation on fracture induced stress

5 周期內(nèi)壓力對(duì)誘導(dǎo)應(yīng)力的影響

裂縫延伸過(guò)程中由于地應(yīng)力和滲透率隨地層沉積韻律的變化,經(jīng)常會(huì)導(dǎo)致縫內(nèi)凈壓力呈現(xiàn)不規(guī)則的分布。為了研究縫內(nèi)凈壓力波動(dòng)對(duì)誘導(dǎo)應(yīng)力分布的影響,采用本文中模型模擬縫內(nèi)凈壓力呈周期性變化時(shí)裂縫周?chē)鷳?yīng)力分布。

圖7為縫內(nèi)壓力存在周期性變化但平均壓力與圖2(b)相等時(shí)的裂縫誘導(dǎo)應(yīng)力模擬結(jié)果。從圖7可以看出,周期性的壓力波動(dòng)僅對(duì)距離裂縫壁面五分之一半縫長(zhǎng)以?xún)?nèi)區(qū)域的應(yīng)力分布具有一定影響,遠(yuǎn)離裂縫區(qū)域的應(yīng)力與圖2(b)基本一致。說(shuō)明當(dāng)平均壓力相等時(shí),內(nèi)壓力波動(dòng)對(duì)應(yīng)力分布影響不大,在常規(guī)計(jì)算中,計(jì)算遠(yuǎn)離裂縫區(qū)域的應(yīng)力時(shí)可以將平均壓力直接帶入經(jīng)典公式計(jì)算。

圖7 凈壓力波動(dòng)對(duì)誘導(dǎo)應(yīng)力分布的影響Fig.7 Influence of periodical inner pressure on fracture induced stress

6 結(jié) 論

(1)使用復(fù)變函數(shù)方法建立了局部受均勻壓力作用下的裂縫誘導(dǎo)應(yīng)力模型,并以該模型作為基本單元,借鑒數(shù)值積分思路,疊加得到了可以計(jì)算任意內(nèi)壓力作用下裂縫誘導(dǎo)應(yīng)力分布的計(jì)算模型。

(2)驗(yàn)證了模型的精度,認(rèn)為經(jīng)典的裂縫誘導(dǎo)應(yīng)力計(jì)算模型僅是局部受均勻內(nèi)壓力模型的特解。

(3)與恒定內(nèi)壓相比,考慮縫內(nèi)壓降對(duì)誘導(dǎo)應(yīng)力分布的影響主要體現(xiàn)在裂縫尖端,該影響會(huì)導(dǎo)致遠(yuǎn)井位置形成縫網(wǎng)的可能性大幅度下降、天然裂縫的穩(wěn)定性顯著提高。

(4)不對(duì)稱(chēng)延伸會(huì)對(duì)誘導(dǎo)應(yīng)力分布產(chǎn)生顯著影響,裂縫附近(二分之一半縫長(zhǎng)以?xún)?nèi))的高壓應(yīng)力區(qū)會(huì)向短半縫偏移,但遠(yuǎn)離裂縫的應(yīng)力分布依然關(guān)于裂縫中垂線(xiàn)對(duì)稱(chēng)。

(5)地應(yīng)力變化引起的凈壓力周期性波動(dòng)不會(huì)對(duì)誘導(dǎo)應(yīng)力產(chǎn)生較大影響。

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(編輯 李志芬)

Modeling of fracture induced stress under arbitrary pressure distribution and its application in hydraulic fracturing

PENG Yu,LI Yongming,ZHAO Jinzhou
(State Key Laboratory of Oil&Gas Reservoir Geology and Exploitation,Southwest Petroleum University,Chengdu 610500,China)

A new fracture induced stress model was established that can calculate the stress distribution around a section of the fracture under local pressure,in which complex analysis and the Westergaard stress function in fracture mechanics were applied.Superposition of the above model to the whole hydraulic fracture,a novel model that can compute the induced stress around the fracture under arbitrary pressures was then developed.The simulation results show that the new model has high calculation accuracy,and the classical model can be as a special case of the new model.In the simulation of the influence of pressure drop,asymmetric propagation and net pressure undulate on the induced stress,the results show that,when the pressure drop is taken into consideration,the possibility for the formation of a fracture net is decreasing rapidly from wellbore to fracture tip,but the stability of natural fractures is increasing significantly.Asymmetric propagation can lead to a compressive stress section skewing to the short half fracture,while the stress distribution far from the fracture is still symmetrical to the mid-perpendicular.The affection of the net pressure undulate is within one fifth of the half fracture length.So in this case,the average pressure can be taken into the classical model to calculate the stress distribution far from the fracture face.

hydraulic fracturing;induced stress;net pressure;asymmetric propagation;mathematical model

TE 357.1

:A

彭瑀,李勇明,趙金洲.考慮任意壓力分布的裂縫誘導(dǎo)應(yīng)力場(chǎng)計(jì)算模型及其應(yīng)用[J].中國(guó)石油大學(xué)學(xué)報(bào)(自然科學(xué)版),2017,41(3):92-97.

PENG Yu,LI Yongming,ZHAO Jinzhou.Modeling of fracture induced stress under arbitrary pressure distribution and its application in hydraulic fracturing[J].Journal of China University of Petroleum(Edition of Natural Science),2017,41(3):92-97.

1673-5005(2017)03-0092-06doi:10.3969/j.issn.1673-5005.2017.03.011

2016-06-28

國(guó)家自然科學(xué)基金重大項(xiàng)目(51490653);國(guó)家“973”重點(diǎn)基礎(chǔ)研究發(fā)展計(jì)劃(2013CB228004)

彭瑀(1988-),男,博士研究生,研究方向?yàn)橛蜌獠貕毫阉峄碚撆c應(yīng)用。E-mail:pengyu_frac@foxmail.com。

趙金洲(1962-),男,教授,博士,博士生導(dǎo)師,研究方向?yàn)橛蜌獠貕毫阉峄碚撆c應(yīng)用。E-mail:zhaojz@swpu.edu.cn。