李海龍,夏玉強(qiáng)

(1.中國地質(zhì)大學(xué)(武漢)環(huán)境學(xué)院,湖北 武漢 430074;2.中國地質(zhì)大學(xué)(北京)水資源與環(huán)境科學(xué)學(xué)院,北京 100083;3.美國天普大學(xué)土木與環(huán)境工程系,費(fèi)城 19122)

重力驅(qū)動(dòng)指流(gravity-driven fingers),又稱優(yōu)勢流,簡稱指流或指狀分支流,是地表水在重力作用下滲入非飽和土壤中時(shí)經(jīng)常發(fā)生的一種現(xiàn)象,其特征是滲入土壤中的水流在土壤中形成固定的指狀優(yōu)勢通道。Gerke[1]指出“指流包括水流和溶質(zhì)不經(jīng)其他土壤部分而沿某些特定通道運(yùn)移的所有現(xiàn)象”。土壤中指流的存在大大加速了地表水在土壤中的滲流速度,提高了各種養(yǎng)分及污染物在包氣帶中的運(yùn)移傳輸速度,加劇了工農(nóng)業(yè)生產(chǎn)排放的污染物對土壤的污染。因此,指流的研究已成為與農(nóng)業(yè)、林業(yè)、生態(tài)環(huán)境保護(hù)等行業(yè)密切相關(guān)的基礎(chǔ)研究課題之一。

1 指流的形成原因

指流的形成原因復(fù)雜多樣,一種眾所周知的簡單情形就是多孔介質(zhì)中孔隙的隨機(jī)分布導(dǎo)致一些較大的孔隙連在一起而形成“優(yōu)勢通道”。當(dāng)水流在重力作用下滲入多孔介質(zhì)時(shí),會傾向于沿著這樣的優(yōu)勢通道形成指流。指流研究的復(fù)雜性在于其產(chǎn)生原因的多樣性[2]。例如,Coppola等[3]在單個(gè)土體尺度上,根據(jù)指流產(chǎn)生機(jī)制的不同劃分了4種指流形式:大孔隙(非毛細(xì)管孔)流,結(jié)構(gòu)性孔隙流,濕鋒面不穩(wěn)定性引起的指流,以及由土壤濕潤度(或斥水性)空間分布的不規(guī)則性和時(shí)間動(dòng)態(tài)性引起的指流?？傮w來說,導(dǎo)致指流的因素除土壤孔隙度和滲透性空間分布不均(非均質(zhì)性)外,還有非飽和土壤中孔隙水分布不均[4]、毛細(xì)作用及其滯后效應(yīng)[5]、土壤質(zhì)地和根系結(jié)構(gòu)[6-9]、孔隙類型和分布[10-11]、施加于土壤表面的入滲流量[12]、非飽和土壤中的氣流以及大氣波動(dòng)效應(yīng)等。一個(gè)簡單而基本的例子是,在實(shí)驗(yàn)室平板玻璃模型中人工選擇的嚴(yán)格均質(zhì)砂樣中同樣會產(chǎn)生指流。這種現(xiàn)象在宏觀均質(zhì)的多孔介質(zhì)中,包括在直徑變化范圍很小的玻璃珠堆成的人工多孔介質(zhì)中也會發(fā)生[13-14]。Wallach等[15]的最新研究結(jié)果表明,由于土壤的斥水性,土壤孔隙含水量的空間分布不一定取決于土壤孔隙和滲透性的非均質(zhì)性。

2 近10年來指流研究的主要內(nèi)容

指流研究始于20世紀(jì)60年代[16-18]。過去的10多年中,在國際土壤和水文環(huán)境領(lǐng)域中涌現(xiàn)了大量關(guān)于指流研究的文獻(xiàn)[4],表明指流的研究是該領(lǐng)域前沿研究的一個(gè)熱點(diǎn)。這些研究的內(nèi)容主要集中在以下幾個(gè)方面:①綜述性文章[1-3,4,10-11,19-36];②確定性理論模型和數(shù)值模擬[5,37-53];③室內(nèi)試驗(yàn)、觀測方法改進(jìn)和機(jī)理定性描述[6-7,12,26,41,54-72];④室內(nèi)試驗(yàn)和機(jī)理分析(包括野外采樣后進(jìn)行室內(nèi)分析)[2,13-15,49,61,73-93];⑤野外現(xiàn)場試驗(yàn)(可能有室內(nèi)試驗(yàn))[9,94-106]。

通過對上述研究的分析整理,得出以下主要認(rèn)識:①由于指流研究的復(fù)雜性,其試驗(yàn)研究目前仍以室內(nèi)分析為主,野外研究較少且其尺度與室內(nèi)試驗(yàn)差不多。②指流研究所涉及的主要數(shù)學(xué)工具仍然是以微分方程為主的確定性理論,很少有以隨機(jī)理論為主要工具來研究指流的文獻(xiàn)。一方面,隨機(jī)理論可以被用來確定數(shù)學(xué)模型中水文地質(zhì)參數(shù)的時(shí)空分布;另一方面,在這種分布被確定后,探討模型解的性質(zhì)如它是如何依賴于多孔介質(zhì)的水文地質(zhì)參數(shù)等,還需要微分方程所表示的確定性理論來完成。

目前國內(nèi)有關(guān)指流的研究主要集中于大孔隙流、試驗(yàn)描述以及指流研究進(jìn)展的介紹等,對于指流的理論研究偏少,尤其是缺少對指流數(shù)學(xué)模型的研究。因此,筆者將著重介紹和總結(jié)重力驅(qū)動(dòng)指流的精細(xì)觀測及其數(shù)學(xué)模擬方面的國際研究進(jìn)展。

3 重力驅(qū)動(dòng)指流的觀測及其數(shù)學(xué)模擬研究的重要進(jìn)展

3.1 新的高時(shí)空分辨率的觀測方法

隨著新的試驗(yàn)觀測方法如X-射線斷層攝影術(shù)(X-raytomography,CT)以及光投射法(light transmission method,LTM)的應(yīng)用[28-29,56-57,62,107],在指流試驗(yàn)中可實(shí)現(xiàn)對孔隙水飽和度的高分辨率測定,其時(shí)空分辨率分別可達(dá)1s和1mm。而Deinert等[41]所采用的中子射線實(shí)時(shí)攝影法的時(shí)空分辨率分別可達(dá)30ms和0.5mm。這種硬件觀測水平的提高為開展高水平的理論研究提供了可靠、翔實(shí)的試驗(yàn)觀測數(shù)據(jù)和以前試驗(yàn)中難以發(fā)現(xiàn)的新現(xiàn)象。例如DiCarlo[60]所做的飽和度觀測試驗(yàn)表明,孔隙尺度水平上的物理過程控制著水飽和度在指流指尖處的峰值,從而控制了指流的進(jìn)一步發(fā)展。又如 Bayer等[54]用X-射線斷層攝影術(shù)測得的毛細(xì)壓力在初始階段的增高過程與傳統(tǒng)描述非飽和帶水流的Richards方程所刻畫的單相流有著顯著的差異。

3.2 有關(guān)Richards方程或傳統(tǒng)的水-氣兩相流模型能否完整地刻畫和描述指流的爭論

Nieber[108]利用帶有毛細(xì)滯后效應(yīng)的Richards方程,在多孔介質(zhì)水飽和度初值為零的情形下模擬了指流的生成和持續(xù)過程。他發(fā)現(xiàn),為了能夠在數(shù)值模擬中得到指流,剖分節(jié)點(diǎn)間非飽和水力傳導(dǎo)系數(shù)的計(jì)算不得不采用下游加權(quán)格式。之后,Ritsema等[109]、Nguyen等[110-111]以及Nieber等[49]在進(jìn)行斥水沙質(zhì)土壤中指流的數(shù)值模擬時(shí)得到了類似結(jié)果,其共同點(diǎn)是都采用了下游加權(quán)格式。此外,Ritsema等[109]給出了在斥水沙質(zhì)土壤中指流形成過程及其復(fù)發(fā)的野外現(xiàn)場圖像,并用水-氣兩相流數(shù)值模型和對液相采用下游加權(quán)格式,重現(xiàn)了某種斥水沙質(zhì)土壤中的指流。但未提及若對水和氣都用上游加權(quán)格式將會得出什么結(jié)果。

Eliassi等[44]考慮了Richards方程是否能夠完整地描述非飽和多孔介質(zhì)中指流的問題,懷疑Richards方程的解是否真的可以產(chǎn)生試驗(yàn)中觀測到的水飽和度在指流指尖處的非單調(diào)峰值。他們認(rèn)為,Nieber[108]和Ritsema等[109]所得到的具有水飽和度非單調(diào)峰值的“指狀”數(shù)值解只是一種由于數(shù)值解的振蕩和過飽和而引起的假象,而數(shù)值解的振蕩和過飽和是由于下游加權(quán)法和網(wǎng)格剖分不夠細(xì)引起的數(shù)值計(jì)算誤差。若利用上游加權(quán)法或者網(wǎng)格剖分足夠細(xì),這種人為產(chǎn)生的“指狀”數(shù)值解就會消失。因此,他們認(rèn)為 Richards方程或者與之等價(jià)的水-氣兩相流模型,加上由Mualem[112-113]和Van Genuchten[114]給出的關(guān)于相對滲透系數(shù)和毛細(xì)效應(yīng)滯后現(xiàn)象的基本關(guān)系式所構(gòu)成的數(shù)學(xué)模型不足以描述指流。

Braddock等[40]對 Eliassi等[44]的工作從數(shù)學(xué)角度進(jìn)行了評論。評論指出:①在二維情形,Eliassi等[44]所提出的關(guān)于Richards方程解的單調(diào)性假設(shè)有待進(jìn)一步驗(yàn)證;②關(guān)于時(shí)間離散的截?cái)嗾`差引起了數(shù)值解的振蕩現(xiàn)象;③振蕩現(xiàn)象對毛細(xì)滯后關(guān)系中切換值的選擇造成了很大的困難。Eliassi等[46]回應(yīng)說雖然在二維情形下沒有Richards方程解的單調(diào)性的理論結(jié)果,但是已有的數(shù)百篇公開發(fā)表的論文所給出的二維情形下的解析和數(shù)值解,在均質(zhì)和初邊值條件為常數(shù)時(shí)都表現(xiàn)出了單調(diào)性質(zhì)。

Deinert等[59]對Eliassi等[44]的工作做了另一番評論。認(rèn)為Eliassi等[44]對Nieber[108]所做工作的評論有些言過其實(shí)。因?yàn)镹ieber[108]用簡單模型所得出的模擬結(jié)果和觀測資料相符。Nieber[108]所用的下游加權(quán)格式,只是在模擬觀測得到的指流及其不穩(wěn)定特征時(shí)所選擇的一個(gè)參數(shù)而已,這與 Deinert等[115]利用試驗(yàn)所得到的一種類似于達(dá)西形式的表達(dá)式所定義的動(dòng)態(tài)毛細(xì)壓力所描述的不穩(wěn)定性是一致的。Eliassi等[47]回應(yīng)Nieber[108]所做的數(shù)值模擬是基于下游加權(quán)格式的,因此只能說是假的指流。下游加權(quán)格式會產(chǎn)生足夠大的數(shù)值誤差,從而導(dǎo)致用數(shù)值方法所求解的方程實(shí)際上不是Richards方程本身。當(dāng)網(wǎng)格被適當(dāng)加密以后,數(shù)值誤差減小,而數(shù)值解所產(chǎn)生的人工指流也隨之消失。

試驗(yàn)觀測和理論研究表明,在均質(zhì)非飽和土壤中,連續(xù)入滲條件下水飽和度曲線會在入滲前鋒處形成一個(gè)非單調(diào)的突起,這個(gè)突起不能用傳統(tǒng)的Richards方程來描述[5,60,82]。Nieber等[5]所做的線性穩(wěn)定性分析表明,傳統(tǒng)的Richards方程對無窮小擾動(dòng)是無條件穩(wěn)定的,即不會產(chǎn)生指流;而把Richards方程和動(dòng)態(tài)毛細(xì)壓力結(jié)合起來所得到的模型卻是有條件穩(wěn)定的,即有可能產(chǎn)生指流。他們對上述動(dòng)態(tài)毛細(xì)壓力機(jī)制作了回顧并將之?dāng)U展到了含水量小于田間持水量的很干的土壤中。

DiCarlo[82]的最新研究進(jìn)展表明,上述動(dòng)態(tài)毛細(xì)壓力機(jī)制(DiCarlo在文中稱之為非平衡毛細(xì)壓力,其實(shí)是一樣的)也不能很好地描述指流。在傳統(tǒng)的Richards方程基礎(chǔ)上,加之以動(dòng)態(tài)毛細(xì)壓力,雖然能使模型描述指流的情況大有改善,例如可以預(yù)測出指流發(fā)生的流量范圍等,但對指流尖端處水飽和度非單調(diào)突起的描述仍差強(qiáng)人意。而且當(dāng)多孔介質(zhì)的固體顆粒尺寸和粗糙程度稍有變化時(shí),為了能擬合指流發(fā)生的流量范圍,就需要對動(dòng)態(tài)毛細(xì)壓力項(xiàng)做大幅度的調(diào)整。這些結(jié)果說明,現(xiàn)有的動(dòng)態(tài)毛細(xì)壓力表達(dá)式尚未包含控制指流尖端處水飽和度非單調(diào)突起的物理機(jī)制[82]。DiCarlo[61]在實(shí)驗(yàn)室中觀測到了在入滲前鋒處形成的毛細(xì)壓力的非單調(diào)突起,且和上述水飽和度曲線的非單調(diào)突起類似。

3.3 有關(guān)新機(jī)制和新效應(yīng)的模型

Eliassi等[37]研究了試驗(yàn)觀測到的水飽和度在指流垂直剖面上呈現(xiàn)的稱之為“后推堆積”(hold-backpile-up)的效應(yīng),這種效應(yīng)是以多孔連續(xù)介質(zhì)為基礎(chǔ)的Richards方程無法描述的。他們對傳統(tǒng)的Richards方程作了若干不同形式的拓展,使之能夠描述這種效應(yīng)。每種拓展都由2項(xiàng)組成:即傳統(tǒng)的Darcy-Buckingham通量項(xiàng)和另一對應(yīng)于“后推堆積”效應(yīng)的項(xiàng)。但正如他們所總結(jié)的那樣,這種拓展在更一般意義上的有效性尚存在疑問,且其表達(dá)式在整個(gè)理論框架中的物理意義無完整的定義,有關(guān)參數(shù)也沒有進(jìn)行測定。為了全面考察對“后推堆積”的模擬能力,有必要利用控制方程的各種拓展形式進(jìn)行直接數(shù)值模擬試驗(yàn),以驗(yàn)證其有效性。在孔隙尺度水平試驗(yàn)觀測研究的基礎(chǔ)上,Glass等[85-86,116]提出了與現(xiàn)有的理論框架完全不同的新機(jī)理模型,即“修正的侵入滲透”模型(modifiedinvasion percolation)。這種模型包含了重力、沿相-相界面上的局部界面彎曲、濕潤和非濕潤流體的同時(shí)入侵以及再次入侵等因素對指流的影響,該模型在描述指流中發(fā)生的各種現(xiàn)象如水飽和度的非單調(diào)性、碎裂、跳動(dòng)等方面取得了初步進(jìn)展。Shiozawa等[13]在室內(nèi)用初始狀態(tài)是干的粒狀多孔介質(zhì)所作的一維向下限流量入滲試驗(yàn)及其分析表明,從孔隙尺度的角度來看,在入滲水流前鋒所在的水氣交界處,孔隙含水量和水壓是不連續(xù)的,因此Darcy定律不能用在這樣的水氣界面上。DiCarlo等[43]在 Eliassi等[37]所提出的拓展形式的基礎(chǔ)上,將傳統(tǒng)的Richards方程擴(kuò)展為帶有非單調(diào)的毛細(xì)壓力-飽和度曲線和雙曲項(xiàng)(水壓關(guān)于時(shí)間的二階導(dǎo)數(shù)項(xiàng))的非線性波動(dòng)方程,并推導(dǎo)出該方程的行波解。該解的水飽和度曲線在入滲前鋒處確實(shí)有非單調(diào)突起。若要保證該解的唯一性,則還需加上1個(gè)正則項(xiàng)。

4 研究體會

Ursino等[117]用 Richards方程討論了潮間帶濕地植物根部呼吸的通風(fēng)狀況、蒸騰作用引起的水分運(yùn)移以及植物生長狀態(tài)之間的相互關(guān)系,其模型忽略了氣流。筆者以TOUGH2數(shù)值模擬程序?yàn)榛A(chǔ),就同一問題用水-氣兩相流模型所做的工作[118]表明,Richards方程不足以完全描述植物根部呼吸的通風(fēng)條件。對于同一類型的水力傳導(dǎo)系數(shù)在10-6～10-5m/s之間的沼澤地土壤,水 氣兩相流模型所給出的潮間帶濕地植物根圍空氣飽和度總是大于零。而根據(jù)Ursino等[117]的計(jì)算,Richards方程所給出的根圍空氣飽和度在海潮周期的某一段時(shí)間內(nèi)為零。這說明包氣帶中氣流對水流的影響非常顯著,不能忽略。基于Richards方程的單相流模型由于忽略了氣流,不能用來定量地描述潮間帶濕地土壤的通風(fēng)條件。

通過文獻(xiàn)搜集和研究整理的結(jié)果[119-120]表明,尚無文獻(xiàn)在氣流對指流影響方面做過專門探討。盡管如此,筆者認(rèn)為氣流對指流的發(fā)生和發(fā)展會有重大影響,并相信在指流研究中氣流效應(yīng)不能被忽視,甚至它主導(dǎo)與指流相關(guān)的某些現(xiàn)象(如碎裂和跳動(dòng))。已有的實(shí)驗(yàn)室研究(如Touma等[121])以及野外現(xiàn)場試驗(yàn)(如Dixon等[122])都表明,當(dāng)水流從土壤表面滲入土壤中時(shí),被水流擠壓的空氣能顯著降低水流的入滲速度。而當(dāng)氣壓充分高時(shí),空氣會沖破土壤表層孔隙水對其的限制,從土壤表面逸出,從而造成土壤內(nèi)氣壓的急劇降低以及增加水流入滲速度[121]。這有可能引起指流的跳動(dòng)和碎裂。

由于指流研究是一個(gè)多學(xué)科的問題,其機(jī)制的研究離不開定量的數(shù)學(xué)描述,而機(jī)制、試驗(yàn)和定量的數(shù)學(xué)模型之間是相輔相成、互相驗(yàn)證的關(guān)系。一個(gè)新的機(jī)制,只有經(jīng)過試驗(yàn)和模型驗(yàn)證之后才能證明它的正確性和適用范圍。因此,尋找既具有最少的模型參數(shù),又能刻畫水氣交界處水流和氣流流動(dòng)的最簡單的、具有物理背景意義的數(shù)學(xué)模型,很可能是解開指流復(fù)雜性的關(guān)鍵所在。

5 幾個(gè)關(guān)鍵的科學(xué)問題

綜上所述,在指流的研究中,尚存在大量亟待解決的又很有意義的挑戰(zhàn)性前沿課題。主要概括為如下幾點(diǎn):①能在更小的時(shí)空尺度上刻畫指流發(fā)生和發(fā)展機(jī)制的試驗(yàn)觀測裝置。②能否以傳統(tǒng)的多孔介質(zhì)理論為基礎(chǔ),結(jié)合新的動(dòng)態(tài)毛細(xì)壓力機(jī)制,來完整地描述已經(jīng)觀測到的但是現(xiàn)有模型還不能滿意地描述的指流尖端處水飽和度和壓力的非單調(diào)突起。③建立和完善有關(guān)新的機(jī)制和效應(yīng)的模型,例如孔隙尺度水平上的微觀模型以及上述“后推堆積”效應(yīng)模型。特別地,如何用數(shù)學(xué)模型簡單而自然地描述在入滲水流前鋒的水氣界面處的水-氣兩相流。④新模型的解,包括解的性質(zhì)分析和數(shù)值求解方法。指流的理論模型及數(shù)值模擬方面的突破,將打破其對相關(guān)學(xué)科發(fā)展的制約,不僅對土壤物理學(xué)和水文地質(zhì)學(xué)的理論基礎(chǔ)本身,而且對其他相關(guān)學(xué)科如應(yīng)用數(shù)學(xué)、環(huán)境資源、農(nóng)業(yè)和生態(tài)學(xué)等,都具有重要的推動(dòng)作用。

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