宋延杰,李曉嬌,唐曉敏,付 健

(1.東北石油大學(xué)地球科學(xué)學(xué)院,黑龍江大慶 163318; 2.非常規(guī)油氣成藏與開(kāi)發(fā)省部共建國(guó)家重點(diǎn)實(shí)驗(yàn)室培育基地,黑龍江大慶 163318)

基于連通導(dǎo)電理論和HB方程的骨架導(dǎo)電純巖石電阻率模型

宋延杰1,2,李曉嬌1,唐曉敏1,2,付 健1

(1.東北石油大學(xué)地球科學(xué)學(xué)院,黑龍江大慶 163318; 2.非常規(guī)油氣成藏與開(kāi)發(fā)省部共建國(guó)家重點(diǎn)實(shí)驗(yàn)室培育基地,黑龍江大慶 163318)

針對(duì)現(xiàn)有導(dǎo)電模型很難描述骨架含導(dǎo)電礦物黃鐵礦的巖石導(dǎo)電規(guī)律的難題,利用骨架完全由導(dǎo)電顆粒組成的人造巖樣以及骨架部分由導(dǎo)電顆粒組成的天然和人造巖樣的巖電實(shí)驗(yàn)數(shù)據(jù),研究水電阻率和導(dǎo)電顆粒體積分?jǐn)?shù)變化對(duì)巖石導(dǎo)電規(guī)律的影響,得出骨架導(dǎo)電純巖石的地層因素與孔隙度及電阻增大系數(shù)與含水飽和度在雙對(duì)數(shù)坐標(biāo)上為非線性關(guān)系,隨水電導(dǎo)率減小或?qū)щ婎w粒體積分?jǐn)?shù)增大,地層因素和電阻增大系數(shù)值降低。根據(jù)骨架導(dǎo)電純巖石的組成,結(jié)合連通導(dǎo)電理論特點(diǎn),將骨架導(dǎo)電純巖石分為不導(dǎo)電骨架相、導(dǎo)電骨架相和自由流體相?；谶B通導(dǎo)電方程只能描述單一導(dǎo)電相的兩相混合介質(zhì)導(dǎo)電規(guī)律而HB方程能描述兩相均導(dǎo)電的混合介質(zhì)導(dǎo)電規(guī)律的特點(diǎn),基于連通導(dǎo)電理和HB方程建立骨架導(dǎo)電純巖石電阻率模型。結(jié)果表明,骨架導(dǎo)電純巖石電阻率模型預(yù)測(cè)的地層因素與孔隙度以及電阻增大系數(shù)與含水飽和度的理論關(guān)系與實(shí)驗(yàn)規(guī)律相符且模型滿足物理約束,該模型能夠描述骨架完全和部分由導(dǎo)電礦物組成的純巖石的導(dǎo)電規(guī)律,可用于定量評(píng)價(jià)骨架導(dǎo)電低阻油層的飽和度。

純巖石;黃鐵礦;連通導(dǎo)電理論;HB方程;電阻率模型

近幾年的勘探開(kāi)發(fā)實(shí)踐表明,低阻油氣藏是復(fù)雜油氣藏中最具潛力的主要研究對(duì)象之一,而且在某些油田的低阻油氣藏發(fā)現(xiàn)了含導(dǎo)電礦物黃鐵礦的低電阻率油層[1-5]。雖然含導(dǎo)電礦物黃鐵礦的低阻油層數(shù)量較少,但是隨著油氣資源的需求與日俱增,這類骨架導(dǎo)電低阻油層將具有一定的開(kāi)采價(jià)值。骨架含導(dǎo)電礦物的油氣層與常規(guī)油氣層相比,導(dǎo)電機(jī)制發(fā)生變化,導(dǎo)電規(guī)律變得更復(fù)雜,而現(xiàn)有電阻率解釋模型多是在Archie公式基礎(chǔ)上建立的,很少能描述骨架含一定量導(dǎo)電礦物的純巖石導(dǎo)電規(guī)律。目前,有關(guān)骨架導(dǎo)電的電阻率模型主要有以下幾種:①巖石骨架導(dǎo)電模型[6-8],雖然在模型中引入了巖石骨架導(dǎo)電概念,但未給出骨架導(dǎo)電項(xiàng)與導(dǎo)電顆粒體積分?jǐn)?shù)和電阻率的具體表達(dá)形式,因此該模型不能用于描述導(dǎo)電顆粒體積分?jǐn)?shù)變化的骨架導(dǎo)電純巖石的導(dǎo)電規(guī)律。②混合泥質(zhì)砂巖有效介質(zhì)電阻率模型[9-12],雖然這些模型考慮了導(dǎo)電礦物引起的巖石骨架導(dǎo)電,但是沒(méi)有將骨架顆粒分成導(dǎo)電的骨架顆粒和不導(dǎo)電的骨架顆粒兩種,因此不適用于描述骨架含一定量導(dǎo)電礦物的純巖石導(dǎo)電規(guī)律,只能用于描述骨架完全由導(dǎo)電礦物組成的純巖石的導(dǎo)電規(guī)律。③含黃鐵礦泥質(zhì)砂巖含水飽和度模型[13],雖然該模型可用于描述骨架含一定量導(dǎo)電礦物的泥質(zhì)巖石導(dǎo)電規(guī)律,但是當(dāng)泥質(zhì)體積分?jǐn)?shù)為零時(shí),該模型轉(zhuǎn)變?yōu)辄S鐵礦顆粒和純砂巖的并聯(lián)導(dǎo)電,這對(duì)于描述黃鐵礦顆粒在巖石骨架中呈分散狀的純巖石導(dǎo)電規(guī)律是不適用的,因此該模型不適用于描述由分散狀導(dǎo)電礦物引起的骨架導(dǎo)電純巖石導(dǎo)電規(guī)律。④修正的Archie公式[14-15],基于飽含水骨架導(dǎo)電人造巖樣的電阻率測(cè)量及考慮邊界條件的傳統(tǒng)Archie公式,提出了適用于飽含水骨架導(dǎo)電巖石的修正Archie公式;在此基礎(chǔ)上給出適用于n相介質(zhì)的通用修正Archie公式,該式可用于描述骨架由部分導(dǎo)電顆粒組成的骨架導(dǎo)電純巖石的導(dǎo)電規(guī)律。筆者基于骨架導(dǎo)電巖石的巖電實(shí)驗(yàn)數(shù)據(jù),研究導(dǎo)電骨架顆粒體積分?jǐn)?shù)和電阻率變化對(duì)巖石導(dǎo)電特性的影響,將連通導(dǎo)電理論和HB方程結(jié)合建立骨架導(dǎo)電純巖石電阻率模型。

1 骨架導(dǎo)電巖石導(dǎo)電規(guī)律實(shí)驗(yàn)

利用骨架完全或部分由導(dǎo)電顆粒組成的人造或天然巖心的巖電實(shí)驗(yàn)數(shù)據(jù)[4-5,14],研究骨架導(dǎo)電特性對(duì)地層因素(F)與孔隙度(φ)關(guān)系以及電阻增大系數(shù)(I)與含水飽和度(Sw)關(guān)系的影響。

圖1 骨架完全由導(dǎo)電顆粒組成人造巖樣的F與φ關(guān)系Fig.1 Relationships between F and φ for artificial samples with matrix composed entirely of conductive grains

Glover等制作了10塊骨架完全由導(dǎo)電顆粒組成的人造巖樣[14],導(dǎo)電骨架顆粒由氧化銅組成,其電阻率近似為32.47 Ω·m,巖樣的孔隙度為4%～44%,水的電導(dǎo)率為0.001 26～14.942 6 S/m,測(cè)量了飽和不同水電導(dǎo)率的巖樣電導(dǎo)率。圖1給出了10塊骨架完全由導(dǎo)電顆粒組成的人造巖樣在6種水礦化度下的地層因素與孔隙度實(shí)驗(yàn)關(guān)系?？梢钥闯?在雙對(duì)數(shù)坐標(biāo)上地層因素與孔隙度之間的關(guān)系為非線性關(guān)系;當(dāng)骨架電阻率大于水電阻率時(shí),地層因素值大于1.0,而當(dāng)骨架電阻率小于水電阻率時(shí),地層因素值小于1.0;當(dāng)孔隙度相同時(shí),地層因素值隨水電導(dǎo)率增大而增大;對(duì)應(yīng)不同水電導(dǎo)率的地層因素值之差隨孔隙度增大而逐漸減小。

Clavier等[4]利用10塊黃鐵礦體積分?jǐn)?shù)不同的全直徑礫巖樣品,測(cè)量了5種頻率(19 Hz,35 Hz, 280 Hz,1 kHz,20 kHz)下飽和地層水電導(dǎo)率為16.7,10,2,1.0 S/m的巖樣復(fù)電阻率,巖樣的孔隙度為4.15%～22.5%,黃鐵礦體積分?jǐn)?shù)為0.0%～27.3%。圖2(a)給出了10塊骨架含有不同體積分?jǐn)?shù)黃鐵礦的全直徑礫巖巖樣的地層因素值與孔隙度實(shí)驗(yàn)關(guān)系。Clennell等利用澳大利亞北部大陸邊緣氣田黃鐵礦體積分?jǐn)?shù)不同的25塊巖心樣品(井1巖心樣品10塊(去掉了3塊黏土體積分?jǐn)?shù)較大的巖樣),井2巖心樣品8塊,井3巖心樣品7塊)和5塊人造巖心樣品進(jìn)行了黃鐵礦對(duì)巖石電阻率影響程度的研究[5],巖樣的孔隙度為3.8%～32.2%,水的電導(dǎo)率1.71～2.75 S/m,黃鐵礦體積分?jǐn)?shù)為0.0%～12%。圖2(b)給出了25塊骨架含有不同體積分?jǐn)?shù)黃鐵礦的砂巖巖樣和5塊人造巖樣的地層因素值與孔隙度實(shí)驗(yàn)關(guān)系。從圖2中可以看到,對(duì)于骨架部分由導(dǎo)電顆粒組成的巖石,在雙對(duì)數(shù)坐標(biāo)上地層因素值與孔隙度為非線性關(guān)系;在低孔隙度處,隨著導(dǎo)電顆粒體積分?jǐn)?shù)的增高,地層因素值明顯降低;對(duì)應(yīng)不同導(dǎo)電顆粒體積分?jǐn)?shù)的地層因素值之差隨孔隙度增大而逐漸減小,這是因?yàn)殡S孔隙度增大,導(dǎo)電骨架顆粒的體積分?jǐn)?shù)減小,因此導(dǎo)電骨架顆粒體積分?jǐn)?shù)變化對(duì)地層因素值的影響變小。

Clavier等利用等粒徑的石英顆粒制作了純石英砂巖樣品[4],在縱向不同位置處測(cè)量了該樣品在飽含水和近似束縛水飽和度情況下的電阻率,該樣品的孔隙度為40.4%,地層水電阻率為0.0599 Ω·m,黃鐵礦顆粒體積分?jǐn)?shù)為0%。同時(shí),利用等粒徑的石英顆粒和黃鐵礦顆粒制作了3種相同黃鐵礦顆粒體積分?jǐn)?shù)的純石英砂巖樣品,在縱向不同位置處分別測(cè)量了3種樣品在飽含3種礦化度水和近似束縛水飽和度情況下的電阻率,3種樣品的孔隙度為40.5%、42%、41%,地層水電阻率為0.0596、0.306、0.51 Ω·m,黃鐵礦顆粒體積分?jǐn)?shù)為16.5%。圖3給出了上述4種樣品的電阻增大系數(shù)與含水飽和度的關(guān)系。從圖中可以看到,隨導(dǎo)電顆粒體積分?jǐn)?shù)增大,電阻增大系數(shù)降低;隨著水電阻率增大,電阻增大系數(shù)降低。

圖2 骨架部分由導(dǎo)電顆粒組成的巖樣的F與φ關(guān)系Fig.2 Relationships between F and φ for samples with matrix composed partially of conductive grains

圖3 骨架部分由導(dǎo)電顆粒組成巖樣的I與Sw關(guān)系Fig.3 Relationships between I and Swfor samples with matrix composed partially of conductive grains

2 導(dǎo)電純巖石電阻率模型

2.1 連通導(dǎo)電方程和HB方程

連通導(dǎo)電方程引入了兩個(gè)新參數(shù):導(dǎo)電指數(shù)和水連通校正系數(shù),這兩個(gè)參數(shù)均與孔隙的幾何形狀和水在介質(zhì)中的分布狀態(tài)有關(guān),而且水連通校正系數(shù)可以解釋水在孔隙中的連通作用,因此連通導(dǎo)電方程可以更好地描述巖石的孔隙結(jié)構(gòu)和水的連通性對(duì)巖石導(dǎo)電性的影響[16-19]。對(duì)于骨架不導(dǎo)電的含油氣純巖石,其連通導(dǎo)電方程為

式中,σt為巖石電導(dǎo)率,S/m;σw為地層水電導(dǎo)率, S/m;φw為含水孔隙度;Xw為水連通性校正系數(shù);μ為導(dǎo)電指數(shù)。

當(dāng)Xw很小時(shí),1－Xw近似為1.0,則式(1)可簡(jiǎn)化為

HB方程可以用于描述導(dǎo)電或不導(dǎo)電連續(xù)相介質(zhì)中聚集著導(dǎo)電或不導(dǎo)電分散相的兩相混合物的導(dǎo)電規(guī)律,在低頻情況下HB方程的形式[20-21]為

式中,σ為混合物的電導(dǎo)率,S/m;σP為分散相的電導(dǎo)率,S/m;σc為連續(xù)相的電導(dǎo)率,S/m;Vp為分散相的體積分?jǐn)?shù);Lp為分散相的去極化因子;mp為分散相的膠結(jié)指數(shù)。

2.2 骨架導(dǎo)電純巖石電阻率模型

對(duì)于骨架部分由導(dǎo)電礦物組成的純巖石,其組分為不導(dǎo)電的骨架顆粒、導(dǎo)電的骨架顆粒、不導(dǎo)電的油氣和水。根據(jù)連通導(dǎo)電理論,將骨架導(dǎo)電純巖石地層劃分為不導(dǎo)電骨架相、導(dǎo)電骨架相和自由流體相。其中,不導(dǎo)電骨架相由不導(dǎo)電骨架顆粒和相應(yīng)的束縛水組成,導(dǎo)電骨架相由導(dǎo)電的骨架顆粒和相應(yīng)的束縛水組成,自由流體相由可動(dòng)水和油氣組成。圖4給出了基于連通導(dǎo)電理論和HB方程的骨架導(dǎo)電純巖石電阻率模型的體積模型,其物質(zhì)平衡方程為

式中,Vmanc、Vmac分別為骨架導(dǎo)電純巖石的不導(dǎo)電骨架顆粒、導(dǎo)電骨架顆粒的體積分?jǐn)?shù);φwi、φf(shuō)分別為骨架導(dǎo)電純巖石的束縛水孔隙度、自由流體孔隙度; φwinc、φwic分別為骨架導(dǎo)電純巖石的不導(dǎo)電骨架相和導(dǎo)電骨架相相應(yīng)的束縛水孔隙度;φw、φh分別為骨架導(dǎo)電純巖石的含水孔隙度、含油氣孔隙度;φ為骨架導(dǎo)電純巖石的有效孔隙度。

假定不導(dǎo)電骨架顆粒與導(dǎo)電骨架顆粒大小相近,則φwinc和φwic計(jì)算公式為

認(rèn)為自由水和束縛水電導(dǎo)率相同,均為地層水電導(dǎo)率[22]。由于連通導(dǎo)電方程適用于描述只有一個(gè)導(dǎo)電相存在的兩相混合介質(zhì)的導(dǎo)電規(guī)律,而HB方程可以描述兩相均導(dǎo)電或不導(dǎo)電的混合介質(zhì)的導(dǎo)電規(guī)律,因此不導(dǎo)電骨架相和自由流體相的導(dǎo)電規(guī)律可用連通導(dǎo)電方程描述,而導(dǎo)電骨架相的導(dǎo)電規(guī)律必須用HB方程描述。對(duì)于不導(dǎo)電骨架相和自由流體相,可認(rèn)為其中的水是完全連通的,即各相的水連通校正系數(shù)等于零,應(yīng)用連通導(dǎo)電方程可得各相電導(dǎo)率為

圖4 骨架導(dǎo)電純巖石電阻率模型的體積模型Fig.4 Volume model of resistivity model for matrix-conducting clean sands

式中,σmancp、σf分別為不導(dǎo)電骨架相和自由流體相的電導(dǎo)率,S/m;φ′winc、φ′wf分別為不導(dǎo)電骨架相和自由流體相的相對(duì)含水體積分?jǐn)?shù);μs和μ分別為不導(dǎo)電骨架相和自由流體相的導(dǎo)電指數(shù)。

根據(jù)地層體積模型和各導(dǎo)電相相對(duì)含水體積分?jǐn)?shù)的定義,可得

將式(9)代入式(8),可得

對(duì)導(dǎo)電骨架相應(yīng)用HB方程得出導(dǎo)電骨架相電導(dǎo)率的表達(dá)式為

式中,σma為導(dǎo)電骨架顆粒的電導(dǎo)率,S/m;σmacp為導(dǎo)電骨架相電導(dǎo)率,S/m;mma為導(dǎo)電骨架顆粒的膠結(jié)指數(shù)。

對(duì)式(11)采用迭代方法求解,可求出σmacp。

假設(shè)σt為巖石總電導(dǎo)率,Xmancp、Xmacp和Xf分別為不導(dǎo)電骨架相、導(dǎo)電骨架相和自由流體相的體積分?jǐn)?shù),根據(jù)混合導(dǎo)電定律[23]得

方程(16)即為基于連通導(dǎo)電理論和HB方程的骨架導(dǎo)電純巖石電阻率模型。

3 模型的理論驗(yàn)證

3.1 邊界條件

由式(16)和地層因素定義式,可得地層因素表達(dá)式為

由式(16)和電阻增大系數(shù)定義式,可得電阻增大系數(shù)表達(dá)式為

(1)當(dāng)φ=1時(shí),即巖石完全由孔隙組成,不含任何顆粒成分,則有Vmanc=0,Vmac=0。將Vmac=0代入式(11)可得σmacp=σw。將上述結(jié)果代入Xw表達(dá)式,則有

將φ=1和Xw=0代入式(17),可得出F=1,與按照F定義在φ=1情況下F==1.0相符。

(2)當(dāng)Sw=1時(shí),即巖石孔隙完全被水飽和。將Sw=1代入式(18),可得出I=1,與按照I定義在Sw=1情況下I==1.0相符。

在已有的文獻(xiàn)中，通常將里根執(zhí)政時(shí)期視為美國(guó)經(jīng)濟(jì)向自由化轉(zhuǎn)折的時(shí)期，從經(jīng)濟(jì)學(xué)說(shuō)的轉(zhuǎn)變來(lái)看，這樣的判斷無(wú)疑是正確的。因?yàn)?，在里根?zhí)政的80年代，不僅現(xiàn)代貨幣主義主導(dǎo)了美國(guó)的緊縮性貨幣政策實(shí)施，同時(shí)供給學(xué)派的減稅主張也得到了實(shí)踐的機(jī)會(huì)，最為重要的是沃克在擔(dān)任美聯(lián)儲(chǔ)主席時(shí)在治理通脹時(shí)運(yùn)用了理性預(yù)期學(xué)派的觀點(diǎn)，確立了理性預(yù)期學(xué)派在西方宏觀經(jīng)濟(jì)學(xué)中的地位，以上三個(gè)學(xué)派的學(xué)說(shuō)共同支撐了新古典宏觀經(jīng)濟(jì)學(xué)的理論體系。如果更準(zhǔn)確地說(shuō)，里根時(shí)期已經(jīng)是新古典經(jīng)濟(jì)學(xué)確立其西方經(jīng)濟(jì)學(xué)主流地位的時(shí)期，而在此之后，特別是1998年?yáng)|亞危機(jī)之后，新古典經(jīng)濟(jì)學(xué)的政策所帶來(lái)的全球治理問(wèn)題日趨嚴(yán)重，保護(hù)主義勢(shì)力開(kāi)始重新抬頭。

(3)當(dāng)φ=0時(shí),即巖石完全由不導(dǎo)電顆粒和導(dǎo)電顆粒組成,不含孔隙,則有φwinc=0,φwic=0,φf(shuō)=0。將φwic=0代入式(11)可得σmacp=σma,將上述結(jié)果代入式(15)可得

當(dāng)Vmac=1.0時(shí),即巖石完全由導(dǎo)電顆粒組成,且不含孔隙,則由式(21)可得σt=σmac,與實(shí)際相符。當(dāng)Vmac=0時(shí),即巖石完全由不導(dǎo)電顆粒組成,且不含孔隙,則由式(21)可得σt=0,與實(shí)際相符。

(4)當(dāng)σw=σma,Vmanc=0,φf(shuō)=φ－φwi時(shí),即巖石由導(dǎo)電顆粒和孔隙組成,且孔隙完全含水,則有φwinc=0,φwic=φwi。將σw=σma代入式(11),可得σmacp= σma。將φwinc=0,φwic=φwi代入式(14),可得Xmancp= 0,Xmacp=Vmac+φwi,Xf=φ－φwi。將上述結(jié)果代入式(13),可得σt=σw,與實(shí)際相符。

3.2 理論分析

在束縛水飽和度Swi=0.25,φ=0.13,mmac= 2.0,μ=1.63,μs=1.4條件下,利用方程(16)對(duì)骨架導(dǎo)電純巖石導(dǎo)電規(guī)律進(jìn)行理論分析。

3.2.1 地層因素與孔隙度關(guān)系

對(duì)于骨架完全或部分由導(dǎo)電顆粒組成的巖石,假設(shè)Vmanc=0,σma=0.03 S/m或Vmac=0.1,σma= 0.83 S/m。圖5給出了不同地層水電導(dǎo)率的F與φ交會(huì)圖,從圖中看出F隨σw減小而減小。

綜上,預(yù)測(cè)的骨架導(dǎo)電純巖石地層因素與孔隙度的理論關(guān)系與實(shí)驗(yàn)規(guī)律相符,說(shuō)明方程(16)能描述飽含水骨架導(dǎo)電純巖石的導(dǎo)電規(guī)律。

3.2.2 地層電阻增大系數(shù)與含水飽和度關(guān)系

對(duì)于骨架完全或部分由導(dǎo)電顆粒組成的巖石,假設(shè)Vmanc=0,σma=0.03 S/m或Vmac=0.165,σma= 0.83 S/m。圖7給出了不同地層水電導(dǎo)率的I與Sw交會(huì)圖,從圖中看出I隨σw減小而降低。

對(duì)于骨架由部分導(dǎo)電顆粒組成的巖石,假設(shè)σma=2.5 S/m,σw=1.89 S/m,圖8(a)給出了不同導(dǎo)電骨架顆粒體積分?jǐn)?shù)的I與Sw交會(huì)圖,從圖中看出,I隨Vmac增大而減小。假設(shè)Vmac=0.1,σw=2.08 S/m,圖8(b)給出了不同導(dǎo)電骨架顆粒電阻率的I與Sw交會(huì)圖,從圖中看出I隨ρmac減小而降低。

綜上所述,方程(16)預(yù)測(cè)的骨架導(dǎo)電純巖石電阻增大系數(shù)與含水飽和度的理論關(guān)系與實(shí)驗(yàn)規(guī)律相符,由此說(shuō)明方程(16)能描述含油氣骨架導(dǎo)電純巖石的導(dǎo)電規(guī)律。

圖5 不同地層水電導(dǎo)率的F與φ理論關(guān)系Fig.5 Theoretical relationships between F and φ for different σw

圖6 骨架部分由導(dǎo)電顆粒組成巖石的F與φ理論關(guān)系Fig.6 Theoretical relationships between F and φ of sands with matrix composed partially of conductive grains

圖7 不同地層水電導(dǎo)率的I與Sw理論關(guān)系Fig.7 Theoretical relationships between I and Swfor different σw

圖8 骨架部分由導(dǎo)電顆粒組成巖石的I與Sw理論關(guān)系Fig.8 Theoretical relationships between I and Swof sands with matrix composed partially of conductive grains

4 模型的實(shí)驗(yàn)驗(yàn)證

4.1 骨架完全由導(dǎo)電顆粒組成的巖樣

對(duì)于10塊骨架完全由導(dǎo)電顆粒組成的人造巖樣[14],有Vmanc=0.0,φwinc=0.0,φwic=φwi,利用最優(yōu)化技術(shù)求解σo－σw的非相關(guān)函數(shù),可優(yōu)化得到模型中各未知參數(shù)值(表1)。從表中可以看出,對(duì)于該組骨架完全由導(dǎo)電礦物組成的巖樣,計(jì)算的導(dǎo)電骨架顆粒電阻率為27.5～39.3 Ω·m,均值為32.83 Ω·m,與真值32.47 Ω·m非常接近。圖9給出了利用優(yōu)化的各參數(shù)值骨架導(dǎo)電純巖石電阻率模型計(jì)算該組巖樣的電導(dǎo)率值與巖心測(cè)量值對(duì)比圖(其中符號(hào)點(diǎn)為巖心測(cè)量數(shù)據(jù),曲線為方程計(jì)算結(jié)果),從圖中可以看到曲線與符號(hào)點(diǎn)的一致性很好,說(shuō)明本文建立的骨架導(dǎo)電純巖石電阻率模型能描述骨架完全由導(dǎo)電礦物組成的含水純巖石的導(dǎo)電規(guī)律。

表1 骨架完全由導(dǎo)電顆粒組成的人造巖樣的電阻率模型優(yōu)化參數(shù)Table 1 Optimization parameters of resistivity model for artificial samples with matrix composed entirely of conductive grains

圖9 計(jì)算骨架完全由導(dǎo)電顆粒組成巖樣電導(dǎo)率值與實(shí)驗(yàn)測(cè)量值對(duì)比Fig.9 Comparison of calculated conductivity with measured conductivity for samples with matrix composed entirely of conductive grains

4.2 骨架部分由導(dǎo)電顆粒組成的巖樣

對(duì)于10塊黃鐵礦體積分?jǐn)?shù)不同的全直徑礫巖巖樣[4],利用最優(yōu)化技術(shù)求解σo－σw的非相關(guān)函數(shù),可優(yōu)化得到模型中各未知參數(shù)值,圖10給出了利用優(yōu)化的各參數(shù)值方程計(jì)算該組巖樣的電導(dǎo)率值與巖心測(cè)量值對(duì)比圖(以頻率35 Hz的為例,其中符號(hào)點(diǎn)為巖心測(cè)量數(shù)據(jù),曲線為方程計(jì)算結(jié)果),從圖中可以看到曲線與符號(hào)點(diǎn)的一致性很好,這說(shuō)明建立的骨架導(dǎo)電純巖石電阻率模型能描述骨架含一定量導(dǎo)電礦物的含水純巖石的導(dǎo)電規(guī)律。

圖10 計(jì)算骨架部分由導(dǎo)電顆粒組成巖樣電導(dǎo)率值與實(shí)驗(yàn)測(cè)量值對(duì)比Fig.10 Comparison of calculated conductivity with measured conductivity for samples with matrix composed partially of conductive grains

5 結(jié) 論

(1)骨架導(dǎo)電純巖石的地層因素值與孔隙度在雙對(duì)數(shù)坐標(biāo)上為非線性關(guān)系。當(dāng)骨架電阻率大于水電阻率時(shí),地層因素值大于1.0,而當(dāng)骨架電阻率小于水電阻率時(shí),地層因素值小于1.0。隨水電導(dǎo)率增大,地層因素值增大。隨導(dǎo)電顆粒體積分?jǐn)?shù)和電導(dǎo)率增大,地層因素值降低。

(2)骨架導(dǎo)電純巖石的電阻增大系數(shù)與含水飽和度在雙對(duì)數(shù)坐標(biāo)上為非線性關(guān)系,無(wú)論骨架電阻率大于或小于水電阻率,電阻增大系數(shù)均大于1.0。隨水電導(dǎo)率減小,電阻增大系數(shù)減小。隨導(dǎo)電顆粒體積分?jǐn)?shù)和電導(dǎo)率增大,電阻增大系數(shù)降低。

(3)建立的骨架導(dǎo)電純巖石電阻率模型滿足一定邊界條件:①當(dāng)孔隙度為100%時(shí),地層因素值等于1;②當(dāng)含水飽和度為100%時(shí),電阻增大系數(shù)等于1;③當(dāng)孔隙度為0%,導(dǎo)電顆粒體積分?jǐn)?shù)為100%時(shí),巖石電導(dǎo)率等于導(dǎo)電顆粒電導(dǎo)率,而當(dāng)孔隙度為0%,導(dǎo)電顆粒體積分?jǐn)?shù)為0%時(shí),巖石電導(dǎo)率等于0;④當(dāng)巖石由導(dǎo)電顆粒和飽含水孔隙組成,且水電導(dǎo)率等于導(dǎo)電顆粒電導(dǎo)率時(shí),巖石電導(dǎo)率等于水電導(dǎo)率。

(4)新建立的模型既能描述骨架完全由導(dǎo)電礦物組成的純巖石導(dǎo)電規(guī)律,又能描述骨架含一定量導(dǎo)電礦物的純巖石導(dǎo)電規(guī)律,適用范圍更廣,可用于定量評(píng)價(jià)骨架導(dǎo)電低阻油層的飽和度。

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(編輯 修榮榮)

Matrix-conducting resistivity model for clean sands based on connectivity conductance theory and HB equation

SONG Yanjie1,2,LI Xiaojiao1,TANG Xiaomin1,2,FU Jian1
(1.College of Geoscience,Northeast Petroleum University,Daqing 163318,China; 2.Accumulation and Development of Unconventional Oil and Gas,State Key Laboratory Cultivation Base Jointly-constructed by Heilongjiang Province and Ministry of Science and Technology,Daqing 163318,China)

Most of the commonly used resistivity models are unable to give a precise description of conductive laws of sands whose rock matrix contains a certain amount of pyrite,therefore it is necessary to study the conductive laws and to propose a matrix-conducting resistivity model.The effects of water resistivity and conductive matrix grain content in matrix-conducting clean sands are first analyzed by using laboratory resistivity measurements of artificial and field samples,in which the rock matrix is composed partially or entirely of conductive grains.The results shown in a log-log graph suggest nonlinear relationships between formation resistivity factor and porosity,and between formation resistivity index and water saturation,respectively.Values of the formation resistivity factor and the index decrease with decreasing water conductivity,or increasing conductive matrix grain content.Second,based on the compositions of matrix-conducting clean sands and the characteristics of connectivitySONG Yanjie,LI Xiaojiao,TANG Xiaomin,et al.Matrix-conducting resistivity model for clean sands based on connectivity conductance theory and HB equation[J].Journal of China University of Petroleum(Edition of Natural Science),2014,38 (5):66-74.conductance theory,matrix-conducting clean sands are divided into non-conducting matrix phase,conductive matrix phase and free fluid phase.Since the connectivity conductance equation applies to only one conducting composition and one non-conducting composition,while the HB equation can describe systems of two conducting compositions,a new matrix-conducting resistivity model for clean sands is proposed combining the connectivity conductance theory and the HB equation.The results show that the theoretical relationships between formation resistivity factor and porosity,and between formation resistivity index and water saturation predicted by the proposed model are consistent with the experimental values,and the proposed model is in compliance with meaningful physical bounds.The matrix-conducting resistivity model for clean sands can describe the conductive law of matrix-conducting clean sands,in which rock matrix is composed entirely or partially of conductive grains.The proposed model can be applied to quantitatively calculate saturation in matrix-conducting low resistivity reservoirs.

clean sand;pyrite;connectivity conductance theory;HB equation;resistivity model

P 631.84

A

1673-5005(2014)05-0066-09

10.3969/j.issn.1673-5005.2014.05.009

2014－02－21

國(guó)家自然科學(xué)基金項(xiàng)目(41274110)

宋延杰(1963－),男,教授,博士,主要從事測(cè)井方法與資料解釋研究。E－mail:syj1963@263.net。

宋延杰,李曉嬌,唐曉敏,等.基于連通導(dǎo)電理論和HB方程的骨架導(dǎo)電純巖石電阻率模型[J].中國(guó)石油大學(xué)學(xué)報(bào):自然科學(xué)版,2014,38(5):66-74.