碳酸溶蚀对砂岩储层物性特征及孔隙结构的影响

窦亮彬; 王婷; 方勇; 程学彬; 陈景杨; 程志林

doi:10.6054/j.jscnun.2024016

碳酸溶蚀对砂岩储层物性特征及孔隙结构的影响

窦亮彬^{1, 2, 3, ,},
王婷^{1, 3},
方勇^{1, 3},
程学彬^{1, 3},
陈景杨^{1, 3},
程志林^{1, 3}

1.
西安石油大学石油工程学院，西安 710065
2.
油气资源与探测国家重点实验室/中国石油大学(北京)，北京 102249
3.
西安石油大学/西部低渗—特低渗透油田开发与治理教育部工程研究中心，西安 710065

基金项目:

国家自然科学基金项目 52074221

陕西省教育厅重点科学研究计划项目 21JY036

陕西省重点研发计划项目 2024GX-YBXM-503

详细信息

通讯作者:
窦亮彬，Email: doulb@xsyu.edu.cn

中图分类号: TE4
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出版历程
- 收稿日期: 2024-01-23
- 网络出版日期: 2024-06-21
- 刊出日期: 2024-04-24

The Influence of Carbonate Dissolution on Physical Properties and Pore Structure of Sandstone Reservoir

DOU Liangbin^{1, 2, 3, ,},
WANG Ting^{1, 3},
FANG Yong^{1, 3},
CHENG Xuebin^{1, 3},
CHEN Jingyang^{1, 3},
CHENG Zhiling^{1, 3}

1.
College of Petroleum Engineering, Xi'an Shiyou University, Xi'an 710065, China
2.
State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, China
3.
Engineering Research Center of Development/Management for Western Low to Extra-Low Permeability Oilfield, Xi'an Shiyou University, Xi'an 710065, China

摘要

摘要:
以鄂尔多斯盆地长6不同渗透率级别砂岩岩心为研究对象，系统研究了碳酸溶蚀前后岩心物性特征、矿物组成以及孔隙结构的变化特征。结果表明：对于较低渗透率的岩样，碳酸水的溶蚀作用造成除石英外的其他矿物组分含量均有所下降，而对于高渗岩样，由于孔隙空间比表面积较小，使得矿物与碳酸水作用强度较弱，少量方解石和长石被溶解而含量下降，其他矿物相对含量有所升高。随着碳酸水的不断注入，渗透率为0.1 mD的岩心孔隙度和渗透率均不断增大，渗透率为1.0 mD的岩心孔渗均是先减小后增大，当岩心渗透率为10 mD时，岩心孔隙度和渗透率均不断减小。总体上，孔隙度相对变化率在-5.77%~-3.68%之间；渗透率相对变化率在-21.87%~-18.47%之间。研究结果揭示了不同孔渗级别砂岩与碳酸水相互作用下孔喉结构变化特征，能够为低渗储层CO₂驱油和埋存方案的制定提供重要理论依据。
- 砂岩 /
- 溶蚀 /
- 孔隙特征 /
- 孔隙度 /
- 渗透率
Abstract:
The change of physical properties, mineral composition and pore structure of Chang 6 sandstone cores with different permeability levels in Ordos Basin before and after carbonic acid dissolution were systematically studied. The results show that for rock samples with low permeability, the content of other mineral components except quartz decreases due to the dissolution of carbonate water, while for rock samples with high permeability, due to the small specific surface area of pore space, the interaction strength between minerals and carbonate water is weak, and a small amount of calcite and feldspar are dissolved and the content decreases, while the relative content of other minerals increases. With the continuous injection of carbonate water, the porosity and permeability of the core with a permeability of 0.1 mD increase continuously, and the porosity and permeability of the core with a permeability of 1.0 mD decrease first and then increase. When the core permeability is 10 mD, the core porosity and permeability decrease continuously. In general, the relative change rate of porosity is between -5.77% and -3.68%. The relative change rate of permeability was between -21.87% and -18.47%. The research results reveal the characteristics of pore throat structure changes under the interaction between sandstone with different porosity and permeability levels and carbonate water, which can provide an important theoretical basis for the formulation of CO₂ displacement and storage schemes in low permeability reservoirs.
- sandstone /
- corrosion /
- pore characteristics /
- porosity /
- permeability

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拓扑指数是一类刻画分子结构的数值不变量. 在理论化学中, 拓扑指数被用于预测化合物的物理化学性质、药理活性和生理活性等. 常用的与距离有关的拓扑指数主要有Wiener指数^[1-2]、Harary指数^[3]和Kirchhoff指数^[4].

环辛烷及其衍生物是有机化学中重要的环烷烃, 可用于药物合成、有机合成等. 而研究一些特殊化学链的拓扑指数及其在化学中的应用也成为化学图论的热点问题^[5-13]. 如：WU等^[5]计算了聚苯链、聚苯蜘蛛链的hyper-Wiener指数, 得到上、下界; YANG和ZHANG^[6]给出了随机聚苯链的Wiener指数的期望值; WEI等^[12]给出了随机环辛烷链的Wiener指数的期望值; ZHANG等^[13]确定了随机聚苯链的Schultz指数、Gutman指数、度积Kirchhoff指数、度和Kirchhoff指数的期望值.

关于Kirchhoff指数的研究已经有比较多的成果. 如：YANG和JIANG^[14]给出了单圈图的Kirchhoff指数的极大、极小值; GUO等^[15]确定了全负荷单圈图的Kirchhoff指数的极大、极小值; FENG等^[16]刻画了具有极大、极小度Kirchhoff指数的全负荷单圈图; FEI和TU^[17]刻画了具有最大、第二大度Kirchhoff指数的双圈图. 关于更多介绍Kirchhoff指数的性质及其应用的文献可参考文献[18-21]. 在文献[12-13]的启发下, 本文首先确定了随机环辛烷链的3类Kirchhoff指数(Kirchhoff指数、度积Kirchhoff指数、度和Kirchhoff指数)的期望表达式, 再由该式得到了具有n个八边形的环辛烷链的3类Kirchhoff指数的极大值与极小值, 并刻画了相应的极图.

1. 预备知识

1947年, WIENER^[1]在研究石蜡的沸点与其分子的结构关系时, 提出了用拓扑指数来研究分子的结构性质. 由于当时化学图论发展的限制, H WIENER并未将这一指数与化学图论结合给出定义. 1971年, HOSOYA^[2]对这一指数给出了图论术语表达式, 即 $W\left( G \right) = \sum\limits_{\left\{ {u, v} \right\} \subseteq V\left( G \right)} {{d_G}\left( {u, v} \right)}$ , 其中d_G(u, v)表示顶点u、v之间最短路的长度. 为了纪念H WIENER对这一指数的贡献, 将其称为Wiener指数.

1993年, KLEIN和RANDI Ć^[22]基于电网理论引入了一个新的距离函数, 称之为电阻距离: 通过用单位电阻器代替图G的每条边来将图G视为电网N, 将电网N中顶点u与顶点v之间的有效电阻值称为图G中顶点u与顶点v之间的电阻距离r_G(u, v).

在Wiener指数的定义中用电阻距离替代距离, 则可得到Kirchhoff指数^[4]：

$\operatorname{Kf}(G)=\sum\limits_{\{u, v\}\subseteq V(G)} r_{G}(u, v).$

度积Kirchhoff指数Kf^*(G)^[23]与度和Kirchhoff指数Kf ⁺(G)^[24]是2类加权版的Kirchhoff指数, 分别定义为:

$\begin{aligned} \mathrm{Kf}^{*}(G) &=\sum\limits_{\{u, v\}\subseteq V(G)}\left(d_{G}(u) d_{G}(v)\right) r_{G}(u, v), \\ \mathrm{Kf}^{+}(G) &=\sum\limits_{\{u, v\} \subseteq V(G)}\left(d_{G}(u)+d_{G}(v)\right) r_{G}(u, v) . \end{aligned}$

图 1为一条具有n个正八边形的环辛烷链COC_n, 可以看成是将环辛烷链COC_n-1通过一条割边连接一个新的终端正八边形O_n所构成的.

图 1 n个正八边形的环辛烷链COC_n

Figure 1. A cyclooctane chain with n regular octagons

下载: 全尺寸图片幻灯片

设COC_n=O₁O₂…O_n为具有n (n≥2)个正八边形的环辛烷链, 其中O_t (t=2, 3, …, n)为图COC_n中通过割边u_t-1v_t与O_t-1相连的第t个正八边形. 在O_t中, 与点v_t距离为1、2、3、4的点分别记为o_t、m_t、w_t、l_t. 例如: 图 1中v_n=z₁; o_n=z₂, z₈; m_n=z₃, z₇; w_n=z₄, z₆; l_n=z₅.

在具有n个正八边形的环辛烷链COC_n中, 有一些特殊类的COC_n. 例如: 对任意的2≤t≤n-1, 若均有u_t=o_t, 则将该环辛烷链COC_n记为CO_n; 若u_t=m_t, 则将其记为CM_n; 如u_t=w_t, 则将其记为CW_n; 若u_t=l_t, 则将其记为CL_n.

当n≥3时, COC_n+1的终端八边形O_n+1可以连接到O_n的o_n、m_n、w_n、l_n点, 即4种连接方式, 不妨将得到的4种不同的图形(同构意义下)分别记为COC_n+1¹、COC_n+1²、COC_n+1³和COC_n+1⁴, 见图 2.

图 2 环辛烷链的4种连接方式

Figure 2. Four connection methods of cyclooctane chains

下载: 全尺寸图片幻灯片

一条具有n个正八边形的随机环辛烷链, 记为COC(n; p₁, p₂, p₃), 是按照以下的连接方式进行n-2步得到的链(每一步t (t=3, 4, …, n)都随机选择其中一种连接方式)：(1)COC_t-1→COC_t¹, 可能性为p₁; (2)COC_t-1→COC_t², 可能性为p₂; (3)COC_t-1→COC_t³, 可能性为p₃; (4)COC_t-1→COC_t⁴, 可能性为1-p₁-p₂-p₃, 其中p₁、p₂、p₃是与参数t无关的常数, 且0≤p₁, p₂, p₃≤1, 0≤p₁+p₂+p₃≤1.

特别地, COC(n; 1, 0, 0)、COC(n; 0, 1, 0)、COC(n; 0, 0, 1)、COC(n; 0, 0, 0)分别为链CO_n、CM_n、CW_n、CL_n.

2. 随机环辛烷链的Kirchhoff指数

用E[Kf(COC(n; p₁, p₂, p₃))]表示随机环辛烷链COC(n; p₁, p₂, p₃)的Kirchhoff指数的期望, 记r(u|G)= $\sum\limits_{v \in V\left( G \right)} {r\left( {u, v} \right)}$ . 为了计算随机环辛烷链的Kirchhoff指数期望的表达式, 首先给出COC(n; p₁, p₂, p₃)的Kirchhoff指数的递归公式.

引理1 设COC_n为具有n (n≥1)个正八边形的环辛烷链, 则

$\begin{array}{l} \mathrm{Kf}\left(\mathrm{COC}_{n}\right)=\mathrm{Kf}\left(\mathrm{COC}_{n-1}\right)+8 r\left(u_{n-1} \mid \mathrm{COC}_{n-1}\right)+ \\ \quad\quad 148(n-1)+42 . \end{array}$

(1)

证明首先介绍如何求图中两顶点之间的Kirchhoff指数, 以图 1中顶点u_n-1与顶点z₇之间的Kirchhoff指数为例. 由物理中电阻串联、并联的规律知, 2个电阻值分别为R₁和R₂的电阻串联的总电阻为R₁+R₂, 并联的总电阻为R₁R₂/(R₁+R₂). 由图 1可以看出顶点u_n-1到顶点z₇的连接情况为：路z₁z₂z₃z₄z₅z₆z₇与路z₁z₈z₇并联, 然后与路z₁u_n-1串联. 从而可得顶点u_n-1与顶点z₇之间的Kirchhoff指数为 $\frac{{2 \times 6}}{{2 + 6}} + 1 = \frac{5}{2}$ . 下面对COC_n中的顶点进行分类，以求解任意顶点对x、y的电阻距离r(x, y).

对于任意顶点v∈V(COC_n-1), 由图 1知r(z_i, v)=r(z_i, u_n-1)+r(u_n-1, v), 且

$r\left(z_{i}, v\right)=r\left(u_{n-1}, v\right)+\left\{\begin{array}{ll} 1 \quad(i=1) , \\ 1+\frac{7}{8}=\frac{15}{8} \quad (i=2,8), \\ 1+\frac{3}{2}=\frac{5}{2} \quad (i=3,7), \\ 1+\frac{15}{8}=\frac{23}{8} \quad(i=4,6), \\ 1+2=3 \quad(i=5) . \end{array}\right.$

(2)

对于z_k ∈V(O_n) (1≤k≤8), 有

$\sum\limits_{i=1}^{8} r\left(z_{k}, z_{i}\right)=\left(\frac{7}{8}+\frac{3}{2}+\frac{15}{8}\right) \times 2+2=\frac{21}{2}.$

由r(x|G)= $\sum\limits_{y \in V\left( G \right)} {r\left( {x, y} \right)}$ 及式(2), 有

$\begin{array}{c} r\left(z_{i} \mid \mathrm{COC}_{n}\right)=\sum\limits_{j=1}^{8} r\left(z_{i}, z_{j}\right)+\sum\limits_{v \in V\left(\operatorname{COC}_{n-1}\right)} r\left(z_{i}, v\right)= \\ \frac{21}{2}+8(n-1) r\left(z_{i}, u_{n-1}\right)+r\left(u_{n-1} \mid \mathrm{COC}_{n-1}\right), \end{array}$

从而

$r\left(z_{i} \mid \mathrm{COC}_{n}\right)=r\left(u_{n-1} \mid \mathrm{COC}_{n-1}\right)+\left\{\begin{array}{ll} 8 n+\frac{5}{2} & (i=1), \\ 15 n-\frac{9}{2} & (i=2,8), \\ 20 n-\frac{19}{2} & (i=3,7), \\ 23 n-\frac{25}{2} & (i=4,6), \\ 24 n-\frac{27}{2} & (i=5) . \end{array}\right.$

(3)

进而

$\begin{array}{c} \mathrm{Kf}\left(\mathrm{COC}_{n}\right)=\mathrm{Kf}\left(\mathrm{COC}_{n-1}\right)+\sum\limits_{i=1}^{8} r\left(z_{i} \mid \mathrm{COC}_{n}\right)- \\ \frac{1}{2} \sum\limits_{i=1}^{8} \sum\limits_{j=1}^{8} r\left(z_{i}, z_{j}\right)=\mathrm{Kf}\left(\mathrm{COC}_{n-1}\right)+ \\ 8 r\left(u_{n-1} \mid \mathrm{COC}_{n-1}\right)+148(n-1)+42. \end{array}$

故式(1)成立. 证毕.

记U_n=E[r(u_n|COC_n)], 下面给出U_n的表达式.

引理2 设COC(n; p₁, p₂, p₃)为具有n (n≥1)个正八边形的随机环辛烷链, 则

$U_{n}=\left(12-\frac{9}{2} p_{1}-2 p_{2}-\frac{1}{2} p_{3}\right) n^{2}+\left(\frac{9}{2} p_{1}+2 p_{2}+\frac{1}{2} p_{3}-\frac{3}{2}\right) n.$

(4)

证明分以下4种情况来计算U_n.

情形1：COC_n→COC_n+1¹. 此时u_n=z₂或z₈, r(u_n|COC_n)为r(z₂|COC_n)的可能性为p₁.

情形2：COC_n→COC_n+1². 此时u_n=z₃或z₇, r(u_n|COC_n)为r(z₃|COC_n)的可能性为p₂.

情形3：COC_n→COC_n+1³. 此时u_n=z₄或z₆, r(u_n|COC_n)为r(z₄|COC_n)的可能性为p₃.

情形4：COC_n→COC_n+1⁴. 此时u_n=z₅, r(u_n|COC_n)为r(z₅|COC_n)的可能性为1-p₁-p₂-p₃.

由情形1至情形4及式(3), 可知

$\begin{aligned} U_{n}=& r\left(z_{2} \mid \mathrm{COC}_{n}\right) p_{1}+r\left(z_{3} \mid \mathrm{COC}_{n}\right) p_{2}+r\left(z_{4} \mid \mathrm{COC}_{n}\right) p_{3}+ \\ & r\left(z_{5} \mid \mathrm{COC}_{n}\right)\left(1-p_{1}-p_{2}-p_{3}\right)= \\ & p_{1}\left[r\left(u_{n-1} \mid \operatorname{COC}\left(n-1 ; p_{1}, p_{2}, p_{3}\right)\right)+15 n-\frac{9}{2}\right]+ \\ & p_{2}\left[r\left(u_{n-1} \mid \operatorname{COC}\left(n-1 ; p_{1}, p_{2}, p_{3}\right)\right)+20 n-\frac{19}{2}\right]+ \\ & p_{3}\left[r\left(u_{n-1} \mid \operatorname{COC}\left(n-1 ; p_{1}, p_{2}, p_{3}\right)\right)+23 n-\frac{25}{2}\right]+ \\ & \left(1-p_{1}-p_{2}-p_{3}\right)\left[r\left(u_{n-1} \mid \operatorname{COC}\left(n-1 ; p_{1}, p_{2}, p_{3}\right)\right)+\right. \\ & \left.24 n-\frac{27}{2}\right]. \end{aligned}$

而E[U_n]=U_n, 则

$\begin{aligned} U_{n}&=p_{1}\left[U_{n-1}+15 n-\frac{9}{2}\right]+p_{2}\left[U_{n-1}+20 n-\frac{19}{2}\right]+ \\ &p_{3}\left[U_{n-1}+23 n-\frac{25}{2}\right]+\left(1-p_{1}-p_{2}-p_{3}\right)\left[U_{n-1}+24 n-\frac{27}{2}\right]= \\ &U_{n-1}+\left(24-9 p_{1}-4 p_{2}-p_{3}\right) n+\left(9 p_{1}+4 p_{2}+p_{3}-\frac{27}{2}\right). \end{aligned}$

(5)

将初始值U₁=E[r(u₁|COC(1;p₁, p₂, p₃))]=( $\frac{7}{8} + \frac{3}{2} + \frac{{15}}{8}$ )×2+2= $\frac{{21}}{2}$ 代入式(5), 递归可得式(4). 证毕.

由引理1和引理2, 易得随机环辛烷链的Kirchhoff指数的期望值.

定理1 设COC(n; p₁, p₂, p₃)为具有n (n≥1)个正八边形的随机环辛烷链, 则

$\begin{array}{l} E\left[\operatorname{Kf}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]= \\ \quad \frac{1}{3}\left(96-36 p_{1}-16 p_{2}-4 p_{3}\right) n^{3}+\left(20+36 p_{1}+16 p_{2}+4 p_{3}\right) n^{2}- \\ \quad \frac{1}{3}\left(30+72 p_{1}+32 p_{2}+8 p_{3}\right) n. \end{array}$

(6)

证明对式(1)两边取期望, 由式(4), 有

$\begin{array}{l} E\left[\operatorname{Kf}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]= \\ \quad E\left[\operatorname{Kf}\left(\operatorname{COC}\left(n-1 ; p_{1}, p_{2}, p_{3}\right)\right)\right]+ \\ \quad 8 E\left[r\left(u_{n-1} \mid \operatorname{COC}\left(n-1 ; p_{1}, p_{2}, p_{3}\right)\right)\right]+ \\ \quad 148(n-1)+42=E\left[\operatorname{Kf}\left(\operatorname{COC}\left(n-1 ; p_{1}, p_{2}, p_{3}\right)\right)\right]+ \\ \quad 8 U_{n-1}+148 n-106. \end{array}$

从而

$\begin{array}{l} E\left[\operatorname{Kf}\left(\operatorname{COC}\left(n+1 ; p_{1}, p_{2}, p_{3}\right)\right)\right]= \\ \quad E\left[\operatorname{Kf}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]+\left(96-36 p_{1}-16 p_{2}-4 p_{3}\right) n^{2}+ \\ \quad \left(136+36 p_{1}+16 p_{2}+4 p_{3}\right) n+42 . \end{array}$

(7)

将初始值E[Kf(COC(1;p₁, p₂, p₃))]=42代入式(7), 递归可得式(6)成立. 证毕.

由于COC(n; 1, 0, 0)、COC(n; 0, 1, 0)、COC(n; 0, 0, 1)、COC(n; 0, 0, 0)分别为链CO_n、CM_n、CW_n、CL_n. 将(p₁, p₂, p₃)=(1, 0, 0)、(p₁, p₂, p₃)=(0, 1, 0)、(p₁, p₂, p₃)=(0, 0, 1)、(p₁, p₂, p₃)=(0, 0, 0)分别代入定理1的期望表达式, 易得下面的推论.

推论1 4种特殊环辛烷链CO_n、CM_n、CW_n、CL_n的Kirchhoff指数为

$\begin{array}{c} \mathrm{Kf}\left(\mathrm{CO}_{n}\right)=20 n^{3}+56 n^{2}-34 n ,\\ \mathrm{Kf}\left(\mathrm{CM}_{n}\right)=\frac{80}{3} n^{3}+36 n^{2}-\frac{62}{3} n ,\\ \mathrm{Kf}\left(\mathrm{CW}_{n}\right)=\frac{92}{3} n^{3}+24 n^{2}-\frac{38}{3} n ,\\ \mathrm{Kf}\left(\mathrm{CL}_{n}\right)=32 n^{3}+20 n^{2}-10 n. \end{array}$

下面给出环辛烷链的Kirchhoff指数达到上、下界的极图刻画.

定理2 在所有具有n (n≥3)个正八边形的环辛烷链中, CL_n是具有最大Kirchhoff指数的惟一的环辛烷链, CM_n是具有最小Kirchhoff指数的惟一的环辛烷链.

证明设COC(n; p₁, p₂, p₃)是具有n (n≥3)个正八边形的随机环辛烷链. 由定理1, 有

$\begin{array}{l} f\left(p_{1}, p_{2}, p_{3}\right)=E\left[\operatorname{Kf}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]= \\ \quad \left(-12 n^{3}+36 n^{2}-24 n\right) p_{1}+\left(-\frac{16}{3} n^{3}+16 n^{2}-\frac{32}{3} n\right) p_{2}+ \\ \quad \left(-\frac{4}{3} n^{3}+4 n^{2}-\frac{8}{3} n\right) p_{3}+32 n^{3}+20 n^{2}-10 n . \end{array}$

当n≥3时, 有

$\begin{array}{c} \frac{\partial f}{\partial p_{1}}=-12 n^{3}+36 n^{2}-24 n<0 ,\\ \frac{\partial f}{\partial p_{2}}=-\frac{16}{3} n^{3}+16 n^{2}-\frac{32}{3} n<0 ,\\ \frac{\partial f}{\partial p_{3}}=-\frac{4}{3} n^{3}+4 n^{2}-\frac{8}{3} n<0. \end{array}$

而0≤p₁, p₂, p₃≤1, 0≤p₁+p₂+p₃≤1, 所以f(p₁, p₂, p₃)≤32n³+20n²-10n, 等号成立当且仅当p₁=p₂=p₃=0, 即CL_n是具有最大Kirchhoff指数的惟一的环辛烷链.

另一方面, 有

$\begin{array}{l} f\left(p_{1}, p_{2}, p_{3}\right)=\left(-12 n^{3}+36 n^{2}-24 n\right)\left(p_{1}+p_{2}\right)+ \\ \quad \left(\frac{20}{3} n^{3}-20 n^{2}+\frac{40}{3} n\right) p_{2}+\left(-\frac{4}{3} n^{3}+4 n^{2}-\frac{8}{3} n\right) p_{3}+ \\ \quad 32 n^{3}+20 n^{2}-10 n \geqslant\left(\frac{20}{3} n^{3}-20 n^{2}+\frac{40}{3} n\right) p_{2}+ \\ \quad \left(-\frac{4}{3} n^{3}+4 n^{2}-\frac{8}{3} n\right) p_{3}+20 n^{3}+56 n^{2}-34 n= \\ \quad \left(-\frac{4}{3} n^{3}+4 n^{2}-\frac{8}{3} n\right)\left(p_{2}+p_{3}\right)+\left(\frac{24}{3} n^{3}-24 n^{2}+16 n\right) p_{2}+ \\ \quad 20 n^{3}+56 n^{2}-34 n \geqslant\left(\frac{24}{3} n^{3}-24 n^{2}+16 n\right) p_{2}+\frac{56}{3} n^{3}+ \\ \quad 60 n^{2}-\frac{110}{3} n . \end{array}$

显见

$f\left(p_{1}, p_{2}, p_{3}\right) \geqslant\left(\frac{24}{3} n^{3}-24 n^{2}+16 n\right) p_{2}+\frac{56}{3} n^{3}+60 n^{2}-\frac{110}{3} n.$

(8)

的等号成立当且仅当p₁+p₂=1, p₂+p₃=1且0≤p₁+p₂+p₃≤1, 即(p₁, p₂, p₃)=(0, 1, 0). 将p₂=1代入式(8), 可得f(p₁, p₂, p₃)≥ $\frac{{80}}{3}$ n³+36n²- $\frac{{62}}{3}$ n, 等号成立当且仅当(p₁, p₂, p₃)=(0, 1, 0), 即CM_n是具有最小Kirchhoff指数的惟一的环辛烷链. 证毕.

3. 随机环辛烷链的度积Kirchhoff指数

在有n (n≥2)个正八边形的环辛烷链COC_n中: 第1个和第n个正八边形中恰有1个3度点, 其余7个点都是2度点; 在其余的n-2个正八边形中, 恰有2个3度点, 其余都是2度点; 在O_n中, d(z₁)=3, d(z_i)=2 (2≤i≤8). 所以

$\sum\limits_{v \in V\left(\mathrm{COC}_{n-1}\right)} d_{\mathrm{COC}_{n}}(v)=6(n-1) \times 2+2(n-1) \times 3-1=18 n-19$

且在O_n中,

$\begin{array}{l} \sum\limits_{j=1}^{8} d\left(z_{j}\right) r\left(z_{j}, z_{i}\right)=2 \times \sum\limits_{j=1}^{8} r\left(z_{j}, z_{i}\right)+r\left(z_{1}, z_{i}\right)= \\ \quad \left\{\begin{array}{l} 21\quad (i=1), \\ 21+\frac{7}{8}=\frac{175}{8} \quad(i=2,8), \\ 21+\frac{3}{2}=\frac{45}{2} \quad(i=3,7), \\ 21+\frac{15}{8}=\frac{183}{8} \quad(i=4,6), \\ 21+2=23 \quad(i=5) . \end{array}\right. \end{array}$

(9)

下面求Kf^* (COC(n; p₁, p₂, p₃))的期望值. 记V=V(COC_n), V₁=V(COC_n-1), V₂=V(COC_n-1)\{u_n-1}.

定理3 设COC(n; p₁, p₂, p₃)为具有n (n≥1)个正八边形的随机环辛烷链, 则

$\begin{array}{l} E\left[\mathrm{Kf}^{*}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]= \\ \quad 6\left(27-\frac{81}{8} p_{1}-\frac{9}{2} p_{2}-\frac{9}{8} p_{3}\right) n^{3}+ \\ \quad 18\left(2+\frac{81}{8} p_{1}+\frac{9}{2} p_{2}+\frac{9}{8} p_{3}\right) n^{2}+ \\ \quad 12\left(-\frac{29}{12}-\frac{81}{8} p_{1}-\frac{9}{2} p_{2}-\frac{9}{8} p_{3}\right) n-1 . \end{array}$

(10)

证明令Kf^*(COC_n)=A₁+B₁+C₁, 其中

$\begin{array}{c} A_{1} =\sum\limits_{\{u, v\} \subseteq V_{1}} d(u) d(v) r(u, v), \\ B_{1}=\sum\limits_{v \in V_{1}} \sum\limits_{z_{i} \in V\left(O_{n}\right)} d(v) d\left(z_{i}\right) r\left(v, z_{i}\right) ,\\ C_{1}= \sum\limits_{\left\{z_{i}, z_{j}\right] \subseteq V\left(O_{n}\right)} d\left(z_{i}\right) d\left(z_{j}\right) r\left(z_{i}, z_{j}\right). \end{array}$

由式(2)、(9)及对称性, 类似地, 有

$\begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {A_1} = \sum\limits_{\{ u,v\} \subseteq {V_2}} d (u)d(v)r(u,v) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{v \in {V_2}} {{d_{{\rm{CO}}{{\rm{C}}_n}}}} \left( {{u_{n - 1}}} \right)d(v)r\left( {{u_{n - 1}},v} \right) = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{K}}{{\rm{f}}^*}\left( {{\rm{CO}}{{\rm{C}}_{n - 1}}} \right) + \sum\limits_{v \in {V_1}} d (v)r\left( {{u_{n - 1}},v} \right),\\ {B_1} = \sum\limits_{v \in {V_1}} d (v)\left[ {3\left( {r\left( {v,{u_{n - 1}}} \right) + 1} \right) + 4\left( {r\left( {v,{u_{n - 1}}} \right) + \frac{{15}}{8}} \right) + } \right.\\ \begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 4\left( {r\left( {v,{u_{n - 1}}} \right) + \frac{5}{2}} \right) + 4\left( {r\left( {v,{u_{n - 1}}} \right) + \frac{{23}}{8}} \right) + }\\ {\left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2\left( {r\left( {v,{u_{n - 1}}} \right) + 3} \right)} \right] = 17\sum\limits_{v \in {V_1}} d (v)r\left( {v,{u_{n - 1}}} \right) + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 38(18n - 19),}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {C_1} = \frac{1}{2}\sum\limits_{i = 1}^8 d \left( {{z_i}} \right)\left( {\sum\limits_{j = 1}^8 d \left( {{z_j}} \right)r\left( {{z_j},{z_i}} \right)} \right) = 189.} \end{array} \end{array}$

因此,

$\begin{array}{l} \mathrm{Kf}^{*}\left(\mathrm{COC}_{n}\right)=\mathrm{Kf}^{*}\left(\mathrm{COC}_{n-1}\right)+18 \sum\limits_{v \in V_{1}} d(v) r\left(u_{n-1}, v\right)+ \\ \ \ \ \ \ \ \ \ 684 n-533 . \end{array}$

(11)

而U_n¹=E[ $\sum\limits_{v \in V\left( {C\left( {n;{p_1}, {p_2}, {p_3}} \right)} \right)} {d\left( v \right)r\left( {{u_n}, v} \right)}$ ]. 与U_n类似, 由式(2)、(9)知

$\begin{array}{l} U_{n}^{1}=p_{1} \sum\limits_{v \in V} d(v) r\left(z_{2}, v\right)+p_{2} \sum\limits_{v \in V} d(v) r\left(z_{3}, v\right)+ \\ \quad p_{3} \sum\limits_{v \in V} d(v) r\left(z_{4}, v\right)+\left(1-p_{1}-p_{2}-p_{3}\right) \sum\limits_{v \in V} d(v) r\left(z_{5}, v\right)= \\ \quad p_{1}\left[\sum\limits_{v \in V_{1}} d(v) r\left(u_{n-1}, v\right)+\frac{135}{4} n-\frac{55}{4}\right]+ \\ \quad p_{2}\left[\sum\limits_{v \in V_{1}} d(v) r\left(u_{n-1}, v\right)+45 n-25\right]+ \\ \quad p_{3}\left[\sum\limits_{v \in V_{1}} d(v) r\left(u_{n-1}, v\right)+\frac{207}{4} n-\frac{127}{4}\right]+ \\ \quad \left(1-p_{1}-p_{2}-p_{3}\right)\left[\sum\limits_{v \in V_{1}} d(v) r\left(u_{n-1}, v\right)+54 n-34\right]=U_{n-1}^{1}+ \\ \quad \left(54-\frac{81}{4} p_{1}-9 p_{2}-\frac{9}{4} p_{3}\right) n+\left(-34+\frac{81}{4} p_{1}+9 p_{2}+\frac{9}{4} p_{3}\right). \end{array}$

导入初始值U₁¹= $\sum\limits_{v \in V\left( {{\rm{CO}}{{\rm{C}}_1}} \right)} {d\left( v \right)r\left( {{u_1}, v} \right)}$ =21, 递归可得

$\begin{aligned} U_{n}^{1}=&\left(27-\frac{81}{8} p_{1}-\frac{9}{2} p_{2}-\frac{9}{8} p_{3}\right) n^{2}+\\ &\left(-7+\frac{81}{8} p_{1}+\frac{9}{2} p_{2}+\frac{9}{8} p_{3}\right) n+1. \end{aligned}$

(12)

再由式(11)、(12)可得

$\begin{array}{l} E\left[\operatorname{Kf}^{*}\left(\operatorname{COC}\left(n+1 ; p_{1}, p_{2}, p_{3}\right)\right)\right]= \\ \ \ \ \ E\left[\mathrm{Kf}^{*}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]+18\left(27-\frac{81}{8} p_{1}-\frac{9}{2} p_{2}-\right. \\ \ \ \ \ \left.\frac{9}{8} p_{3}\right) n^{2}+18\left(31+\frac{81}{8} p_{1}+\frac{9}{2} p_{2}+\frac{9}{8} p_{3}\right) n+169, \end{array}$

导入初始值E[Kf^* (COC(1;p₁, p₂, p₃))]=168, 递归可得式(10)成立. 证毕.

将(p₁, p₂, p₃)=(1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 0, 0)代入式(10), 易得下面的推论.

推论2 4种特殊环辛烷链CO_n, CM_n, CW_n, CL_n的度积Kirchhoff指数为

$\begin{array}{c} {\rm{K}}{{\rm{f}}^*}\left( {{\rm{C}}{{\rm{O}}_n}} \right) = \frac{{405}}{4}{n^3} + \frac{{873}}{4}{n^2} - \frac{{301}}{2}n - 1{\rm{ , }}\\ {\rm{K}}{{\rm{f}}^*}\left( {{\rm{C}}{{\rm{M}}_n}} \right) = 135{n^3} + 117{n^2} - 83n - 1,\\ {\rm{K}}{{\rm{f}}^*}\left( {{\rm{C}}{{\rm{W}}_n}} \right) = \frac{{621}}{4}{n^3} + \frac{{225}}{4}{n^2} - \frac{{85}}{2}n - 1,\\ {\rm{K}}{{\rm{f}}^*}\left( {{\rm{C}}{{\rm{L}}_n}} \right) = 162{n^3} + 36{n^2} - 29n - 1. \end{array}$

下面给出环辛烷链的度积Kirchhoff指数达到上、下界的极图刻画.

定理4 在所有具有n (n≥3)个正八边形的环辛烷链中, CL_n是具有最大度积Kirchhoff指数的惟一的环辛烷链, CM_n是具有最小度积Kirchhoff指数的惟一的环辛烷链.

证明设COC(n; p₁, p₂, p₃)为具有n (n≥3)个正八边形的随机环辛烷链. 由定理3, 有

$\begin{array}{l} f_{1}\left(p_{1}, p_{2}, p_{3}\right)=E\left[\mathrm{Kf}^{*}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]= \\ \quad \begin{array}{l} \left(-\frac{243}{4} n^{3}+\frac{729}{4} n^{2}-\frac{243}{2} n\right) p_{1}+\left(-27 n^{3}+81 n^{2}-54 n\right) p_{2}+ \\ \left(-\frac{27}{4} n^{3}+\frac{81}{4} n^{2}-\frac{27}{2} n\right) p_{3}+162 n^{3}+36 n^{2}-29 n-1. \end{array} \end{array}$

当n≥3时, 有

$\begin{array}{c} \frac{\partial f_{1}}{\partial p_{1}}=-\frac{243}{4} n^{3}+\frac{729}{4} n^{2}-\frac{243}{2} n<0 \\ \frac{\partial f_{1}}{\partial p_{2}}=-27 n^{3}+81 n^{2}-54 n<0 ,\\ \frac{\partial f_{1}}{\partial p_{3}}=-\frac{27}{4} n^{3}+\frac{81}{4} n^{2}-\frac{27}{2} n<0. \end{array}$

而0≤p₁, p₂, p₃≤1, 0≤p₁+p₂+p₃≤1, 所以f₁(p₁, p₂, p₃)≤162n³+36n²-29n-1, 等号成立当且仅当p₁=p₂=p₃=0, 即CL_n是具有最大度积Kirchhoff指数的惟一的环辛烷链.

另一方面, 有

$\begin{array}{l} f_{1}\left(p_{1}, p_{2}, p_{3}\right)=\left(-\frac{243}{4} n^{3}+\frac{729}{4} n^{2}-\frac{243}{2} n\right)\left(p_{1}+p_{2}\right)+ \\ \quad \left(\frac{135}{4} n^{3}-\frac{405}{4} n^{2}+\frac{135}{2} n\right) p_{2}+\left(-\frac{27}{4} n^{3}+\frac{81}{4} n^{2}-\frac{27}{2} n\right) p_{3}+ \\ \quad 162 n^{3}+36 n^{2}-29 n-1 \geqslant\left(\frac{135}{4} n^{3}-\frac{405}{4} n^{2}+\frac{135}{2} n\right) p_{2}+ \\ \quad \left(-\frac{27}{4} n^{3}+\frac{81}{4} n^{2}-\frac{27}{2} n\right) p_{3}+\frac{405}{4} n^{3}+\frac{873}{4} n^{2}-\frac{301}{2} n-1= \\ \quad \left(-\frac{27}{4} n^{3}+\frac{81}{4} n^{2}-\frac{27}{2} n\right)\left(p_{2}+p_{3}\right)+\left(\frac{81}{2} n^{3}-\frac{243}{2} n^{2}+81 n\right) p_{2}+ \\ \quad \frac{405}{4} n^{3}+\frac{873}{4} n^{2}-\frac{301}{2} n-1 \geqslant\left(\frac{81}{2} n^{3}-\frac{243}{2} n^{2}+81 n\right) p_{2}+ \\ \quad \frac{189}{2} n^{3}+\frac{477}{2} n^{2}-164 n-1. \end{array}$

显见

$\begin{array}{l} f_{1}\left(p_{1}, p_{2}, p_{3}\right) \geqslant\left(\frac{81}{2} n^{3}-\frac{243}{2} n^{2}+81 n\right) p_{2}+\frac{189}{2} n^{3}+ \\ \ \ \ \ \ \ \frac{477}{2} n^{2}-164 n-1 \end{array}$

(13)

的等号成立当且仅当p₁+p₂=1, p₂+p₃=1且0≤p₁+p₂+p₃≤1, 即(p₁, p₂, p₃)=(0, 1, 0). 将p₂=1代入式(13)可得f₁(p₁, p₂, p₃)≥135n³+117n²-83n-1, 等号成立当且仅当(p₁, p₂, p₃)=(0, 1, 0), 即CM_n是具有最小度积Kirchhoff指数的惟一的环辛烷链. 证毕.

4. 随机环辛烷链的度和Kirchhoff指数

本节求Kf⁺(COC(n; p₁, p₂, p₃))的期望值. 仍记V=V(COC_n), V₁=V(COC_n-1), V₂=V(COC_n-1)\{u_n-1}.

定理5 设COC(n; p₁, p₂, p₃)为具有n (n≥1)个正八边形的随机环辛烷链, 则

$\begin{array}{r} E\left[\mathrm{Kf}^{+}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]=6\left(24-9 p_{1}-4 p_{2}-p_{3}\right) n^{3}+ \\ 18\left(\frac{61}{18}+9 p_{1}+4 p_{2}+p_{3}\right) n^{2}-12\left(\frac{37}{12}+9 p_{1}+4 p_{2}+p_{3}\right) n. \end{array}$

(14)

证明令Kf⁺(COC_n)=A₂+B₂+C₂, 其中, A₂= $\sum\limits_{\left\{ {u, v} \right\} \subseteq {V_1}} {\left( {d\left( u \right) + d\left( v \right)} \right)r\left( {u, v} \right)}$ , B₂= $\sum\limits_{v \in {V_1}} {\sum\limits_{{z_i} \in V\left( {{O_n}} \right)} {\left( {d\left( v \right) + d\left( {{z_i}} \right)} \right)r\left( {v, {z_i}} \right)} }$ , C₂= $\sum\limits_{\left\{ {{z_i}, {z_j}} \right\} \subseteq V\left( {{O_n}} \right)} {\left( {d\left( {{z_i}} \right) + d\left( {{z_j}} \right)} \right)r\left( {{z_i}, {z_j}} \right)}$ .

由式(2)、(9), 类似地, 有

$\begin{array}{l} \quad A_{2}=\sum\limits_{\{u, v\} \subseteq V_{2}}(d(u)+d(v)) r(u, v)+ \\ \quad \quad \quad \sum\limits_{v \in V_{2}}\left(d_{\mathrm{Coc}_{n}}\left(u_{n-1}\right)+d(v)\right) r\left(u_{n-1}, v\right)= \\ \quad \quad \quad \mathrm{Kf}^{+}\left(\mathrm{COC}_{n-1}\right)+r\left(u_{n-1} \mid \mathrm{COC}_{n-1}\right) \\ B_{2}=\sum\limits_{v \in V_{1}} \sum\limits_{z_{i} \in V\left(\sigma_{n}\right)} d(v) r\left(v, z_{i}\right)+\quad \sum\limits_{v \in V_{1}} \sum\limits_{z_{i} \in V\left(O_{n}\right)} d\left(z_{i}\right) r\left(v, z_{i}\right)= \\ \quad \quad 8 \sum\limits_{v \in V_{1}} d(v) r\left(v, u_{n-1}\right)+17 r\left(u_{n-1} \mid \mathrm{COC}_{n-1}\right)+ \\ \quad \quad \frac{37}{2}(18 n-19)+304(n-1), \\ \quad \quad C_{2}=\frac{1}{2} \sum\limits_{i=1}^{8} \sum\limits_{j=1}^{8}\left(d\left(z_{i}\right)+d\left(z_{j}\right)\right) r\left(z_{i}, z_{j}\right)=\frac{357}{2} . \end{array}$

因此,

$\begin{array}{l} \mathrm{Kf}^{+}\left(\mathrm{COC}_{n}\right)=\mathrm{Kf}^{+}\left(\mathrm{COC}_{n-1}\right)+18 r\left(u_{n-1} \mid \mathrm{COC}_{n-1}\right)+ \\ \quad 8 \sum\limits_{v \in V_{1}} d(v) r\left(u_{n-1}, v\right)+637 n-477 . \end{array}$

(15)

记U_n²=E[r(u_n|COC(n; p₁, p₂, p₃))], 与U_n类似, 由式(2)、(3)知

$\begin{aligned} U_{n}^{2}=& r\left(z_{2} \mid \mathrm{COC}_{n}\right) p_{1}+r\left(z_{3} \mid \mathrm{COC}_{n}\right) p_{2}+r\left(z_{4} \mid \mathrm{COC}_{n}\right) p_{3}+\\ & r\left(z_{5} \mid \mathrm{COC}_{n}\right)\left(1-p_{1}-p_{2}-p_{3}\right)=\\ & U_{n-1}^{2}+\left(24-9 p_{1}-4 p_{2}-p_{3}\right) n+\left(-\frac{27}{2}+9 p_{1}+4 p_{2}+p_{3}\right). \end{aligned}$

代入初始值U₁²=21/2, 递归可得

$U_{n}^{2}=\frac{1}{2}\left(24-9 p_{1}-4 p_{2}-p_{3}\right) n^{2}+\frac{1}{2}\left(-3+9 p_{1}+4 p_{2}+p_{3}\right) n .$

(16)

由式(15)、(16), 可得

$\begin{array}{l} E\left[\mathrm{Kf}^{+}\left(\operatorname{COC}\left(n+1 ; p_{1}, p_{2}, p_{3}\right)\right)\right]=\\ \ \ \ \ E\left[\mathrm{Kf}^{+}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]+18\left(24-9 p_{1}-4 p_{2}-p_{3}\right) n^{2}+\\ \ \ \ \ 18\left(\frac{277}{9}+9 p_{1}+4 p_{2}+p_{3}\right) n+168 . \end{array}$

代入初始值E[Kf⁺(COC(1;p₁, p₂, p₃))]=168, 递归可得式(14)成立. 证毕.

将(p₁, p₂, p₃)=(1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 0, 0)分别代入式(14), 易得如下推论.

推论3 4种特殊环辛烷链CO_n, CM_n, CW_n, CL_n的度和Kirchhoff指数为

$\begin{array}{c} \mathrm{Kf}^{+}\left(\mathrm{CO}_{n}\right)=90 n^{3}+223 n^{2}-145 n, \\ \mathrm{Kf}^{+}\left(\mathrm{CM}_{n}\right)=120 n^{3}+133 n^{2}-85 n, \\ \mathrm{Kf}^{+}\left(\mathrm{CW}_{n}\right)=138 n^{3}+79 n^{2}-49 n, \\ \mathrm{Kf}^{+}\left(\mathrm{CL}_{n}\right)=144 n^{3}+61 n^{2}-37 n. \end{array}$

下面给出环辛烷链的度和Kirchhoff指数达到上、下界的极图刻画.

定理6 在所有具有n (n≥3)个正八边形的环辛烷链中, CL_n是具有最大度和Kirchhoff指数的惟一的环辛烷链, CM_n是具有最小度和Kirchhoff指数的惟一的环辛烷链.

证明设COC(n; p₁, p₂, p₃)为具有n (n≥3)个正八边形的随机环辛烷链. 由定理5, 有

$\begin{array}{l} f_{2}\left(p_{1}, p_{2}, p_{3}\right)=E\left[\mathrm{Kf}^{+}\left(\operatorname{COC}\left(n ; p_{1}, p_{2}, p_{3}\right)\right)\right]= \\ \ \ \ \ \left(-54 n^{3}+162 n^{2}-108 n\right) p_{1}+\left(-24 n^{3}+72 n^{2}-48 n\right) p_{2}+ \\ \ \ \ \ \left(-6 n^{3}+18 n^{2}-12 n\right) p_{3}+144 n^{3}+61 n^{2}-37 n. \end{array}$

当n≥3时, 有

$\begin{array}{c} \frac{\partial f_{2}}{\partial p_{1}}=-54 n^{3}+162 n^{2}-108 n<0, \\ \frac{\partial f_{2}}{\partial p_{2}}=-24 n^{3}+72 n^{2}-48 n<0 ,\\ \frac{\partial f_{2}}{\partial p_{3}}=-6 n^{3}+18 n^{2}-12 n<0 . \end{array}$

而0≤p₁, p₂, p₃≤1, 0≤p₁+p₂+p₃≤1, 所以f₂(p₁, p₂, p₃)≤144n³+61n²-37n, 等号成立当且仅当p₁=p₂=p₃=0, 即CL_n是具有最大度和Kirchhoff指数的惟一的环辛烷链.

另一方面, 有

$\begin{array}{l} f_{2}\left(p_{1}, p_{2}, p_{3}\right)=\left(-54 n^{3}+162 n^{2}-108 n\right)\left(p_{1}+p_{2}\right)+ \\ \ \ \ \ \left(30 n^{3}-90 n^{2}+60 n\right) p_{2}+\left(-6 n^{3}+18 n^{2}-12 n\right) p_{3}+ \\ \ \ \ \ 162 n^{3}+144 n^{3}+61 n^{2}-37 n \geqslant\left(30 n^{3}-90 n^{2}+60 n\right) p_{2}+ \\ \ \ \ \ \left(-6 n^{3}+18 n^{2}-12 n\right) p_{3}+90 n^{3}+223 n^{2}-145 n= \\ \ \ \ \ \left(-6 n^{3}+18 n^{2}-12 n\right)\left(p_{2}+p_{3}\right)+\left(36 n^{3}-108 n^{2}+72 n\right) p_{2}+ \\ \ \ \ \ 90 n^{3}+223 n^{2}-145 n \geqslant\left(36 n^{3}-108 n^{2}+72 n\right) p_{2}+84 n^{3}+ \\ \ \ \ \ 241 n^{2}-157 n . \end{array}$

显见

$\begin{array}{l} f_{2}\left(p_{1}, p_{2}, p_{3}\right) \geqslant\left(36 n^{3}-108 n^{2}+72 n\right) p_{2}+ \\ \ \ \ \ 84 n^{3}+241 n^{2}-157 n. \end{array}$

(17)

的等号成立当且仅当p₁+p₂=1, p₂+p₃=1且0≤p₁+p₂+p₃≤1, 即(p₁, p₂, p₃)=(0, 1, 0). 将p₂=1代入式(17)，可得f₂(p₁, p₂, p₃)≥120n³+133n²-85n, 等号成立当且仅当(p₁, p₂, p₃)=(0, 1, 0), 即CM_n是具有最小度和Kirchhoff指数的惟一的环辛烷链. 证毕.

5. 随机环辛烷链的3类Kirchhoff指数的平均值

记 $\mathcal{H}$ _n为所有具有n个正八边形的环辛烷链的集合, 则 $\mathcal{H}$ _n中所有环辛烷链的3类Kirchhoff指数的平均值记为 $\mathcal{H}$ _n的3类Kirchhoff指数的平均值, 其表达式分别为

$\begin{array}{c} \mathrm{Kf}_{\mathrm{avr}}\left(\mathcal{H}_{n}\right)=\frac{1}{\left|\mathcal{H}_{n}\right|} \sum\limits_{G \in \mathcal{H}_{n}} \mathrm{Kf}(G) ,\\ \mathrm{Kf}_{\mathrm{avr}}^{*}\left(\mathcal{H}_{n}\right)=\frac{1}{\left|\mathcal{H}_{n}\right|} \sum\limits_{G \in \mathcal{H}_{n}} \mathrm{Kf}^{*}(G) ,\\ \mathrm{Kf}_{\mathrm{avr}}^{+}\left(\mathcal{H}_{n}\right)=\frac{1}{\left|\mathcal{H}_{n}\right|} \sum\limits_{G \in \mathcal{H}_{n}} \mathrm{Kf}^{+}(G). \end{array}$

由于 $\mathcal{H}$ _n中每条随机链的产生都是等可能的, 故而当(p₁, p₂, p₃, 1-p₁-p₂-p₃)=(1/4, 1/4, 1/4, 1/4)时, 可以得到其平均值. 将(p₁, p₂, p₃)=(1/4, 1/4, 1/4)代入定理1、定理3、定理5的期望表达式, 易得下面的结论.

定理7 环辛烷链集 $\mathcal{H}$ _n的3类Kirchhoff指数的平均值分别为

$\begin{array}{c} \mathrm{Kf}_{\text {avr }}\left(\mathcal{H}_{n}\right)=\frac{82}{3} n^{3}+34 n^{2}-\frac{58}{3} n ,\\ \mathrm{Kf}_{\text {avr }}^{*}\left(\mathcal{H}_{n}\right)=\frac{1107}{8} n^{3}+\frac{855}{8} n^{2}-\frac{305}{4} n-1, \\ \mathrm{Kf}_{\text {avr }}^{+}\left(\mathcal{H}_{n}\right)=123 n^{3}+124 n^{2}-79 n. \end{array}$

我们发现

$\begin{array}{l} \mathrm{K} \mathrm{f}_{\mathrm{av}}\left(\mathcal{H}_{n}\right)=\frac{1}{4}\left(\mathrm{Kf}\left(\mathrm{CO}_{n}\right)+\mathrm{Kf}\left(\mathrm{CM}_{n}\right)+\mathrm{Kf}\left(\mathrm{CW}_{n}\right)+\mathrm{Kf}\left(\mathrm{CL}_{n}\right)\right),\\ \ \ \ \ \ \ \ \ \mathrm{Kf}_{\text {avr }}^{*}\left(\mathcal{H}_{n}\right)=\frac{1}{4}\left(\mathrm{Kf}^{*}\left(\mathrm{CO}_{n}\right)+\mathrm{Kf}^{*}\left(\mathrm{CM}_{n}\right)+\right. \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left.\mathrm{Kf}^{*}\left(\mathrm{CW}_{n}\right)+\mathrm{Kf}^{*}\left(\mathrm{CL}_{n}\right)\right), \\ \mathrm{Kf}_{\text {avr }}^{+}\left(\mathcal{H}_{n}\right)=\frac{1}{4}\left(\mathrm{Kf}^{+}\left(\mathrm{CO}_{n}\right)+\mathrm{Kf}^{+}\left(\mathrm{CM}_{n}\right)+\mathrm{Kf}^{+}\left(\mathrm{CW}_{n}\right)+\right. \\ \ \ \ \ \ \ \ \ \left.\mathrm{Kf}^{+}\left(\mathrm{CL}_{n}\right)\right). \end{array}$

这意味着可以用这4条特殊的链CO_n、CM_n、CW_n、CL_n的Kirchhoff指数(度积Kirchhoff指数、度和Kirchhoff指数)的平均值来表示 $\mathcal{H}$ _n的Kirchhoff指数(度积Kirchhoff指数、度和Kirchhoff指数)整体的平均值.

图 1 实验装置示意图

Figure 1. Schematic diagram of experimental apparatus

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图 2 岩心驱替前后矿物组分变化

Figure 2. Changes of mineral composition before and after core displacement

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图 3 驱替前后岩心孔隙度的变化

Figure 3. Changes of core porosity before and after displacement

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图 4 岩心驱替前后孔隙度变化率

Figure 4. Change rate of porosity before and after core displacement

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图 5 岩心驱替后渗透率变化

Figure 5. Permeability changes after core displacement

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图 6 0.1 mD岩心驱替前后核磁共振T₂谱

Figure 6. Nuclear magnetic resonance T₂ spectra of 0.1 mD core before and after displacement

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图 7 1.0 mD岩心驱替前后核磁共振T₂谱

Figure 7. Nuclear magnetic resonance T₂ spectra of 1.0 mD core before and after displacement

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图 8 10 mD岩心驱替前后核磁共振T₂谱

Figure 8. Nuclear magnetic resonance T₂ spectra of 10 mD core before and after displacement

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图 9 渗透率为0.1 mD时岩心驱替前后孔隙占比变化

Figure 9. Pore ratio changes before and after core displacement with a permeability of 0.1 mD

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图 10 渗透率为1.0 mD时岩心驱替前后孔隙率的变化

Figure 10. Pore ratio changes before and after core displacement with a permeability of 1.0 mD

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图 11 渗透率为10 mD时岩心驱替前后孔隙率的变化

Figure 11. Pore ratio changes before and after core displacement with a permeability of 10 mD

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表 1 实验岩样物性

Table 1 Physical properties of experimental rock samples

岩心编号	直径/cm	长度/cm	孔隙体积/mL	孔隙度/%	渗透率/mD
0.1-2	2.5	6	4.050 6	15.79	0.1
0.1-3			4.030 4	13.69
0.1-4			4.557 0	15.47
0.1-5			4.293 7	14.48
1-2	2.5	6	4.260 7	14.47	1.0
1-3			4.200 3	14.26
1-4			2.074 7	7.04
1-5			3.177 8	10.79
10-2	2.5	6	4.678 5	15.89	10
10-3			4.800 2	16.31
10-4			4.749 3	16.13
10-5			4.719 0	16.03

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表 2 模拟地层水配方

Table 2 Simulated formation water formulations

盐类化学式	摩尔质量/(g·mol^-1)	渗透率为0.1 mD		渗透率为1.0 mD		渗透率为10 mD
盐类化学式	摩尔质量/(g·mol^-1)	c/(mol·L^-1)	m/(g·L^-1)	c/(mol·L^-1)	m/(g·L^-1)	c/(mol·L^-1)	m/(g·L^-1)
MgCl₂-6H₂O	203	0.036 0	7.311 5	0.000 1	0.027 1	0.000 5	0.103 2
CaCl₂	111	0.138 6	15.386 2	0.000 7	0.072 4	0.001 0	0.111 3
NaHCO₃	84	0.003 6	0.301 9	0.033 1	2.777 0	0.049 3	4.144 9
Na₂SO₄	142	0.031 6	4.492 9	0.011 2	1.588 6	0.002 1	0.301 8
NaCl	58.5	1.555 4	90.990 4	0.111 5	6.522 7	0.119 2	6.975 1

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表 3 实验方案

Table 3 Experimental scheme

岩心编号	驱替温度/℃	驱替压力/MPa	驱替流体	注入PV数
0.1-2	无	无	无	0
0.1-3	75.8	19	碳酸水	1
0.1-4	75.8	19	碳酸水	2
0.1-5	75.8	19	碳酸水	6
1-2	无	无	无	0
1-3	75.8	19	碳酸水	1
1-4	75.8	19	碳酸水	2
1-5	75.8	19	碳酸水	6
10-2	无	无	无	0
10-3	75.8	19	碳酸水	1
10-4	75.8	19	碳酸水	2
10-5	75.8	19	碳酸水	6

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图(11) / 表(3)

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1. 预备知识
2. 随机环辛烷链的Kirchhoff指数
3. 随机环辛烷链的度积Kirchhoff指数
4. 随机环辛烷链的度和Kirchhoff指数
5. 随机环辛烷链的3类Kirchhoff指数的平均值

碳酸溶蚀对砂岩储层物性特征及孔隙结构的影响

通讯作者: 窦亮彬，Email: doulb@xsyu.edu.cn

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The Influence of Carbonate Dissolution on Physical Properties and Pore Structure of Sandstone Reservoir

1. 预备知识

2. 随机环辛烷链的Kirchhoff指数

3. 随机环辛烷链的度积Kirchhoff指数

4. 随机环辛烷链的度和Kirchhoff指数

5. 随机环辛烷链的3类Kirchhoff指数的平均值

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目录

1. 预备知识

2. 随机环辛烷链的Kirchhoff指数

3. 随机环辛烷链的度积Kirchhoff指数

4. 随机环辛烷链的度和Kirchhoff指数

5. 随机环辛烷链的3类Kirchhoff指数的平均值

通讯作者:
窦亮彬，Email: doulb@xsyu.edu.cn