**Figure 1.**
Schematic of capillary pores

$\phi $, lined with C-S-H [

62].

**Figure 1.**
Schematic of capillary pores

$\phi $, lined with C-S-H [

62].

**Figure 2.**
Schematic diagram of (**a**) oil well cementing and (**b**) cement slurry flow in a tilted channel.

**Figure 2.**
Schematic diagram of (**a**) oil well cementing and (**b**) cement slurry flow in a tilted channel.

**Figure 3.**
Effect of ${\varphi}_{m}$ on (**a**) the velocity; and (**b**) the volume fraction profiles, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.4,$\text{}{R}_{0}=$ 0.1, ${R}_{1}=$ −2.5,$\text{}{R}_{2}=$ 0.1, ${R}_{3}=$ 0.01, ${R}_{4}=$ 0.07,$\text{}{R}_{5}=$ 1.8, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, $m=$ 1, $\theta =$ 45^{o}.

**Figure 3.**
Effect of ${\varphi}_{m}$ on (**a**) the velocity; and (**b**) the volume fraction profiles, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.4,$\text{}{R}_{0}=$ 0.1, ${R}_{1}=$ −2.5,$\text{}{R}_{2}=$ 0.1, ${R}_{3}=$ 0.01, ${R}_{4}=$ 0.07,$\text{}{R}_{5}=$ 1.8, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, $m=$ 1, $\theta =$ 45^{o}.

**Figure 4.**
Distribution of the velocity for (**a**) ${\varphi}_{m}$ = 0.45; (**b**) ${\varphi}_{m}$= 0.50; (**c**) ${\varphi}_{m}$= 0.55; (**d**) ${\varphi}_{m}$= 0.60; (**e**) ${\varphi}_{m}$= 0.65; and the cement volume concentration for (**f**) ${\varphi}_{m}$= 0.45; (**g**) ${\varphi}_{m}$= 0.50; (**h**) ${\varphi}_{m}$= 0.55; (**i**) ${\varphi}_{m}$= 0.60; (**j**) ${\varphi}_{m}$= 0.65.

**Figure 4.**
Distribution of the velocity for (**a**) ${\varphi}_{m}$ = 0.45; (**b**) ${\varphi}_{m}$= 0.50; (**c**) ${\varphi}_{m}$= 0.55; (**d**) ${\varphi}_{m}$= 0.60; (**e**) ${\varphi}_{m}$= 0.65; and the cement volume concentration for (**f**) ${\varphi}_{m}$= 0.45; (**g**) ${\varphi}_{m}$= 0.50; (**h**) ${\varphi}_{m}$= 0.55; (**i**) ${\varphi}_{m}$= 0.60; (**j**) ${\varphi}_{m}$= 0.65.

**Figure 5.**
Effect of $\frac{{K}_{c}}{{K}_{\mu}}$ on (**a**) the velocity; and (**b**) the volume fraction profiles, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3,$\text{}{R}_{0}=$ 0.1, ${R}_{1}=$ −2.5,$\text{}{R}_{2}=$ 0.1, ${R}_{3}=$ 0.01, ${R}_{4}=$ 0.07,$\text{}{R}_{5}=$ 1.8, ${\varphi}_{m}=$ 0.65, $m=$ 1, $\theta =$ 45^{o}.

**Figure 5.**
Effect of $\frac{{K}_{c}}{{K}_{\mu}}$ on (**a**) the velocity; and (**b**) the volume fraction profiles, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3,$\text{}{R}_{0}=$ 0.1, ${R}_{1}=$ −2.5,$\text{}{R}_{2}=$ 0.1, ${R}_{3}=$ 0.01, ${R}_{4}=$ 0.07,$\text{}{R}_{5}=$ 1.8, ${\varphi}_{m}=$ 0.65, $m=$ 1, $\theta =$ 45^{o}.

**Figure 6.**
Distribution of the velocity for (**a**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0; (**b**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0.02; (**c**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0.04; (**d**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0.06; (**e**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0.08; and the cement volume concentration for (**f**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0; (**g**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0.02; (**h**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0.04; (**i**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0.06; (**j**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0.08.

**Figure 6.**
Distribution of the velocity for (**a**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0; (**b**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0.02; (**c**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0.04; (**d**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0.06; (**e**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0.08; and the cement volume concentration for (**f**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0; (**g**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0.02; (**h**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0.04; (**i**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0.06; (**j**) $\frac{{K}_{c}}{{K}_{\mu}}$ = 0.08.

**Figure 7.**
Effect of $\theta $ on (**a**) the velocity; and (**b**) the volume fraction profiles, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3,$\text{}{R}_{0}=$ 0.1, ${R}_{1}=$ -2.5,$\text{}{R}_{2}=$ 10, ${R}_{3}=$ 0.01, ${R}_{4}=$ 0.07,$\text{}{R}_{5}=$ 1.8, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, ${\varphi}_{m}=$ 0.65, $m=$ 1.

**Figure 7.**
Effect of $\theta $ on (**a**) the velocity; and (**b**) the volume fraction profiles, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3,$\text{}{R}_{0}=$ 0.1, ${R}_{1}=$ -2.5,$\text{}{R}_{2}=$ 10, ${R}_{3}=$ 0.01, ${R}_{4}=$ 0.07,$\text{}{R}_{5}=$ 1.8, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, ${\varphi}_{m}=$ 0.65, $m=$ 1.

**Figure 8.**
Distribution of the velocity for (**a**) $\theta $ = 0°; (**b**) $\theta $ = 30°; (**c**) $\theta $ = 45°; (**d**) $\theta $ = 60°; (**e**) $\theta $ = 90°; and the cement volume concentration for (**f**) $\theta $ = 0°; (**g**) $\theta $ = 30°; (**h**) $\theta $ = 45°; (**i**) $\theta $ = 60°; (**j**) $\theta $ = 90°.

**Figure 8.**
Distribution of the velocity for (**a**) $\theta $ = 0°; (**b**) $\theta $ = 30°; (**c**) $\theta $ = 45°; (**d**) $\theta $ = 60°; (**e**) $\theta $ = 90°; and the cement volume concentration for (**f**) $\theta $ = 0°; (**g**) $\theta $ = 30°; (**h**) $\theta $ = 45°; (**i**) $\theta $ = 60°; (**j**) $\theta $ = 90°.

**Figure 9.**
Effect of m on (**a**) the velocity; and (**b**) the volume fraction profiles, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3,$\text{}{R}_{0}=$ 0.1, ${R}_{1}=$ −2.5,$\text{}{R}_{2}=$ 10, ${R}_{3}=$ 0.01, ${R}_{4}=$ 0.07,$\text{}{R}_{5}=$ 1.8, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, ${\varphi}_{m}=$ 0.65, $\theta =$ 90°.

**Figure 9.**
Effect of m on (**a**) the velocity; and (**b**) the volume fraction profiles, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3,$\text{}{R}_{0}=$ 0.1, ${R}_{1}=$ −2.5,$\text{}{R}_{2}=$ 10, ${R}_{3}=$ 0.01, ${R}_{4}=$ 0.07,$\text{}{R}_{5}=$ 1.8, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, ${\varphi}_{m}=$ 0.65, $\theta =$ 90°.

**Figure 10.**
Effect of ${R}_{0}$ on (**a**) the velocity; and (**b**) the volume fraction profiles, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3, ${R}_{1}=$ −2.5,$\text{}{R}_{2}=$ 0.1, ${R}_{3}=$ 0.01, ${R}_{4}=$ 0.07,$\text{}{R}_{5}=$ 1.8, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, ${\varphi}_{m}=$ 0.65, $m=$ 1, $\theta =$ 45^{o}.

**Figure 10.**
Effect of ${R}_{0}$ on (**a**) the velocity; and (**b**) the volume fraction profiles, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3, ${R}_{1}=$ −2.5,$\text{}{R}_{2}=$ 0.1, ${R}_{3}=$ 0.01, ${R}_{4}=$ 0.07,$\text{}{R}_{5}=$ 1.8, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, ${\varphi}_{m}=$ 0.65, $m=$ 1, $\theta =$ 45^{o}.

**Figure 11.**
Distribution of the velocity for (**a**) ${R}_{0}$ = 0.01; (**b**) ${R}_{0}$ = 0.1; (**c**) ${R}_{0}$ = 1; (**d**) ${R}_{0}$ = 10; and the cement volume concentration for (**e**) ${R}_{0}$ = 0.01; (**f**) ${R}_{0}$ = 0.1; (**g**) ${R}_{0}$ = 1; (**h**) ${R}_{0}$ = 10.

**Figure 11.**
Distribution of the velocity for (**a**) ${R}_{0}$ = 0.01; (**b**) ${R}_{0}$ = 0.1; (**c**) ${R}_{0}$ = 1; (**d**) ${R}_{0}$ = 10; and the cement volume concentration for (**e**) ${R}_{0}$ = 0.01; (**f**) ${R}_{0}$ = 0.1; (**g**) ${R}_{0}$ = 1; (**h**) ${R}_{0}$ = 10.

**Figure 12.**
Effect of ${R}_{1}$ on (**a**) the velocity; and (**b**) the volume fraction profiles, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3, ${R}_{0}=$ 0.1,$\text{}{R}_{2}=$ 0.1, ${R}_{3}=$ 0.01, ${R}_{4}=$ 0.07,$\text{}{R}_{5}=$ 1.8, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, ${\varphi}_{m}=$ 0.65, $m=$ 1, $\theta =$ 45°.

**Figure 12.**
Effect of ${R}_{1}$ on (**a**) the velocity; and (**b**) the volume fraction profiles, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3, ${R}_{0}=$ 0.1,$\text{}{R}_{2}=$ 0.1, ${R}_{3}=$ 0.01, ${R}_{4}=$ 0.07,$\text{}{R}_{5}=$ 1.8, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, ${\varphi}_{m}=$ 0.65, $m=$ 1, $\theta =$ 45°.

**Figure 13.**
Distribution of the velocity for (**a**) ${R}_{1}$ = 0; (**b**) ${R}_{1}$ = −1.5; (**c**) ${R}_{1}$ = −2.5; (**d**) ${R}_{1}$ = −3.5; and the cement volume concentration for (**e**) ${R}_{1}$ = 0; (**f**) ${R}_{1}$ = −1.5; (**g**) ${R}_{1}$ = −2.5; (**h**) ${R}_{1}$ = −3.5.

**Figure 13.**
Distribution of the velocity for (**a**) ${R}_{1}$ = 0; (**b**) ${R}_{1}$ = −1.5; (**c**) ${R}_{1}$ = −2.5; (**d**) ${R}_{1}$ = −3.5; and the cement volume concentration for (**e**) ${R}_{1}$ = 0; (**f**) ${R}_{1}$ = −1.5; (**g**) ${R}_{1}$ = −2.5; (**h**) ${R}_{1}$ = −3.5.

**Figure 14.**
Effect of ${R}_{1}$ on (**a**) the velocity; and (**b**) the volume fraction profiles, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3, ${R}_{0}=$ 0.1, ${R}_{1}=$ −2.5, ${R}_{3}=$ 0.01, ${R}_{4}=$ 0.07, ${R}_{5}=$ 1.8, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, ${\varphi}_{m}=$ 0.65, $m=$ 1, $\theta =$ 45°.

**Figure 14.**
Effect of ${R}_{1}$ on (**a**) the velocity; and (**b**) the volume fraction profiles, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3, ${R}_{0}=$ 0.1, ${R}_{1}=$ −2.5, ${R}_{3}=$ 0.01, ${R}_{4}=$ 0.07, ${R}_{5}=$ 1.8, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, ${\varphi}_{m}=$ 0.65, $m=$ 1, $\theta =$ 45°.

**Figure 15.**
(**a**) Effect of ${R}_{3}$ on the velocity and the volume fraction profiles, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3,$\text{}{R}_{0}=$ 0.1, ${R}_{1}=$ −2.5,$\text{}{R}_{2}=$ 0.1, ${R}_{4}=$ 0.07,$\text{}{R}_{5}=$ 1.8, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, ${\varphi}_{m}=$ 0.65, $m=$ 1, $\theta =$ 45°; (**b**) effect of ${R}_{4}$ on the velocity and volume fraction profile, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3, ${R}_{0}=$ 0.1, ${R}_{1}=$ −2.5,$\text{}{R}_{2}=$ 0.1, ${R}_{3}=$ 0.01, ${R}_{5}=$ 1.8, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, ${\varphi}_{m}=$ 0.65, $m=$ 1, $\theta =$ 45°; and (**c**) effect of ${R}_{5}$ on the velocity and volume fraction profile, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3,$\text{}{R}_{0}=$ 0.1, ${R}_{1}=$ −2.5,$\text{}{R}_{2}=$ 0.1, ${R}_{3}=$ 0.01, ${R}_{4}=$ 0.07, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, ${\varphi}_{m}=$ 0.65, $m=$ 1, $\theta =$ 45°.

**Figure 15.**
(**a**) Effect of ${R}_{3}$ on the velocity and the volume fraction profiles, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3,$\text{}{R}_{0}=$ 0.1, ${R}_{1}=$ −2.5,$\text{}{R}_{2}=$ 0.1, ${R}_{4}=$ 0.07,$\text{}{R}_{5}=$ 1.8, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, ${\varphi}_{m}=$ 0.65, $m=$ 1, $\theta =$ 45°; (**b**) effect of ${R}_{4}$ on the velocity and volume fraction profile, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3, ${R}_{0}=$ 0.1, ${R}_{1}=$ −2.5,$\text{}{R}_{2}=$ 0.1, ${R}_{3}=$ 0.01, ${R}_{5}=$ 1.8, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, ${\varphi}_{m}=$ 0.65, $m=$ 1, $\theta =$ 45°; and (**c**) effect of ${R}_{5}$ on the velocity and volume fraction profile, with $\beta =$ 1.82, ${\varphi}_{avg}=$ 0.3,$\text{}{R}_{0}=$ 0.1, ${R}_{1}=$ −2.5,$\text{}{R}_{2}=$ 0.1, ${R}_{3}=$ 0.01, ${R}_{4}=$ 0.07, $\frac{{K}_{c}}{{K}_{\mu}}=$ 0.05, ${\varphi}_{m}=$ 0.65, $m=$ 1, $\theta =$ 45°.

**Table 1.**
Values of the dimensionless numbers and other parameters.

**Table 1.**
Values of the dimensionless numbers and other parameters.

Parameters | Range of Values |
---|

${\varphi}_{m}$ | 0.45, 0.5, 0.55, 0.6, 0.65 |

${K}_{c}/{K}_{\mu}$ | 0, 0.02, 0.04, 0.06, 0.08 |

$\theta $ | 0°, 30°, 45°, 60°, 90° |

$m$ | −0.3, −0.1, 0, 0.1, 0.3, 0.7 |

${R}_{0}$ | 0.01, 0.1, 1, 10 |

${R}_{1}$ | 0, −1.5, −2.5, −3.5 |

${R}_{2}$ | 0, 0.5, 1, 1.5 |

${R}_{3}$ | 0.01, 0.1, 1 |

${R}_{4}$ | 0.01, 0.1, 1 |

${R}_{5}$ | 0.01, 0.1, 1 |