# Experimental Development Process of a New Fluid–Solid Coupling Similar-Material Based on the Orthogonal Test

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Method

#### 2.1. Materials

#### 2.1.1. Fluid–Solid Coupling Similarity Theory

_{x}, K

_{y}, K

_{z}are the permeability coefficients in x, y, and z directions (cm/s), K

_{x}= K

_{y}= K

_{z}; p is the water pressure (MPa); S is the storage coefficient; e is the volume strain; W is the source sink term.

_{ij,i}is the stress tensor; X

_{j}is the volume force (N/m

^{3}); ρ is the density (g/cm

^{3}); u

_{i}is the displacement (cm).

_{ij}is the total stress tensor; $\overline{{\sigma}_{ij}}$ is the effective stress tensor; α is the effective stress coefficient of Biolt; δ is the Kronker mark; ρ is the density (g/cm

^{3}).

_{G}, C

_{u}, C

_{l}, C

_{λ}, C

_{γ}, C

_{e}, C

_{ρ}, and C

_{t}are the similarity ratios of the shear modulus, displacement, model size, Lame constant, bulk density, volumetric strain, density, and time, respectively.

_{l}, C

_{λ}, and C

_{k}are the similarity ratios of the model size, Lame constant, and permeability coefficient, respectively.

#### 2.1.2. Similar-Material Components

#### 2.2. Methodology

#### 2.2.1. Orthogonal Test Schemes of Similar-Material Proportion

_{25}(5

^{4}). The level values of each factor were set as input in the orthogonal experimental design module of SPSS software, leading to the schemes, as listed in Table 2.

#### 2.2.2. Fabricating Specimens

- (1)
- Aggregate, cementing agent and regulator were weighed proportionately.
- (2)
- The aggregate and cement were mixed evenly, followed by adding water 0.5 times of cement.
- (3)
- Vaseline was heated to a liquid state and poured into the above mixture.
- (4)
- Antiwear hydraulic oil was added and stirred.
- (5)
- The well-mixed materials were loaded into a mold and compacted. The mold for testing specimen tensile strength is a PVC tube with a height of 25 mm and an inner diameter of 45 mm (Figure 2h).
- (6)
- Demolded and labeled, specimens were maintained for three days at room temperature.

#### 2.2.3. Testing Index Parameters of Specimens

_{water}is the specimen weight after immersion (g); m

_{dry}is the specimen weight before immersion (g).

_{c}and brittle behavior (BB)

_{t}

_{v}is the failure load value of the specimen (N); and L

_{h}is the thickness of the specimen (mm).

^{2}); A is the specimen sectional area (cm

^{2}); L is the specimen height (cm); $\Delta {h}_{1}$, $\Delta {h}_{2}$ are initial water head difference and water head difference after t time (cm), respectively. In Table 3, the variation range of the permeability coefficient of similar material specimen is 8.79 × 10

^{−8}–2.95 × 10

^{−4}cm/s.

## 3. Results and Discussion

#### 3.1. Results

#### 3.1.1. Density Analysis

_{jm}is the sum of the experimental indexes corresponding to the j factor, m level is in the range analysis. K

_{jm}

_{-a}is the average value of K

_{jm}and R

_{j}is the range of the j column factor, reflecting the variation range of the test index. The larger the R

_{j}, the greater the effect of the factors on the test indicators, which can determine the primary and secondary factors.

_{1}of factor A are as follows: K

_{A}

_{1}= 1.766 + 1.761 + 1.884 + 1.879 + 1.832 = 9.122, K

_{A}

_{1}

_{-a}= K

_{A}

_{1}/5 = 1.824. Similarly, K

_{A}

_{2}= 9.346, K

_{A}

_{2}

_{-a}= K

_{A}

_{2}/5 = 1.869; K

_{A}

_{3}= 9.013, K

_{A}

_{3}

_{-a}= K

_{A}

_{3}/5 = 1.803; K

_{A}

_{4}= 9.058, K

_{A}

_{4}

_{-a}= K

_{A}

_{4}/5 = 1.812; K

_{A}

_{5}=8.703, K

_{A}

_{5}

_{-a}= K

_{A}

_{5}/5 = 1.741.

_{A}= K

_{A}

_{2}

_{-a}− K

_{A}

_{5}

_{-a}= 0.128, R

_{B}= 0.047, R

_{C}= 0.082, R

_{D}= 0.036, as shown in Table 4.

_{A}> R

_{C}> R

_{B}> R

_{D}. Therefore, the order of the factors that affects the specimen density is A > C > B > D.

_{e}), the percentage of the regression model error in the total error (R-Sq) and the adjusted R-Sq. R-Sq is used to show that the model is in line with the data, and the larger the value, the better the regression model and the data. The larger the R-Sq value, the better the fit between the regression model and the data.

_{e}= 0.0632119, R-Sq = 80.46%, the adjusted R-Sq = 75.39%, verifying the reliability of the similar-material density regression model.

#### 3.1.2. Compressive Strength Analysis

_{C}> R

_{A}> R

_{D}> R

_{B}. Therefore, the order of the factors affecting the specimen density is C > A > D > B. Figure 6 shows the intuitive analysis chart of effective factors of specimen compressive strength. Specimen compressive strength increases when increasing the percentage of river sand in the aggregate and the mass ratio of cement and vaseline.

_{c}= −2.04562 + 3.2768A − 0.13096B + 0.469376C − 5.75D

_{e}= 0.173737, R-Sq =94.31%, the adjusted R-Sq =82.94%, indicating the reliability of the similar-material compressive strength regression model.

#### 3.1.3. Tensile Strength Analysis

_{C}> R

_{A}> R

_{D}> R

_{B}. Therefore, the order of the factors affecting the specimen density is C > A > D > B. Figure 7 shows the intuitive analysis chart of factors affecting the specimen tensile strength. Specimen tensile strength increases with increasing the percentage of river sand in aggregate and the mass ratio of cement and vaseline.

_{t}= −0.19218 + 0.2916A − 0.00784B + 0.0439676C − 0.424D

_{e}= 0.0146990, R-Sq = 95.06%, the adjusted R-Sq = 85.17%, confirming the reliability of the similar-material tensile strength regression model.

#### 3.1.4. Permeability Coefficient Analysis

_{D}> R

_{C}> R

_{B}> R

_{A}. Therefore, the order of the factors affecting the specimen density is D > C > B > A. Figure 8 shows the intuitive analysis chart of influence factors affecting the specimen permeability coefficient. With increasing the percentage of antiwear hydraulic oil in the total mass of similar materials, the mass ratio of cement and vaseline, and the mass ratio of calcium carbonate and talc powder, the permeability coefficient first increases and then decreases.

_{e}= 0.0000412355, R-Sq = 86.20%, the adjusted R-Sq = 78.60%, indicating the reliability of similar-material permeability coefficient regression model.

#### 3.2. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The raw materials of similar-material: (

**a**) River sand; (

**b**) Calcium carbonate; (

**c**) Talc powder; (

**d**) White cement; (

**e**) Vaseline; (

**f**) Antiwear hydraulic oil.

**Figure 2.**The fabrication process of similar-material specimen: (

**a**) Weighing; (

**b**) Stirring; (

**c**) Heating vaseline; (

**d**) Adding hydraulic oil; (

**e**) Compacting; (

**f**) Demolding; (

**g**) Partial specimen; (

**h**) PVC tube.

**Figure 5.**Intuitive analysis chart of density influencing factor: (

**a**) Factor A; (

**b**) Factor B; (

**c**) Factor C; (

**d**) Factor D.

**Figure 6.**Intuitive analysis chart of compressive strength influencing factor: (

**a**) Factor A; (

**b**) Factor B; (

**c**) Factor C; (

**d**) Factor D.

**Figure 7.**Intuitive analysis chart of tensile strength influencing factor: (

**a**) Factor A; (

**b**) Factor B; (

**c**) Factor C; (

**d**) Factor D.

**Figure 8.**Intuitive analysis chart of permeability coefficient influencing factor: (

**a**) Factor A; (

**b**) Factor B; (

**c**) Factor C; (

**d**) Factor D.

Level | Factors | |||
---|---|---|---|---|

A (%) | B | C | D (%) | |

1 | 75 | 3:4 | 3:7 | 2 |

2 | 80 | 4:4 | 4:6 | 3 |

3 | 85 | 5:4 | 5:5 | 4 |

4 | 90 | 6:4 | 6:4 | 5 |

5 | 95 | 7:4 | 7:3 | 6 |

Schemes | A (%) | B | C | D (%) |
---|---|---|---|---|

S1 | 1 (75) | 1 (3:4) | 1 (3:7) | 1 (2) |

S2 | 1 | 2 (4:4) | 2 (4:6) | 2 (3) |

S3 | 1 | 3 (5:4) | 3 (5:5) | 3 (4) |

S4 | 1 | 4 (6:4) | 4 (6:4) | 4 (5) |

S5 | 1 | 5 (7:4) | 5 (7:3) | 5 (6) |

S6 | 2 (80) | 1 | 2 | 3 |

S7 | 2 | 2 | 3 | 4 |

S8 | 2 | 3 | 4 | 5 |

S9 | 2 | 4 | 5 | 1 |

S10 | 2 | 5 | 1 | 2 |

S11 | 3 (85) | 1 | 3 | 5 |

S12 | 3 | 2 | 4 | 1 |

S13 | 3 | 3 | 5 | 2 |

S14 | 3 | 4 | 1 | 3 |

S15 | 3 | 5 | 2 | 4 |

S16 | 4 (90) | 1 | 4 | 2 |

S17 | 4 | 2 | 5 | 3 |

S18 | 4 | 3 | 1 | 4 |

S19 | 4 | 4 | 2 | 5 |

S20 | 4 | 5 | 3 | 1 |

S21 | 5 (95) | 1 | 5 | 4 |

S22 | 5 | 2 | 1 | 5 |

S23 | 5 | 3 | 2 | 1 |

S24 | 5 | 4 | 3 | 2 |

S25 | 5 | 5 | 4 | 3 |

**Table 3.**The index parameters of specimens (PrePEM: pre-peak elastic modulus; PostPEM: post peak elastic modulus; BB: brittle behavior; RTC: the ratio of uniaxial tensile strength and uniaxial compressive strength).

Schemes | σ_{c} (MPa) | PrePEM (MPa) | PostPEM (MPa) | BB | σ_{t} (MPa) | RTC | ρ (g/cm^{3}) | K (cm/s) |
---|---|---|---|---|---|---|---|---|

S1 | 0.268 | 24.15 | 9.05 | 2.67 | 0.028 | 1/9.6 | 1.766 | 3.15 × 10^{−5} |

S2 | 0.228 | 22.39 | 10.38 | 2.16 | 0.021 | 1/10.9 | 1.761 | 2.08 × 10^{−5} |

S3 | 0.250 | 23.06 | 9.00 | 2.56 | 0.025 | 1/10.0 | 1.884 | 2.95 × 10^{−4} |

S4 | 0.278 | 24.55 | 9.15 | 2.68 | 0.026 | 1/10.7 | 1.879 | 2.85 × 10^{−6} |

S5 | 0.461 | 40.31 | 14.21 | 2.84 | 0.048 | 1/9.6 | 1.832 | 1.23 × 10^{−6} |

S6 | 0.272 | 24.38 | 9.33 | 2.61 | 0.024 | 1/11.3 | 1.778 | 1.09 × 10^{−4} |

S7 | 0.400 | 43.92 | 15.17 | 2.90 | 0.033 | 1/12.1 | 1.851 | 2.58 × 10^{−6} |

S8 | 0.435 | 51.20 | 19.89 | 2.57 | 0.042 | 1/10.4 | 1.886 | 8.79 × 10^{−5} |

S9 | 0.859 | 103.13 | 43.00 | 2.40 | 0.095 | 1/9.1 | 1.892 | 3.25 × 10^{−6} |

S10 | 0.472 | 45.18 | 16.09 | 2.81 | 0.050 | 1/9.4 | 1.939 | 2.06 × 10^{−6} |

S11 | 0.500 | 55.23 | 18.79 | 2.94 | 0.045 | 1/11.1 | 1.731 | 8.13 × 10^{−5} |

S12 | 1.005 | 98.76 | 34.58 | 2.86 | 0.101 | 1/10.0 | 1.837 | 8.45 × 10^{−6} |

S13 | 1.103 | 112.40 | 42.00 | 2.68 | 0.099 | 1/11.1 | 1.843 | 1.25 × 10^{−7} |

S14 | 0.521 | 39.87 | 15.05 | 2.65 | 0.046 | 1/11.3 | 1.809 | 5.17 × 10^{−5} |

S15 | 0.365 | 40.26 | 16.35 | 2.46 | 0.040 | 1/9.1 | 1.792 | 3.09 × 10^{−5} |

S16 | 0.910 | 106.90 | 40.61 | 2.63 | 0.080 | 1/11.4 | 1.895 | 2.00 × 10^{−7} |

S17 | 1.223 | 121.20 | 49.88 | 2.43 | 0.102 | 1/12.0 | 1.894 | 1.56 × 10^{−7} |

S18 | 0.538 | 40.26 | 16.35 | 2.46 | 0.053 | 1/10.2 | 1.728 | 4.25 × 10^{−5} |

S19 | 0.502 | 42.29 | 17.23 | 2.45 | 0.053 | 1/9.5 | 1.755 | 8.09 × 10^{−6} |

S20 | 0.786 | 55.23 | 18.79 | 2.94 | 0.070 | 1/11.2 | 1.786 | 1.02 × 10^{−6} |

S21 | 1.311 | 135.60 | 52.40 | 2.59 | 0.111 | 1/11.8 | 1.736 | 8.79 × 10^{−8} |

S22 | 0.531 | 39.83 | 15.05 | 2.65 | 0.057 | 1/9.3 | 1.798 | 7.59 × 10^{−6} |

S23 | 0.656 | 45.92 | 17.17 | 2.67 | 0.058 | 1/11.3 | 1.722 | 9.93 × 10^{−5} |

S24 | 0.715 | 66.20 | 29.20 | 2.23 | 0.079 | 1/9.1 | 1.725 | 1.21 × 10^{−6} |

S25 | 1.116 | 108.69 | 43.60 | 2.49 | 0.101 | 1/11.1 | 1.722 | 2.29 × 10^{−7} |

Factors | A | B | C | D | Sum of Test Results | |
---|---|---|---|---|---|---|

ρ | K_{1} | 9.122 | 8.906 | 9.040 | 9.003 | ∑ = 45.241 |

K_{2} | 9.346 | 9.141 | 8.808 | 9.163 | ||

K_{3} | 9.013 | 9.063 | 8.977 | 9.087 | ||

K_{4} | 9.058 | 9.060 | 9.219 | 8.986 | ||

K_{5} | 8.703 | 9.071 | 9.197 | 9.002 | ||

K_{1}_{-a} | 1.824 | 1.781 | 1.808 | 1.801 | ||

K_{2}_{-a} | 1.869 | 1.828 | 1.762 | 1.833 | ||

K_{3}_{-a} | 1.803 | 1.813 | 1.795 | 1.817 | ||

K_{4}_{-a} | 1.812 | 1.812 | 1.844 | 1.797 | ||

K_{5}_{-a} | 1.741 | 1.814 | 1.839 | 1.800 | ||

R | 0.128 | 0.047 | 0.082 | 0.036 | ||

σ_{c} | K_{1} | 1.485 | 3.261 | 2.330 | 3.574 | ∑ = 20.705 |

K_{2} | 2.438 | 3.387 | 2.023 | 3.428 | ||

K_{3} | 3.494 | 2.982 | 2.651 | 3.382 | ||

K_{4} | 3.959 | 2.875 | 3.744 | 2.892 | ||

K_{5} | 4.329 | 3.200 | 4.957 | 2.429 | ||

K_{1}_{-a} | 0.297 | 0.652 | 0.466 | 0.715 | ||

K_{2}_{-a} | 0.488 | 0.677 | 0.405 | 0.686 | ||

K_{3}_{-a} | 0.699 | 0.596 | 0.530 | 0.676 | ||

K_{4}_{-a} | 0.792 | 0.575 | 0.749 | 0.578 | ||

K_{5}_{-a} | 0.866 | 0.640 | 1.191 | 0.686 | ||

R | 0.378 | 0.102 | 0.586 | 0.229 | ||

σ_{t} | K_{1} | 0.148 | 0.288 | 0.234 | 0.352 | ∑ = 1.487 |

K_{2} | 0.244 | 0.314 | 0.196 | 0.329 | ||

K_{3} | 0.331 | 0.277 | 0.252 | 0.298 | ||

K_{4} | 0.358 | 0.299 | 0.350 | 0.263 | ||

K_{5} | 0.406 | 0.309 | 0.455 | 0.245 | ||

K_{1}_{-a} | 0.0296 | 0.0576 | 0.0468 | 0.0704 | ||

K_{2}_{-a} | 0.0488 | 0.0628 | 0.0392 | 0.0658 | ||

K_{3}_{-a} | 0.0662 | 0.0554 | 0.0504 | 0.0596 | ||

K_{4}_{-a} | 0.0716 | 0.0598 | 0.07 | 0.0526 | ||

K_{5}_{-a} | 0.0812 | 0.0618 | 0.091 | 0.049 | ||

R | 0.0516 | 0.0074 | 0.0518 | 0.0214 | ||

K | K_{1} | 3.51 × 10^{−4} | 2.22 × 10^{−4} | 1.35 × 10^{−4} | 1.44 × 10^{−4} | ∑ = 1.19 × 10^{−3} |

K_{2} | 2.05 × 10^{−4} | 3.96 × 10^{−5} | 2.68 × 10^{−4} | 2.44 × 10^{−5} | ||

K_{3} | 1.72 × 10^{−4} | 5.25 × 10^{−4} | 6.81 × 10^{−4} | 7.56 × 10^{−4} | ||

K_{4} | 5.20 × 10^{−5} | 6.71 × 10^{−5} | 9.96 × 10^{−5} | 7.89 × 10^{−5} | ||

K_{5} | 1.08 × 10^{−5} | 3.54 × 10^{−5} | 4.85 × 10^{−6} | 1.86 × 10^{−4} | ||

K_{1}_{-a} | 7.02 × 10^{−5} | 4.44 × 10^{−5} | 2.70 × 10^{−5} | 2.88 × 10^{−5} | ||

K_{2}_{-a} | 4.10 × 10^{−5} | 7.92 × 10^{−6} | 5.36 × 10^{−5} | 4.88 × 10^{−6} | ||

K_{3}_{-a} | 3.44 × 10^{−5} | 1.05 × 10^{−4} | 1.36 × 10^{−4} | 1.51 × 10^{−4} | ||

K_{4}_{-a} | 1.04 × 10^{−5} | 1.34 × 10^{−5} | 1.99 × 10^{−5} | 1.58 × 10^{−5} | ||

K_{5}_{-a} | 2.16 × 10^{−6} | 7.08 × 10^{−6} | 9.70 × 10^{−7} | 3.72 × 10^{−5} | ||

R | 7.00 × 10^{−5} | 9.79 × 10^{−5} | 1.35 × 10^{−4} | 1.46 × 10^{−4} |

**Table 5.**The variance analysis of similar-material density (Seq SS: the sum of the squares of deviations; Adj SS: adjusted squares sum of deviations; Adj MS: adjusted squares sum of mean-square error).

Variance Sources | Free Degree | SeqSS | Adj SS | Adj MS | F | p |
---|---|---|---|---|---|---|

A | 4 | 0.042940 | 0.042940 | 0.010735 | 2.69 | 0.109 |

B | 4 | 0.005942 | 0.005942 | 0.001486 | 0.37 | 0.823 |

C | 4 | 0.022829 | 0.022829 | 0.005707 | 1.43 | 0.309 |

D | 4 | 0.004546 | 0.004546 | 0.001137 | 0.28 | 0.880 |

Error | 8 | 0.031966 | 0.031966 | 0.003996 |

Stratum | ρ (g/cm^{3}) | σ_{c} (MPa) | σ_{t} (MPa) | K (cm/s) | |
---|---|---|---|---|---|

Mudstone | Protolith | 1.815 | 54.25 | 9.65 | 3.55 × 10^{−6} |

Model | 1.801 | 0.261 | 0.048 | 2.52 × 10^{−7} | |

Sandstone | Protolith | 1.903 | 92.30 | 16.25 | 1.69 × 10^{−5} |

Model | 1.893 | 0.458 | 0.081 | 1.20 × 10^{−6} |

Stratum | A (%) | B | C | D (%) | Sand: Calcium Carbonate: Talc Powder: White Cement: Vaseline: Antiwear Hydraulic Oil |
---|---|---|---|---|---|

Mudstone | 87.45 | 1.27 | 1.19 | 4.78 | 12.46:1.00:0.79:0.69:0.89:0.79 |

Sandstone | 75.32 | 1.43 | 1.74 | 3.65 | 5.19:1.00:0.70:0.51:0.29:0.29 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Liu, S.; Liu, W. Experimental Development Process of a New Fluid–Solid Coupling Similar-Material Based on the Orthogonal Test. *Processes* **2018**, *6*, 211.
https://doi.org/10.3390/pr6110211

**AMA Style**

Liu S, Liu W. Experimental Development Process of a New Fluid–Solid Coupling Similar-Material Based on the Orthogonal Test. *Processes*. 2018; 6(11):211.
https://doi.org/10.3390/pr6110211

**Chicago/Turabian Style**

Liu, Shiliang, and Weitao Liu. 2018. "Experimental Development Process of a New Fluid–Solid Coupling Similar-Material Based on the Orthogonal Test" *Processes* 6, no. 11: 211.
https://doi.org/10.3390/pr6110211