Experimental study of non-Darcian flow characteristics in permeable stones

Abstract. This study provides experimental evidence of Forchheimer flow and transition between different flow regimes from the perspective of pore size of permeable stone. We have firstly carried out the seepage experiments of permeable stones with four different mesh sizes, including 24 mesh size, 46 mesh size, 60 mesh size, and 80 mesh size, which corresponding to mean particle sizes (50 % by weight) of 0.71 mm, 0.36 mm, 0.25 mm, and 0.18 mm. The seepage experiments show that obvious deviation from Darcian flow regime is visible. In addition, the critical specific discharge corresponding to the transition of flow regimes (from pre-Darcian to post-Darcian) increases with the increase of particle sizes. When the “pseudo” hydraulic conductivity (K) (which is computed by the ratio of specific discharge and the hydraulic gradient) increases with the increase of specific discharge (q), the flow regime is denoted as the pre-Darcian flow. After the specific discharge increases to a certain value, the “pseudo” hydraulic conductivity begins to decrease, and this regime is called the post-Darcian flow. In addition, we use the mercury injection experiment to measure the pore size distribution of four permeable stones with different particle sizes, and the mercury injection curve is divided into three stages. The beginning and end segments of the mercury injection curve are very gentle with relatively small slopes, while the intermediate mercury injection curve is steep, indicating that the pore size in permeable stones is relatively uniform. The porosity decreases as the mean particle sizes increases, and the mean pore size can faithfully reflect the influence of particle diameter, sorting degree and arrangement mode of porous medium on seepage parameters. This study shows that the size of pores is an essential factor for determining the flow regimes. In addition, the Forchheimer coefficients are also discussed in which the coefficient A (which is related to the linear term of the Forchheimer equation) is linearly related to 1/d 2 as A = 0.0025 (1/d 2) + 0.003; while the coefficient B (which is related to the quadratic term of the Forchheimer equation) is a quadratic function of 1/d as B =1.14E-06 (1/d)2 − 1.26E-06 (1/d). The porosity (n) can be used to reveal the effect of sorting degree and arrangement on seepage coefficient. The larger porosity leads to smaller coefficients A and B under the condition of the same particle size.


different flow regimes from the perspective of pore size of permeable stone. We have firstly 23 carried out the seepage experiments of permeable stones with four different mesh sizes, 24 including 24 mesh size, 46 mesh size, 60 mesh size, and 80 mesh size, which corresponding 25 to mean particle sizes (50% by weight) of 0.71 mm, 0.36 mm, 0.25 mm, and 0.18 mm. The 26 seepage experiments show that obvious deviation from Darcian flow regime is visible. In 27 addition, the critical specific discharge corresponding to the transition of flow regimes (from 28 pre-Darcian to post-Darcian) increases with the increase of particle sizes. When the "pseudo" 29 hydraulic conductivity (K) (which is computed by the ratio of specific discharge and the 30 hydraulic gradient) increases with the increase of specific discharge (q), the flow regime is 31 denoted as the pre-Darcian flow. After the specific discharge increases to a certain value, the 32 "pseudo" hydraulic conductivity begins to decrease, and this regime is called the post-33 Darcian flow. In addition, we use the mercury injection experiment to measure the pore size 34 distribution of four permeable stones with different particle sizes, and the mercury injection 35 curve is divided into three stages. The beginning and end segments of the mercury injection 36 curve are very gentle with relatively small slopes, while the intermediate mercury injection 37 curve is steep, indicating that the pore size in permeable stones is relatively uniform. The 38 porosity decreases as the mean particle sizes increases, and the mean pore size can faithfully 39 reflect the influence of particle diameter, sorting degree and arrangement mode of porous 40 medium on seepage parameters. This study shows that the size of pores is an essential factor 41 for determining the flow regimes. In addition, the Forchheimer coefficients are also discussed 42 in which the coefficient A (which is related to the linear term of the where q is the specific discharge, J is the hydraulic gradient, and K is the hydraulic 55 conductivity. However, when the specific discharge increases above a certain threshold, 56 deviation from Darcy's law is evident and the flow regime changes from Darcian flow regime 57 to the so called non-Darcian flow regime (Bear, 1972), which was first observed by 58 Forchheimer (1901)

Experimental Materials and Procedures 173
Four different particle sizes of permeable stones are selected to carry out the seepage 174 experiment in this study. It is necessary to make a brief overview of the preparation process 175 of permeable stone, which is a type of artificially made tight porous medium formed by sand 176 grains and cementing compound. In the process of preparing permeable stone, a certain 177 particle size of sand and cementing compound is put in a mold, and is consolidated at room 178 temperature. We have carried out the seepage experiments of permeable stones with four 179 different mesh sizes, including 24 mesh size, 46 mesh size, 60 mesh size, and 80 mesh size, 180 and the mesh size is defined as the number of mesh elements (all in square shapes) in a one 181 inch by one inch square, thus a greater number of mesh size implies a smaller particle size. For instance, we can convert above four different mesh sizes of permeable stones into 183 corresponding particle sizes of 0.71 mm, 0.36 mm, 0.25 mm and 0.18 mm, respectively. The 184 pore structure of permeable rock will not change in the process of the seepage experiment 185 under room temperature, and the physical diagrams of four kinds of permeable stones with 186 different particle sizes are shown in Fig. 2  It is worth mentioning that the contact surface of the sample and the plexiglass column 196 is sealed to prevent any preferential flow through the wall of the plexiglass column. After the 197 permeable stone is inserted into the plexiglass column, both ends are sealed with silicone glue. 198 The water passing through the permeable stone is then collected by a cylindrical tank. 199 Moreover, the ratio of the internal diameter of the column to the particle size of permeable 200 stone is greater than 12, which can eliminate any possible wall effect on the seepage 201 according to Beavers et al. (1972). When carrying out the experiment, it usually takes about 202 two hours to saturate the permeable stone. For each packed sample, more than 25 tests with 203 different constant inlet pressures were conducted under steady-state flow condition. In 204 addition, for each group of permeable stone, repeated tests under the same experimental 205 condition were carried out 3-4 times to ensure the accuracy of the results. 206

Permeable stone seepage experiment 208
In this study, we selected permeable stone with four different particle sizes as the 209 research objects, including 24 mesh size, 46 mesh size, 60 mesh size and 80 mesh size. The 210 mesh size is the number of holes per inch of screen mesh and the particle size is inversely 211 proportional to the mesh size. The mean particle sizes corresponding to the four different 212 mesh sizes are 0.71 mm, 0.36 mm, 0.25 mm, and 0.18 mm, respectively, where the mean 213 particle size is corresponding to 50% by weight hereinafter in this study. Such a definition of 214 mean particle size may be different from some other studies such as Fetter (2001) which has 215 used 10% by weight as the mean particle size. The relationship between the specific 216 discharge (q) and the hydraulic gradient (J) of permeable stones is plotted in Fig. 4. The units 217 of specific discharge mentioned in this study are all converted to meters per day (m/d). satisfactory. To analyze the influence of pore size on seepage flow regimes, we have obtained 227 the relationship between q and the "pseudo" hydraulic conductivity (K) (which is computed 228 using q/J) of four permeable stones with different particle sizes, as shown in Fig. 5. We 229 should point out that the "pseudo" hydraulic conductivity term discussed here for non-230 Darcian flow is usually not a constant, thus it is different from the hydraulic conductivity 231 term used in Darcy's law, which is a constant. It is obvious that the hydraulic conductivity is 232 not a constant with the increase of specific discharge, so it is called the "pseudo" hydraulic 233 conductivity . We can divide the q-K curve into two segments: for the first segment, K increases with 237 the increase of q, which is denoted as the pre-Darcian flow. For the second segment, after q 238 increases to a certain value, K begins to decrease with q, which is called the post-Darcian 239 flow. When the hydraulic gradient is small (and q is small as well), a great portion of water is 240 bounded (or becomes immobile) on the surface of solids due to the solid-liquid interfacial 241 force, and only a small fraction of the water is mobile and free to flow through the pores. As 242 the hydraulic gradient increases (and q increases as well), the initial threshold for mobilizing 243 the previously immobile water near the solid-liquid surface is overcome and more water 244 participates in the flow. For this reason, the "pseudo" hydraulic conductivity increases with 245 the increase of hydraulic gradient and the specific discharge in the first segment. When the 246 specific discharge increases to the critical specific discharge (qc), the "pseudo" hydraulic 247 conductivity is maximized. According to we will use the mercury injection experiment to measure the pore size distribution of the four 267 permeable stones with different particle sizes and use the information to describe the flow 268 regimes. 269 To quantitatively study the pore size and pore throat distribution, we need to envisage a 270 physically based conceptual model to describe the pore structures of permeable stones. The 271 commonly used model is the so-called capillary model (Pittman, 1992 where Pc is the capillary force,  is the solid-liquid interfacial tension, θ is the wet angle 277 between the liquid and the solid surface, and r is the radius of curvature in capillary. 278 Since mercury is a nonwetting phase to solids, so to get mercury into the pores of the 279 permeable stone, an external force (or displacement pressure) must be applied to overcome 280 the capillary force. When a greater pressure is applied, mercury can enter smaller pores. 281 When a certain pressure is applied, the injection pressure is equivalent to the capillary 282 pressure in the corresponding pore. Then we can calculate the corresponding capillary radius 283 according to Eq. (3-1), and the volume of mercury injected is the pore volume. pressure and the volume of injected mercury, from which one can also obtain the pore-throat 289 distribution curve and capillary pressure curve. According to the amount of mercury injected 290 at different injection pressures, the relation between the injection pressure and the injection 291 saturation is shown in Fig. 6. 292 Fig. 6 shows that the mercury injection curve can be divided into three stages. Firstly, 293 during the initial stage (A-B) which has a very mild slope, the intake pressure is very small 294 and the intake saturation is also very low. With the increasing of the injection pressure, the 295 intake saturation slowly increases. Secondly, during the intermediate mercury entry stage (B-296 C) which has a steep slope, a small pressure change will lead to a significant saturation 297 change. This means that the pores are relatively uniform and the differences in pore sizes are 298 small. Hence, we can use the pressure ratios of B and C (PC/PB) to reflect the inhomogeneity 299 of the pore size in the porous media. Besides, when the saturation reaches 50%, the 300 corresponding pressure value (P50) reflects the characteristics of the mean pore size, and a 301 larger P50 leads to a larger mean pore size. Finally, during the end stage (C-D) which has a 302 very mild slope as well, the amount of mercury will not increase considerably when the 303 injection pressure increases. This indicates that nearly all the pores are essentially filled with 304 mercury, and the mercury injection experiment is completed. After completing the mercury 305 injection experiments, we have obtained the mercury injection curves of four permeable 306 stones with different particle sizes, as shown in Fig. 7. 307 We can make a number of interesting observations based on Fig. 7. Firstly, the pressure 308 at the starting point (when the saturation begins to increase), denoted as PA, increases as the 309 mean particle size decreases. This means that the maximum pore size in permeable stone 310 decreases with the decrease of the mean particle sizes. Secondly, the mercury injection curves C increase as the mean particle sizes decreases. Moreover, the pressure ratios corresponding 314 to points B and C (PC/PB) also decrease with the decrease of particle sizes, suggesting even 315 more uniform pore size distributions with decreasing particle sizes. Thirdly, the intermediate 316 mercury entry stages gradually shift to the right with the decrease of particle sizes. When the 317 saturation reaches 50%, the corresponding pressure (the median pressure) decreases with the 318 increase of the mean particle sizes. Fourthly, the mercury injection curves of these four 319 permeable stones with different particle sizes all approach 100% saturation with very mild 320 slopes, indicating that there are few small pores in the permeable stones. We have extracted 321 the key pressure characteristic values of mercury injection experiment of Fig. 7, and  To observe the pore size distributions of the four permeable stones with different particle 329 sizes in more details, we can calculate the percentages of different pore sizes in permeable 20 different permeable stones, we best-fit the Gaussian curve of the pore distribution of four 353 permeable stones with different particle sizes, and the best-fitted parameters are shown in 354 Table 2. We can make several interesting observations from Table 2  The pore size distributions fall within ever narrower ranges with mesh sizes become 370 larger. Moreover, the cumulative percentage frequency curves of the pore size distributions 371 with different particle sizes are exhibited in Fig. 12 and the results are shown in Table 3.  Table 3. We find that the 385 porosity decreases as the particle size increases while the mean pore diameter increases. And 386 the mean pore size can reflect the influence of particle diameter, sorting degree and 387 arrangement mode of porous medium on seepage parameters. 388 Note: Rm is the mean pore diameter, R50 is the pore diameter corresponding to the median 390 pressure P50. 391 We can see from Eq. (3-8) that the Jn is inversely proportional to the square of the particle 421 size, and the Jr is inversely proportional to the particle size when the specific discharge 422 remains the same. Both Jn and Jr are closely related to specific surface area and sizes of pores. 423 Therefore, the particle size is an important factor affecting the Forchheimer coefficient, 424 where  and  are constants related to the shape, sorting, and arrangement of the particles, 429 and the specific derivation process is detailed in the previous study (Huang, 2012). The 430 experimental results showed that the coefficient A was inversely proportional to the particle 431 diameter square (d 2 ) and coefficient B was inversely proportional to the particle size (d) 432  Table 4. Furthermore, we can identify nice correlations between the Forchheimer 442 coefficient A and 1/d 2 and between the Forchheimer coefficient B and 1/d, which are shown 443 in Fig. 13 and Fig. 14 That is to say, the relationship between coefficient A and 1/d 2 is consistent with the law of 454 simple cubic arrangement porous media, but the relationship between coefficient B and 1/d is 455 not consistent with the law of simple cubic arrangement porous media. The structure of 456 porous medium arranged in cubes is different from the permeable stone. The porosity of the 457 porous media with spheres arranged in cubic is close to 0.48, independent of the diameter of 458 spheres. While the particle shape, arrangement and tightness of permeable stone are different, 459 and the porosity of permeable stone with different particle size is also different (see Table 3).

Influence of porosity on equation coefficient 462
In above sections, we have analyzed the influence of particle sizes on seepage 463 coefficient. Furthermore, the pore size and pore specific surface area are also related to the 464 arrangement and sorting degree of particles, that is, to the porosity of porous media. To 465 explore the effect of sorting degree on seepage coefficient, we draw a schematic diagram of 466  Fig. 15. The degree of particle sorting is one 467 of the important factors affecting the pore size. In porous media with a poor sorting degree, 468 the pore size is usually determined by the diameter of the smallest particle. We can see from 469 Furthermore, we have also provided the schematic diagrams of spherical particles with 476 equal size in two simple arrangements, namely cube arrangement and hexahedron 477 arrangement, as shown in Fig. 16. And the cube arrangement is the less compact arrangement 478 with a pore diameter of 0.414d, while the hexahedron arrangement is the more compact 479 arrangement with a pore diameter of 0.155d. The characteristic value of pore structure in 480 different arrangement with the same particle size are shown in Table 5. We can see that 481 different arrangement modes will substantially affect the pore specific surface area and pore 482 size of porous media. The more compactly packed particles lead to the larger pore specific 483 surface area and stronger viscous force. Meanwhile, the smaller pore diameter is associated 484 with stronger effect of viscous force and inertia force. In summary, the better sorting degree 485 of particles leads to the weaker viscous and inertial forces, then the coefficients A and B will 486 be smaller. As the better sorting degree and the less compact (or looser) arrangement particles 487 mean the larger porosity, so we can conclude that the larger porosity leads to the smaller 488 coefficients A and B under the condition of the same particle size. 489 490 Fig. 16. The schematic diagram of particle arrangement in different types. 491 However, the structure of natural porous media is much more complex and 494 heterogeneous than what has been shown in Figure 16, so it is difficult to quantitatively 495 describe the effect of sorting degree and arrangement on seepage law. 496 Fig. 17. Variation of A with n of six gravels with different particle sizes. 498 In view of this, we can use a macro parameter porosity (n) to reveal the effect of sorting 499 degree and arrangement on seepage coefficient. In order to verify the correctness of the above 500 analysis results, we selected the seepage experiment results of Niranjan (1973)  are discussed. In addition, the mercury injection experiment is proposed to investigate the 516 pore distribution of the permeable stones. In addition, the Forchheimer coefficients are 517 specifically discussed. The main conclusions can be summarized as follows: 518 1) The relationships between specific discharge (q) and the "pseudo" hydraulic conductivity 519 (K) (which is computed as a ratio of q and hydraulic gradient, J) of permeable stones show 520 that deviation from Darcian flow regime is clearly visible. In addition, the critical specific 521 discharge corresponding to the transition of flow regimes (from pre-Darcian to post-Darcian) 522 increases with the increase of mean particle size. 523 2) When the specific discharge is small, only a small fraction of the water flowing through 524 the pores. The rest of the water adheres to the surface of the solid particles (immobile), 525 partially blocking the flow pathways. As the specific discharge increases, more water 526 becomes mobile and participates in flow. Hence, the "pseudo" hydraulic conductivity 527 increases with the increase of specific discharge. When the specific discharge increases to the 528 critical specific discharge (qc), the "pseudo" hydraulic conductivity is maximized, and then it 529 begins to decrease as the specific discharge continues to increase. 530 3) The mercury injection experiment results show that the mercury injection curve can be 531 divided into three segments. The beginning and end segments of the mercury injection curve 532 of the four permeable stones with different particle sizes are very gentle, while the main (or 4) The porosity decreases as the mean particle size of permeable stone increases while the 536 mean pore diameter increases. And the porosity can reflect the influence of particle diameter, 537 sorting degree and arrangement mode of porous medium on seepage parameters. The larger 538 porosity leads to the smaller coefficients A and B under the condition of the same particle size. 539 5) The coefficient A is linearly related to 1/d 2 and the relationship between coefficient A and 540 1/d 2 is given as The specific discharge, m/d. 546

K
The Hydraulic conductivity, m/d. 547

J
The dimensionless parameter defined as hydraulic gradient. 548

Pc
The capillary force, Pa. 551

P50
The corresponding pressure value when the saturation reaches 50%, MPa. 552 PA, PB, PC The pressure corresponding to different stages on mercury injection curve, MPa. 553 σ The solid-liquid interfacial tension. 554 θ The wet angle between the liquid and the solid surface. 555 r The radius of curvature in capillary, mm. 556 d The particle size, mm. 557 https://doi.org/10.5194/hess-2021-588 Preprint. Discussion started: 30 November 2021 c Author(s) 2021. CC BY 4.0 License.