A Simple Water Retention Model Based on Grain Size Distribution
Abstract
:1. Introduction
2. Key Phenomena for Water Retention Curve Models
2.1. Pore Dimension and Pore Filling
2.2. Contact Angle between Water and Soil Surface
3. Proposed Model
- Achieve reasonably accurate predictions without requiring calibration to WRC data;
- Preferably use input parameters that are relatively easy to measure;
- Consider pore filling;
- Be capable of considering variation in contact angle with respect to particle size.
- The GSD of a material is used to calculate its clay, silt, and sand and larger fractions.
- The sizes in the GSD are then adjusted using an aspect ratio, such that smaller particles (e.g., clay or silt) are reduced in size to a greater extent than larger particles.
- A series of primary pore sizes are then created, by assuming the particles in each particle size bin of the adjusted GSD pack together to create a total volume of pores of a proportional size.
- The volume of primary pores associated with each pore size is estimated by assuming the pore volume fraction is proportional to the mass fraction of the associated particle size bin. That is, for a particle size bin accounting for 10% of a soil’s mass, the volume of pores associated with this bin account for 10% of its total pore volume, as determined from its adopted saturated volumetric water content.
- A set of criteria are applied to each primary pore size to determine what degree of pore filling should occur within each pore size. Any required pore filling is done so to achieve the packing densities at different levels of Apollonian packing.
- Whether pores are unfilled, partially filled, or filled is determined by considering whether there is a sufficient volume of smaller particle fractions that could fill each set of primary pores in a binary mixture to achieve the packing density corresponding to Apollonian packing.
- Having determined the degree of filling for each set of primary pores, they are then redistributed (as required) into a larger number of smaller pores, according to the appropriate Apollonian geometry, and combined to give a final pore size distribution.
- Each pore size in the final distribution is transformed to an equivalent area circular pore, with a drainage suction, creating the final pore size distribution.
- The predicted WRC is the theoretical WRC of the pore size distribution, which is estimated by assuming a number of arbitrary suctions, and calculating the volumetric water content of the water filled or undrained pores.
3.1. Inferring a Primary Pore Size Distribution from Grain Size Distribution (GSD)
3.2. Primary Pore Filling
3.2.1. Smallest Particle Criterion
3.2.2. Filling of Pores Associated with Clay Sized Particles
3.2.3. Filling of Pores Associated with Silt Sized Particles
- What is the size ratio that defines a suitably smaller particle, to form a binary mixture with a particular degree of filling (packing density)? That is, what size particles (dss,i) are small enough to effectively fill the pores between the larger particles di,adjusted?
- What is the maximum fraction xi of larger particles di,adjusted that still ensures that there are enough smaller particles (dss,i) to achieve the assumed packing density in the coarser particle fraction? That is, how to decide when there are too few smaller particles to effectively fill the pores between larger particles?
3.2.4. Filling of Pores Associated with Sand and Larger Sized Particles
3.3. Pore Size Distribution to Water Retention Curve
3.4. Summary of Model Inputs and Outputs
- Sand fraction;
- Silt fraction;
- Clay fraction;
- Saturated volumetric water content;
- Receding contact angle;
- Particle size bins from the grain size distribution (GSD).
4. Materials and Methods
4.1. Materials
4.2. Methods
4.3. Validation of the Proposed WRC Prediction Model
5. Results and Discussion
Retention Curves of the Mine Materials
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
Material | Gs | Liquid Limit | Plastic Limit | Receding CA (°) | Initial Porosity | Source |
---|---|---|---|---|---|---|
Tailings BE | 2.65 2 | 0.17 | - 1 | 0 2 | 0.403 | [58,59] |
Silty clay | 2.65 2 | 0.404 | 0.248 | 0 2 | 0.438 | [60] |
Silt Loam 10861 | 2.65 2 | - 1 | - 1 | 0 2 | 0.402 | [61] |
Loam 1260 | 2.77 | - 1 | - 1 | 0 2 | 0.4 | UNSODA [62] |
Maryland clay | 2.65 | 0.7 | 0.24 | 0 2 | 0.563 | [63] 3 |
Loamy sand 10741 | 2.65 2 | N/A | N/A | 0 2 | 0.355 | [61] |
Grenoble sand | 2.65 2 | N/A | N/A | 0 2 | 0.385 | [13] |
Appendix B
Material | Sand Fraction | Silt Fraction | Clay Fraction | Receding CA (°) | Saturated Volumetric Water Content, θs |
---|---|---|---|---|---|
BT2 topsoil | 0.28 | 0.54 | 0.18 | 3 | 0.52 |
M1 tailings | 0.12 | 0.82 | 0.06 | Varied 1 | 0.441 |
M1 topsoil | 0.38 | 0.48 | 0.14 | 6 | 0.39 |
PD tailings | 0.3 | 0.64 | 0.06 | Varied 1 | 0.45 |
Tailings BE | 0.3 | 0.65 | 0.05 | 0 2 | 0.403 |
Silty clay | 0.31 | 0.35 | 0.34 | 0 2 | 0.438 |
Silt Loam 10861 | 0.06 | 0.78 | 0.16 | 0 2 | 0.402 |
Loam 1260 | 0.36 | 0.52 | 0.12 | 0 2 | 0.4 |
Maryland clay | 0 | 0.29 | 0.71 | 0 2 | 0.563 |
Loamy sand 10741 | 0.78 | 0.15 | 0.7 | 0 2 | 0.355 |
Grenoble sand | 1 | 0 | 0 | 0 2 | 0.385 |
Material | Initial Void Ratio, e | Saturated Volumetric Water Content, θs | Receding CA (°) |
---|---|---|---|
BT2 topsoil | 1.083 | 0.52 | 3 |
M1 tailings | 0.789 | 0.441 | Varied 1 |
M1 topsoil | 0.652 | 0.39 | 6 |
PD tailings | 0.82 | 0.45 | Varied 1 |
Tailings BE | 0.674 | 0.403 | 0 2 |
Silty clay | 0.78 | 0.438 | 0 2 |
Silt Loam 10861 | 0.67 | 0.402 | 0 2 |
Loam 1260 | 0.665 | 0.4 | 0 2 |
Maryland clay | 1.29 | 0.563 | 0 2 |
Loamy sand 10741 | 0.55 | 0.355 | 0 2 |
Grenoble sand | 0.625 | 0.385 | 0 2 |
Material | Initial Void Ratio, e | Specific Gravity, Gs | α | Receding CA (°) |
---|---|---|---|---|
BT2 topsoil | 1.083 | 2.63 | 1.35 | 3 |
M1 tailings | 0.789 | 1.96 | 1.35 | Varied 1 |
M1 topsoil | 0.652 | 2.66 | 1.35 | 6 |
PD tailings | 0.82 | 2.22 | 1.35 | Varied 1 |
Tailings BE | 0.674 | 2.65 2 | 1.35 | 0 2 |
Silty clay | 0.78 | 2.65 2 | 1.35 | 0 2 |
Silt Loam 10861 | 0.67 | 2.65 2 | 1.35 | 0 2 |
Loam 1260 | 0.665 | 2.77 | 1.35 | 0 2 |
Maryland clay | 1.29 | 2.65 | 1.35 | 0 2 |
Loamy sand 10741 | 0.55 | 2.65 2 | 1.5 | 0 2 |
Grenoble sand | 0.625 | 2.65 2 | 1.5 | 0 2 |
Material | Clay Fraction | β | Saturated Volumetric Water Content, θs | Receding CA (°) |
---|---|---|---|---|
BT2 topsoil | 0.18 | 0.516 | 0.52 | 3 |
M1 tailings | 0.06 | 0.516 | 0.441 | 0 1 |
M1 topsoil | 0.14 | 0.516 | 0.39 | 6 |
PD tailings | 0.06 | 0.516 | 0.45 | 0 1 |
Tailings BE | 0.05 | 0.516 | 0.403 | 0 2 |
Silty clay | 0.34 | 0.516 | 0.438 | 0 2 |
Silt Loam 10861 | 0.16 | 0.516 | 0.402 | 0 2 |
Loam 1260 | 0.12 | 0.516 | 0.4 | 0 2 |
Maryland clay | 0.71 | 0.516 | 0.563 | 0 2 |
Loamy sand 10741 | 0.7 | 0.516 | 0.355 | 0 2 |
Grenoble sand | 0 | 0.516 | 0.385 | 0 2 |
Material | Initial Void Ratio, e | Specific Gravity, Gs | Saturated Volumetric Water Content, θs | Receding CA (°) |
---|---|---|---|---|
BT2 topsoil | 1.083 | 2.63 | 0.52 | 3 |
M1 tailings | 0.789 | 1.96 | 0.441 | Varied 1 |
M1 topsoil | 0.652 | 2.66 | 0.39 | 6 |
PD tailings | 0.82 | 2.22 | 0.45 | Varied 1 |
Tailings BE | 0.674 | 2.65 2 | 0.403 | 0 2 |
Silty clay | 0.78 | 2.65 2 | 0.438 | 0 2 |
Silt Loam 10861 | 0.67 | 2.65 2 | 0.402 | 0 2 |
Loam 1260 | 0.665 | 2.77 | 0.4 | 0 2 |
Maryland clay | 1.29 | 2.65 | 0.563 | 0 2 |
Loamy sand 10741 | 0.55 | 2.65 2 | 0.355 | 0 2 |
Grenoble sand | 0.625 | 2.65 2 | 0.385 | 0 2 |
Material | D10 (μm) | D60 (μm) | ac | Liquid Limit | Specific Gravity, Gs | m | Initial Void Ratio, e | λ (m2/kg) | Receding CA (°) |
---|---|---|---|---|---|---|---|---|---|
BT2 topsoil | N/A | N/A | 0.0007 | 0.462 | 2.63 | 0.00003 | 1.083 | 200 | 3 |
M1 tailings | N/A | N/A | 0.0007 | 0.497 | 1.96 | 0.00003 | 0.789 | 200 | Varied 1 |
M1 topsoil | N/A | N/A | 0.0007 | 0.274 | 2.66 | 0.00003 | 0.652 | 200 | 6 |
PD tailings | N/A | N/A | 0.0007 | 0.482 | 2.22 | 0.00003 | 0.82 | 200 | Varied 1 |
Tailings BE | N/A | N/A | 0.0007 | 0.17 | 2.652 | 0.00003 | 0.674 | 200 | 0 2 |
Silty clay | N/A | N/A | 0.0007 | 0.404 | 2.652 | 0.00003 | 0.78 | 200 | 0 2 |
Silt Loam 10861 | - 4 | - 4 | - 4 | - 3,4 | - 4 | - 4 | - 4 | - 4 | - 4 |
Loam 1260 | 1 | 50 | 0.01 | N/A | 2.77 | N/A | 0.665 | 200 | 0 2 |
Maryland clay | N/A | N/A | 0.0007 | 0.7 | 2.65 | 0.00003 | 1.29 | 200 | 0 2 |
Loamy sand 10741 | 7.4 | 265 | 0.01 | N/A | 2.652 | N/A | 0.55 | 200 | 0 2 |
Grenoble sand | 140 | 310 | 0.01 | N/A | 2.652 | N/A | 0.625 | 200 | 0 2 |
Material | Saturated Volumetric Water Content, θs | ξ1 | p | β1 | β2 | ψmax (kPa) | θr_max | α1 | α2 | λ | Receding CA (°) |
---|---|---|---|---|---|---|---|---|---|---|---|
BT2 topsoil | 0.52 | 11.07 | 2 | 0.09 | 0.95 | 1600 | 0.25 | 16.02 | 2.01 | 2 | 3 |
M1 tailings | 0.441 | 11.07 | 2 | 0.09 | 0.95 | 1600 | 0.25 | 16.02 | 2.01 | 2 | Varied 1 |
M1 topsoil | 0.39 | 11.07 | 2 | 0.09 | 0.95 | 1600 | 0.25 | 16.02 | 2.01 | 2 | 6 |
PD tailings | 0.45 | 11.07 | 2 | 0.09 | 0.95 | 1600 | 0.25 | 16.02 | 2.01 | 2 | Varied 1 |
Tailings BE | 0.403 | 11.07 | 2 | 0.09 | 0.95 | 1600 | 0.25 | 16.02 | 2.01 | 2 | 0 2 |
Silty clay | 0.438 | 11.07 | 2 | 0.09 | 0.95 | 1600 | 0.25 | 16.02 | 2.01 | 2 | 0 2 |
Silt Loam 10861 | 0.402 | 11.07 | 2 | 0.09 | 0.95 | 1600 | 0.25 | 16.02 | 2.01 | 2 | 0 2 |
Loam 1260 | 0.4 | 11.07 | 2 | 0.09 | 0.95 | 1600 | 0.25 | 16.02 | 2.01 | 2 | 0 2 |
Maryland clay | 0.563 | 11.07 | 2 | 0.09 | 0.95 | 1600 | 0.25 | 16.02 | 2.01 | 2 | 0 2 |
Loamy sand 10741 | - 3 | - 3 | - 3 | - 3 | - 3 | - 3 | - 3 | - 3 | - 3 | - 3 | - 3 |
Grenoble sand | - 3 | - 3 | - 3 | - 3 | - 3 | - 3 | - 3 | - 3 | - 3 | - 3 | - 3 |
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Material | Gs | Liquid Limit | Plastic Limit | Receding CA (°) |
---|---|---|---|---|
BT2 topsoil | 2.63 | 0.462 | 0.372 | 3 |
M1 tailings | 1.96 | 0.497 | 0.353 | 25 |
M1 topsoil | 2.66 | 0.274 | 0.191 | 6 |
PD tailings | 2.22 | 0.482 | 0.208 | 21 |
M1 coal particles | - * | - * | - * | 47 |
M1 mineral particles | - * | - * | - * | 4 |
PD coal particles | - * | - * | - * | 44 |
PD mineral particles | - * | - * | - * | 15 |
Model | Input Parameters | Equation Related to Volumetric Water Content | Equation for Suction or Suction Head | Outputs | Source |
---|---|---|---|---|---|
Proposed | Sand, silt and clay fraction, θs, GSD | A PSD and its theoretical WRC | N/A | ||
MV | e, θs, GSD | Volumetric water content and suction head for each GSD bin | [16] | ||
AP | e, Gs, α, GSD | Volumetric water content and suction head for each GSD bin | [11] | ||
CCQ | Clay fraction, β, θs, GSD | Volumetric water content and suction head for a range of GSD bins | [12] | ||
NSMC | e, Gs, θs, GSD | Volumetric water content and suction head for pores between an arbitrary particle size are water filled | [20] | ||
MK | D10, D60, ac, LL, Gs, m, e, λ | N/A | Saturation degree at an arbitrary suction | [70] | |
IMP | θs, ξ1, p, β1, β2, ψmax, θr_max, α1, α2, λ, GSD | Volumetric water content and suction for each GSD bin | [71] |
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Vidler, A.; Buzzi, O.; Fityus, S. A Simple Water Retention Model Based on Grain Size Distribution. Appl. Sci. 2021, 11, 9452. https://doi.org/10.3390/app11209452
Vidler A, Buzzi O, Fityus S. A Simple Water Retention Model Based on Grain Size Distribution. Applied Sciences. 2021; 11(20):9452. https://doi.org/10.3390/app11209452
Chicago/Turabian StyleVidler, Andrew, Olivier Buzzi, and Stephen Fityus. 2021. "A Simple Water Retention Model Based on Grain Size Distribution" Applied Sciences 11, no. 20: 9452. https://doi.org/10.3390/app11209452
APA StyleVidler, A., Buzzi, O., & Fityus, S. (2021). A Simple Water Retention Model Based on Grain Size Distribution. Applied Sciences, 11(20), 9452. https://doi.org/10.3390/app11209452