# 8.15: Graded Sands and Gravels

## 8.15.1 Introduction

The im curve for graded sands and gravels can be determined by:

1. Determine the fraction fines X. The limiting particle diameter for the fines can be determined with equation (7.13-2).
2. Determine the PSD and split the PSD in n fractions. Correct the PSD so the fines are not part of the PSD anymore. Adjust the volumetric concentration by deducting the fines fraction X.
3. Adjust the pseudo liquid dynamic viscosity μx, density ρx and relative submerged density Rsd,x for the presence of fines and determine the resulting hydraulic gradient curve for the pseudo liquid, il,x.
4. Determine curves related to the pseudo liquid.
1. Determine the im,x,i curve for each ith fraction individually for both the spatial volumetric concentration and the delivered volumetric concentration, using the adjusted pseudo liquid properties.
2. Sum the im,x,i curves for the n fractions multiplied by the fraction fi to determine the total hydraulic gradient im,x, for both the spatial volumetric concentration and the delivered volumetric concentration, in the pseudo liquid.
3. Determine the resulting Erhg,x curve, for both the spatial volumetric concentration and the delivered volumetric concentration.
5. Determine curves related to the carrier liquid.
1. Determine the im,i curves for the n fractions by multiplying the im,x,i curves by the ratio of the pseudo liquid density to the carrier liquid density ρxl.
2. Sum the im,i curves for the n fractions multiplied by the fraction fi to determine the total hydraulic gradient im, for both the spatial volumetric concentration and the delivered volumetric concentration, in the carrier liquid.
3. Determine the resulting Erhg curve, for both the spatial volumetric concentration and the delivered volumetric concentration.
6. Determine the bed fraction curves for each fraction multiplied by the fraction fi.
7. Sum the bed fraction curves for the n fractions multiplied by the fraction fi to obtain the total bed fraction curve.
8. Determine the slip ratio curves for each fraction multiplied by the fraction fi.
9. Sum the slip ratio curves for the n fractions multiplied by the fraction fi to obtain the total slip ratio curve.

Step 2 needs some clarification. Suppose we have a sand with 3 fractions, each 1/3 by weight. The first fraction consists of fines, the second fraction of particles with a d=0.5 mm and the third fraction of particles with d=1 mm. The spatial volumetric concentration of the sand in the carrier liquid is 30%.

Now the fines form a pseudo homogeneous liquid together with the carrier liquid. So in terms of solids effect they do not take part in the solids effect and have to be removed from the PSD. What is left is a PSD with 50% particles with a d=0.5 mm and 50% particles with d=1 mm. The spatial volumetric concentration of this sand is now 20%. So the hydraulic gradients have to be determined for this remaining sand and not for the original sand.

If a sand does not contain fines, the liquid properties and the PSD do not have to be adjusted.

## 8.15.2 The Adjusted Pseudo Liquid Properties

First the limiting particle diameter is determined, based on a Stokes number of 0.03. The value of 0.03 is found based on many experiments from literature. Since the Stokes number depends on the line speed, here the Limit Deposit Velocity is used as an estimate of the operational line speed.

The LDV is approximated by:

$\ \mathrm{v}_{\mathrm{l s}, \mathrm{ld v}}=\mathrm{7 .5} \cdot \mathrm{D}_{\mathrm{p}}^{\mathrm{0 . 4}}$

Giving for the limiting particle diameter:

$\ \mathrm{d}_{\mathrm{lim}}=\sqrt{\frac{\mathrm{S t k} \cdot \mathrm{9} \cdot \rho_{\mathrm{l}} \cdot v_{\mathrm{l}} \cdot \mathrm{D}_{\mathrm{p}}}{\rho_{\mathrm{s}} \cdot \mathrm{v}_{\mathrm{ls}, \mathrm{ldv}}}} \approx \sqrt{\frac{\mathrm{S t k} \cdot \mathrm{9} \cdot \rho_{\mathrm{l}} \cdot v_{\mathrm{l}} \cdot \mathrm{D}_{\mathrm{p}}}{\rho_{\mathrm{s}} \cdot 7.5 \cdot \mathrm{D}_{\mathrm{p}}^{0.4}}}$

The fraction of the sand in suspension, resulting in a homogeneous pseudo fluid is named X. This gives for the density of the homogeneous pseudo fluid:

$\ \rho_{\mathrm{x}}=\rho_{\mathrm{l}}+\rho_{\mathrm{l}} \cdot \frac{\mathrm{X} \cdot \mathrm{C}_{\mathrm{v} \mathrm{s}} \cdot \mathrm{R}_{\mathrm{s} \mathrm{d}}}{\left(1-\mathrm{C}_{\mathrm{v s}}+\mathrm{C}_{\mathrm{v s}} \cdot \mathrm{X}\right)} \quad\text{ if }\mathrm{X}=\mathrm{1} \quad \Rightarrow \quad \rho_{\mathrm{x}}=\rho_{\mathrm{m}}=\rho_{\mathrm{l}}+\rho_{\mathrm{l}} \cdot \mathrm{C}_{\mathrm{v s}} \cdot \mathrm{R}_{\mathrm{s d}}$

So the concentration of the homogeneous pseudo fluid is not Cvs,x=X·Cvs, but:

$\ \mathrm{C}_{\mathrm{vs}, \mathrm{x}}=\frac{\mathrm{X} \cdot \mathrm{C}_{\mathrm{vs}}}{\left(1-\mathrm{C}_{\mathrm{vs}}+\mathrm{C}_{\mathrm{vs}} \cdot \mathrm{x}\right)}$

This is because part of the total volume is occupied by the particles that are not in suspension. The remaining spatial concentration of solids to be used to determine the individual hydraulic gradients curves of the fractions is now:

$\ \mathrm{C}_{\mathrm{v s}, \mathrm{r}}=(\mathrm{1}-\mathrm{X}) \cdot \mathrm{C}_{\mathrm{v s}}$

The dynamic viscosity can now be determined according to Thomas (1965):

$\ \mu_{\mathrm{x}}=\mu_{\mathrm{l}} \cdot\left(1+2.5 \cdot \mathrm{C}_{\mathrm{v s}, \mathrm{x}}+\mathrm{1 0 .0 5} \cdot \mathrm{C}_{\mathrm{v s}, \mathrm{x}}^{\mathrm{2}}+\mathrm{0 . 0 0 2 7 3 \cdot} \mathrm{e}^{\mathrm{1 6 . 6} \cdot \mathrm{C}_{\mathrm{vs}, \mathrm{x}}}\right)$

The kinematic viscosity of the homogeneous pseudo fluid is now:

$\ v_{\mathrm{x}}=\frac{\mu_{\mathrm{x}}}{\rho_{\mathrm{x}}}$

One should realize however that the relative submerged density has also changed to:

$\ \mathrm{R}_{\mathrm{sd}, \mathrm{x}}=\frac{\rho_{\mathrm{s}}-\rho_{\mathrm{x}}}{\rho_{\mathrm{x}}}$

With the new homogeneous pseudo liquid density, kinematic viscosity, relative submerged density and volumetric concentration the hydraulic gradient can be determined for each fraction of the adjusted PSD.

## 8.15.3 Determination of the Hydraulic Gradient

After adjusting for the new homogeneous pseudo liquid density ρx, kinematic viscosity $$\ v_{\mathrm{x}}$$ and relative submerged density Rsd,x the hydraulic gradient can be determined for each fraction of the adjusted PSD using the volumetric concentration of the remaing solids. It is important to determine the hydraulic gradient curve for the full velocity range for both spatial and delivered concentrations and not the relative excess hydraulic gradient curves. The reason is, that the hydraulic gradient curves include the liquid curve for the adjusted homogeneous pseudo fluid properties, while the relative excess hydraulic gradient curves don’t. Later the relative excess hydraulic gradient curves can be determined using the hydraulic gradient of the pure carrier liquid and the relative submerged density of the solids in the pure carrier liquid.

The resulting hydraulic gradient im,x based on the pseudo liquid and relative excess hydraulic gradient Erhg,x are:

$\ \begin{array}{ll}\mathrm{i}_{\mathrm{m}, \mathrm{x}}=\sum_{\mathrm{i}=\mathrm{1}}^{\mathrm{n}} \mathrm{f}_{\mathrm{i}} \cdot \mathrm{i}_{\mathrm{m}, \mathrm{x}, \mathrm{i}} \cdot \mathrm{w}_{\mathrm{i}} \quad \text { with: }\quad \sum_{\mathrm{i}=\mathrm{1}}^{\mathrm{n}} \mathrm{f}_{\mathrm{i}}=\mathrm{1} \quad \text { and } \quad \frac{\mathrm{1}}{\mathrm{n}} \cdot \sum_{\mathrm{i}=\mathrm{1}}^{\mathrm{n}} \mathrm{w}_{\mathrm{i}}=\mathrm{1} \\ \mathrm{E}_{\mathrm{r h g}, \mathrm{x}}=\frac{\mathrm{i}_{\mathrm{m}, \mathrm{x}}-\mathrm{i}_{\mathrm{l}, \mathrm{x}}}{\mathrm{R}_{\mathrm{s d}, \mathrm{x}} \cdot \mathrm{C}_{\mathrm{v s}}}\quad \mathrm{o r} \quad \mathrm{E}_{\mathrm{r h g}, \mathrm{x}}=\frac{\mathrm{i}_{\mathrm{m}, \mathrm{x}}-\mathrm{i}_{\mathrm{l}, \mathrm{x}}}{\mathrm{R}_{\mathrm{s d}, \mathrm{x}} \cdot \mathrm{C}_{\mathrm{v t}}}\end{array}$

The resulting hydraulic gradient im based on the original carrier liquid and relative excess hydraulic gradient Erhg are:

$\ \begin{array}{left}\mathrm{i}_{\mathrm{m}}=\frac{\rho_{\mathrm{x}}}{\rho_{\mathrm{l}}} \cdot \mathrm{i}_{\mathrm{m}, \mathrm{x}}=\frac{\rho_{\mathrm{x}}}{\rho_{\mathrm{l}}} \cdot \sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{f}_{\mathrm{i}} \cdot \mathrm{i}_{\mathrm{m}, \mathrm{x}, \mathrm{i}} \cdot \mathrm{w}_{\mathrm{i}} \quad\text{ with: }\quad \sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{f}_{\mathrm{i}}=\mathrm{1} \quad\text{ and }\quad \frac{\mathrm{1}}{\mathrm{n}} \cdot \sum_{\mathrm{i}=\mathrm{1}}^{\mathrm{n}} \mathrm{w}_{\mathrm{i}}=\mathrm{1}\\ \mathrm{E}_{\mathrm{r h} \mathrm{g}}=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s} \mathrm{d}} \cdot \mathrm{C}_{\mathrm{v} \mathrm{s}}} \quad\text{ or }\quad \mathrm{E}_{\mathrm{r h} \mathrm{g}}=\frac{\mathrm{i}_{\mathrm{m}}-\mathrm{i}_{\mathrm{l}}}{\mathrm{R}_{\mathrm{s} \mathrm{d}} \cdot \mathrm{C}_{\mathrm{v} \mathrm{t}}}\end{array}$

The variable wi is a weighing factor, enabling to give certain particle diameters more weight in the total hydraulic gradient. Here the weighing factors are set to 1.

The resulting bed fraction is:

$\ \zeta=\tilde{\mathrm{A}}_{\mathrm{b}}=\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{f}_{\mathrm{i}} \cdot \tilde{\mathrm{A}}_{\mathrm{b}, \mathrm{i}}=\sum_{\mathrm{i}=1}^{n} \mathrm{f}_{\mathrm{i}} \cdot \zeta_{\mathrm{i}} \quad\text{ with: }\sum_{\mathrm{i}=1}^{n} \mathrm{f}_{\mathrm{i}}=\mathrm{1}$