Sediment Transport in River Flows: New Approaches and Formulas

Abstract

A new method for estimating the total sediment discharge, as built on balance of power acting to moving sediment particle in “water stream-bottom sediments—sediments” system, enables consideration of interrelated influence of hydraulic variables state of flow and sediment. At the same time, the basic sticking point of river hydraulics, that is, interaction of fluid and bottom, is specified not from the part of fluid boundary, but from that of bottom sediments and their properties, well studied in soil science. Setting the size of bottom sediments by means of their qualitative characteristics allows avoiding calculation errors that occur when using specific values of quantiles of bottom sediments in calculations. Consideration of the critical velocities and the phase hydraulic space of the flow allowed obtaining the equations for transporting capacity of the flow, suspended, and bed load discharges.

1. Introduction

Solid runoff is one of the important state variables’ indicators of two-phase water mass circulation in a water object. Solid runoff of a water object represents solid matter, available in river flow or moving lake water masses, and having different genesis: ground (mineral solids) or organic matter. Solid runoff of water objects can be considered in different applications: static (water turbidity), dynamic (suspended and bed load, total sediment discharge), and indirect dynamic (bed mark changes and banks transformation).

Any irregular non-stationary two-phase flow is characterized by the processes of redistribution of solid matter in the riverbed or water area of a water body. At the same time, both the processes of sedimentation and bottom erosion and the transit of sediment can be observed.

Sediment can move in river flow by means of drawing or rolling over a bed (bed load sediment), saltation, in suspended state (suspended sediment), and over flow surface due to water surface tension (flotation). And maximum possible amount of solid matter, which specified water discharge can carry over, is called flow transporting capaсity. Transporting capaсity of a flow defines a process of redistribution of sediment in a channel—the main factor of channel processes. If sediment discharges lesser than transporting capacity of a flow, bottom sediment is engaged into the movement, and bottom erosion occurs. When sediment discharge starts exceeding transporting capacity of a flow, sedimentation of moving sediment occurs and bed marks increase. If sediment discharge in river flow conforms to its transporting capacity, then dynamic balance is observed between suspension and settling processes.

Currently, many formulas exist for suspended and bed load sediment discharges, total sediment discharge, and flow transporting capacity calculation. At the same, the high order of arguments degree in the sediment transport formulas leads to greater calculation error. One of the most significant calculation errors for such formulas is using the bottom sediment size, which is featured by high variability in the river channel, as an argument. For objective estimation of this value, bottom sediment samples taking for the whole watercourse cross section are required that seems to be impossible in some cases. Known formulas of sediment discharge do not consider interrelated effect of flow hydraulics and transportable solid particles. Also, the issue of friction parameters setting on solid boundary of river flow, despite multi-year study of this process, still remains not enough investigated.

Sediment transport calculation algorithms, developed separately for suspended and bed load sediment in the river flow, also bring known issues in calculation accuracy. Sediment transport is caused by flow energy and the size of transportable particles, only river flow is characterized by the transport potential in accordance with which the amount (mass) of transported particles is determined. And, depending on particle size distribution of transportable material, a part of sediment uprises in water layer and migrates with water mass of flow, and a part is dragged and rolled over a bed. Moreover, the ratio of suspended and bed load kind of transport is highly variable. It depends on hydrodynamic flow pattern and solid matter income from outside (e.g., from river basin or following the result of dumping of soil in the river channel). Hydrodynamic flow pattern, in queue, is defined by channel irregularity, slope variability, and bottom sediment size, as well as water and sediment discharges of upstream.

Creating a method for calculating total sediment discharge based on the balance of forces acting in the two-phase river flow will allow for the interrelated effects of hydraulically variables of a flow state and solid matter carried by the flow. At the same time, the main stumbling block of river hydraulics is the interaction of a moving stream, and the bottom should be considered not from the side of the boundary layer of the liquid, but from the side of bottom sediment, their well-studied properties in soil science. Setting the bottom sediment size through their qualitative characterization (by categories that include wide ranges of particle coarseness variation) allows us to avoid calculation errors that arise when using specific values of bottom sediment quantiles.

2. Phase hydraulic space and critical velocities

In a river flow, it can be considered the change in the flow velocity within one water discharge (phase hydraulic space) or its average value (average flow velocity within the phase hydraulic space) for the current water discharge. In the first case, we may speak of phase hydraulic space [1]. It seems clear that the same water discharge can carry the following amount of sediment G = [0; G_max], where G_max is max possible amount of sediment transportable by this water discharge per unit of time or, in other words, transporting capacity of a flow. At G = 0, the flow will be clarified, and its velocity will be minimal, and the depth will increase. At G = G_max, velocity increases and flow depth decreases [2, 3]. Moreover, the same water discharge (Q = const) conforms to each uttermost case. It is evident that specified water discharge at constant flow width can be defined from different combinations of depth and velocity. By plotting velocity along one axle and flow depth along another axle and indicating the points corresponding to one flow discharge value h_i·v_i = Q/B = const at flow width В = const in the graph, we get the function representing phase hydraulic space of a flow (system state space). Sediment discharge G_i corresponds to each depth-velocity combination (h_i·v_i = Q/B = const). Sediment discharge value for each combination is defined by transporting potential of a flow. That is, transporting potential of a flow represents the mass of solid matter transportable per unit of time through the cross section of a flow at constant water discharge. And this mass of solid matter, in turn, determines the ratio of velocity to depth of flow. Transporting potential of a flow is lesser than or equals to flow transporting capacity (at the same water discharge). The uttermost points of presented function are defined by process physics and conform to clarified flow (h_max, v_min) from one side and transporting capacity a flow (h_min, v_max) from another side.

Within possible flow velocity change range at specified water discharge, velocity can conform to different critical values. Critical river flow velocities traditionally are called such flow velocities at which the conditions of movement of the water flow and sediment carried by the flow change. Critical flow velocities predefine movement pattern of both liquid phase (laminar, turbulent, subcritical, and supercritical flow and so on) and solid phase of a flow. Depending on this, different types of critical velocities predefining sediment transport and bed mark change regimens are classified. For example [4]:

• non-eroding velocity (extreme velocity, whereas the basic part of bottom sediment is at rest state);

• breakaway velocity (start of mass movement of particles);

• non-silting velocity (extreme velocity, whereas the particles stay in suspended state).

Inter alia, Goncharov provides such definition for breakaway velocity: “the least average flow velocity, whereat unhampered breakaway of individual protruding grains on the bed occurs and average level of pulsation uplift forces is approximately equal to grain weight in water” [5]. Such definition enables concluding that breakaway velocity conforms to the start of particles movement both in suspended and bed load sediment forms. According to Zamarin [6], “non-silting velocity is the least average flow velocity, whereat suspended sediment, contained in water, do not leave a flow.” Obviously, the area of the clarified flow will be characterized by the lower limit of non-eroding velocity, while the area of maximum suspended load on the flow will be characterized by the non-silting velocity. Moreover, it is implied that the start of particles movement in river flow, predominantly, by saltation and in suspended form, conforms to velocity, being within the range between non-eroding and breakaway velocities. The latter can be explained by the fact that when the transporting potential of the river flow falls, particles of the largest size are the first to be deposited on the bottom during the movement of multi-factional sediment. Upper layer of deposited sediment will be represented by fine-grain fractions. Then, if transporting potential increases, at first, the upper layer of bottom sediment, represented by lesser size particles, starts movement. In support of the latter, let us consider the expressions, defined for critical velocities of movement start by drawing or rolling over a bed, v_{cr bed}, and suspensing, v_{cr suspend} [7]:

$$v_{\mathit{cr}\mathit{bed}}=5.75lg\left(2\frac{h}{d_{50}}\right)\sqrt{{\mathrm{\theta }}_{\mathit{cr}\mathit{bed}}\left(\frac{{\rho }_s}{{\rho }_w}-1\right){\mathit{gd}}_{50}},$$E1

$$v_{\mathit{cr}\mathit{\text{suspend}}}=5.75lg\left(\frac{12h}{6d_{50}}\right)\sqrt{{\mathrm{\theta }}_{\mathit{cr}\mathit{\text{suspend}}}\left(\frac{{\rho }_s}{{\rho }_w}-1\right){\mathit{gd}}_{50}},$$E2

$${\mathrm{\theta }}_{\mathit{cr}\mathit{bed}}=\frac{0.3}{1+1.2D^{\ast }}+0.055\left(1-exp\left[-0.02D^{\ast }\right]\right),$$E3

$${\mathrm{\theta }}_{\mathit{cr}\mathit{\text{suspend}}}=\frac{0.3}{1+D^{\ast }}+0.1\left(1-exp\left[-0.05D^{\ast }\right]\right),$$E4

$$D^{\ast }=d_{50}{\left[g\frac{\frac{{\rho }_s}{{\rho }_w}-1}{v^2}\right]}^{\frac{1}{3}},$$E5

where d₅₀—particle diameter with probability 50%, m; g—acceleration of gravity, m/s²; ν—kinematic viscosity coefficient, m²/s; ρ_s and ρ_w—densities of soil and water, respectively, kg/m³; h—average flow depth, m.

From these ratios, it follows that the movement of suspended particles, whose particle size is less than 30% of the size of the particles moving by drawing, is determined by a smaller value of the critical velocity. In a context of big grain size variety of mineral particles, available in river channel, this means that the start of sediment movement falls to suspended form.

The phase hydraulic space is characterized by the morphometry of the channel and the nature of the underlying surface, and the transporting potential of the flow is determined in accordance with the amount of solid matter entering the flow. Type of the function approximating the phase hydraulic space is defined by cross section form, and the function itself represents the velocity-to-depth ratio for constant water discharge at studied cross section. Figure 1 provides an example of phase hydraulic space for a channel with rectangular shape. In this case, change range for flow velocities and depth is defined by water discharge. It seems to be clear that water discharge predefines opportunity to reach this or that critical velocity. Thus, within the limits of one water discharge, a different ratio of hydraulic variables of a flow state can be established. This ratio is strongly predefined by solid matter ingress from catchment.

Figure 1.

Phase hydraulic space of a flow in the channel cross section for Q = const: 1—the area of the maximum suspended load on the flow; 2—the area of physically possible values; 3—clarified flow area; 4—non-eroding velocity (start of the movement of suspended sediment); 5—breakaway velocity (movement of suspended and bed load sediment); 6—non-silting velocity.

The flow velocities achieved within the phase hydraulic space for a fixed water discharge were considered above. Let us now stop at the average velocity, which corresponds to a given water discharge. If sediment in the river flow is formed solely by means of channel deformation, then we speak of channel-forming water flows. In this case, the average rates of the beginning and end of the process of channel deformation correspond to certain critical values. Then, the ratio of critical velocities and sediment transport pattern comes to the following schematic (Figure 2) (the view of this dependence is conditional).

Figure 2.

Correlation of critical velocities and type of sediment transport.

At the same time, both for the phase hydraulic space (within a fixed water discharge) and for the average velocity for a given water discharge, there is a ratio of critical flow rates and the nature of transport of multi-fractional sediment:

• v₁—velocity lesser than minimal non-eroding velocity (no sediment movement, G = 0);

• v₂—non-eroding velocity (start of suspended sediment movement, G_suspend);

• v₃—breakaway velocity (start of bed load and suspended sediment movement, G_{suspend + bedload});

• v₄—non-silting velocity (transporting capacity of a flow, G_max).

3. Review of sediment discharge formulas

In the early days of research into the movement of solid material in river flow, a separation of total sediment into a suspended and a bed load sediment part was adopted. Such posing of the question was justified by measurement base opportunities and useful to form general representation on regularities of involvement into movement and solid matter transfer in river flow. However, amid modern views on process physics, vast experience of full-scale and laboratory experiment, instrument opportunities, and all-round interdisciplinary integration, such vision of the issue seems slightly limited.

3.1 Suspended sediment discharge formulas

Development of suspended sediment discharge formulas, unfortunately, failed to find enough place in hydraulic calculations practice. At the same time in investigation of suspended sediment, the big attention was paid to turbidity distribution along vertical and along river length. Furthermore, vast investigations with another averaging scale were conducted on generalized materials analysis of spatiotemporal turbidity distribution (Karaushev [8], Shamov [9]). And as formulas for the consumption of suspended sediment in the standards and applied works, formulas for calculating the transporting capacity of a flow are proposed. Admixture propagation equation based on diffusion theory of sediment movement (Taylor [10], Schmidt [11], Makkaveev [12], Karaushev [8]) is represented in educational and scientific literature for suspended sediment discharge calculation. However, neither in the first case (formulas of transporting capacity of a flow) nor in the second case (admixture propagation equation), the task of suspended matter concentration estimation in a flow is not solved finally. It is evident that suspension-bearing load of river flow not always conforms to its transporting capacity. And admixture propagation equation, being an elementary continuity equation at known suspended matter concentration in a flow (boundary condition specification), leaves the question of estimating this concentration open.

In the last century, some suspended sediment discharge formulas were also based on knowledge on suspended matter concentration in a near-bottom layer and came down to integral calculation of suspended matter concentration distribution epure. In 1937, Rouse [13] provided theoretic equation for vertical distribution of suspended particles in turbulent flow. Suspended matter concentration formulas by Karaushev [8], Van Rijn [14], Sedaei et al. [15], Bagnold [16], Karasev [4], etc., are used in hydraulic calculations practice.

3.2 Bed load sediment discharge formulas

As is commonly known, assessment of bed load sediment discharge for natural water objects is one of the most complicated hydraulic tasks. Disconcertingly, lately activity in developing new approaches to solve this issue is not enough. Moreover, the main encumbrance lays in the absence of reliable verification for proposed calculation formulas according to field studies data. Bed load sediment discharge formulas can be focused both on the movement of separate solid particles directly or ridge form of sediment movement. Formulas describing ridge form of sediment movement consider geometric parameters of ridges, their length, height, etc., and can be used for rivers with sand bed. Such formulas are supported with relatively true data of sediment discharge observation that enables optimization of both formula’s structure and its parameters. Bed load sediment in rivers with gravel bed represents the biggest complexity in sediment discharge measurements. This, accordingly, hampers probation of bed load sediment discharge formulas and optimization of structure and parameters of such formulas.

Some researchers classify the following groups of bed load sediment discharge formulas:

1. Dependence of sediment discharge on hydraulic flow characteristics (Shamov [9], Levi [17], Goncharov [5], Grishanin [18], Egiazarov [19], Van Rijn [20]);

2. Relation of sediment discharge to water content (water discharge) of a river (Meyer-Peter et al. [21], Meyer-Peter and Müller [22], Schoklitsch [23, 24], Gilbert [25]);

3. Relation of the sediment discharge to the attractive force of the flow (Egiazarov [19], Bagnold [16]);

4. Formulas, where stochastic nature of sediment movement is expressed (Einstein [26, 27], Velikanov [28], Shen and Hung [29]).

Given enough great quantity of bed load sediment discharge formulas, but total sediment discharge formulas are not as common. However, at empirical nature of bed load sediment discharge formulas, the formulas for total sediment discharge are often more physically based. Total sediment discharge is a function of hydraulic flow parameters, such as average flow velocity, depth, water discharge, slope, size, hydraulic size, and density of the particles, as well as shear stress on solid boundary of a flow. Some formulas are developed based on dimensional analysis and almost all of them based on main concept of shear force of a flow.

The formula by Yang and Lim is defined using dimensional analysis for rivers with sand bed [30]. The formula by Ackers and White is also derived from dimensional analysis [31, 32]. The transport of fine-dispersed material is associated with a shear velocity, and the transport of larger particles is associated with an average flow velocity. Karim and Kennedy also obtained a formula for the total sediment through the theory of dimensions, making the total sediment flow dependent on the average and dynamic flow velocity, hydraulic size, and average particle size [33, 34]. Yang hypothesized that the determining factor in the concentration of sediment in alluvial channels is the specific power of flow, which can be defined as the calculated per unit time dissipation of potential energy per unit weight of water [35, 36]. The formula by Engelund and Hansen, defined in the middle of last century [37], is based on Bagnold’s flow power concept and theory of similarity. The formula by Molinas and Wu is based on shearing force of a flow [38]. In this formula, the Darcy-Weisbach equation is solved together with an expression for the frictional force, which gives the relationship between the total sediment concentration and the resulting flow force. The formula by Bagnold [16, 39] is based on energy balance concept, where flow power predefines energy for sediment transport. The formula of total sediment discharge developed by Karasev is based on two dependences for suspended and bed load sediment discharges [4]. The commonality of the mechanisms of movement consists of a single process of interaction between a liquid and a solid medium characterized by turbidity of ascent [4].

3.3 Formulas of transporting capacity of a flow

The values of empirical coefficients in the formulae of sediment discharge can be determined by minimizing the deviations between the results of calculations and observational data. But when deriving the formulas of flow transporting capacity, it’s hard to focus on observation data for definite river, as limit flow saturation with sediment is not achieved on all rivers and not for all water content periods.

One of the determining factors for the involvement of sediment particles in the flow is the turbulence regime of the river—velocity pulsations have a suspending and supporting effect on the particles in the flow. But it is known that the presence of a solid substance in the flow significantly reduces the pulsations of velocities, the flow becomes relatively orderly. Whereas all other things being equal, the clarified stream, having a large erosion capacity of the channel, has a more turbulent mode of movement.

In can therefore be concluded that the degree of flow saturation with sediment has nonlinear dependence from average flow velocity and, among others, depends on flow movement pattern.

Therefore, when derivation of the formulas of flow transporting capacity, not only dependence on the amount of transported matter from hydraulic variables of flow state shall be considered, but the factors predefining limit fluid saturation with suspensions. Such factors can include suspending capacity of a flow, Froude and Reynolds criteria, as well as presence of small fractions in a flow. It is known that high content of finest particles increases water viscosity and, therefore, impacts onto flow capacity to transport coarser fractions. Accordingly, “the limit of flow saturation with sediment depends on both flow hydraulics and transported sediment content” [8]. The formulas by Zamarin [6], Bagnold [39], Karaushev [8], and others are known from assessment practice of flow transporting capacity G_mах.

4. New approaches to sediment transport assessment

4.1 Formula of total sediment discharge

Main equation of mathematic model for water and solid matter movement in river flow is based on balance of forces acting to moving sediment particle in “water flow—bottom sediment—sediment” system. The forces acting on the particle side are counteracted by the forces acting on the flow side:

$${\overrightarrow{F}}_{\mathit{\text{flow}}}+{\overrightarrow{F}}_s=0.$$E6

Total force balance equation in “water flow—bottom sediment—sediment” system has the following view:

$${\overrightarrow{F}}_{\mathit{\text{grav}}}+{\overrightarrow{F}}_{\mathit{\text{inert}}}+{\overrightarrow{F}}_A+{\overrightarrow{F}}_{\mathit{\text{grav}}s}+{\overrightarrow{F}}_{\mathit{\text{resist}}}+{\overrightarrow{F}}_{\mathit{\text{inert}}s}=0,$$E7

where ${\overrightarrow{F}}_{\mathit{\text{grav}}}$—shear projection of the flow gravity; ${\overrightarrow{F}}_{\mathit{\text{inert}}}$—the force of inertia of the moving volume of water enclosed between the sections; ${\overrightarrow{F}}_A$—Archimedes force; ${\overrightarrow{F}}_{\mathit{\text{grav}}s}$—a retaining projection of gravity acting on a sediment particle moving in a flow; ${\overrightarrow{F}}_{\mathit{\text{resist}}}$—bottom resistance force; ${\overrightarrow{F}}_{\mathit{\text{inert}}s}$—the force of inertia of a particle moving in a flow.

Bottom resistance force F_resist is written by analogy with the well-known formula of the linear relationship between soil shear resistance and normal load [40]:

$$F_{\mathit{\text{resist}}}=\mathit{Rf}+\mathit{cS},$$E8

where f—internal friction coefficient, dimensionless; c—adhesion of soil particles during shear kg/(m·s²) (for disconnected soil c = 0); S—the area of force application, m²; in this case, the load R represents the system of loads (F_flow) acting to soils particles from flow side:

$$R=F_{\mathit{\text{flow}}}={\overrightarrow{F}}_{\mathit{\text{grav}}}+{\overrightarrow{F}}_{\mathit{\text{inert}}}+{\overrightarrow{F}}_A$$E9

Therefore, main equation of water and solid matter movement has the following view:

$$\left(1-f\right)\left(\mathit{mg}\left[I-\frac{\partial h}{\partial x}\right]-m\frac{\mathit{dv}}{\mathit{dt}}\right)-N_{\mathit{act}}{m}_s\frac{{\mathit{dv}}_s}{\mathit{dt}}+N_{\mathit{act}}{m}_{s}g-\mathit{cS}=0,$$E10

$$\frac{\partial h}{\partial t}+h\frac{\partial v}{\partial x}+v\frac{\partial h}{\partial x}=0,$$E11

$$v_s=\sqrt{v^2+{\omega }^2},$$E12

$$\frac{{\mathit{dE}}_{\mathit{\text{flow}}}}{\mathit{dt}}-\frac{{\mathit{dE}}_s}{\mathit{dt}}=0,$$E13

where m—the mass of the volume of water enclosed between the two cross sections, kg; g—acceleration of gravity, m/s²; I—bottom slope, dimensionless; v—flow velocity, m/s; h—flow depth, m; х—longitudinal coordinate, m; S—force application area, m²; m_s—particle mass, kg; v_s—particle velocity, m/s; N_act—the number of moving particles in the flow; ω—hydraulic particle size, m/s; E_flow—kinetic energy of the flow, kg·m²/s²; E_s—kinetic energy of moving particles, kg·m²/s²; f—coefficient of internal friction, dimensionless; c—the parameter of adhesion of soil particles during shear, kg/(m·s²).

Water and solid matter movement equation (Eq. (10)) is closed by flow continuity equations (Eq. (11)), equations for particle velocity (Eq. (12)), and the equation of flow and particles kinetic power balance (Eq. (13)).

For conditions for uniform steady movement after some transformations of the equation (Eq. (10)), we can acquire that sediment discharge G, that is, the mass of solid matter passing through flow cross section per unit of time (kg/s), shall be equal to:

$$G=\frac{N_{\mathit{act}}{m}_s}{\mathrm{\Delta }t}=\frac{\left[\frac{\mathit{cBv}\cdot \mathrm{\Delta }t}{g}-\left(1-f\right)\mathit{mI}\right]}{\mathrm{\Delta }t},$$E14

where B—the width of the stream, m. Or, if we write that m = ρ_w·h·B·v·∆t and water discharge Q = v·h·B, (m³/s) we get:

$$G=Q\left[\frac{c}{\mathit{hg}}-\left(1-f\right)I{\rho }_w\right],$$E15

Thus, upon reductions and transformations, two basic groups remain in the formula: gravitational component (ρ_w‧m‧I) and soil shear strength or friction force (c/(h‧g) + ρ_w‧f‧m‧I).

It worth noting that formula (Eq. (15)) calculates the mass of solid matter in water and this mass shall be brought to real mass of solid matter:

$$G^{\text{'}}=G\frac{{\rho }_s}{{\rho }_s-{\rho }_w}.$$E16

Therefore, the formula (Eq. (15)) takes the following view:

$$G=\frac{{\rho }_s}{{\rho }_s-{\rho }_w}Q\left[\frac{c}{\mathit{hg}}-\left(1-f\right)I{\rho }_w\right].$$E17

Parameters f and c of the formula (Eq. (17)) depend on water content phase of a river and size of bottom sediment. Dependences on different water content periods can be used to define the values of parameter f (for the group of the studied rivers):

$$f_1=-0.1293D+1.7143,$$E18

$$f_2=-0.0477D+1.2937,$$E19

$$f_3=-0.0114D+1.0556,$$E20

or in common view

$$f_i=a_{i}D+b_i,$$E21

where i—the water content index (1—maximum, 2—average, 3—minimum water content); D—a qualitative sign of the size of bottom sediment: 1—loam; 2—sand; 3—sand-pebbles; 4—gravel; 5—pebbles.

Following the optimization, the values of parameter с for connected soil (clay loams) for the studied rivers are equal to 0.385, 0.505, and 1.55 kg/(m‧s²) in average for the periods of low-, medium-, and high-water content, respectively. For loose soils for the most of studied rivers within medium- and high-water content obtained values of parameter с following the optimization were equal to zero.

The given values of the parameters f and c are preliminary and may be useful for unexplored rivers. For rivers where hydraulic variables of flow state were measured, the values of these parameters are estimated by minimizing the deviation of the calculated and observed values of the sediment discharges.

4.2 Formula of flow transporting capacity

Let us consider deriving the formula of flow transporting capacity. Based on phase hydraulic space concept, we assume that flow transporting capacity G_maх is equal to sediment discharge at max flow velocity (and minimal depth) for specified water discharge, that is,

$$G_{\max }=\frac{{\rho }_s}{{\rho }_s-{\rho }_w}Q\left[\frac{c}{h_{\min }g}-\left(1-f\right){\rho }_{w}I\right],$$E22

where h_min—minimum possible depth at fixed water discharge, slope and bottom sediment size, m.

Minimum possible depth is predefined by channel’s morphometry and water discharge. A decrease in h_min is already practically impossible, since within the value of this depth there will be an intensive suspend of solid matter to the bottom.

Thus, the calculation of the transporting capacity of the flow is preceded by an estimate of the values of h_min or v_max.

To estimate the boundary velocity v_max (so-called silting velocity), at which the suspend in the flow either precipitates or the qualitative state of the flow changes—the flow becomes viscous, we can use, for example, the expression [41]:

$$v_{\max }^2=\frac{{\rho }_s}{{\rho }_s-{\rho }_w}\frac{1}{2}\mathit{gh}.$$E23

According to the formula (Eq. (23)), minimal flow depth will be:

$$h_{\min }=\sqrt[3]{2\frac{{\rho }_s-{\rho }_w}{{\rho }_s}\frac{Q^2}{B^{2}g}}.$$E24

Then the expression for transporting capacity of a flow (Eq. (22)), considering the formula (Eq. (24)) will be as follows:

$$G_{\max }=\frac{{\rho }_s}{{\rho }_s-{\rho }_w}Q\left[\frac{c}{g\sqrt[3]{2\frac{{\rho }_s-{\rho }_w}{{\rho }_s}\frac{Q^2}{B^{2}g}}}-\left(1-f\right){\rho }_{w}I\right].$$E25

Therefore, defined analytical formula for transporting capacity of a flow (Eq. (25)) is based on balance of forces acting in the system “water flow—bottom sediment—sediment” [1], the formula of soil shear resistance [40], and the formula of the boundary velocity of particle deposition in the water flow [41]. Probation of formula (Eq. (25)) and comparative analysis of some formulas of flow transporting capacity are provided in [1].

4.3 Suspended and bed load sediment discharge formulas

Consider now the derivation of formulas for suspended and bed load sediment. The start of sediment suspension is due to increase in average flow velocity against non-eroding velocity. The formula of non-eroding velocity v_cr, defined based on multiple studies, is as follows [42]:

$$v_{\mathit{cr}}=1.15\sqrt{g}{\left(h_{\mathit{cr}}d\right)}^{0.25}.$$E26

By expressing the depth from (Eq. (26)), we will get:

$$h_{\mathit{cr}}=\frac{v_{\mathit{cr}}^4}{{1.15}^4{g}^{2}d}.$$E27

By substituting the defined expression (Eq. (27)) to total sediment discharge formula (Eq. (17)) we’ll get suspended sediment discharge formula, G_suspend

$$G_{\mathit{\text{suspend}}}=\frac{{\rho }_s}{{\rho }_s-{\rho }_w}Q\left({1.15}^4\frac{\mathit{cgd}}{v^4}-\left(1-f\right)I{\rho }_w\right).$$E28

Since total sediment discharge represents the sum of suspended and bed load sediment discharges, we’ll get the expression for bed load sediment discharge, G_bedload

$$G_{\mathit{\text{bedload}}}=G-G_{\mathit{\text{suspend}}},$$E29

$$G_{\mathit{\text{bedload}}}=\frac{{\rho }_s}{{\rho }_s-{\rho }_w}Q\left(\frac{c}{\mathit{gh}}-\left(1-f\right)I{\rho }_w\right)-\frac{{\rho }_s}{{\rho }_s-{\rho }_w}Q\left({1.15}^4\frac{\mathit{cgd}}{v^4}-\left(1-f\right)I{\rho }_w\right),$$E30

$$G_{\mathit{\text{bedload}}}=\frac{{\rho }_s}{{\rho }_s-{\rho }_w}\mathit{Qc}\left(\frac{1}{\mathit{gh}}-{1.15}^4\frac{\mathit{gd}}{v^4}\right).$$E31

This section thus presents the derivation of four formulas: total sediment discharge formula (Eq. (17)), flow transport capacity formula (Eq. (25)), suspended sediment discharge (Eq. (28)), and bed load sediment discharge (Eq. (31)) formulas. All derived formulas are based on the equation of water and solids motion (Eq. (10)), the concept of phase hydraulic space, and the relationships describing the critical flow states.

5. Calculation results for derived formulas

Consider the results of approbation of the new sediment discharge formulas. Observation data on 15 hydrometric stations, located on American rivers in the states Alaska, Idaho, Colorado, Washington, and Wisconsin, were used as calculation material. Observations on these rivers and creeks were conducted in 70s–80s years of the last century, and investigation results are represented in the report “Measured total sediment loads (suspended and bed load sediment) for 93 United States streams.” This report is published on official website of Geological Survey at the US Home Department and is in public domain [43]. The report represents the data on suspended and bed load sediment measured nearly simultaneously. Besides this, the report provides hydraulic variables of flow state and и grain-size analysis of sediment and bottom sediment. “The data, most of which were not published earlier, were measured by means different individuals and entities… Despite known sampling issues, the data are, probably, the best of those available for the moment” [43] (for 1989).

The report provides observation results in 93 rivers and creeks; however, the most comprehensive data required for calculation are represented for 15 rivers only. Totally, 252 measurement data for medium water content period were used in calculations. The range of main hydraulic characteristics of the studied rivers, wherein calculations were conducted, is represented in Table 1.

No	Hydrometric section	I, non-dim.	Q, m^3/s	v, m/s	h, m	B, m	Gbedload, kg/s	Gsuspend, kg/s	G, kg/s
1	Susitna River near Talkeetna, Alaska	0.00146	612–1160	1.8–2.7	1.7–2.3	183–202	197–849	2.26–10.4	199–859
2	Chulitua River below Canyon near Talkeetna, Alaska	0.00131	564–946	2.0–2.5	2.5–3.1	108–123	462–1690	26.9–145	488–1835
3	Tanana River at Fairbanks, Alaska	0.000467	1010–2020	1.3–1.9	2–2.9	296–469	2410–10,030	16.3–89.4	2426–10,119
4	Snake River near Anatone, Wash	0.001033	1990–3770	2.4–3.3	4.7–5.8	181–197	61.6–1270	1.07–58.7	62.67–1328
5	Toutle River at Tower Road near Silver Lake, Wash.	0.00311	112–248	1.5–3.1	0.77–1.5	61–70	538–5820	33–232	682–5901
6	Fork Toutle River near Kid Valley, Wash.	0.0037	110–185	2.4–2.8	0.85–1.1	56–59	1590–4980	110–338	1883–5090
7	Clearwater River at Spalding, Idaho	0.000312	847–1810	1.4–2.5	4.3–5.1	135–143	15.8–124	0.578–6.54	16.38–130.54
8	Yampa River at Deerloge Park, Colo	0.000673	108–447	0.81–1.3	1.5–3.9	90–93	113–998	3.6–13.2	122.3–1008
9	Wisconsin River at Muscods, Wis.	0.000311	114–714	0.49–0.88	0.71–2.6	278–310	1.42–20	1.76–23.3	3.18–43.3
10	Black River near Galesville, Wis.	0.000221	20.1–80.7	0.44–0.54	0.55–1.4	72–122	0.422–5.5	1.28–4.09	1.702–9.59
11	Chippewa River at Durand, Wis.	0.000326	132–884	0.77–1.1	1.3–3.2	215–244	2.78–64.5	5.52–23.3	8.3–87.8
12	Chippewa River near Pepin, Wis.	0.000309	118–391	0.57–0.86	10.76–1.8	229–274	2.24–45.7	2.89–14.7	5.13–60.4
13	North Fork of Lick Creek near Yellow Pine, Idаho	0.00666	1.28–4.25	0.52–0.95	0.32–0.51	7.3–8.8	0.00294–0.0978	0.0005–0.0292	0.00344–0.127
14	South Fork of Salmon River near Cascade, Idaho	0.00695	22–77.5	0.62–1.3	1.1–1.7	31.5–34.5	0.132–4.11	0.00632–6.42	0.138–10.53
15	Chippewa River near Caryville, Wis.	0.000213	117–779	0.45–1.1	1.4–2.8	185–247	0.936–16.4	0–13.5	0.936–29.9

Table 1.

The main hydraulic characteristics of the studied rivers.

On average, the deviations between the observed and calculated values for the medium water content period have been: according to the formula of total sediment discharge (Eq. (17))—41%; according to the formula of suspended sediment discharge (Eq. (28))—48%; according to the formula of bed load sediment discharge (Eq. (31))—46%. The results obtained are quite acceptable and confirm the operability of the above formulas.

It should be noted that calculation results for the formulas derived by other authors, as provided in this work, were published earlier many times, including [1]. At the same, the best results are shown by the formulas [1]:

• Engelund and Hansen’s [37]:

$$G={\rho }_s\cdot B\frac{0.05{\left|v\right|}^5}{\left({\rho }_s/{\rho }_w-1\right)\sqrt{g}{d}_{50}{C}^3},$$E32

• Karim and Kennedy’s [34]:

$$G=B\cdot k{\left[\frac{v}{\sqrt{g\left({\rho }_s/{\rho }_w-1\right)d_{50}}}\right]}^{2.97}{\left(\frac{u^{\ast }}{\omega }\right)}^{1.47}\sqrt{g\left({\rho }_s/{\rho }_w-1\right)d_{50}^3},$$E33

• Bagnold’s [39]:

$$G={\frac{\left|v\right|}{C^2}}^3\left(0.24+0.01\frac{u^{\ast }C}{\omega g^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right){\rho }_w,$$E34

where k—coefficient of proportionality equal to 0.00139; τ—shear stress at the bottom, kg/(m·s²); C—Shezi’s coefficient, m^0.5/s.

Figure 3 shows the relationship between the total sediment discharges observed and the calculated according to the above formulas for the study rivers. As can be seen from the graphs, the points of the observed and formula-calculated sediment discharges are almost bisecting each other. Small values of sediment load discharge are better calculated using the Karim-Kennedy’s (Eq. (33)) and Engelund-Hansen’s (Eq. (32)) formulas, but at the same time, the points of observed and calculated sediment discharge using the Engelund and Hansen’s formula (Eq. (32)) have a greater scatter and a systematic bias toward underestimation of calculated sediment discharge for large values of sediment load. At the same time, Shmakova’s formula (Eq. (17)) and Bagnold’s formula (Eq. (34)) showed the best result in the area of large values and the worst in the area of small values.

Figure 3.

Observed G_observ and calculated G_calc total sediment discharge values using Shmakova’s formula (Eq. (17))—1, Karim-Kennedy’s formula (Eq. (33))—2, Engelund-Hansen’s formula (Eq. (32))—3, and Bagnold’s formula—(Eq. (34))—4.

Now consider the results of the validation of formula (Eq. (25)) and conduct a comparative analysis of some formulas for the transport capacity of the flow [1]:

• Zamarin’s formula for hydraulic particle size 0.002 < ω < 0.008 m/s [6]:

$$G_{\max }=Q0.022{\left[\frac{v}{\omega }\right]}^{\frac{3}{2}}\sqrt{\mathit{hI}};$$E35

• Goncharov’s formula [5] for u^*/ω < 2.5:

$$G_{\max }={\rho }_{s}Q\frac{1+\varphi }{800}\frac{d}{h}\frac{v_{\mathit{cr}}}{v}\left(\frac{v^3}{v_{\mathit{cr}}^3}-1\right)\left(\frac{v}{v_{\mathit{cr}}}-1\right),$$E36

$$\varphi =\sqrt{\frac{{\rho }_s-{\rho }_w}{0.9{\rho }_w}\frac{\mathit{gd}}{{\omega }^2},}$$E37

$$v_{\mathit{cr}}=0.96\sqrt{{\mathit{gd}}^{0.4}{\left(d+0.0014\right)}^{0.6}}{\left(\frac{h}{d}\right)}^{0.2};$$E38

• Bagnold’s formula [39]:

$$G_{\max }=Q{\rho }_s\frac{{\rho }_w}{{\rho }_s-{\rho }_w}\frac{C_f{v}^2}{\mathit{gh}}\left(\frac{0.13}{f-I}+\frac{0.01}{\raisebox{1ex}{$\omega $}\!\left/ \!\raisebox{-1ex}{$v$}\right.-I}\right),$$E39

$$C_f={\left(\frac{k}{ln\left(\raisebox{1ex}{$12h$}\!\left/ \!\raisebox{-1ex}{$\mathrm{\Delta }$}\right.\right)}\right)}^2,$$E40

$$\mathrm{\Delta }=\left\{\begin{array}{l}3d_{90}\text{for}\frac{{\rho }_w}{{\rho }_s-{\rho }_w}\frac{u_{\ast }^2}{\mathit{gd}}<1\\ 3\frac{{\rho }_w}{{\rho }_s-{\rho }_w}\frac{u_{\ast }^2}{\mathit{gd}}{d}_{90}\text{for}\frac{{\rho }_w}{{\rho }_s-{\rho }_w}\frac{u_{\ast }^2}{\mathit{gd}}\ge 1\end{array}\right.,$$E41

where v_cr—non-eroding velocity, m/s; Q—water discharge, m³/s; I—bottom slope, dimensionless; v—average flow velocity, m/s; h—average flow depth, m; f—coefficient of internal friction, dimensionless; ω—hydraulic particle size, m/s; С_f—coefficient of friction; k—Carman constant, equal to 0.41; Δ—effective roughness height, m; d, d₉₀ и d₉₅—particle diameter with probability 50, 90 и 95%, respectively, m; ρ_s и ρ_w—densities of soil and water, respectively, kg/m³; g—acceleration of gravity, m/s²; u*—dynamic velocity, m/s.

The main criterion for the quality of the calculations will be the condition that the observed values of the total sediment discharge do not exceed the calculated values. This somewhat tentative and rather qualitative assessment of the correctness of the formulas can be explained by the fact that there is no information for the study rivers on how much suspended load of the flow reaches its maximum possible values. Or, in other words, it is not clear whether the measured sediment discharge G_observ corresponds to the flow transport capacity G_max. A similar qualitative analysis is also allowed, for example, in [6]. This qualitative assessment is supplemented by another requirement—the calculated values must deviate reasonably from the measured values. That is, the calculated maximum sediment discharges G_{max calc} in its value should correspond to the hydraulic conditions of the flow. In general terms, the conditions of conformity set out can be written as:

$$\left\{\begin{array}{l}{G}_{\mathit{\text{observ}}}\le G_{max\mathit{\text{calc}}}\le k\cdot G_{\mathit{\text{observ}}}\\ k\in \left[1,\frac{G_{\max }}{G_{\mathit{\text{observ}}}}\right]\end{array}\right.,$$E42

where k—coefficient determining the degree to which the canal is filled with moving sediment (based on observations), dimensionless.

For formula (Eq. (25)), the friction parameters were assumed to be equal to those optimized for the high and medium water periods in the calculations using the analytical sediment flow formula (Eq. (17)).

Table 2 shows the relative number of cases of non-exceeding δ, % of the observed values of sediment discharge by calculated ones, average relative deviations σ_total, % (between the calculated values of the flow transport capacity and the observed total sediment discharge) and σ_min, % (only between the calculated values of the flow transport capacity, not exceeding the observed values, and the observed total sediment discharge). The latter indicator illustrates the degree of deviation toward a clearly erroneous calculation—as the calculation of G_max assumes that condition (Eq. (42)) is fulfilled.

Formula	δ, %	σtotal, %	σmin, %
Zamarin’s (Eq. (35))	45	555	671
Goncharov’s (Eq. (36))	38	819	882
Bagnold’s (Eq. (39))	24	849	434
Shmakova’s (Eq. (25))	33	87	66

Table 2.

Results of calculations using the flow capacity formulas.

Results of the calculations according to the four formulas are shown on the Figure 4. Degree of qualitative correspondence between the calculated values of total sediment discharge and the observed values is demonstrated by the excess of calculated points over the bisecting lines.

Figure 4.

Observed G_{max observ} and calculated G_{max calc} flow transport capacity values using Zamarin’s formula (Eq. (35))—1, Goncharov’s formula (Eq. (36))—2, Bagnold’s formula (Eq. (39))—3, and Shmakova’s formula (Eq. (25))—4.

The results of the calculations above illustrate, in summary form, that in general (55–76%) the calculated sediment discharges are higher than the observed values. However, the values calculated using Shmakova’s formula (Eq. (25)) are the most consistent with the order of magnitude of the observed sediment discharges, with a deviation σ_total of about 87%. For the same formula, the most adequate values σ_min (66%) were also obtained.

The calculations were based on a quantitative assessment of the quality of the calculation formulas by comparing the calculated values from the above formulas and the observed values of the transport capacity of the flow. For the study watercourses at the observed average flow depth, the silting velocities v_max were calculated using formula (Eq. (23)). These velocities were compared with the observed velocities v, to which the average depths correspond. If the velocities v and v_max are approximately equal, it can be assumed that the measured sediment flow rate is the transport capacity of the flow G_{max observ}. The detected G_{max observ} values were compared with the G_max values calculated from the flow capacity formulas above.

Table 3 shows the calculation data and the results of the calculations for the station Fork Toutle River near Kid Valley. The v and v_max values for this station, the only one among all the study stations, showed sufficient proximity. The discrepancy between these values is between 1 and 15%. G_max was calculated using three of the formulas (except Goncharov’s formula (Eq. (36)) due to non-compliance with the conditions of applicability of this formula for the study river).

Q, m^3/s	vmax, m/s	v, m/s	В, m	h, m	Gmax observ, kg/s	Gmax, kg/s
						Zamarin’s (Eq. (35))	Bagnold’s (Eq. (39))	Shmakova’s (Eq. (25))
185	2.9	2.8	56	1.1	5090	110	352	3367
110	2.6	2.4	56	0.9	1883	53	181	1953
123	2.6	2.6	56	0.85	2045	64	269	2417
147	2.6	2.4	59	0.85	1714	69	260	2607
153	2.8	2.4	59	0.98	1978	71	225	2348

Table 3.

Calculation data and calculation results for transport capacity, Fork Toutle River near Kid Valley.

As can be seen from Table 3, the Shmakova’s formula (Eq. (25)) shows the best agreement between the observed and calculated values of the flow transport capacity. The average relative deviation for this formula was 29%. For Zamarin’s (Eq. (35)) and Bagnold’s (Eq. (39)) formulas, the calculated G_max values were significantly lower than the observed values.

Many of the formulas for flow transport capacity and minimum non-silting velocity have been derived experimentally for canals at maximum sediment load. In the case of natural watercourses, however, selecting the formula for the transport capacity and optimizing its parameters is extremely difficult due to the lack of reliable observational data on the maximum possible of sediment load in the flow. For this reason, the calculations in this chapter are somewhat tentative. However, the results obtained demonstrate the incompleteness of theoretical research in the study of the transport capacity of rivers. Concurrently, an important advantage of the formulas (Eqs. (17), (25), (28), and (31)) is analytical conclusion from the equation of basic two-phase mass carryover in river flow (Eq. (10)). In this equation, the forces are written not in relation to the water flow, but in relation to a moving solid (the shearing projection of the gravity of the water flow, the retaining projection of the gravity of moving particles, the inertia forces of the water flow and moving particles, the force of the soil resistance to shear). Also in equation (Eq. (10)), the interaction of the water flow, and the bottom is represented by the resistance of the bottom sediment to the tangential load from the flow side. Therefore, formulas (Eqs. (17), (25), (28), and (31)) are based on interrelated calculation of water flow and solid matter and supported by qualitative, but not quantitative characteristic of bottom sediment size. Friction parameters are derived functionally from bottom sediment size categories, which are represented by wide ranges of bottom sediment sizes.

6. Conclusions

Two-phase river flow represents a complicated system of liquid and solid phases and underlying surface interaction. Its main particulars are as follows:

1. Sediment transport in the river flow is, from one hand, predefined by flow hydrodynamics and, from the other hand, impacts to hydraulic variables of flow state;

2. Bottom sediment size is highly variable along the channel and in water flow and depends on water content phase;

3. Resistance on solid boundary of a flow in cross section is not constant within a year;

4. Practice of calculating solid runoff shows insufficient study of the interaction of river flow and river bed.

Existing methods of sediment transport calculations are not always universal and do not fit for flows of any scale and different hydraulic conditions. In this case, sediment transport process in any kind (suspended or bed load) and any degree of flow saturation with solid phase (from clarified to transporting capacity) is the same for all types of channels and for any water content periods. This process is based on the power of the river flow, which determines the amount of solid matter transported. Accordingly, assessment algorithms of any kind sediment discharge (suspended and bed load) shall be the result of theoretic equations describing two-phase river flow hydrodynamics. This means that these algorithms (formulas) shall be fully interconnected each other and follow one from another. And the structure of sediment transport formula shall be fully coordinated with measuring base opportunities. In particular, the average size of bottom sediment, highly variable in cross section, or its quantile values with specified occurrence decrease calculation accuracy. In this case, integral river bed characteristics, such as bottom sediment size categories, are more convenient for operation. Algorithms, provided in this work, can avoid the above deficiencies in a whole and increase calculation accuracy of sediment discharge for different types of rivers.

Acknowledgments

The research was funded by the study № 0154-2019-0003 of the state research plan for the Institute of Limnology RAS.