5.4. 1 theoretical basis of hydrodynamic dispersion
In porous media, when there are two or more miscible fluids, under the action of fluid movement, a transition zone is generated between them, and the concentration tends to be average. This phenomenon is called hydrodynamic dispersion in porous media, which is called dispersion phenomenon for short. The role of dispersion phenomenon is called dispersion for short.
We use the following simple experiment to illustrate the existence of dispersion phenomenon.
Take a cylinder, fill it with uniform fine sand, make it saturated with water, and form a stable flow field in the cylinder. At this time (t=0), the tracer solution with the concentration of c0 is continuously injected into the upper end of the barrel (the tracer does not react with the substances in the barrel), and the tracer concentration in the barrel is guaranteed to be zero before injecting the tracer. The whole device is shown in Figure 5.3(a). Assuming that the tracer concentration measured at the end of cylindrical sand column is ct, the relationship between tracer concentration ct and time t is expressed in the form of breakthrough curve [Figure 5.3 (b)]. If there is no dispersion, the concentration curve should be as shown by the dotted line in Figure 5.3(b). In fact, the concentration curve is shown as a solid line in fig. 5.3(b) due to dispersion.
Fig. 5.3 Schematic diagram of indoor diffusion experiment
In fact, when pollutants migrate in aquifers, diffusion usually occurs.
The causes of dispersion can be summarized as follows: water flows in the medium, the complex micro-morphology of the pore system of the medium, the molecular diffusion caused by the solute concentration gradient, and the influence of the change of water properties (such as viscosity and density) on the velocity distribution (velocity field). Interaction between solute and solid particles in water-adsorption, precipitation, degradation, ion exchange, biochemistry and other processes.
The dispersion process is mainly the result of the combination of molecular diffusion and mechanical diffusion, which are introduced respectively below.
5.4. 1. 1 molecular diffusion
Molecular diffusion is a phenomenon of material movement caused by different concentrations under the action of physical chemistry, and it is a process from heterogeneity to homogeneity. Molecular diffusion exists not only when the liquid is at rest, but also when the liquid is moving, including vertical diffusion along the moving direction and horizontal diffusion perpendicular to the moving direction. In other words, molecular diffusion always exists in the whole dispersion process of porous media. Therefore, when there is no relative flow between local sewage and sewage, pollutants in sewage will also enter groundwater due to molecular diffusion. In a static fluid, the molecular diffusion of solute can be described by Fickian law ⅰ:
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Among them: φ-diffusion flux, that is, the mass flux of solute per unit area per unit time, and its dimension is m L-2t-1;
D0—— molecular diffusion coefficient, and a negative value indicates that diffusion proceeds from a high concentration area to a low concentration area;
C-volume concentration of solute;
DC/dx- concentration gradient of solute in x direction.
In porous media, the diffusion process is very slow. Although the diffusion of polluted groundwater in aquifer can be realized by simple molecular diffusion in principle, it depends on the concentration gradient of pollutants. If the concentration gradient is not large, this dispersion is actually very slow.
Therefore, it is considered that if the migration distance is more than several meters or the required forecast period is less than 100 ~ 200 a, molecular diffusion can be ignored when calculating and forecasting the distribution of pollutants. Molecular diffusion should be considered only when this process lasts for a long time (more than hundreds of years), or when studying short-distance migration without seepage, or when studying radioactive waste pollution.
5.4. 1.2 convection (diffusion)
In fact, convection and dispersion have always been linked and inseparable, and we only distinguish them for the convenience of research. Convective diffusion refers to the phenomenon that pollutant particles spread in aquifers at the average actual velocity of groundwater (also called average velocity), which can be determined according to Darcy's law:
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Where: ux——x the average actual velocity of groundwater in X direction;
K-permeability coefficient;
N- effective porosity;
DH/dl- hydraulic gradient.
5.4. 1.3 mechanical dispersion
When pollutant particles move in porous media, the velocity and direction of each point in the flow field are different because of the viscosity of fluid and the existence of solid particles. The inhomogeneity of this velocity vector is so obvious that the average velocity vector can not describe the real motion of solute particles well. That is to say, there are a lot of movements in the flow field that deviate from the average velocity. Therefore, the migration of solute naturally exceeds the range predicted by average velocity, as shown in Figure 5.4. This velocity vector inhomogeneity is mainly related to the characteristics of porous media, which can be divided into the following situations: ① Due to the existence of fluid viscosity, the velocity near the particle surface in a single pore channel is zero, while the velocity at the center of the channel is the largest, as shown in Figure 5.5(a); ② The maximum velocity and average velocity of channels with different apertures are different, as shown in Figure 5.5(b); ③ When fluid flows in porous media, it is blocked by solid particles and detours. Sometimes, the velocity branches and divides in other directions, and the streamline fluctuates and deviates from the average flow direction, as shown in Figure 5.5(c). All these make solute particles spread not only in the direction of water flow, but also in the direction perpendicular to water flow. People call the discrete movement of solute particles relative to the average flow rate caused by the change of flow rate on the microscopic scale as mechanical dispersion.
Fig. 5.4 Propagation of Continuous Point Source Tracer in Uniform Flow
Figure 5.5 Several situations of dispersion
In heterogeneous aquifer, the dispersion phenomenon caused by the uneven distribution of seepage velocity is called macro-mechanical dispersion, and its mechanism is consistent with mechanical dispersion in principle. The main reason is still the uneven velocity, but the research unit is larger. For example, in layered aquifers with different permeability, the polluted water will invade in a tongue shape along the stratum with good permeability and extend far away. In fractured aquifers with different fissure widths, sewage moves faster in wider fissures and can reach a long distance. On the contrary, in narrow cracks, sewage moves slowly.
It is generally believed that mechanical dispersion is an irreversible process. For the convenience of operation, we can use a mathematical expression similar to Fei's affirmation law to describe it mathematically.
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Where: φ-diffusion flux;
C-volume concentration of solute in the flow field;
Dn- constant, called mechanical dispersion coefficient, has the dimension of [L2t- 1], and the negative sign indicates that the solute propagates in the direction of low concentration.
5.4. 1.4 hydrodynamic dispersion
Hydrodynamic dispersion is the diffusion and migration of solute relative to the average velocity due to the inhomogeneity of velocity distribution and solute concentration distribution in porous media. This is an irreversible process.
Under the action of hydrodynamic dispersion, the concentration of pollutants decreases with the increase of the distance from the pollution source, and the longitudinal velocity is greater than the lateral velocity in the direction of groundwater flow, that is, the longitudinal diffusion is greater than the lateral diffusion.
5.4.2 Other factors affecting the migration of pollutants in groundwater
There are two pollutants in groundwater, reactive and non-reactive. Non-reactive substances (such as chloride) do not react with groundwater and aquifer, and their migration is only the result of the comprehensive action of convection and hydrodynamic dispersion; However, for reactive substances, it must be considered that they will undergo adsorption-desorption, ion exchange, precipitation-dissolution, oxidation-reduction, biological reaction and other reactions in the aquifer.
adsorbing
When pollutants move in the aquifer, the amount of some pollutants decreases due to the adsorption of the medium. The functions that belong to this aspect mainly include:
(1) Mechanical filtration: Due to the different pore sizes of media, suspended solids, colloids and emulsions in groundwater are trapped in small pores or "blind holes" by mechanical filtration, which reduces the content of these substances in water.
(2) Physical adsorption: In porous media, because rock particles have surface energy, they can adsorb cations in water, especially highly dispersed clay particles, which have large surface energy and can adsorb a large number of ions. Cation exchange will also occur, which will reduce some ions in water and increase others.
(3) Chemical adsorption: Some ions in sewage are absorbed into the crystal lattice of the medium and become a part of the crystal lattice of the medium, so it is impossible for them to return to the solution, thus reducing the concentration of these ions in water.
(4) Biosorption: The migration of microorganisms in groundwater depends on the survival time of microorganisms on the one hand and the adsorption of rock particles on it on the other. Because of the surface energy and electrostatic force of rock particles, a large number of microorganisms can be adsorbed. Therefore, the concentration of organisms (especially bacteria) decreases rapidly in the process of groundwater migration, and the migration distance is generally within a few hundred meters.
In the mathematical description of solute transport, all kinds of adsorption are often combined and expressed by a coefficient to distinguish it from hydrodynamic dispersion.
Influence of density and viscosity of 5.4.2.2 liquid
Fig. 5.6 Inclined interface of liquids with different densities in layered formation (ρ 1 > ρ 2)
When sewage moves in the rock stratum, the formation of dispersion zone is mainly caused by various dispersion effects, and the development and evolution of dispersion transition zone is also affected by liquid density and viscosity.
When the density of sewage is different from that of clean groundwater, at the interface of horizontal strata, the vertical interface will gradually incline due to the action of gravity, and the heavy liquid with high density will be below the inclined plane, while the lighter liquid will "float" on the inclined plane. When the density difference between them is large, heavy liquid can form long finger-like or tongue-like intrusion along the bottom of the layer under the inclined plane, such as the intrusion of salt water. Many researchers believe that the projected length Lp (Figure 5.6) of the interface between two liquids on the X-axis is related to the relative density difference, formation properties and permeability time, and the following empirical relationship can be obtained:
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Where:-relative density difference, = (ρ1-ρ 2)/ρ 2;
ρ 1, ρ2—— pushed liquid and density of pushed liquid in aquifer;
K, m, n-permeability coefficient, thickness and porosity of aquifer;
φ —— dip angle of rock stratum;
T—— pushing duration;
X coefficient, generally 1.4 ~ 2.2.
If the time is long, heavy (high salinity) liquid can travel along the bottom of the layer for a long distance. For example, when k=600m/d, m=25m, n=0. 1, ρ1=1g/cm3, ρ2= 1.00g/cm3, take X. If the salinity of sewage is not large, the density difference between them is small, and when Δ ρ < 0.001,the interface length will not be too large, only increasing by several meters every year.
The density difference not only affects the shape of the interface, but also has a certain influence on the moving speed of the interface, which can be expressed by the following formula:
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Where: q-single-width flow, which is constant flow; The remaining symbols are the same as before.
In the horizontal formation, φ=0 and sinφ=0, then Vρ=q/mn, which is equal to the moving speed v of the homogeneous liquid piston; When tilting, if the sign is the same as sinφ, and the second term in the above formula is negative, then V ρ < V, and vice versa.
Then v ρ > v; When the sewage moves vertically downwards, if it leaks vertically from the sewage storage tank, φ = π and sinφ=- 1, then
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Here is the additional permeability gradient Iρ= caused by different densities. Therefore, when the heavy sewage is located above the fresh groundwater, under the condition of hydrodynamic static (I=0, q=0), the migration of polluted water will also occur, that is, it has speed under the action of gravity.
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Use data: K=20m/d, m=25m, n=0. 1, =0.0 1 to calculate the time t required for heavily polluted water to sink from the water surface to the bottom of the aquifer;
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It can be seen that the speed of heavy polluted water settling out of fresh water is very fast.
The effect of liquid viscosity on dispersion can be evaluated by the viscosity ratio M=(μ 1/μ2), where μ 1 is the viscosity of the pushed liquid (sewage) and μ2 is the viscosity of the pushed liquid (clean groundwater). Many experimental data show that when the density is constant, the larger the m is, the smaller the dispersion length is, and the greater the liquid velocity is, the more obvious the influence of viscosity is. It has no effect on the diffusion of pure molecules. When the viscosity increases with the decrease of water temperature, the dispersion coefficient also increases.
5.4.2.3 decay and degradation.
When studying the migration of radioactive pollutants and organic substances in groundwater, the physical decay of radioactive substances and the biodegradation of organic substances will lead to the continuous decrease of their concentrations in groundwater, which should be considered.
The concentration of radioactive substances changes as follows:
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Where: C0-initial concentration, in milliliter (mL-3);
CT——t concentration at time t, with the dimension of mL-3;
λr- decay constant of radioactive material, t-1;
Time is up.
The above formula can also be used to describe the biodegradation process of organic pollutants, but the biodegradation constant λ c should be used instead of λ r.
5.4.3 Basic mathematical model of solute transport in groundwater
Mathematical models describing groundwater solute transport can be divided into three categories: deterministic model, stochastic model and "black box" model. They correspond to modern mathematical models describing groundwater movement, but the movement of solute is more complicated than that of groundwater. This section mainly discusses the deterministic model of solute transport.
Considering the convection and hydrodynamic dispersion of solute in the process of migration, combined with the law of conservation of mass, we can derive the so-called convection-dispersion equation by using the method of spatial average, which is the basic mathematical model to describe solute migration. The specific derivation process is as follows.
Fig. 5.7 schematic diagram of micro-components.
Take any tiny mass balance body (infinitesimal element) in the studied seepage field (as shown in Figure 5.7). The change of solute mass in trace elements in dt time is caused by three aspects.
Hydrodynamic dispersion
Hydrodynamic dispersion consists of mechanical dispersion and molecular diffusion. Let φ 1 be the mechanical diffusion flux, φ2 be the molecular diffusion flux, P be the hydrodynamic diffusion flux, and C be the solute concentration in the seepage field, then:
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Where: D=Dn+D0, d is the hydrodynamic dispersion coefficient, and the anisotropy is the second-order tensor.
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Formula (5- 10) can also be expressed as:
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The change of solute mass in the infinitesimal due to dispersion in the X direction is the difference m ′ x between the solute mass flowing into the X section and the solute mass flowing out of the X+δ X section. Namely:
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In the formula (5- 12), n is the effective porosity.
Similarly, in the y and z directions:
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Due to the dispersion of Δ t time, the change of solute mass in the whole element is as follows:
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5.4.3.2 convection
Let u represent the actual average velocity of groundwater. In the x direction, the change of solute mass in the microelement caused by the average overall movement of water m "x is:
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Therefore, the change m "of solute mass in microelement caused by convection is:
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5.4.3.3 adsorption and others.
Suppose that due to chemical reactions (such as adsorption, etc. ) or other reasons, the change of solute mass in groundwater per unit volume per unit time is W, then the change of solute mass in trace elements in δ t time is:
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Assuming that the change rate of solute concentration at time t near point (x, y, z) is, the change amount m of solute mass in the infinitesimal element is:
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According to the law of conservation of mass, there should be:
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Substitute the above items into (5-20) at the same time to obtain the product:
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Assuming that the main direction of dispersion is consistent with the coordinate axis, then:
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Substituting (5-22) for (5-2 1) includes:
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Equation (5-23) is called convection dispersion equation (or hydrodynamic dispersion equation).
The convection-dispersion equation can be expressed as:
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Where: K=D/n, ρ is the liquid density, and u is the actual average velocity of groundwater.
The above hydrodynamic dispersion equation is a basic mathematical model to quantitatively describe solute transport in groundwater, and it is a second-order nonlinear partial differential equation. Its solution methods can generally be divided into analytical solution, semi-analytical solution and numerical solution. The actual groundwater pollution problem is very complicated (such as point source, line source, non-point source and multi-point source, etc.). ); Its injection method is instantaneous and continuous; The aquifer environment is also changeable. There are many related works on this subject for reference, and I won't introduce them here because of the space limitation.