Exploration Of The Basic Seepage Equation In Porous Media

Seepage equation is the basis of any quantitative description of groundwater flow and solute transport model and its core is to answer the hydraulic resistance variation of groundwater movement in porous media. Traditional theory of flow in porous media considers that: When the flow rate is very low, the inertial force is negligible, the velocity and hydraulic gradient to obey Darcy's law (u=KJ). With increasing flow velocity, the inertial force increasing and playing a dominant role, seepage gradually deviate from Darcy's law and the relationship between the velocity and the hydraulic gradient obey the Forchheimer quadratic equation (J=Au+Bu2). When the flow rate increases to a certain value that turbulence emerges, the viscous force can be neglected, and the hydraulic gradient is proportional to the square of flow rate (J=Bu2). By analogy to theory of flow in tubes, Reynolds number (Re) was introduced to determine the flow regime in porous medium and piecewise function was used to portray the seepage law under different flow regime.Despite the use of the piecewise function can research the seepage law of different flow regime, at the same time, it brought out the following questions:(1) To portray a specific flow field, Re value must be first determined for every point in the flow field so that the seepage equation can be choose, but in fact the value of Re is often not known in advance.(2) If there are different flow regimes in the same flow field, there must be different equations for them, but how to determine the equation for different location? For example, in the flow field near the pumping well, there must be a Darcy flow zone with small hydraulic gradient far from the well. Near the pumping well, there may be a turbulence flow zone. Between the two flow zones, it is a non-Darcy flow zone. However the location and extent of the above three kinds of flow pattern is difficult to determine.(3) The flow pattern at the same location may change over time, so different equations have to be chosen at different times. The same question is how to choose the equation? For example, a point near the pumping wells, drawdown is small in the early pumping moment, the flow pattern is Darcy flow. After longer pumping time, Darcy flow transforms of non-Darcy flow gradually. However, it is often unknown when would the flow pattern transformed.Research the uniform equation for groundwater seepage is the basic theoretical issues of groundwater quantitative study, so it has important theoretical significance to explore a uniform equation for flow in porous medium.In1933, Nikuradse carried out experiments in six rough tubes to research the resistance coefficient variation with Reynolds number and roughness. Inspired by the experiments of Nikuradse, more rougher pipes were developed. When the size of the grain used to produce roughness becomes to the limit, the grains touch and rough pipes become to single-porosity medium.11kinds of different rough tubes were made to conduct flow resistance experiments. The results show that: laminar flow (linear flow region) gradually decreases with the increase of roughness of the pipes. When roughness increased to the limit, laminar linear flow region disappears. In fact, the nature porous media structure should be more complex than the above limit roughness tube. Thus, we may suppose there's no linear flow exist in porous medium. In order to verify our guess, we conducted experiments in porous medium. According to Darcy's Law, if the value of Re<10, the hydraulic conductivity K should be a constant. On the other hand, if the hydraulic conductivity K is not a constant, then the seepage is not a linear flow. In order to test this speculation, experiments in permeable stone (permeable stone can guarantee that the pore structure remained relatively stable during the experiment) were conducted. The experiments data showed that: when the value of Re<10, K was not a constant (K≠C). Furthermore, we produced a porous media model with pellets(d=3mm) arranged in cubes to carry out the seepage resistance experiments. The results also show that: as the value of Re increase, was a continuous trend of in permeability coefficient, K was not a constant (K≠C) but decrease gradually. At the same time, we collected the original data of the Darcy's experiments. Darcy's experimental data showed K is not constant too, which indicated flow in porous media is not linear flow. All these results indicate that linear flow would only exit when water movements were very simple. Due to the complicated structure of porous media, the flow resistance and seepage velocity can no longer satisfy the linear flow conditions. Darcy's Law is only an approximation equation of groundwater movement.By the ideal model of porous media with pellets arranged in cubic, we derived the basic equations of groundwater flow.The equation has been verified by experiments in porous media with pellets arranged in cubes. The experimental results show that with the flow rate (or Re) increases gradually, the permeability coefficient K is not equal to a constant (K≠C). The relationship between the flow rate(u) and hydraulic gradient (J)can be characterized by a complete quadratic equation (J=Au+Bu2).According to the causes of flow resistance, head loss is composed of viscous force and inertial energy loss. The viscous energy loss is proportional to the velocity and inertia energy loss is proportional to the square of the velocity. Only when the flow is linear and constant with no inertial energy loss, the flow can be described by linear Darcy's Law. Otherwise, inertial energy loss must be exit and there would no linear flow in porous medium. So Darcy's Law is just an approximate description of the groundwater movement in porous medium.The proportion of the viscous and inertial energy loss is affected by both flow rate and penetration characteristics. As the flow rate increases, the proportion of viscous energy loss gradually decreases but the inertia energy loss increases. The viscous energy loss is inversely proportional to the particle diameter. The inertial energy loss is inversely proportional to the square of particle diameter. So the inertial term is more easily influenced by the particle diameter. When the flow rate is equal, smaller the particle diameter is, greater the proportion of viscous term is. On the contrary, greater the particle diameter is, smaller the proportion of viscous term is. Thus, when the flow rate is very small, the error of Darcy's Law is small.Finally, on the basis of analysis of seepage mechanics, influencing factors of parameters A and B were discussed. Results show that diameter, arrangement and sorting of the particles are the three main factors, which has a greater impact on the parameters A.(1) The particle diameter (d) is the most important factors. As d increases, A and B decrease. A is inversely proportional to d2and B is inversely proportional to d.(2) The influence of sorting and arrangement of the particles to A and B could be described by porosity n. The worse the sorting of the particles, the more tightly the particles were packed, the higher the value of porosity n and A and B. Otherwise, the values of A and B decrease.(3) Pore size R is the function of d and n, which can be written as R=f (d,n). As the value of the pore size R increases, the values of parameter A and B decreace. Othervise, as the value of the pore size R decreases, the values of parameter A and B increace.(4) Besides, parameter A is also related to the viscosity of liquids. The stronger the viscosity of liquids is, the greater the proportion of viscous term is. To the same kind of liquid, as the t temperature decreases, the value of parameter A increases.By lots of experiments and theoretical analysis, we considered that Darcy flow is only possible exit in uniform linear flow. But there will be no uniform linear flow in porous media due to the complex structure of groundwater aquifer. So flow in porous medium does not obey the linear law but the quadratic equation, and Darcy's Law is only the approximate equation when the flow rate doesn't change significantly. Groundwater movement can be described by the quadratic equation. Diameter, arrangement and sorting of the particles are the three main factors, in which diameter is the most important factor. According to the experiments of porous media with pellets arranged in cubes, we put forward the equation of flow in porous media, and the relationship among d, A and B was discussed. But the flow in more complex media still needs to be investigated, such as media composed of different sizes of particles arranged in cubes or other arrangements. In addition, mechanism of low-speed non-Darcy flow in fine particles media needs further investigation, and these are very important for perfect the groundwater percolation theory.