P1,(2: water levels in the left and right reservoirs, LI length of the sand column(distance), LI Equation(2. 1)is the famous Darcys From basic hydraulics it is known that (1 is also the groundwater head at the left end of the sand column, and (2 the groundwater head at the right end of the sand column Groundwater head In general, groundwater head is defined as the elevation head plus the pressure head, i.e Z pg Z elevation of the point concerned above the reference level, [ L] pressure in the fluid at that point, MLTI p: density of fluid(mass per unit volume), [ML] g: acceleration of gravity, [LT ]g=9.81 The quantities z and p/pg are usually called the elevation head and pressure head, respectivel Specific discharge The specific discharge is defined as the discharge per unit of cross-sectional area and denoted by q. It follows from Equation(2.2)that =-K By taking the limit Al>0, the equation (2.3)becomes (24) where q: specific discharge,[LT1 do / dl: gradient of groundwater head(hydraulic gradient),[-1
φ1, φ2: water levels in the left and right reservoirs, [L]; L: length of the sand column (distance), [L]. Equation (2.1) is the famous Darcy's law. From basic hydraulics it is known that φ1 is also the groundwater head at the left end of the sand column, and φ2 the groundwater head at the right end of the sand column. Groundwater head In general, groundwater head is defined as the elevation head plus the pressure head, i.e. g p = z + ρ ϕ (2.2) where z : elevation of the point concerned above the reference level, [L]; p : pressure in the fluid at that point, [ML-1T-2]; ρ : density of fluid (mass per unit volume), [M L-3]; g : acceleration of gravity, [LT-1]. g = 9.81 m/sec2 . The quantities z and p/ρg are usually called the elevation head and pressure head, respectively. Specific discharge The specific discharge is defined as the discharge per unit of cross-sectional area and denoted by q. It follows from Equation (2.2) that, l q = - K ∆ ∆ϕ (2.3) By taking the limit ∆l → 0, the equation (2.3) becomes, dl d q = - K ϕ (2.4) where q : specific discharge, [LT-1]; dφ /dl: gradient of groundwater head (hydraulic gradient), [-]. 10
Equation (2.4)is the differential formulation of the Darcy's law. It expresses a linear relationship between the specific discharge and hydraulic gradient Velocity Although the specific discharge has the dimension of the velocity, it is not the actual velocity of the groundwater flow. The total cross-sectional area is a, but the area through which the groundwater can flow is only n*A. n is the porosity of the Hence. in Darcy's experiment, the mean velocity, v, of the flow can be computed na n (25) The actual velocity is always greater than the specific discharge since the porosity is smaller than one Validity of Darcys law The linear relationship between the specific discharge and hydraulic gradient suggests that the Darcy's law can be applied to the laminar fluid flow. Therefore, Renolds number can be as an indicator of validity of the Darcy's law. The Renolds number for porous medium is defined as d Re (26) d: average grain diameter, LI v: kinematic viscosity, [LT ] it is related to dynamic viscosity with v=n/p n: dynamic viscosity, [ML T] Experiments show that when Re <10, flow is laminar fluid flow and Darcy's law can be applied. Practical experiences show that Darcy's law can be applied to most cases of groundwater flow in porous medium Intrinsic permeability In Darcy's experiments, if using different fluids flowing through the same sand column, different values of the coefficient of permeability will be measured. It means that the coefficient of permeability does not only depend on the characteristics of the medium, but also on the properties of fluid. Experiments show that the property of fluids influencing the coefficient of permeability is the kinematic viscosity. The following relation is usually used
Equation (2.4) is the differential formulation of the Darcy's law. It expresses a linear relationship between the specific discharge and hydraulic gradient. Velocity Although the specific discharge has the dimension of the velocity, it is not the actual velocity of the groundwater flow. The total cross-sectional area is A, but the area through which the groundwater can flow is only n*A. n is the porosity of the sand. Hence, in Darcy's experiment, the mean velocity, v, of the flow can be computed as n q = nA Q v = (2.5) The actual velocity is always greater than the specific discharge since the porosity is smaller than one. Validity of Darcy's law The linear relationship between the specific discharge and hydraulic gradient suggests that the Darcy's law can be applied to the laminar fluid flow. Therefore, Renolds number can be used as an indicator of validity of the Darcy's law. The Renolds number for porous medium is defined as ν q d Re = (2.6) where d: average grain diameter, [L], ν: kinematic viscosity, [L2 T-1]; it is related to dynamic viscosity with ν = η/ρ; η : dynamic viscosity, [ML-1T-1]. Experiments show that when Re < 10, flow is laminar fluid flow and Darcy's law can be applied. Practical experiences show that Darcy's law can be applied to most cases of groundwater flow in porous medium. Intrinsic permeability In Darcy's experiments, if using different fluids flowing through the same sand column, different values of the coefficient of permeability will be measured. It means that the coefficient of permeability does not only depend on the characteristics of the medium, but also on the properties of fluid. Experiments show that the property of fluids influencing the coefficient of permeability is the kinematic viscosity. The following relation is usually used: 11
where K is called the intrinsic permeability with a dimension of [L ] depending only on the characteristics of the medium It is known that the kinematic viscosity varies with the temperature and density of fluids. In the cases of flow of hot groundwater or salt groundwater, coefficient of permeability will be a function depending on the variation of the kinematic viscosity. However, the intrinsic permeability is a constant. Therefore, in these cases, the following alternative formulation of Darcy s law is more convenient g (28) For the case of fresh groundwater flow, the introduction of the intrinsic permeability has no advantages since the kinematic viscosity is a constant. Therefore, in the case of fresh groundwater flow, the coefficient of permeability is used The analogy of laminar flow in tubes with groundwater flow in porous medium indicates that the intrinsic permeability is proportional to squared diameter of grain size(d)and porosity (n). In practice, empirical formula may be used to calculate the value of intrinsic pe One of such formula is Kozeny-Carman's formula k-cd (29) where average diameter of particle size, L] porosity, [1 c: constant to be determined with the experiment Equation(2.9)explains why the permeability of clay is small and gravel is large. Although the porosity of clay is very large, the pore space is very small resulting of low permeability.On contrary, the pore space of gravel is very large and therefore, the permeability of gravel is high. Table 2. 1 gives range of permeability for common porous medium In real ity, the structure of porous medium is so complicated that no direct formula can be used to calculate the permeability. Its value is usually determined with pumping tests which are introduced in chapters 4 and 6
ν κg K = (2.7) where κ is called the intrinsic permeability with a dimension of [L2 ], depending only on the characteristics of the medium. It is known that the kinematic viscosity varies with the temperature and density of fluids. In the cases of flow of hot groundwater or salt groundwater, coefficient of permeability will be a function depending on the variation of the kinematic viscosity. However, the intrinsic permeability is a constant. Therefore, in these cases, the following alternative formulation of Darcy's law is more convenient: dl g d = - ϕ ν q κ (2.8) For the case of fresh groundwater flow, the introduction of the intrinsic permeability has no advantages since the kinematic viscosity is a constant. Therefore, in the case of fresh groundwater flow, the coefficient of permeability is used. The analogy of laminar flow in tubes with groundwater flow in porous medium indicates that the intrinsic permeability is proportional to squared diameter of grain size (d2 ) and porosity (n). In practice, empirical formula may be used to calculate the value of intrinsic permeability. One of such formula is Kozeny-Carman's formula: (1- n ) n = c d 2 3 2 κ (2.9) where d : average diameter of particle size, [L]; n : porosity, [-]; c : constant to be determined with the experiment. Equation (2.9) explains why the permeability of clay is small and gravel is large. Although the porosity of clay is very large, the pore space is very small resulting of low permeability. On contrary, the pore space of gravel is very large and therefore, the permeability of gravel is high. Table 2.1 gives range of permeability for common porous medium. In reality, the structure of porous medium is so complicated that no direct formula can be used to calculate the permeability. Its value is usually determined with pumping tests which are introduced in chapters 4 and 6. 12
Table 2.1 Range of permeability for common porous medium Medium K(m/day) Clay 101-10 1053-10-3 Silt 1042-10 101-102 Gravel 10-10-8 104-102 2.2 Generalization of Darcy's Law Darcy's law in the differential form of Equation(2. 4)expresses the relationship between the specific discharge and hydraulic gradient for the uniform flow in one dimension. Darcy's law can be extended to more general cases of groundwater flow For three dimensional flow in isotropic porous medium, Darcy's law will be written as qx (210) where qx, qy, and qz are three flow components in x, y, and z directions, respectively. The groundwater head will be a function of x, y, and z coordinates and is defined as (p(x,y, z)=z+ p(x,y, z) (2.11) where p is assumed a constant In cases of three dimensional groundwater flow in anisotropic medium with principle directions of the permeability in x, y, and z directions, Darcy's law can be generalized q=-K
Table 2.1 Range of permeability for common porous medium Medium κ (m2 ) K (m/day) Clay 10-17 -10-15 10-5 -10-3 Silt 10-15 -10-13 10-3 -10-1 Sand 10-12 -10-10 10-1 -102 Gravel 10-9 -10-8 104 -102 2.2 Generalization of Darcy's Law Darcy's law in the differential form of Equation (2.4) expresses the relationship between the specific discharge and hydraulic gradient for the uniform flow in one dimension. Darcy's law can be extended to more general cases of groundwater flow. For three dimensional flow in isotropic porous medium, Darcy's law will be written as: x = - K x ∂ ∂ϕ q z q = - K z ∂ ∂ϕ (2.10) y q = - K y ∂ ∂ϕ where qx, qy, and qz are three flow components in x, y, and z directions, respectively. The groundwater head will be a function of x, y, and z coordinates and is defined as: g p(x, y, z) (x, y, z) = z + ρ ϕ (2.11) where ρ is assumed a constant. In cases of three dimensional groundwater flow in anisotropic medium with principle directions of the permeability in x, y, and z directions, Darcy's law can be generalized as: x qx = - Kxx ∂ ∂ϕ 13
where Kxx, Kyy, and Kz are anisotropic permeability in three principle directions, respectively The total specific discharge q can be considered as a vector consisting of three components qx, qy, and qz. Therefore, q can be expressed as K△ (2.1 where Kx 0 0K 00 is so-called permeability tensor and is the hydraulic gradient The geometric illustration of Equation(2. 13)is shown in Figure 2.2
y qy = - Kyy ∂ ∂ϕ (2.12) z qz = - Kzz ∂ ∂ϕ where Kxx, Kyy, and Kzz are anisotropic permeability in three principle directions, respectively. The total specific discharge q can be considered as a vector consisting of three components qx, qy, and qz. Therefore, q can be expressed as: q = - K ∆ϕ (2.13) where: 0 0 K 0 K 0 K 0 0 K = zz yy xx is so-called permeability tensor and ∂ ∂ϕ ∂ ∂ϕ ∂ ∂ϕ ∆ϕ z y x = is the hydraulic gradient. The geometric illustration of Equation (2.13) is shown in Figure 2.2. 14