§2.2 Basic Law
2.2.1 Law of Energy Conservation
In 1856，a French engineer Henry Darcy conducted his famous experiment as sketched in Figure 2.3.A one-dimensional steady flow in the sand column was created by keeping water level constant in the left and right reservoirs.
Figure 2.3 Darcy's experiment
Darcy measured the total discharge through a sand column when changing the difference between water levels in two reservoirs.He found the total discharge was proportional to the difference of water levels：
Equation（2.3）is the famous Darcy's law.
Where：Q is total discharge，［L3T-1］；A is cross-sectional area，［L2］；K is hydraulic conductivity or coefficient of permeability，［LT-1］；φ1 and φ2 are water levels in the left and right reservoirs，［L］，φ1 is also the groundwater head of the left sand column，and φ2 is the groundwater head of the right sand column；L is length of the sand column，［L］；i is hydraulic gradient.
1.Groundwater head
In general，groundwater head is defined as the elevation head plus the pressure head：
Where：z is elevation of the point concerned above the reference level，［L］；p is pressure in the fluid at that point，［ML-1T-2］；ρ is density of fluid（mass per unit volume），［ML-3］；g is acceleration of gravity，［LT-1］，g=9.81m/sec2.
And the quantities z and p/ρg are usually called the elevation head and pressure head，respectively.
2.Hydraulic gradient
The rate of groundwater movement depends on the hydraulic gradient.The hydraulic gradient is the change in head per unit of distance in a given direction.If the direction is not specified，it is understood to be the direction in which the maximum rate of decrease of head occurs.
3.Hydraulic conductivity and intrinsic permeability
In Darcy's experiment，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.Hydraulic conductivity replaces the term“field coefficient of permeability”and should be used in referring to the water transmitting characteristic of material in quantitative terms.It is still common practice to refer in qualitative terms to“permeable”and“impermeable”material.
Hydraulic conductivity is not only different in different types of rocks but may also be different from place to place in the same rock.If the hydraulic conductivity is essentially the same in any area，the aquifer in that area is said to be homogeneous.If，on the other hand，the hydraulic conductivity differs from one part of the area to another，the aquifer is said to be heterogeneous.Hydraulic conductivity may also be different in different directions at any place in an aquifer.If the hydraulic conductivity is essentially the same in all directions，the aquifer is said to be isotropic.If it is different in different directions，the aquifer is said to be anisotropic.
Experiments show that the property of fluids influencing the hydraulic conductivity is the kinematic viscosity.The following relation is usually used：
Where：k is called the intrinsic permeability with a dimension of，［L2］，depending only on the characteristics of the medium；ρ is the mass density，［ML-3］；μ is the dynamic viscosity，［ML-1T-1］，and measures the resistance of fluid to shearing that is necessary for flow；ν is kinematic viscosity，［L2T-1］，it is related to dynamic viscosity with
.
It is known that the kinematic viscosity varies with the temperature and density of fluids.In the cases 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：
For the case off 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 analog of laminar flow in tubes with groundwater flow in porous medium indicates that the intrinsic permeability is proportional to squared diameter of pore space（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：
Where：d is diameter of pore space，［L］；n is porosity；c is constant to be determined with the experiment.
Equation（2.7）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.
Table 2.1 Range of permeability for common porous medium
4.Specific discharge
The specific discharge is defined as the discharge per unit of cross-sectional area and denoted by q.It follows from equation（2.3）that，
By taking the limit Δl→0，the equation（2.8）becomes，
Where：q is specific discharge，［LT-1］；-dφ/dL is gradient of groundwater head（hydraulic gradient）.
Equation（2.9）is the differential formulation of the Darcy's law.It expresses a linear relationship between the specific discharge and hydraulic gradient.
5.Velocity
Although the specific discharge has the same 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 nA.n is the porosity of the sand.Hence，in Darcy's experiment，the actual velocity（v）of the flow can be computed as：
Where：v is actual velocity，［LT-1］；v is seepage velocity，［LT-1］.
The actual velocity is always greater than the seepage velocity since the porosity is smaller than one.
6.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 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：
Where：d is average grain diameter，［L］；ν is kinematic viscosity，［L2T-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.
Darcy's law in the differential form of equation（2.9）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 groundwater flow in isotropic porous medium，the hydraulic conductivity（K）having the same value in all directions and is a scalar.Darcy's law will be written as：
Where：qx，qy and qz are three flow components in x，y and z directions in isotropy medium respectively.
The groundwater head will be a function of x，y and z coordinates and is defined as：
Where：ρ is assumed a constant.
In cases of three-dimensional groundwater flow in anisotropic medium，the hydraulic conductivity（K）having directional properties and is really a tensor in 3D.Darcy's law will be written as：
The conductivity ellipse（anisotropic vs.isotropic）shown as Figure 2.4.
Figure 2.4 Conductivity ellipse（anisotropic vs.isotropic）
If Kyy=Kxx then the media is isotropic and ellipse is a circle.It is convenient to describe Darcy's law as：
Where：▽ is called del and is a gradient operator，so ▽φ is the gradient in all three di rections（in 3D），.
The magnitudes of K in the principal directions are known as the principal conductivities.If the coordinate axes are aligned with the principal directions of the conductivity tensor，then Darcy's law can be generalized as：
Where：Kxx，Kyy and Kzz are anisotropic hydraulic conductivity in three principal directions，respectively.
2.2.2 Law of Mass Conservation
Groundwater flow also satisfies the principle of mass conservation，or mass balance.Taking a small parallelepiped in porons medium，groundwater flow through the parallelepiped must obey the mass balance stated as：
Total mass in-total mass out=change of mass storage
If the density of groundwater is a constant，mass balance will be identical to water balance.Water balance states that
Total flow in-total flow out=change of water storage
From the principle of mass balance，the so-called equation of continuity can be derived.
Figure 2.5 shows the mass fluxes through six sides of a parallelepiped in porous medium.
First，consider flow in x-direction only and qx［L3/L2-T］is specific discharge in x-direction（volume flux per area）at a point（x，y，z）.The mass flow through plane y-z at（x，y，z）is ρqxdydz［M/T］.Figure 2.6 shows the mass fluxes through 6 sides of a parallelepiped in porous medium.
Figure 2.5 Rectangular parallelepipeds in porous medium
Figure 2.6 Mass fluxes through 6 sides of a parallelepiped
Rate of change of mass flux in the x-direction per unit time per cross-section is
.
Mass flow into the entry plane y-z is：
And mass flow out of the exit plane y-z is：
In the x-direction，the flow in minus the flow out is：
Similarly，the flow in the y-direction through the plane dxdz can be derived as：
Similarly，we get for the net mass flux in the z-direction：
The total mass in the parallelepiped dxdydz is：
M=ρ（dxdydz）n
The change（increase）of mass storage per unit time in the parallelepiped is：
According to the mass balance，the change of mass storage must be equal to total increase of mass due to flow in x，y，and z directions.Thus，the summation of equations（2.17），（2.18）and（2.19）equals equation（2.20）as：
Equation（2.21）is the general equation of continuity of groundwater flow.The equation of continuity is second fundamental equation of groundwater flow.The equation of continuity combined with Darcy's law will result in basic equations describing groundwater flow in porous medium.