Z q Figure 2.2 Geometric illustration of composition of specific discharge 2.3 Equation of continuity Groundwater flow also satisfies the principle of conservation of mass, or mass balance Taking a small parallelepiped in medium, groundwater flow through the arallelepiped 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 2.3 shows the mass fluxes through six sides of a parallelepiped in porous medium
x qx qy y z q qz Figure 2.2 Geometric illustration of composition of specific discharge 2.3 Equation of continuity Groundwater flow also satisfies the principle of conservation of mass, or mass balance. Taking a small parallelepiped in porous 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 2.3 shows the mass fluxes through six sides of a parallelepiped in porous medium. 15
D2+)(D1) +)(a) D+)(D1) Figure 2.3 Mass fluxes through a parallelepiped in porous medium First, consider flow in y-direction only. The mass carried into the left face of the element by the specific discharge qy is Ax△z The mass carried out from the right face of the element is a(pq ay-Ay AxAz Hence the increase of mass(per unit time) due to the flow in y-direction will be pq△xAz-pq,+oc q a(pq Ax△Z Z Similarly, the increase of mass due to the flow in x and z directions can be derived pqx△y△z-|pq a(paxAx AyAZ a(pq) Ax△V△Z 16
)x )y )z Dqy Dqx Dqz Dqz +)(Dqz) Dqy +)(Dqy) Dqx+)(Dqx) Figure 2.3 Mass fluxes through a parallelepiped in porous medium First, consider flow in y-direction only. The mass carried into the left face of the element by the specific discharge qy is: q x z y ρ ∆ ∆ The mass carried out from the right face of the element is: y x z y ( q ) q + y y ∆ ∆ ∆ ∂ ∂ ρ ρ Hence the increase of mass (per unit time) due to the flow in y-direction will be: x y z y ( q ) y x z = - y ( q ) q x z - q + y y y y ∆ ∆ ∆ ∂ ∂ ρ ∆ ∆ ∆ ∂ ∂ ρ ρ ∆ ∆ ρ (2.14) Similarly, the increase of mass due to the flow in x and z directions can be derived as: (2.15) 16 x y z x ( q ) x y z = - x ( q ) q y z - q + x x x x ∆ ∆ ∆ ∂ ∂ ρ ∆ ∆ ∆ ∂ ∂ ρ ρ ∆ ∆ ρ
0(pq2) a(pq) pq2Ax△y-pq2 Az AxA △xV△Z (2.16) The total mass in the parallelepiped△x△y△zis △M=p(Ax△y△zn The change(increase)of mass storage per unit time in the parallelepiped is a(△M) at QlP(△xAy△z)n] (217) 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. 14), (2.15 )and (2.16) equals(2.17)as: p(△xy△zn]=- a(pq a(pqy). a(p Axy△z Equation(2. 18)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 2.4 Basic equations for steady incompressible flow The groundwater flow will be in steady state when there is no change of mass storage. In this case. the mass balance states that total mass in= total mass out Since d(AM)ot=0 in the case of steady flow, the equation of continuity become a(pq apqv)a(p (2.19)
and x y z z ( q ) z x y = - z ( q ) q x y - q + z z z z ∆ ∆ ∆ ∂ ∂ ρ ∆ ∆ ∆ ∂ ∂ ρ ρ ∆ ∆ ρ (2.16) The total mass in the parallelepiped ∆x ∆y ∆z is: ∆M = ρ(∆x∆y∆z)n The change (increase) of mass storage per unit time in the parallelepiped is: [ ( x y z)n] t = t ( M) ρ ∆ ∆ ∆ ∂ ∂ ∂ ∂ ∆ (2.17) 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 (2.14), (2.15) and (2.16) equals (2.17) as: ] x y z z ( q ) + y ( q ) + x ( q ) [ ( x y z)n]= -[ t x y z ∆ ∆ ∆ ∂ ∂ ρ ∂ ∂ ρ ∂ ∂ ρ ρ ∆ ∆ ∆ ∂ ∂ (2.18) Equation (2.18) 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. 2.4 Basic equations for steady incompressible flow The groundwater flow will be in steady state when there is no change of mass storage. In this case, the mass balance states that total mass in = total mass out. Since ∂(∆M)/∂t = 0 in the case of steady flow, the equation of continuity become: (2.19) 17 = 0 z ( q ) + y ( q ) + x ( q )x y z ∂ ∂ ρ ∂ ∂ ρ ∂ ∂ ρ
Furthermore, if the fluid is incompressible, the density p is a constant and equation(2. 19) q Equation(2.20)is the equation of continuity for steady incompressible groundwater flow Substitution of Darcy's law, equation(2.10)into(2. 20)gives d2+29+92q=0 2 which is the basic differential equation of steady incompressible flow in homogeneous isotropic porous medium. It is noted that Equation(2.2 1)is the standard Laplace equation and s often written in abbreviation form Substitution of Darcy' s law, equation(2. 11)into(2.20)will give the differential equation of steady incompressible flow in anisotropic porous medium, which is (KxC)+(K∞)+2(Ka∞)=0 (223) When the porous medium is anisotropic but homogeneous, equation(2.23)reduces to a2 K Kz 2.5 Basic equations for non-steady compressible flow In the case of non-steady flow, the storage of mass is changed since the total mass in is not equal to the total mass out 18
Furthermore, if the fluid is incompressible, the density ρ is a constant and equation (2.19) reduces to: = 0 z q + y q + x qx y z ∂ ∂ ∂ ∂ ∂ ∂ (2.20) Equation (2.20) is the equation of continuity for steady incompressible groundwater flow. Substitution of Darcy's law, equation (2.10) into (2.20) gives: = 0 z + y + x 2 2 2 2 2 2 ∂ ∂ ϕ ∂ ∂ ϕ ∂ ∂ ϕ (2.21) which is the basic differential equation of steady incompressible flow in homogeneous isotropic porous medium. It is noted that Equation (2.21) is the standard Laplace equation and is often written in abbreviation form: ∆ = 0 2ϕ (2.22) Substitution of Darcy's law, equation (2.11) into (2.20) will give the differential equation of steady incompressible flow in anisotropic porous medium, which is: ) = 0 z (K z ) + y (K y ) + x (K x xx yy zz ∂ ∂ϕ ∂ ∂ ∂ ∂ϕ ∂ ∂ ∂ ∂ϕ ∂ ∂ (2.23) When the porous medium is anisotropic but homogeneous, equation (2.23) reduces to: = 0 z + K y + K x 2 2 zz 2 2 yy 2 2 xx ∂ ∂ ϕ ∂ ∂ ϕ ∂ ∂ ϕ K (2.24) 2.5 Basic equations for non-steady compressible flow In the case of non-steady flow, the storage of mass is changed since the total mass in is not equal to the total mass out. 18
From the principles of Soil Mechanics, it is known that the total pressure, Oz, from overlying geological strata is balanced by the effective pressure, oi, of grains and pore pressure, p, of water. le oz-Oz, tp (225) Since the total pressure can be assumed to be independent of time, any increase of pore pressure will result in a decrease of effective pressure. The consequences of such a change (1) water is compressed due to the increase of pore pressure. Therefore, the density of water is a function of time (2) grain skeleton is expanded due to the decrease of effective pressure. Hence, the porosity and size of grain skeleton are functions of time Taking the above effects into consideration, the change of mass storage in equation (2. 17)can be expressed as a△x)=(△xyna+p△xya)a+pnat(△xyA2) (226) Compressibility of water From the elastic theory, the relative change of density of water is proportional to the change of pore pressure, 1.e, d p (2.27) where Bw is the coefficient of compressibility of water. Its value is about 0.5x10 m/N From(2.27)it follows that dpdp ap (228) ot dp at Compressibility of soil From the theory of elasticity, soil will be compressed under the pressure. When sample is onfined horizontally, the vertical relative compression is proportional to the increase of pressure. le
From the principles of Soil Mechanics, it is known that the total pressure, σz, from overlying geological strata is balanced by the effective pressure, σz', of grains and pore pressure, p, of water. i.e. σz = σz′ + p (2.25) Since the total pressure can be assumed to be independent of time, any increase of pore pressure will result in a decrease of effective pressure. The consequences of such a change are: (1) water is compressed due to the increase of pore pressure. Therefore, the density of water is a function of time; (2) grain skeleton is expanded due to the decrease of effective pressure. Hence, the porosity and size of grain skeleton are functions of time. Taking the above effects into consideration, the change of mass storage in equation (2.17) can be expressed as: ( x y z) t + n t n + ( x y z) t [ ( x y z)n]= ( x y z)n t ∆ ∆ ∆ ∂ ∂ ρ ∂ ∂ ρ ∆ ∆ ∆ ∂ ∂ρ ρ ∆ ∆ ∆ ∆ ∆ ∆ ∂ ∂ (2.26) Compressibility of water From the elastic theory, the relative change of density of water is proportional to the change of pore pressure, i.e, = dp d βw ρ ρ (2.27) where βw is the coefficient of compressibility of water. Its value is about 0.5x10-9 m2 /N. From (2.27) it follows that: t p = t p dp d = t w ∂ ∂ β ρ ∂ ρ ∂ ∂ ∂ρ (2.28) Compressibility of soil From the theory of elasticity, soil will be compressed under the pressure. When sample is confined horizontally, the vertical relative compression is proportional to the increase of pressure, i.e 19