dAz do where vertical pressure on soil with area(Ax Ay)and height Az coefficient of compressibility of soil. Common values are 10-10 m/N for sand, and 10--10-m/N for clay In many practical cases usually the horizontal deformations of the soil skeleton is much smaller than the vertical deformation. Therefore, it can be assumed that Ax, Ay are constant and only Az changes. Under this assumption 「△x△v△zl= at Axdys a(△)0oxy=a△x△ yAz-Oz' 30 at Since oz=0i+ pis assumed to be independent of time, i.e =( that at at Substitution of (2.31)into(2.30)yields [△x△y△z]=aAxy△z Change of porosity Although the size Ax Ay Az changes with compression, it is assumed that only pore space is changed, but the total value of grains, Vs, is kept constant since grains can be considered completely incompressible. The assumption of Vs=(I-n) Ax Ay Az constant gives
' z = - d z d z α σ ∆ ∆ (2.29) where: σz': vertical pressure on soil with area (∆x ∆y) and height ∆z; α : coefficient of compressibility of soil. Common values are 10-8-10-7 m2 /N for sand, and 10-7-10-6 m2 /N for clay. In many practical cases usually the horizontal deformations of the soil skeleton is much smaller than the vertical deformation. Therefore, it can be assumed that ∆x, ∆y are constant and only ∆z changes. Under this assumption, t x y = - x y z t ( z) x y = t ( z) [ x y z] = t z z z ∂ ∂ σ ∆ ∆ α∆ ∆ ∆ ∂ ∂ σ ∂ σ ∂ ∆ ∆ ∆ ∂ ∂ ∆ ∆ ∆ ∆ ∂ ∂ ′ ′ ′ (2.30) Since σz = σz' + p is assumed to be independent of time, i.e. = 0 t z ∂ ∂ σ so that t p = - t z ∂ ∂ ∂ ∂ σ ′ (2.31) Substitution of (2.31) into (2.30) yields: t p [ x y z] = x y z t ∂ ∂ ∆ ∆ ∆ α∆ ∆ ∆ ∂ ∂ (2.32) Change of porosity Although the size ∆x ∆y ∆z changes with compression, it is assumed that only pore space is changed, but the total value of grains, Vs, is kept constant since grains can be considered completely incompressible. The assumption of Vs = (1-n) ∆x ∆y ∆z constant gives: 20
avs a(1-n) O( Az) △x△y△z+(1-n)△x△ (2.33) t From equation(.33)with(2.32), the relation between the changes of the porosity with the change of the pore pressure is found as On I-n a(Az) (234) =(1-no p t△zat Equation of continuity of non-steady compressible flow Substitutions of equations(2.28), (2.32)and(2. 34)into(2.26) gives p(△xNy△z)n= Ax Ay Az p[a+n阝 (2.35) Substitution of (2. 35)into(2. 18)gives the general equation of continuity for non-steady ible flo 02 q a(pqy). a(pq,) Basic equation of non-steady compressible flow In case of compressible flow in homogeneous isotropic porous medium, Darcy's law of equation(2. 10)will be replaced by (2.37) =-K Substitution of (2.37)into(2.36)results in g(a+nB) ot kop+ap+O2 P (238)
= 0 t ( z) x y z + (1- n) x y t (1- n) = t Vs ∂ ∂ ∆ ∆ ∆ ∆ ∆ ∆ ∂ ∂ ∂ ∂ (2.33) From equation (2.33) with (2.32), the relation between the changes of the porosity with the change of the pore pressure is found as: t p = (1- n) t ( z) z 1- n = t n ∂ ∂ α ∂ ∂ ∆ ∂ ∆ ∂ (2.34) Equation of continuity of non-steady compressible flow Substitutions of equations (2.28), (2.32) and (2.34) into (2.26) gives: t p [ ( x y z)n]= x y z [ + n ] t w ∂ ∂ ρ ∆ ∆ ∆ ∆ ∆ ∆ ρ α β ∂ ∂ (2.35) Substitution of (2.35) into (2.18) gives the general equation of continuity for non-steady compressible flow as: ] z ( q ) + y ( q ) + x ( q ) = -[ t p ( + n ) x y z w ∂ ∂ ρ ∂ ∂ ρ ∂ ∂ ρ ∂ ∂ ρ α β (2.36) Basic equation of non-steady compressible flow In case of compressible flow in homogeneous isotropic porous medium, Darcy's law of equation (2.10) will be replaced by: x p g K q = - x ∂ ∂ ρ y p g K q = - y ∂ ∂ ρ (2.37) z p g K q = - K - z ∂ ∂ ρ Substitution of (2.37) into (2.36) results in: z ] + g z p + y p + x p = K [ t p g( + n ) 2 2 2 2 2 w ∂ ∂ρ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ρ α β (2.38) 21
If the vertical gradient of fluid density can be neglected (for example in one fluid flow) Equation(.38)reduces to apap. ap pg(a+nBw) Op ax a k at (239) which is a basic equation of non-steady compressible flow in homogeneous isotropic medium Using the relation of an equation in terms of head can be found as a0, a0 820S where Ss=pg(a +nBw) is usually called the storativity or specific storage coefficient. It is defined as the volume of water stored in a unit volume of soil by a unit increase of the head. It has a dimension of [L']. The physical explanation of the storativity is that with the increase of head(1)more water is stored in pore space since water is compressed; and (2) pore space enlarged since the grain skeleton is expanded. Since the expansion and compression of water and soil skeleton is elastic. S is called elastic storage
If the vertical gradient of fluid density can be neglected (for example in one fluid flow), Equation (2.38) reduces to: t p K g( + n ) = z p + y p + x p w 2 2 2 2 2 ∂ ρ α β ∂ ∂ ∂ ∂ ∂ ∂ ∂ (2.39) which is a basic equation of non-steady compressible flow in homogeneous isotropic medium. Using the relation of g p = z + ρ ϕ an equation in terms of head can be found as: K t S = z + y + x s 2 2 2 2 2 ∂ ∂ϕ ∂ ∂ ϕ ∂ ∂ ϕ ∂ ∂ ϕ (2.40) where is usually called the storativity or specific storage coefficient. It is defined as the volume of water stored in a unit volume of soil by a unit increase of the head. It has a dimension of [L S = g( + n ) s α βw ρ -1]. The physical explanation of the storativity is that with the increase of head (1) more water is stored in pore space since water is compressed; and (2) pore space is enlarged since the grain skeleton is expanded. Since the expansion and compression of water and soil skeleton is elastic, Ss is called elastic storage. 22
3. Steady Groundwater Flow in Aquifers 3. 1 Groundwater flow in a confined aquifer 3.1.1 Conceptual hydrogeological model The aquifer is confined by two impermeable layers on the top and bottom The aquifer consists of homogeneous porous medium(K=constant) The aquifer is of uniform thickness(h); The aquifer is bounded with two parallel rivers on the left and right with constant river stage A---- H K Figure 3.1 A confined aquifer Under these assumptions, groundwater flow in the aquifer is one-dimensional steady flow from the left river through the aquifer to the right river 3.1.2 Mathematical model A mathematical model describing steady groundwater flow in the aquifer consists of a partial differential equation which is a governing equation of groundwater flow, and boundary conditions which are external influences on groundwater flow in aquifers The governing equation of groundwater flow can be derived with equation of continuity and Darcy's law. In section 2. 4 the basic equation of steady groundwater flow in homogeneous
3. Steady Groundwater Flow in Aquifers 3.1 Groundwater flow in a confined aquifer 3.1.1 Conceptual hydrogeological model - The aquifer is confined by two impermeable layers on the top and bottom; - The aquifer consists of homogeneous porous medium ( K = constant); - The aquifer is of uniform thickness (H); - The aquifer is bounded with two parallel rivers on the left and right with constant river stage. n H H0 HL x L 0 x K Figure 3.1 A confined aquifer Under these assumptions, groundwater flow in the aquifer is one-dimensional steady flow from the left river through the aquifer to the right river. 3.1.2 Mathematical model A mathematical model describing steady groundwater flow in the aquifer consists of: - a partial differential equation which is a governing equation of groundwater flow; and - boundary conditions which are external influences on groundwater flow in aquifers. The governing equation of groundwater flow can be derived with equation of continuity and Darcy's law. In section 2.4 the basic equation of steady groundwater flow in homogeneous 23
medium is derived as in(2.21), which is applicable to the steady flow in this confined aquifer However, since flow in this case is only in X-direction, equation(2.2 1)is reduced to a2=0 ax2 which is the governing equation of one-dimensional steady flow in the confined aquifer Furthermore, groundwater head in the aquifer has to satisfy the following two boundary conditions φ H (3.2) Equations(3. 1)and (3.2)form the mathematical model of one-dimensional steady flow in the confined aquifer 3.1.3 Analytical solution The solution of the mathematical model gives the distribution of groundwater head in the The integration of equation (3. 1)results in CI and c2 are constants of the integration and must be determined by the boundary conditions Application of the boundary conditions of (3.2)gives CI=Ho The final solution becomes Ho-HL which shows that the distribution of groundwater head in the cross-section is a straight line from left river stage to the right river stage
medium is derived as in (2.21), which is applicable to the steady flow in this confined aquifer. However, since flow in this case is only in x-direction, equation (2.21) is reduced to = 0 x 2 2 ∂ ∂ ϕ (3.1) which is the governing equation of one-dimensional steady flow in the confined aquifer. Furthermore, groundwater head in the aquifer has to satisfy the following two boundary conditions: | = H | = H x=L L x=0 0 ϕ ϕ (3.2) Equations (3.1) and (3.2) form the mathematical model of one-dimensional steady flow in the confined aquifer. 3.1.3 Analytical solution The solution of the mathematical model gives the distribution of groundwater head in the aquifer. The integration of equation (3.1) results in: ϕ = c1 x + c2 c1 and c2 are constants of the integration and must be determined by the boundary conditions. Application of the boundary conditions of (3.2) gives: L H - H c = - c = H 0 L 2 1 0 The final solution becomes: x L H - H = H - 0 L ϕ 0 (3.3) which shows that the distribution of groundwater head in the cross-section is a straight line from left river stage to the right river stage. 24