Tunnel Face Stability & New CPT Applications - Geo-Engineering

A.1. Semi-confined System of Aquifers 187in each aquifer, k i the permeability and H i the layer thickness. The aquitard above layer i hasconductivity ˜c i and the specific discharge q i through this aquitard isq i = ϕ i − ϕ i−1˜c i= ϕ i − ϕ 0˜c i− ϕ i−1 − ϕ 0˜c i+1with ϕ 0 the constant head above the top layer.The potential for layer i is i = k i H i (ϕ i − ϕ 0 )(A.11)(A.12)Introducing a i,j = 1/ ( k i H i ˜c j)the differential equation for each layer is given by∇ 2 i = q i − q i+1 = −a i−1,i i−1 + ( a i,i + a i,i+1)i − a i+1,i+1 i+1(A.13)where due to boundary conditions for the lowest aquifer{ai,n+1 = 0a n+1,i = 0∀i = 1, 2...n(A.14)For one-dimensional and axi-symmetrical problems the particular solutions satisfy the differentialequation∇ 2 W(x, y, ω) = ω 2 W(x, y, ω)(A.15)and it follows that i = A i W(x, y, ω)(A.16)is a suitable set of solutions to this problem. Substituting this solution in the set of differentialequations for this system leads toA i ω 2 = −a i−1,i A i−1 + (a i,i + a i,i+1 )A i − a i+1,i+1 A i+1 ,(A.17)with the requirement that A 0 = 0 in order to include 0 = 0. This system of equations interms of A i has nontrivial solutions only if the determinant vanishes. That condition leads to anequation in terms ω 2 which has n roots ω 2 j with accompanying constants A i,j . As the equationfor i = 1 has only two constants, it can be divided by A 1,j to find the ratio A 2,j /A 1,j , which inturn can be used to find A 3,j /A 1,j and so on. This leads to a set of constantsα i,j = A i,j /A 1,j(A.18)for each value of ω j .Now the set of particular solutions for each value of ω j can be written asn∑ i = α i,j A 1,j W(x, y, ω j )j=1(A.19)where A 1,j is an unknown parameter. For the problem of one-dimensional flow there are twoparticular solutions: e −ω jx and e ω jx . For this case the particular solution can be rewritten asn∑ [ i = α i,j Bj e ωjx + C j e −ω jx ]j=1(A.20)