7. chapter 7 - steady state flow _a4

Chapter 7 Steady-state Flow through Permeable Media________________________

74

CHAPTER 7 STEADY – STATE FLOW THROUGH PERMEABLE MEDIA The problem stands to determine the hydraulic head distribution, the hydraulic gradients, the seepage velocities and the discharge due to the water flow through a permeable medium. The seepage phenomenon is due to the difference between the hydraulic head values, specified on various boundaries of the domain (in this case, between the upstream and downstream sides of a retaining structure). The water head has the general expression

γpzH += (7.1)

while the hydraulic gradient is the ratio between the hydraulic head difference in two points along the flow line and the flow line’s length

12

12

LHHgradHi −

== (7.2)

The equations governing the phenomenon are the equation of continuity:

0=∂∂

+∂

∂+

∂∂

zq

yq

xq zyx (7.3)

where qx, qy and qz are the discharge components through area unit (sometimes called flux), and the generalized Darcy low:

______________________Basics of the Finite Element Method Applied in Civil Engineering

75

Fig. 7.1 Seepage phenomenon problem and the corresponding finite elements

model

D1

D2

H1

H2

γp

z1

2

3

4

5

6

7

8

D1

D2 z

xy

C

C

qr

1 2

6 5

vr


76

∂∂

+∂∂

+∂∂

−=zHk

yHk

xHkv xzxyxx

∂∂

+∂∂

+∂∂

−=zHk

yHk

xHkv yzyyxy (7.4)

∂∂

+∂∂

+∂∂

−=zHk

yHk

xHkv zzyzxz

where vx, vy and vz are the seepage velocity components and ki the hydraulic conductivity (permeability) components of the medium. In matrix form

[ ]gradHqqq

z

y

x

k q −=

= (7.5)

with k a 3 × 3 conductivity matrix and the gradient operator

∂∂∂∂∂∂

=

z

y

xgrad . (7.6)

The seepage equations yield the general form:

[ ][ ] 0=gradHgrad k (7.7) to which the boundary conditions are added: prescribed water head

HH = on ΓH and prescribed specific discharge (flux) qqn = on Γq, where qn is the normal flux through the boundary. For the specific example, the prescribed boundary conditions are H = H1 on D1 and H = H2 on D2, while the imposed discharge qn is due to a forced drainage on boundary C.


77

In the differential approach, the unknown function is the hydraulic head H(x, y, z), the general form corresponds to the system A(u) = 0 and the boundary conditions represent the B(u) = 0 equations. In the variational approach, the associated functional is:

[ ] ∫∫ ΓΓ−−= HdqdVgradHE

T

Vq

21 (7.8)

in which the first term is the hydraulic energy dissipated in the seepage process. For a certain element, the hydraulic head H(x, y, z) is expressed by the approximate (shape) functions according to the nodal values Hi:

882211 ),,(...),,(),,(),,( HzyxNHzyxNHzyxNzyxH +++= (7.9) or, in vector form

ezyxH NH=),,( (7.10) Note that in this type of problem the hydraulic head (the unknown) is a scalar, with no components along the Cartesian coordinate system directions, while N is a vector of shape functions. Hence, the dedicated finite element has only one degree of freedom (DOF) per node. The hydraulic gradient associated to the element can also be expressed by the nodal hydraulic heads:

[ ] [ ] eqgradgradH HBNH == (7.11) where Bq is the vector with the first derivatives of the shape functions. From the Darcy low, the specific discharge as a function of nodal water heads, yields:

eqHkBq −= (7.12)


78

The elemental functional yields

∫∫ ΓΓ

=

ee

dqdVE TTeV q

Tq

Tee NH - HkBBH

21 (7.13)

In this form the functional depends only on the nodal values of the hydraulic head. The first integral identifies as the seepage matrix of an element while the second one is the boundary conditions vector:

∫=eV q

Tqs dVkBBk ; ∫Γ Γ=

e

dqTNr (7.14)

where q is the prescribed discharge through the elements faces. The functional expression yields:

rHHkH Tees

TeeE −=

21 (7.15)

For the entire seepage domain, the functional is the sum of elemental contributions:

RHHKHrHHkH Ts

Tm

Tm

sT

m

eeEE −=−

== ∑∑∑

= 21

21

111

(7.16) In this way, the continuous domain was replaced by a discrete model with the functional expressed according to the nodal values of the hydraulic head. The stationary condition 0/

,1=∂∂

= niiHE leads to a linear algebraic system:

RHK =s (7.17) The boundary conditions, as prescribed hydraulic head, are applied using the same algorithm as for the prescribed displacements. If in a node j lying on the boundary Γ the hydraulic head has a prescribed value H , in the algebraic system the following equality is added to equation j:


79

HkHk j = (7.18)

where k is an arbitrary number. If k is chosen with some orders of magnitude above the diagonal coefficients of the system, the solution of equation j becomes HH j = . By solving the equation system with the prescribed boundary conditions, the nodal hydraulic head vector is determined. Consequently, the hydraulic spectrum (the equipotential lines) can be drawn. The hydraulic gradients and the discharge can be assessed by returning to the element level. The discharge through an element face is

∫=eS ne dSqQ (7.19)

where qn is the water particle velocity normal to the element face Se,

[ ] q n=

=

z

y

x

zyxn

qqq

nnnq (7.20)

with n the director cosine vector of the normal to the element face in the global coordinate system

eqnq HB k n−= (7.21)

eqeS qee

dSQ HCHB k n =

−= ∫ (7.22)

The total discharge through a certain cross section (or boundary) yields by adding the contributions of all element faces lying on that surface.

7. chapter 7 - steady state flow _a4

Documents