Steady interface flow towards the coast¶

Mark Bakker, TU Delft, The Netherlands

In [1]:

# these are needed so that the notebook can also be run on Python 2
from __future__ import division, print_function
from pylab import *
%matplotlib notebook
from semi_interface import *

Interface flow towards the coast¶

Flow is confined below the land and semi-confined below the sea. The leaky sea bottom extends a distance $L_s$ below the sea. The freshwater is moving and the saltwater is at rest. Input variables:

k: hydraulic conductivity [m/d]
H: aquifer thickness [m]
c: resistance of leaky layer [d]
grad: absolute value of head gradient towards coast upstream of interface toe
rhof: density of fresh water [kg/m3]
rhos: density of salt water [kg/m3]
Ls: length of leaky sea bottom below sea (may also be inf) [m]
ztop: elevation of top of aquifer with respect to datum [m]
sealevel: elevation of sealevel with respect to datum [m]

Case I of Bakker (2006)¶

In [2]:

sc1 = SemiCoast(k=10, H=10, c=100, grad=0.0005, 
                rhof=1000, rhos=1025, Ls=inf, 
                ztop=0, sealevel=0)
print('toe of interface at:', sc1.toe())
print('tip of interface at:', sc1.tip())
sc1.plot(xmin=-300, xmax=300);

toe of interface at: -211.68452838
tip of interface at: 153.261886479

Case II of Bakker (2006)¶

In [3]:

sc2 = SemiCoast(k=10, H=10, c=100, grad=0.00375, 
                rhof=1000, rhos=1025, Ls=inf, 
                ztop=0, sealevel=0)
print('toe of interface at:', sc2.toe())
print('tip of interface at:', sc2.tip())
sc2.plot(xmin=-300, xmax=300);

toe of interface at: 53.6836486682
tip of interface at: 298.632622947

Case III of Bakker (2006)¶

In [4]:

sc3 = SemiCoast(k=10, H=10, c=100, grad=0.0005, 
                rhof=1000, rhos=1025, Ls=80, 
                ztop=0, sealevel=0)
print('toe of interface at:', sc3.toe())
print('tip of interface at:', sc3.tip())
sc3.plot(xmin=-300, xmax=300);

toe of interface at: [-212.41139369]
tip of interface at: 80.0

Case IV of Bakker (2006)¶

In [5]:

sc4 = SemiCoast(k=10, H=10, c=100, grad=0.00375, 
                rhof=1000, rhos=1025, Ls=150, 
                ztop=0, sealevel=0)
print('toe of interface at:', sc4.toe())
print('tip of interface at:', sc4.tip())
sc4.plot(xmin=-300, xmax=300);

toe of interface at: 50.3243452459
tip of interface at: 150.0

Specification of upstream head (onshore, so $x<0$)¶

Rather than specifying the gradient upstream of the toe, the head h is specified at some point x in the aquifer below the land (so x is negative).

In [6]:

schead = SemiCoastHead(k=10, H=10, c=100, x=-1000, h=1, 
                       rhof=1000, rhos=1025, Ls=1000, 
                       ztop=0, sealevel=0)
print('toe of interface at:', schead.toe())
print('tip of interface at:', schead.tip())
print('location of h=1 (should be -1000)', schead.onshorex(h=1))
print('computed gradient:', schead.grad)
schead.plot();

toe of interface at: [-103.86297425]
tip of interface at: [ 181.97278682]
location of h=1 (should be -1000) [-1000.]
computed gradient: [ 0.00083693]

Figure for three different sealevels¶

In [7]:

schead = SemiCoastHead(k=10, H=10, c=100, x=-1000, h=0.25, 
                       rhof=1000, rhos=1025, Ls=1000, 
                       ztop=0, sealevel=0, label='h=0.25')
schead.plot(xmin=-500, xmax=200)
schead = SemiCoastHead(k=10, H=10, c=100, x=-1000, h=0.5, 
                       rhof=1000, rhos=1025, Ls=1000, 
                       ztop=0, sealevel=0, label='h=0.5')
schead.plot(newfig=False);
schead = SemiCoastHead(k=10, H=10, c=100, x=-1000, h=1, 
                       rhof=1000, rhos=1025, Ls=1000, 
                       ztop=0, sealevel=0, label='h=1')
schead.plot(newfig=False);
legend(loc='best');

Interface flow below an island¶

Consider steady Dupuit interface flow below a long island. The width of the island is $2W$. The infiltration on the island is uniform and equal to $N$. Flow is unconfined below the island and semi-confined below the sea. The leaky sea bottom extends a distance $L_s$ below the sea. The freshwater is moving and the saltwater is at rest. If the interface intersects the bottom of the aquifer, the toe of the interface is below the island. Input variables:

k: hydraulic conductivity [m/d]
D: depth of aquifer bottom below sealevel [m]
c: resistance of leaky layer [d]
rhof: density of fresh water [kg/m3]
rhos: density of salt water [kg/m3]
W: half the width of the island [m]
N: areal infiltration rate [m/d]
Nstreamlines: number of streamlines to be drawn

In [8]:

ii = IslandInterface(k=10, D=100, c=100, rhof=1000, rhos=1025, W=1000, N=0.001, Nstreamlines=10)
ii.plot()

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