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seepage_earth_dams_isotropic_soil.m
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seepage_earth_dams_isotropic_soil.m
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%=========================================================================
% MATLAB script file to draw the flownets of a homogeneous isotropic earth
% embankment dam using the conformal mapping technique
% For further details, go to the link below:
% https://github.com/SubhadipN/Conformal_Mapping#seepage-analysis
%=========================================================================
% Prepared by SUBHADIP NASKAR, RESEARCH SCHOLAR, IIT GUWAHATI
%=========================================================================
clear all; clc; warning('off','all')
%=========================================================================
% INPUTS::
hd = -20; % Height of the dam (m)
hw = -19; % Height of the water level (m)
twd = 12; % Top width of the dam (m)
fw = 6; % Width of the filter (m)
bd = 32; % Bottom width of the dam (m)
k = 3.5*10^(-8); % Coefficient of permeability (m/s)
%=========================================================================
% required for optimize scaling of the plot area
fl = 15; % Numbers of flow lines (odd number)
yf = 1.3; % Depth factor
xf = 1.2; % Width factor
%=========================================================================
% DERIVED PARAMETERS::
C = [-bd+fw+0.5*(bd-twd)*hw/hd -hw];
xo = 0.5*(C(1)+sqrt(C(1)^2+C(2)^2));
q = 2*k*xo;
C = -1/(2*k*q);
ss = -2*hd/(bd-twd);
%=========================================================================
% PLOT OF THE MODIFIED EMBANKMENT DAM IN Z-PLANE::
p1 = [-bd+fw 0];
p2 = [p1(1)-hd/ss -hd];
p3 = [p2(1)+twd -hd];
p4 = [fw 0];
for i = 1:6
if i == 1
x1 = p1(1);
y1 = p1(2);
x2 = p2(1);
y2 = p2(2);
yy = linspace(y1,y2,2000);
xx = 0.5*(q*yy.^2/(k*hw^2)-k*hw^2/q);
p1(1) = min(xx);
p2(1) = max(xx);
elseif i == 2
x1 = p2(1);
y1 = p2(2);
x2 = p3(1);
y2 = p3(2);
elseif i == 3
x1 = p3(1);
y1 = p3(2);
x2 = p4(1);
y2 = p4(2);
elseif i == 4
x1 = 0;
y1 = 0;
x2 = p1(1);
y2 = p1(2);
elseif i == 5
x1 = p4(1);
y1 = p4(2);
x2 = 0;
y2 = 0;
else
x1 = p2(1)+hw/ss;
y1 = -hw;
y2 = -hw;
x2 = 0.5*(q*y2^2/(k*hw^2)-k*hw^2/q);
end
if i ~= 1
xx = linspace(x1,x2,2000);
m = (y2-y1)/(x2-x1);
yy = m.*xx-m*x1+y1;
end
figure (1);
xlim([-bd/xf xf^3*fw]);
ylim([0 -yf*hd]);
hold on;
set(gca,'FontSize',20);
box on;
if i <= 4
if i == 1
leg1 = plot(xx,yy,'b','linewidth',1.5);
end
plot(xx,yy,'b','linewidth',1.5);
elseif i == 5
leg2 = plot(xx,yy,'m','linewidth',6);
else
leg3 = plot(xx,yy,'b-.','linewidth',1.5);
end
end
%=========================================================================
% FLOWNETS IN W-PLANE::
x = linspace(k*hw,0,fl);
y = linspace(0,q,fl);
for i = 1:fl
xx = linspace(0,k*hw,100);
yx = y(i)*diag(eye(100)); % '||' to x-axis
xy = x(i)*diag(eye(100));
yy = linspace(0,q,100); % '||' to y-axis
wx(i,1:100) = complex(xx,yx');
wy(i,1:100) = complex(xy',yy);
end
%=========================================================================
% FLOWNETS IN Z-PLANE::
zx = C*wx.^2;
zy = C*wy.^2;
%=========================================================================
% PLOT OF THE FLOWNETS IN Z-PLANE::
for i = 1:fl
leg4 = plot(real(zx(i,:)),imag(zx(i,:)),'k--','linewidth',1.5);
leg5 = plot(real(zy(i,:)),imag(zy(i,:)),'r','linewidth',1.5);
end
xlabel('Width (m)','fontsize',20);
ylabel('Height (m)','fontsize',20)
% title('Flownets for homogeneous isotropic earth dam','fontsize',20)
legend([leg1 leg2 leg3 leg4 leg5],{'Dam outline','Filter',...
'Water level','Flow lines','Equipotential lines'},'fontsize',20)
for i = 1:2:fl
if i == fl
t1 = ['\phi_{',num2str(i),'} = ',num2str(-x(i))];
else
t1 = ['\phi_{',num2str(i),'} = ',num2str(-x(i),'%10.1e')];
end
if i == 1
t2 = ['\psi_{',num2str(i),'} = ',num2str(y(i))];
else
t2 = ['\psi_{',num2str(i),'} = ',num2str(y(i),'%10.1e')];
end
xc1 = max(real(zy(i,:)));
yc1 = max(imag(zy(i,:)));
ht1 = text(xc1+0.4,yc1,t1,'HorizontalAlignment','left',...
'Color','r','fontsize',20);
set(ht1,'Rotation',30);
xc2 = min(real(zx(i,:)));
yc2 = max(imag(zx(i,:)));
ht2 = text(xc2-0.25,yc2,t2,'HorizontalAlignment','right',...
'Color','k','fontsize',20);
if (i == fl)
set(ht2,'Rotation',-45);
end
end
t = ['z = f(w) = -0.25w^2/(k^2x_0)'];
text(-10.5,24,t,'HorizontalAlignment','left','Color','k','fontsize',20);
axis equal; grid on; grid minor;
%=========================================================================