Code covered by the BSD License  

Highlights from
Environmental Modeling

image thumbnail

Environmental Modeling

by

 

20 Jul 2007 (Updated )

Introduction to Modelling and Simulation for Environmental Problems

gw_flow
function gw_flow
% groundwater flow 
%    using MATLAB analytical solutions                   
%
%   $Ekkehard Holzbecher  $Date: 2007/02/03 $
%--------------------------------------------------------------------------
% Baseflow
H = 6.;             % thickness [L]
h0 = 5.5;           % reference piezometric head [L] 
K = 5.e-5;          % hydraulic conductivity [L/T] 
por = 0.25;         % porosity []
Qx0 = 1.e-6;        % baseflow in x-direction [L^2/T]
Qy0 = 0;            % baseflow in y-direction [L^2/T]

% Wells
x0 = [99 145];      % x-coordinates well position [L]
y0 = [50 78];       % y-coordinates well position [L]
Q = [1.e-4 1.e-4];  % pumping / recharge rates [L^3/T]
R = [0.3 0.2];      % well radius [L]

% Mesh
xmin = 0;           % minimum x-position of mesh [L]
xmax = 200;         % maximum x-position of mesh [L]
ymin = 0;           % minimum y-position of mesh [L]
ymax = 200;         % maximum y-position of mesh [L]

% Reference point position in mesh
iref = 1; jref = 1;

% Graphical output options
gsurfh = 0;         % piezometric head surface plot
gcontf = 10;        % no. filled contour lines (=0: none)
gquiv = 0;          % arrow field plot
gflowp_fit = 10;    % flowpaths forward in time
gflowp_bit = 0;     % no. flowpaths backward in time (=0: none)
gflowp_dot = 5;     % flowpaths with dots indicating speed
gstream = 10;       % streamfunction plot

%----------------------------------------execution-------------------------------
xvec = linspace(xmin,xmax,100);
yvec = linspace(ymin,ymax,100);
[x,y] = meshgrid (xvec,yvec);                      % mesh
phi = -Qx0*x - Qy0*y;                              % baseflow potential
for j = 1:size(Q,2)
    r = sqrt((x-x0(j)).*(x-x0(j))+(y-y0(j)).*(y-y0(j)));   % distances to well
    phi = phi + (Q(j)/(2*pi))*log(r);              % potential
end
if h0 > H
    phi0 = -phi(iref,jref) + K*H*h0 - 0.5*K*H*H; 
else
    phi0 = -phi(iref,jref) + 0.5*K*h0*h0;          % reference potential 
end                                                
hc = 0.5*H+(1/K/H)*(phi+phi0);                     % head confined
hu = sqrt ((2/K)*(phi+phi0));                      % head unconfined
phicrit = phi0 + 0.5*K*H*H;                        % transition confined / unconfined
confined = (phi>=phicrit);                         % confined / unconfined indicator
h = confined.*hc+~confined.*hu;                    % head
psi = -Qx0*y + Qy0*x;
for j = 1:size(Q,2)
    psi = psi + (Q(j)/(pi+pi))*atan2((y-y0(j)),(x-x0(j)));  % streamfunction 
end
%---------------------------------------display messages-------------------
if all(all(confined))
    display ('aquifer confined');
else
    if all(all(~confined)) 
        display ('aquifer unconfined'); 
    else
        display ('aquifer partially confined and unconfined'); 
    end
end    
if any(any(h<0)) 
    display ('aquifer falls partially dry'); 
    h = max(0, h);
end

%--------------------------------------------------------------------------
[u,v] = gradient (-phi);                           % discharge vector  
Hh = confined.*H + ~confined.*h;                   % aquifer depth  
u = u./Hh/(xvec(2)-xvec(1))/por; v = v./Hh/(yvec(2)-yvec(1))/por;

%--------------------------------------graphical output--------------------
if gsurfh 
    figure; surf (x,y,h);                          % surface 
end 
if gcontf | gquiv | gflowp_fit | gflow_bit | gflowp_dot | gstream figure; end
if gcontf                                          % filled contours  
    colormap(winter); 
    contourf (x,y,h,gcontf,'w'); 
    colorbar; hold on;
end
if gquiv 
    quiver (x,y,u,v,'y'); hold on;                 % arrow field
end
if gflowp_fit                                      % flowpaths 
    xstart = []; ystart = [];
    for i = 1:100
        if v(1,i) > 0 xstart = [xstart xvec(i)];...
                ystart = [ystart yvec(1)]; end
        if v(100,i) < 0 xstart = [xstart xvec(i)];...
                ystart = [ystart yvec(100)]; end
        if u(i,1) > 0 xstart = [xstart xvec(1)];...
                ystart = [ystart yvec(i)]; end
        if u(i,100) < 0 xstart = [xstart xvec(100)];...
                ystart = [ystart yvec(i)]; end
    end
    h = streamline (x,y,u,v,xstart,ystart);
    set (h,'Color','r'); 
end
if gflowp_bit          
    for j=1,size(Q,2)
        if Q(j)>0           % only for pumping wells
            xstart = x0(j) + R(j)*cos(2*pi*[1:1:gflowp_bit]/gflowp_bit); 
            ystart = y0(j) + R(j)*sin(2*pi*[1:1:gflowp_bit]/gflowp_bit);
            h = streamline (x,y,-u,-v,xstart,ystart);
            set (h,'Color','y')
        end
    end
end
if gflowp_dot
    [verts averts] = streamslice(x,y,u,v,gflowp_dot);
    sc = 10/mean(mean(sqrt(u.*u+v.*v)));
    iverts = interpstreamspeed(x,y,u,v,verts,sc);  
    h = streamline(iverts);
    set (h,'Marker','.','Color','y','MarkerSize',18)
end
if gstream
    h = contour (x,y,psi,gstream,'k','LineWidth',1);  
end

Contact us