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fbt_delta_settling_fieldapp.m
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fbt_delta_settling_fieldapp.m
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%close all, clc
%% Getting input parameters first:
tic
%% Compute Mud Fall Velocity
% This subroutine implements the Dietrich relation for fall velocity.
Rep = ((Rm * g * Dm) ^ 0.5) * Dm / nu;
xx = log(Rep ^ 2) / log(10);
yy = -3.76715 + 1.92944 * xx - 0.09815 * (xx ^ 2) - 0.00575 * (xx ^ 3) + 0.00056 * (xx ^ 4);
Rf = ((10 ^ yy) / Rep) ^ (1 / 3);
vsm = Rf * ((Rm * g * Dm) ^ 0.5);
% vsm=0.34*0.001; % floc velocity
% %% Compute Normal Flow
% % This subroutine computes the Shields stress, slope and depth of the normal flow
% % of the fluvial region associated with the volume sand feed rate per unit wifth qso,
% % the water discharge per unit width qw and the sand grain size Ds.
%
% tsn = (qso / ((Rs * g * Ds) ^ 0.5 * Ds * al)) ^ (1 / nl) + tsc;
% Sn = Cza * (Rs * Ds * tsn) ^ 1.5 * g ^ 0.5 / qw;
% Hn = Rs * Ds * tsn / Sn;
%% Set Initial bed
%This subroutine sets up the initial bed so that the fluvial reach has
% initial length rsi and slope Sfbi, and the subaqueous reach has initial
% length rmax-rbi and slope Stbi.
for i = 1:1:N + 1
sfbar(i) = dsbar * (i-1); % originally i-1, but changed to i because etafluv devided by 0 ;
rfluv(i) = rsi * sfbar(i);
etafluv(i) = etatI + rsi * Sfbi - rsi * sfbar(i) * Sfbi;
r(i) = rfluv(i);
eta(i) = etafluv(i);
end
rbi = rsi + (etatI - etabI) / Sa;
for i = 1:1:Ns + 1
stbar(i) = dstbar * (i - 1);
rturb(i) = rbi + (rmax - rbi) * stbar(i);
etaturb(i) = etabI - Stbi * (rturb(i) - rbi);
jj = N + 1 + i;
r(jj) = rturb(i);
eta(jj) = etaturb(i);
end
% rturbratio= [0.5 ; rturb];
etaturbi=etaturb;
rss = rsi;
rbb = rbi;
etatop = etatI;
etabot = etabI;
% etasdot = 0;
% etabdot = 0;
Hturb=Hturb-etaturb; %actually not turbidity zone, settling zone
Uturb = qw./Hturb;
time = 0;
ntrack = 0;
j = 0;
% Finished = False;
% Bombed = False;
ystrata(:,j+1)=eta(:,1);
xstrata(:,j+1)=r(:,1);
shoreline(j+1,1)=rss;
etashoreline(j+1,1)=etatop;
fbb(j+1,1)=rbb;
etafbb(j+1,1)=etabot;
timeplot(j+1,1)=time;
%%
for j=1:1:Nprint
% Inner loop
for w = 1:1:Ntoprint
%% Do Fluvial Backwater
%This subroutine implements a numerical solution of the backwater
% equation on both the sand-bed and gravel-bed reaches
Hfluv(N + 1) = xil - etafluv(N + 1);
for i = 1:1:N
fr2p = qw ^ 2 / g / Hfluv(N + 2 - i) ^ 3; % Fr^2
fnp = (etafluv(N + 1 - i) - etafluv(N + 2 - i) - Cfa * fr2p * rss * dsbar) / (1 - fr2p); %dH/dx^
Hpred = Hfluv(N + 2 - i) - fnp; % H predictor corrector
fr2 = qw ^ 2 / g / Hpred ^ 3; %?
fn = (etafluv(N + 1 - i) - etafluv(N + 2 - i) - Cfa * fr2 * rss * dsbar) / (1 - fr2);
Hfluv(N + 1 - i) = Hfluv(N + 2 - i) - 0.5 * (fnp + fn); % Eueler step used
if Hfluv(N + 1 - i) < Hn
Hfluv(N + 1 - i) = Hn; % Normal flow : Hn
end
end
xil = xil + xiddot * dt;
%% Find SandLoad and rsdot %rsdot = S_s dot,shoreline moving rate (52)
% This subroutine determines the sand transport rate qs(i) at every node in accordance with
% (1), (5) and (12) of the notes, and also gets a first estimate of the migration speed
% of the topset-foreset interface in accordance with (52).
for i = 1:1:N+1
ts(i) = Cfa * (qw / Hfluv(i)) ^ 2 / (Rs * g * Ds);
if ts(i) <= tsc
qs(i) = 0;
else
qs(i) = (Rs * g * Ds) ^ 0.5 * Ds * al * (ts(i) - tsc) ^ nl;
end
qs=qs';
% if i==1
% qms(i)=qmo;
% else
% if i<N+1
% qms(i)=qms(i-1) - Lamms*(qs(i-1)-qs(i));
% if qms(i)<0
% qms(i)=0;
% end
% end
% end %% mud deposited per sand in topset
end
% rsdot(j,1) = 1 / Sa * (qs(N + 1)*(1+Lamms) / ((1 - lps) * (rbb - rss))); %1d S_s dot (52) :shock condition
rsdot(j,1)= 1/Sa * (-etasdot(j,1)+intM*Bc*qs(N+1)*(1+Lamms)/0.5/thetaF/(1-lps)/(rbb^2-rss^2)); % (97) : 2d shock condition
%-1/Sa*xiddot+
% qms(N+1) =qms(N) - qs(N+1) * Lamms; %% mud deposited per sand in foreset
% if qms(N)<0
% qms(N)=0;
% end
if (j == 1) && (w == 1)
rbdot(j,1) = rsdot(j,1) * Sa / (Sa - Sturb(1));
else
rbdot(j,1) = rsdot(j,1) + 1 / Sa * (etasdot(j,1) - etabdot(j,1));
end
%% Find New etafluv and rss
% This subroutine implements the Exner equation (47) of the notes over the fluvial
% region to find the new bed elevation. In addition, it updates the migration speed
% and position of the topset-foreset interface according to (52) by adding a term.
etatopold = etatop;
for i = 1:1: N + 1
if i == 1
qsdev(i,1) = (qs(i) - qso) / dsbar;
etafdev(i,1) = (etafluv(i + 1) - etafluv(i)) / dsbar;
else
if i == N + 1
qsdev(i,1) = (qs(i) - qs(i - 1)) / dsbar;
etafdev(i,1) = (etafluv(i) - etafluv(i - 1)) / dsbar;
else
qsdev(i,1) = (qs(i) - qs(i - 1)) / dsbar;
etafdev(i,1) = (etafluv(i + 1) - etafluv(i)) / dsbar;
end
end
if i==1
etafluv(i) = etafluv(i) + (dt * (-qsdev(i,1) / rss / (1 - lps)*(1+Lamms)*intM)) + (sfbar(i) * rsdot(j,1) / rss * etafdev(i,1));
else
etafluv(i) = etafluv(i) + dt * ((sfbar(i)*rsdot(j,1)/rss*etafdev(i,1)) -(Bc/rss*qsdev(i,1)*intM/(1-lps)*(1+Lamms)/thetaF/rss/sfbar(i))); % 3D fluvial Exner (91)
end
eta(i) = etafluv(i);
end
etatop = etafluv(N + 1);
etasdot(j,1) = (etatop - etatopold) / dt;
rsdot(j,1) = rsdot(j,1) - etasdot(j,1) / Sa;
rss = rss + rsdot(j,1) * dt;
for i = 1:1: N + 1
rfluv(i) = sfbar(i) * rss;
r(i) = rfluv(i);
end
%
% % Excel
% lambda_cal
% Lamms=lambda;
%% Find New etaturb and rbb
% This subroutine implements the Exner equation (51) of the notes over the turbidity
% current region to find the new bed elevation. It also updates the migration speed
% and position of the foreset-bottomset interface with the aid of (53) of the notes.
etabotold = etabot;
coef2=1/(rmax-rbb);
for i = 1:1: Ns + 1 % to line 215 : etatdev & cibar dqm
jj = N + 1 + i;
if i == Ns + 1
etatdev(i) = (etaturb(i) - etaturb(i - 1)) / dstbar;
cibardev(i)=(cibar(i)-cibar(i-1))/dstbar;
else
if i == 1
etatdev(i) = (etaturb(i + 1) - etaturb(i)) / dstbar;
cibardev(i)=(cibar(i+1)-cibar(i))/dstbar;
else
etatdev(i) = (etaturb(i + 1) - etaturb(i)) / (dstbar);
cibardev(i)=(cibar(i+1)-cibar(i))/dstbar;
end
end
Qm_ghost(1)=qmo*Bc; %L3/T
% qm(1)=qms(N+1)*(rbb-rt)/(rss-rt);
coef(i)=rbdot(j,1)*(1-stbar(i))/(rmax-rbb);
end
for i = 1 : 1 : Ns+1
%% cibar volume ratio
% dqm needs dt ? ? ? or not // dsbar -> ((rmax-rbb)/Ns)
if i == 1 % dt : 0.00025 * timeyr , too large ? when dt : 0.00025 is better
% cibar(i) = cibar(i) + dt*(((Qm_ghost(1)/thetaF)-(rturb(1)*cibar(1)*Hturb(1)*Uturb(1))-(vsm*ro*cibar(1)*((rmax-rbb)/Ns)*(rturb(1)+rturb(2))/2) )/ (Hturb(1)*((rmax-rbb)/Ns)*(rturb(1)+rturb(2))/2));
% cibar(i) = cibar(i) + dt*( rbdot(j,1)*(1-stbar(i))*cibardev(i)/(rmax-rbb))...
% +dt*(((rturb(1)*Qm_ghost(1))-(rturb(1)*cibar(1)*Hturb(1)*Uturb(1))-(vsm*ro*cibar(1)*((rmax-rbb)/Ns)*(rturb(1)+rturb(2))/2) )/ (Hturb(1)*((rmax-rbb)/Ns)*(rturb(1)+rturb(2))/2));
cibar(i) = cibar(i) + (rbdot(j,1)*(1-stbar(i))*cibardev(i)/(rmax-rbb))*dt+...
dt/Hturb(i)*(-vsm*ro*cibar(i)-(cibar(i)*Uturb(i)*Hturb(i)-qmo)/((rmax-rbb)*dstbar));
% cibar(residual_) = input - output - deposition
else
% cibar(i) = cibar(i) + dt*(((rturb(i)*cibar(i-1)*Uturb(i-1)*Hturb(i-1))-(rturb(i+1)*cibar(i)*Hturb(i)*Uturb(i))-(vsm*ro*cibar(i)*((rmax-rbb)/Ns)*(rturb(i)+rturb(i+1))/2)) / (Hturb(i)*((rmax-rbb)/Ns)*(rturb(i)+rturb(i+1))/2));
% cibar(i) = cibar(i) + dt*( rbdot(j,1)*(1-stbar(i))*cibardev(i)/(rmax-rbb))...
% +dt*(((rturb(i)*cibar(i-1)*Uturb(i-1)*Hturb(i-1))-(rturb(i+1)*cibar(i)*Hturb(i)*Uturb(i))-(vsm*ro*cibar(i)*((rmax-rbb)/Ns)*(rturb(i)+rturb(i+1))/2)) / (Hturb(i)*((rmax-rbb)/Ns)*(rturb(i)+rturb(i+1))/2));
cibar(i) = cibar(i) + (rbdot(j,1)*(1-stbar(i))*cibardev(i)/(rmax-rbb))*dt+...
dt/Hturb(i)*(-vsm*ro*cibar(i)-(cibar(i)-cibar(i-1))*Uturb(i)*Hturb(i)/((rmax-rbb)*dstbar));
end
end
for i = 1:1: Ns+1 % line 247 : Exner equation to elevation change
jj = N + 1 + i;
% qm(i)=-(Uturb(i+1)*Hturb(i+1)*cibar(i+1)-Uturb(i)*Hturb(i)*cibar(i));
qm(i)=cibar(i)*Uturb(i)*Hturb(i);
% qm(i)=0; % for retention rate = 0 case !
% @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
% qm(Ns)=0;
% if i==1
% qm(i) = qm(i);
% else
% qm(i) = qm(i-1)+dqm(i);
% end % qm is higher with this code
% etaturb(i) = etaturb(i) + rbdot(j,1) * (1 - stbar(i)) / (rmax - rbb) * etatdev * dt; %term 2
% deta(i)=(rbdot(j,1) * (1 - stbar(i)) / (rmax - rbb) * etatdev * dt);
% etaturb(i) = etaturb(i) + 1 / (1 - lpm) * vsm * ro * qm(i) / Uturb(i) / Hturb(i) * dt; %term 1
% deta(i) = deta(i) + (1 / (1 - lpm) * vsm * ro * qm(i) / Uturb(i) / Hturb(i) * dt);
% 3d exner turbidity :thetaP(r-rt), turbidity current width
% etaturb(i) = etaturb(i) + rbdot(j,1)*(1-stbar(i))/(rmax-rbb)*etatdev * dt; % term 2
% etaturb(i) = etaturb(i) + ro*vsm*thetaP*qm(i)*(rbb+stbar(i)*(rmax-rbb)-rt(i))/(1-lpm)/Uturb(i)/Hturb(i)/thetaF/(rbb+stbar(i)*(rmax-rbb))*dt; % term 1
% 3d exner turbidity : instead of thetaP(r-rt), Bc (river
% width)
etaturb(i) = etaturb(i) + (ro*vsm*qm(i)*Bc/Uturb(i)/Hturb(i)/(1-lpm)/thetaF/(r(N+1+i)) + rbdot(j,1)*(1-stbar(i))/(rmax-rbb)*etatdev(i) )*dt;
% r = rbb + (rmax-rbb)/Ns*i
deta(i)=(ro*vsm*qm(i)*Bc/Uturb(i)/Hturb(i)/(1-lpm)/thetaF/(r(N+1+i)) + rbdot(j,1)*(1-stbar(i))/(rmax-rbb)*etatdev(i) )*dt;
eta(jj) = etaturb(i);
Hturb(i) = Hturb(i)-deta(i);
Uturb(i)=qw/Hturb(i);
end
etabot = etaturb(1);
etabdot(j,1) = (etabot - etabotold) / dt; % always positive in purely depositional
rbdot(j,1) = rsdot(j,1) + 1 / Sa * (etasdot(j,1) - etabdot(j,1));
rbb = rbb + rbdot(j,1) * dt;
if rbb > rmax
disp('Toe prograded past rmax');
break;
else
for i = 1:1: Ns + 1
jj = N + 1 + i;
rturb(i) = rbb + (rmax - rbb) * stbar(i);
r(jj) = rturb(i);
end
time = time + dt;
ntrack = ntrack + 1;
end
washload=washload + qm(Ns+1)*dt;
% capturemud=capturemud + (qmo-qms(N+1))*dt;
end % Ntoprint w
ystrata(:,j+1)=eta(:,1);
xstrata(:,j+1)=r(:,1);
shoreline(j+1,1) = rss;
etashoreline(j+1,1) = etatop;
fbb(j+1,1) = rbb;
etafbb(j+1,1) = etabot;
timeplot(j+1,1) = time/timeyr;
% timeplot(j+1,1)= time/timeyear;
end % NPrint j
%% Plot output
figure, plot(xstrata, ystrata, 'k');
hold; plot(rfluv, Hfluv+etafluv, 'b');
titletext1 = ['Strata ' num2str(time/timeyr) ' yr-run ' num2str(xiddot*1000*timeyr) ' mm/yr-SLR'];
title(titletext1); xlabel('distance [m]'); ylabel('elevation [m]');
plot(shoreline, etashoreline, '-o');
plot(fbb, etafbb, '-v');
% saveas(gcf,figurename2,'jpg')
% time versus shoreline location
% titletext2 = ['Shoreline and Foreset-Bottomset Trajectories'];
% title(titletext2); xlabel('distance [m]'); ylabel('elevation [m]');
figure, plot(timeplot, shoreline, '-o');
hold; plot(timeplot, fbb, '-v');
titletext2 = ['Shoreline and Foreset-Bottomset Trajectories'];
title(titletext2); xlabel('time [yr]'); ylabel('distance [m]');
toc
% % Dimensionless shoreline fbb figure
Lscale = ((qso+qmo)) / (xiddot) ; % m
Tscale = Lscale / (xiddot) / (1/Sfbi - 1/Sa ) /timeyr ; % yr
figure, plot(shoreline./Lscale, timeplot./Tscale,'-k');
hold; plot(fbb./Lscale, timeplot./Tscale,'--k');
titletext2 = ['Shoreline and Foreset-Bottomset Transition Trajectories'];
title(titletext2); ylabel('time [t*]'); xlabel('distance [L*]');
% saveas(gcf,figurename1,'jpg')
%% wash load cal
washload=washload/((qmo*dt*Ntoprint*Nprint)); % advection settling reach
% .. 'qm' of last node/input-captured ...
%% topset foreset length cal
for l = 1 : Nprint
topsetlength(l)=sqrt((xstrata(N+1,l)-xstrata(1,l))^2 + (ystrata(N+1,l)-ystrata(1,l))^2);
foresetlength(l)=sqrt((xstrata(N+2,l)-xstrata(N+1,l))^2 + (ystrata(N+2,l)-ystrata(N+1,l))^2);
end