Evaluating Settlement of Clay Soils Due To Water Deportation Through Explicit And Implicit Approach

ACADEMIC INTEGRITY PLEDGE: “We declare upon our honor that we have neither given nor received unauthorized help in solving this problem.”

Timothy John Acosta 2007- 16445

Arlene A. Gavino

2008-78469

John Philip Cabañero 2009-41743

MP 1 Submitted to: Sir Juan Michael U.V. Sargado

2011

10/3/2011

Finite Difference Method

2 Abstract

The long-term settlement of clay soils due to expulsion of water is described in an

equation similar to the Heat-Conduction Equation. In this paper, we have studied the

behaviour of the soil as the water flows through it. The water has excess pore pressure

that affects the stability of the soil. For this case, we have considered two methods in

determining the governing equation of Terzaghi’s 1-D Consolidation Theory – Forward-

Time Central-Space (FTCS) Method and the Crank-Nicholson. In order to solve the

explicit scheme, the equation requires approximations of the time derivative with the

use of forward difference and also approximations of the space derivative with the use

of central difference. The implicit scheme (Crank-Nicolson) method also uses

approximations that are second-order accurate in both space and time.

Problem Statement

The problem wishes to solve for the time it takes for the whole settlement to be 90%

consolidated. The solution to a boundary-value problem from the partial differential

equations can be expressed as an explicit or implicit form. The former is a

straightforward method because the values are just evaluated directly from the finite-

difference equation. The latter is quite more complicated because for you to be able

to obtain a solution, you must reduce the system of linear equations in the problem.

In the finite-difference method, the replacement of first and second order derivatives

using central-difference rules is necessary to solve for the governing equation. The rules

for partial derivatives are merely the extension from the full derivatives derived from the

Taylor series expansion of the terms. A point u(x, y) is written as u(xi ,yj), where the

subscripts i and j denote its x and y positions, respectively. For u(x, y) continuous and

defined in a rectangular domain whose horizontal and vertical grids have the widths of

h and k, respectively, the derivatives at the finite point (xi ,yj) are approximated using

the central-difference rules:

Finite Difference Method

3

Partial derivatives in Rectangular Solids

The two-dimensional extension of full derivatives is shown in the equations (1.1), (1.2),

(1.3), and (1.4). The partial derivative of u(x, y) with respect to the variable x is obtained

by differentiating that variable only. This means only the i subscript is involved. Similarly,

the partial derivative with respect to y involves the subscript j only.

Finite Difference Method

4 Theoretical Background

Finite Difference Method is used to solve boundary value Ordinary Differential Equations

(ODEs). In the finite-difference method, solutions are obtained by replacing the given

first and second derivatives into their approximated form using the central-difference

rules. The approximation means that the numerical solution is known only at a finite

number of points in the physical domain. The rules for partial derivatives are merely the

extension from the full derivatives derived from the Taylor series expansion of the terms.

The Finite Difference Method starts with the discretization of space and time such that

there is a number of points in space and an integer number of times at which we

calculate the field variables.

Forward-Time Central-Space Method

The time dependent one dimensional consolidation equation is expressed as:

where u is the pore water pressure distribution in the system, t is the instantaneous time

from the application of a total stress increment, and x is the 1-D space coordinate or

the depth below the top of the clay layer, and C is the coefficient of consolidation. It

describes the spatial-temporal variation of pore pressures (u) through the primary

consolidation coefficient.

In this method, the time derivative can be approximated with the use of forward

difference and the space derivatives can be approximated with second-order central

differences. The differential equation can now be written as:

letting ,

Finite Difference Method

5

the equation will yield us:

Crank-Nicholson Method [a]

Developed by John Crank and Phyllis Nicolson, the Crank-Nicolson Method is an implicit

method of numerically solving partial differential equations. It solves both the accuracy

and the stability problem. Implicit method means in order to find the next value of u,

simultaneous algebraic equations should be determined first.

The Crank-Nicholson equation can be written as:

where u is the pore water pressure distribution in the system, t is the time coordinate,

and F is function of u, t, and x, which is the space coordinate.

In this method, the time derivative can be approximated using the trapezoidal rule and

the space derivative can also be approximated with the use of central difference. This

method also is the mean of the forward Euler method at n and the backward Euler

method at n+1. The Crank-Nicolson discretization can now be written as:

letting

the

equation

will yield us:

Our differential equation can now be solved as a simple Tridiagonal Matrix Algorithm or

Thomas Algorithm, a simplified form of Gaussian elimination.

Finite Difference Method

6

Implementation of Numerical Model

Calculate for the Initial value of U

Computing for Initial Area (Area1)

Implementation of Boundary Conditions

for Dirichlet type and Neumann BC

Computing using FCTS

Given 1 equation and 1 unknown,

we can compute for the value of U at time t = n.

Compute for the Final Area

(Area2)

Compute for Consolidation

(1- Area2) / Area1

Check if the solution achieves 90% consolidation

IF NOT go back in computing using FCTS at t = n+1

until it reaches 90% consolidation.

Print the values in Microsoft Excel

Plot the obtained Data

Finite Difference Method

7

Calculate for the Initial Value of U at t=0

Computing for Initial Area

(Area1 at t=0)

Setup tridiagonal matrix

[Array 1] [Array 2] = [Array 3]

[Array 1] = constants of future values at t=n+1 (unknown)

[Array 2] = unknown values at future t=n+1

[Array 3] = values of present (known) (t=n)

Implementation of Boundary Conditions

for Dirichlet type and Neumann BC

Dirichlet has same initial values as FCTS

(topmost values in tridiagonal matrix)

Neumann, we implement the new BC that gives a ghost node w/c is equal in

magnitude to n-1th node

Compute for the Final Area

(Area2)

Compute for Consolidation

Check if the solution achieves 90% consolidation

IF NOT, Go back in Setting up the Tridiagonal matrix,

BUT your computed new value for the “future” becomes your present value for the next iteration.

Print the values in Microsoft Excel

Plot the obtained Data

Recompute everything by changing the Cv

Finite Difference Method

8

Results and Discussion

Explicit (FCTS)

Implicit (Crank-Nicholson)

0

20000

40000

60000

80000

100000

120000

140000

0 2 4 6 8 10 12

Val

ue

of

U

Nodes

FCTS

Series1

Series2

Series3

Series4

Series5

0

20000

40000

60000

80000

100000

120000

140000

0 2 4 6 8 10 12

Val

ue

of

U

Nodes

Crank-Nicholson

Series1

Series2

Series3

Series4

Series5

Series6

Finite Difference Method

9

FCTS (reversed Cv)

Crank-Nicholson (reversed Cv)

0

20000

40000

60000

80000

100000

120000

140000

0 2 4 6 8 10 12

Val

ue

of

U

Nodes

FCTS (reversed Cv)

Series1

Series2

Series3

Series4

Series5

0

20000

40000

60000

80000

100000

120000

140000

0 2 4 6 8 10 12

Val

ue

of

U

Nodes

Crank-Nicholson (reversed Cv)

Series1

Series2

Series3

Series4

Series5

Series6

Finite Difference Method

10

Based on the results obtained For the given original problem, the computed days for

90% consolidation were 2453 days for FCTS scheme while 2454 days for Crank-Nicholson

scheme. For the reversed value of coefficient of velocity, we got 2358 days for the FCTS

scheme and 2359 days Crank-Nicholson scheme. We can say that FCTS method has a

lesser days when 90% consolidation.

Conclusions and Recommendations

We will discuss the results that we have obtained from the solution using Finite

Difference method. Here we will give the conclusions regarding the soil that is

composed of two saturated clay layers and underlain by impervious bedrock. Both the

Explicit and Crank-Nicolson schemes produced almost the same results for the given

problem.

We also have the plot of the average degree consolidation with respect to time. We

can see in the Excel file that the FCTS method is faster to reach the 90% consolidation

than the Crank-Nicholson. But it is recommended that the Crank-Nicolson scheme be

used rather than the Explicit scheme because Crank-Nicholson satisfy the assumptions

made that the value that must be obtain for U were getting smaller.

Finite Difference Method

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We have computed for the reversed value for the Coefficient of Velocity (Cv), where,

the upper layer has the lower value for Cv and the lower layer has a higher value of Cv.

Based on the given plots we conclude that the reversed Cv has a lesser days of

consolidation compare to the original set-up.

In programming for FCTS we always iterate from t=0 until we reaches the 90%

consolidation. But in using the Crank-Nicholson method, the iteration starts at t=n until

the 90% consolidation have been achieved.

References

Chapra, Steven. Numerical Methods for engineers 4th Edition. New York: McGraw-Hill, 2002.

Hoffman, J. D. Numerical Methods for Engineers and Scientists 2nd Edition, New York: McGraw-Hill, Inc., 1992.

Sir Juan Michael U.V. Sargado lecture notes

Larsson St., Partial Differential Equations With Numerical Methods, ISBN, 2009

Pinchover, Y. and Rubinstein J., Introduction to Partial Differential Equations, Cambridge

University Press, 2005

Powell, A., Finite Difference Solution of the Heat Equation,

http://dspace.mit.edu/bitstream/handle/1721.1/35256/22-00JSpring-

2002/NR/rdonlyres/Nuclear-Engineering/22-00JIntroduction-to-Modeling-and-

SimulationSpring2002/55114EA2-9B81-4FD8-90D5-5F64F21D23D0/0/lecture_16.pdf,

September 2011

Salleh S. and Zomaya A. Computing for Numerical Methods Using Visual C++

[a]Wilmott,P.,Crank–Nicolson,Method, http://www.ps.uci.edu/~markm/numerical_methods/Crank-Nicolson_method.pdf, September 2011

Finite Difference Method

12 Source Code

#include<stdio.h> #include<stdlib.h> #include "files.h" #include<math.h> void main(){ int i; int ctr=0, nodes=10; double p=225000; double r=1.2; double deltaz=10/nodes; double deltat=86400; double alpha1=0.0000003*deltat/pow(deltaz,2); double alpha2=0.0000002*deltat/pow(deltaz,2); double Area1=0, Area2=0; double Consolidation; sarray Z=defsarray(nodes); sarray U=defsarray(nodes); sarray Unew=defsarray(nodes); for(i=0;i<nodes;i++) Z.elems[i]=deltaz*i; U=StressCalc(nodes,p,r,Z,deltaz); FILE*fcts=fopen("fcts.txt","w"); for(i=0;i<nodes-1;i++) Area1=Area1+(0.5*(U.elems[i]+U.elems[i+1])*deltaz); sarrayprintf(nodes,fcts,U); do{ ctr++; /*Boundary Conditions*/ U.elems[0]=0; U.elems[nodes]=U.elems[nodes-1]; /*FCTS*/ Unew=FCTS(nodes,alpha1,alpha2,deltaz,U); sarrayprintf(nodes,fcts,Unew); for(i=0;i<nodes;i++) U.elems[i]=Unew.elems[i]; Area2=0; for(i=0;i<nodes-1;i++) Area2=Area2+(0.5*(U.elems[i]+U.elems[i+1])*deltaz);

Finite Difference Method

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Consolidation=1-(Area2/Area1); }while(Consolidation<0.9); printf("FCTS: %.2lf seconds\n",ctr*deltat); fclose(fcts); ctr=0; /*CRANK*/ matrix TriDiagonal=defmatrix(nodes,3); sarray RH=defsarray(nodes); sarray Unknowns=defsarray(nodes); alpha1=0.0000003*deltat/(2*pow(deltaz,2)); alpha2=0.0000002*deltat/(2*pow(deltaz,2)); ctr=0; FILE*crank=fopen("crank.txt","w"); U=StressCalc(nodes,p,r,Z,deltaz); do{ /*TriDiagonal*/ for (i=1; i<(nodes-1); i++){ if (i*deltaz<4){ TriDiagonal.elems[i][0] = -alpha1; TriDiagonal.elems[i][1] = 1 + 2*alpha1; TriDiagonal.elems[i][2] = -alpha1; }else { TriDiagonal.elems[i][0] = -alpha2; TriDiagonal.elems[i][1] = 1 + 2*alpha2; TriDiagonal.elems[i][2] = -alpha2; } } /*RIGHT HAND SIDE*/ for (i=1; i<(nodes-1); i++){ if (i*deltaz<4) RH.elems[i] = alpha1*U.elems[i+1] + (1 - 2*alpha1)*U.elems[i] + alpha1*U.elems[i-1]; else RH.elems[i] = alpha2*U.elems[i+1] + (1 - 2*alpha2)*U.elems[i] + alpha2*U.elems[i-1]; } /*Boundary Conditions*/ TriDiagonal.elems[0][0] = 0.0; TriDiagonal.elems[0][1] = 1.0; TriDiagonal.elems[0][2] = 0.0; TriDiagonal.elems[nodes-1][0] = -2*alpha2; TriDiagonal.elems[nodes-1][1] = 1 + 2*alpha2; TriDiagonal.elems[nodes-1][2] = 0.0; RH.elems[0] = 0.0; RH.elems[nodes-1] = (1 - 2*alpha2)*U.elems[nodes-1] + 2*alpha2*U.elems[nodes-2]; Unknowns = ThomasAlg(nodes,RH,TriDiagonal); for(i=0; i<nodes; i++){

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U.elems[i] = Unknowns.elems[i]; } ctr++; sarrayprintf(nodes,crank,U); Area2=0; for(i=0;i<nodes-1;i++) Area2=Area2+(0.5*(U.elems[i]+U.elems[i+1])*deltaz); Consolidation=1-(Area2/Area1); }while(Consolidation<0.9); printf("CRANK: %.2lf seconds\n",deltat*ctr); fclose(crank); }