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Water infiltration in porous materials
Water InJiltration in Porous Materials by
S. H. LIN
Department of Chemical Engineering, Taoyuan, Taiwan, Republic of China
Yuan Ze Institute
of Technology,
Neili,
ABSTRACT: Water infiltration in porous materials is investigated in this study. A highly nonlinear hydraulic dtflusivity function is considered in the physical model. The orthogonal collocation method is employedfor solving the governing nonlinear d@erential equation. The present numerical solutions for porous materials are in good agreement with the experimental data of Gummerson et al. Because of its high accuracy, computational stability and easy application, the orthogonal collocation provides a good alternative for tackling the nonlinear water infiltration problem.
I. Introduction In past decades, water infiltration in porous materials has been a problem of common interest to many investigators in building science, chemical engineering, civil engineering, petroleum engineering as well as soil science. The water infiltration problem occurs in underground disposal of sewage and waste water, water removal from petroleum reservoirs, wetting and drying of building materials, water movement in soils, just to mention a few examples. A recent review of water infiltration and sorption in porous materials has been given by Hall (1). In spite of its wide applications, the water infiltration problem remains a difficult one to deal with primarily because of the highly nonlinear nature of the governing differential equation. This difficulty arises from a strong dependence of hydraulic diffusivity on the water content of porous materials. Since the early publications of Klute (2) and Philips (3), various approximate methods have been proposed for solving this highly nonlinear differential equation (410). In the present communication, orthogonal collocation is employed for tackling the nonlinear infiltration problem. As will be seen later, this method proves to be very accurate and highly versatile, and provides a handy alternative for handling the nonlinear differential equation. ZZ. Transient Flow in Unsaturated Porous Materials Unsteady state one-dimensional water infiltration described by the following differential equation :
ae
ae
in porous
materials
L 1
z=ax We) g with D(O) = D,,f (d) and the initial and boundary
~TheFranklinlnstitute0016-0032/92 $5.00+0.00
conditions
can be
(1) given by
85
S. H. Lin
t=o;
e=eo
.x=o;
e=e,
x=L;
-=o
ae
ax
where 8 is the water content, x the axial distance from the porous material. The above the following dimensionless
D(e) the hydraulic diffusivity of the porous material, surface, t the time, and L the thickness of unsaturated equations can be cast in dimensionless form by using variables :
e-e,
C=e,_eo’
Equations
z=y’
tDo L
Y=T
L-x
(1) to (4) become
(5) subject to
c=o
(6)
y=o;
;+=o
(7)
Y= 1;
c=o.
(8)
5 =
0;
The hydraulic diffusivity D(C) has been known to be a highly water contentdependent function. Various forms of the functian f(C) have been suggested by many investigators (7). They range from simple linear function of C to a highly nonlinear one. Two of the most general and widely used water content-dependent functions of f(C) are represented by f(C)
= a,exp(a,+a2C+a3C2+
. . .)
(9)
and f(C)
= ao+a,C+a,C2+
..
(10)
where the constant polynomial coefficients, a, can be obtained by directly fitting Eq. (9) or (10) to the experimental data. From a practical viewpoint, both equations offer the same degree of flexibility. However, Eq. (10) is a little easier to use because it does not need any modification of the equation and/or data in the curve fitting process. Hence Eq. (10) is adopted in this work to represent the hydraulic diffusivity. The coefficients a, in Eq. (10) were obtained by using the experimental data of Gummerson et al. (11) who determined the water content in porous materials and the hydraulic diffusivity by a NMR imaging technique. The fitted coefficients Journal
86
of the Franklin Pergamon
Institute Press plc
Water Infiltration in Porous Materials TABLE I
Polynomial curvefit of hydraulic difsusivity data f(C) = aO(l-a,C+a&?--a,C3+c,C4-c5C5)
Material
a0
Portland limestone Corsehill sandstone Gypsum building plaster Locharbriggs sandstone Cement, lime and sand Gypsum building plaster plus hydrated lime
6
a3
al
- 6.586
-20.814
124.763
444.134
-214.453
-9.568
-8.554
18.197
- 26.750
-226.714
- 20.335
131.391
- 383.247
486.644
-232.272
- 176.424
- 17.993
88.656
-230.036
251.548
- 100.502
- 148.845
-21.154
139.336
-424.009
581.505
-327.143
-238.342
- 12.447
- 151.338
181.607
-91.602
60.015
-355.667
of the polynomial are listed in Table I. It appears that except for one case, all other hydraulic diffusivity data can be well represented by a fifth-order polynomial. Figures 1, 2 and 3 show a comparison of those fitted curves (the continuous lines) with the measured data of Gummerson et al. (11). The fit is seen to be excellent and the hydraulic diffusivity polynomials can be used for the present simulation.
I
-0
02
I
I
I
@4
O-6
06
IQ
Water Content
FIG. 1. Hydraulic
diffusivity as a function B-Corsehill
Vol. 329, No. I, pp. 8%91, 1992 Printed in Great Britain
of water content. sandstone.
A-Portland
limestone,
87
S. H. Lin
0
FIG. 2. Hydraulic
diffusivity
02
04 06 Water Content
as a function of water content. B-Locharbriggs sandstone.
0.0
I.0
A-Gypsum
building
plaster,
Water Content FIG.
88
3. Hydraulic
diffusivity as a function of water content. A-Cement, B-gypsum building plaster plus hydrated lime.
lime and sand,
Journal of the Franklin lnstrtute Pergamon Press plc
Water Injiltration
in Porous Materials
To solve the nonlinear governing equation, i.e. Eq. (5), the orthogonal collocation method (12) is employed here. This method is in essence an approximation of the spatial differentials in the preceding equations by an orthogonal polynomial. The orthogonal polynomial can be of the Jacobi, Legendre or even Chebyshev type. The Jacobi type has been found (12) to be the most suitable for the present problem because of better convergence and accuracy. The dimensionless water content is assumed to be represented by the following orthogonal expansion : C(r, Y) = C(T, l)+(l-
Y’) i Lgz)P,_ ,(Y2) j- 1
(11)
in which d,(r) are the expansion coefficients and P,_ 1( Y’) the Jacobi polynomial of order j. In terms of this expansion at the ith collocation point, Eq. (5) can be rewritten as
$$ = f[C(i)] nt’
B,,,C(j)
+
Yi!!E!$
j= I
[l%: A,,iCW~.(12)
It should be noted that Eq. (11) automatically satisfies Eq. (7) because of symmetric expansion in Y. The second boundary condition then becomes C(n+l) which is used to eliminate
(13)
C(n + 1) from Eq. (12) leading 1+ jI,C B,,,C(j) forj=
and the initial condition
= 1
] +
df
to
[l+,$,Aj,iC(j)]~ (14)
1,2,...,n
becomes C(i) = 0.
(15)
The n ordinary differential equations were integrated simultaneously in this present work by the Runge-Kutta-Gill method. The accuracy of the orthogonal collocation method has been shown to depend on the order of the orthogonal polynomial approximation (12). For general applications, a sixth-order of approximation can yield highly accurate results and was adopted for the present computations.
III. Discussion of Results Gummerson et al. (11) measured water infiltration by a NMR imaging technique. The data were collected for the water infiltrations in six different porous materials. As a typical example, Fig. 4 shows a comparison of the experimental data for gypsum building plaster plus hydrated lime with the present model predictions. Apparently the agreement between them is excellent, testifying to the validity of the physical model. Vol. 329, No. I, pp. 8591, Printed in Great Britain
1992
89
S. H. Lin
Distance from Surfacr. mm
FIG. 4. Dimensionless
According approximated
to Crank by
water content
profiles during water infiltration. 0 measured.
(7), water absorption
into the dry porous
~
Predicted,
materials
x(C, t) = f’;* g(C).
can be
(16)
This implies that the water content profiles can be plotted against xtr ‘I* to yield certain correlation functions of g(C). Figure 5 is such a plot for three different porous solid materials. Also shown in this figure are the present model predictions. Again, the agreement is very good for all three cases. This further ascertains the validity of the physical model and the argument represented by Eq. (16).
0
4
6
12
xt -I/* FIG. 5. Dimensionless water content A-Portland limestone, B- Cement, 90
.mm/min
16
20
“Z
versus xt ‘I’. ~ Predicted, 0 measured. lime and sand, C-Gypsum building plaster. Journalof the
Frankhn Institute Pergamon Press plc
Water Infiltration
in Porous Materials
Z V. Conclusions
Water infiltration in porous materials was investigated in the present study by using a water content-dependent physical model. The hydraulic diffusivity in porous materials represents a highly nonlinear function of water content. The model equation was solved by the orthogonal collocation method which proved to be very stable and accurate in tackling the nonlinear problem and offers a good alternative for solving the water infiltration problem. The predicted results were compared with the experimental measurements and the agreement between them was found to be very good.
References (1) C. Hall, “Water sorptivity of mortars and concretes: a review”, Msg. Concr. Res., Vol. 41, No. 147, pp. 51-61, 1989. (2) A. Klute, “A numerical method for solving the flow equation for water in unsaturated materials”, Soil Sci., Vol. 73, No. 1, pp. 105-l 16, 1952. (3) J. R. Philip, “Theory of infiltration : 1. The infiltration equation and its solution”, Soil Sci., Vol. 83, No. 4, pp. 345-357, 1957. (4) R. Singh, “Solution of a diffusion equation”, J. Hydraulics Div., ASCE, Vol. 91, No. HYl, pp. 55-74, 1967. (5) J. Y. Parlange, “Theory of water movement in soils”, Soil Sci., Vol. 111, No. 2, pp. 134137, 1973. (6) J. C. Bruch and G. Zyvolski, in “Mathematics of Finite Elements and Applications” (Edited by J. R. Whiteman), Academic Press, New York, 1973. (7) J. Crank, “Mathematics of Diffusion”, 2nd edn, Clarendon Press, Oxford, 1975. (8) C. Hall, “Water movement in porous building materials-unsaturated flow theory and its application”, Sldg Envir., Vol. 12, No. 2, pp. 117-125, 1977. (9) R. J. Gummerson, C. Hall and W. D. Hoff, “Capillary water transport in masonry structures : building construction application of Darcy’s law”, Construct. Pap., Vol. 1, No. 1, pp. 17-27, 1980. (10) A. N. Kalimeris, “Water flow processes in porous building materials”, Ph.D. thesis, University of Manchester, U.K., 1984. (11) R. J. Gummerson, C. Hall, W. D. Hoff, R. Hawkes, G. N. Holland and W. S. Moore, “Unsaturated water flow within porous materials observed by NMR imaging”, Nature, Vol. 281, No. 5726, pp. 5657, 1979. (12) B. A. Finlayson, “Nonlinear Analysis in Chemical Engineering”, McGraw-Hill, New York, 1980.
Vol. 329, No. I, pp. E-91, Printed m Great Britain
1992
91
S. H. LIN
Department of Chemical Engineering, Taoyuan, Taiwan, Republic of China
Yuan Ze Institute
of Technology,
Neili,
ABSTRACT: Water infiltration in porous materials is investigated in this study. A highly nonlinear hydraulic dtflusivity function is considered in the physical model. The orthogonal collocation method is employedfor solving the governing nonlinear d@erential equation. The present numerical solutions for porous materials are in good agreement with the experimental data of Gummerson et al. Because of its high accuracy, computational stability and easy application, the orthogonal collocation provides a good alternative for tackling the nonlinear water infiltration problem.
I. Introduction In past decades, water infiltration in porous materials has been a problem of common interest to many investigators in building science, chemical engineering, civil engineering, petroleum engineering as well as soil science. The water infiltration problem occurs in underground disposal of sewage and waste water, water removal from petroleum reservoirs, wetting and drying of building materials, water movement in soils, just to mention a few examples. A recent review of water infiltration and sorption in porous materials has been given by Hall (1). In spite of its wide applications, the water infiltration problem remains a difficult one to deal with primarily because of the highly nonlinear nature of the governing differential equation. This difficulty arises from a strong dependence of hydraulic diffusivity on the water content of porous materials. Since the early publications of Klute (2) and Philips (3), various approximate methods have been proposed for solving this highly nonlinear differential equation (410). In the present communication, orthogonal collocation is employed for tackling the nonlinear infiltration problem. As will be seen later, this method proves to be very accurate and highly versatile, and provides a handy alternative for handling the nonlinear differential equation. ZZ. Transient Flow in Unsaturated Porous Materials Unsteady state one-dimensional water infiltration described by the following differential equation :
ae
ae
in porous
materials
L 1
z=ax We) g with D(O) = D,,f (d) and the initial and boundary
~TheFranklinlnstitute0016-0032/92 $5.00+0.00
conditions
can be
(1) given by
85
S. H. Lin
t=o;
e=eo
.x=o;
e=e,
x=L;
-=o
ae
ax
where 8 is the water content, x the axial distance from the porous material. The above the following dimensionless
D(e) the hydraulic diffusivity of the porous material, surface, t the time, and L the thickness of unsaturated equations can be cast in dimensionless form by using variables :
e-e,
C=e,_eo’
Equations
z=y’
tDo L
Y=T
L-x
(1) to (4) become
(5) subject to
c=o
(6)
y=o;
;+=o
(7)
Y= 1;
c=o.
(8)
5 =
0;
The hydraulic diffusivity D(C) has been known to be a highly water contentdependent function. Various forms of the functian f(C) have been suggested by many investigators (7). They range from simple linear function of C to a highly nonlinear one. Two of the most general and widely used water content-dependent functions of f(C) are represented by f(C)
= a,exp(a,+a2C+a3C2+
. . .)
(9)
and f(C)
= ao+a,C+a,C2+
..
(10)
where the constant polynomial coefficients, a, can be obtained by directly fitting Eq. (9) or (10) to the experimental data. From a practical viewpoint, both equations offer the same degree of flexibility. However, Eq. (10) is a little easier to use because it does not need any modification of the equation and/or data in the curve fitting process. Hence Eq. (10) is adopted in this work to represent the hydraulic diffusivity. The coefficients a, in Eq. (10) were obtained by using the experimental data of Gummerson et al. (11) who determined the water content in porous materials and the hydraulic diffusivity by a NMR imaging technique. The fitted coefficients Journal
86
of the Franklin Pergamon
Institute Press plc
Water Infiltration in Porous Materials TABLE I
Polynomial curvefit of hydraulic difsusivity data f(C) = aO(l-a,C+a&?--a,C3+c,C4-c5C5)
Material
a0
Portland limestone Corsehill sandstone Gypsum building plaster Locharbriggs sandstone Cement, lime and sand Gypsum building plaster plus hydrated lime
6
a3
al
- 6.586
-20.814
124.763
444.134
-214.453
-9.568
-8.554
18.197
- 26.750
-226.714
- 20.335
131.391
- 383.247
486.644
-232.272
- 176.424
- 17.993
88.656
-230.036
251.548
- 100.502
- 148.845
-21.154
139.336
-424.009
581.505
-327.143
-238.342
- 12.447
- 151.338
181.607
-91.602
60.015
-355.667
of the polynomial are listed in Table I. It appears that except for one case, all other hydraulic diffusivity data can be well represented by a fifth-order polynomial. Figures 1, 2 and 3 show a comparison of those fitted curves (the continuous lines) with the measured data of Gummerson et al. (11). The fit is seen to be excellent and the hydraulic diffusivity polynomials can be used for the present simulation.
I
-0
02
I
I
I
@4
O-6
06
IQ
Water Content
FIG. 1. Hydraulic
diffusivity as a function B-Corsehill
Vol. 329, No. I, pp. 8%91, 1992 Printed in Great Britain
of water content. sandstone.
A-Portland
limestone,
87
S. H. Lin
0
FIG. 2. Hydraulic
diffusivity
02
04 06 Water Content
as a function of water content. B-Locharbriggs sandstone.
0.0
I.0
A-Gypsum
building
plaster,
Water Content FIG.
88
3. Hydraulic
diffusivity as a function of water content. A-Cement, B-gypsum building plaster plus hydrated lime.
lime and sand,
Journal of the Franklin lnstrtute Pergamon Press plc
Water Injiltration
in Porous Materials
To solve the nonlinear governing equation, i.e. Eq. (5), the orthogonal collocation method (12) is employed here. This method is in essence an approximation of the spatial differentials in the preceding equations by an orthogonal polynomial. The orthogonal polynomial can be of the Jacobi, Legendre or even Chebyshev type. The Jacobi type has been found (12) to be the most suitable for the present problem because of better convergence and accuracy. The dimensionless water content is assumed to be represented by the following orthogonal expansion : C(r, Y) = C(T, l)+(l-
Y’) i Lgz)P,_ ,(Y2) j- 1
(11)
in which d,(r) are the expansion coefficients and P,_ 1( Y’) the Jacobi polynomial of order j. In terms of this expansion at the ith collocation point, Eq. (5) can be rewritten as
$$ = f[C(i)] nt’
B,,,C(j)
+
Yi!!E!$
j= I
[l%: A,,iCW~.(12)
It should be noted that Eq. (11) automatically satisfies Eq. (7) because of symmetric expansion in Y. The second boundary condition then becomes C(n+l) which is used to eliminate
(13)
C(n + 1) from Eq. (12) leading 1+ jI,C B,,,C(j) forj=
and the initial condition
= 1
] +
df
to
[l+,$,Aj,iC(j)]~ (14)
1,2,...,n
becomes C(i) = 0.
(15)
The n ordinary differential equations were integrated simultaneously in this present work by the Runge-Kutta-Gill method. The accuracy of the orthogonal collocation method has been shown to depend on the order of the orthogonal polynomial approximation (12). For general applications, a sixth-order of approximation can yield highly accurate results and was adopted for the present computations.
III. Discussion of Results Gummerson et al. (11) measured water infiltration by a NMR imaging technique. The data were collected for the water infiltrations in six different porous materials. As a typical example, Fig. 4 shows a comparison of the experimental data for gypsum building plaster plus hydrated lime with the present model predictions. Apparently the agreement between them is excellent, testifying to the validity of the physical model. Vol. 329, No. I, pp. 8591, Printed in Great Britain
1992
89
S. H. Lin
Distance from Surfacr. mm
FIG. 4. Dimensionless
According approximated
to Crank by
water content
profiles during water infiltration. 0 measured.
(7), water absorption
into the dry porous
~
Predicted,
materials
x(C, t) = f’;* g(C).
can be
(16)
This implies that the water content profiles can be plotted against xtr ‘I* to yield certain correlation functions of g(C). Figure 5 is such a plot for three different porous solid materials. Also shown in this figure are the present model predictions. Again, the agreement is very good for all three cases. This further ascertains the validity of the physical model and the argument represented by Eq. (16).
0
4
6
12
xt -I/* FIG. 5. Dimensionless water content A-Portland limestone, B- Cement, 90
.mm/min
16
20
“Z
versus xt ‘I’. ~ Predicted, 0 measured. lime and sand, C-Gypsum building plaster. Journalof the
Frankhn Institute Pergamon Press plc
Water Infiltration
in Porous Materials
Z V. Conclusions
Water infiltration in porous materials was investigated in the present study by using a water content-dependent physical model. The hydraulic diffusivity in porous materials represents a highly nonlinear function of water content. The model equation was solved by the orthogonal collocation method which proved to be very stable and accurate in tackling the nonlinear problem and offers a good alternative for solving the water infiltration problem. The predicted results were compared with the experimental measurements and the agreement between them was found to be very good.
References (1) C. Hall, “Water sorptivity of mortars and concretes: a review”, Msg. Concr. Res., Vol. 41, No. 147, pp. 51-61, 1989. (2) A. Klute, “A numerical method for solving the flow equation for water in unsaturated materials”, Soil Sci., Vol. 73, No. 1, pp. 105-l 16, 1952. (3) J. R. Philip, “Theory of infiltration : 1. The infiltration equation and its solution”, Soil Sci., Vol. 83, No. 4, pp. 345-357, 1957. (4) R. Singh, “Solution of a diffusion equation”, J. Hydraulics Div., ASCE, Vol. 91, No. HYl, pp. 55-74, 1967. (5) J. Y. Parlange, “Theory of water movement in soils”, Soil Sci., Vol. 111, No. 2, pp. 134137, 1973. (6) J. C. Bruch and G. Zyvolski, in “Mathematics of Finite Elements and Applications” (Edited by J. R. Whiteman), Academic Press, New York, 1973. (7) J. Crank, “Mathematics of Diffusion”, 2nd edn, Clarendon Press, Oxford, 1975. (8) C. Hall, “Water movement in porous building materials-unsaturated flow theory and its application”, Sldg Envir., Vol. 12, No. 2, pp. 117-125, 1977. (9) R. J. Gummerson, C. Hall and W. D. Hoff, “Capillary water transport in masonry structures : building construction application of Darcy’s law”, Construct. Pap., Vol. 1, No. 1, pp. 17-27, 1980. (10) A. N. Kalimeris, “Water flow processes in porous building materials”, Ph.D. thesis, University of Manchester, U.K., 1984. (11) R. J. Gummerson, C. Hall, W. D. Hoff, R. Hawkes, G. N. Holland and W. S. Moore, “Unsaturated water flow within porous materials observed by NMR imaging”, Nature, Vol. 281, No. 5726, pp. 5657, 1979. (12) B. A. Finlayson, “Nonlinear Analysis in Chemical Engineering”, McGraw-Hill, New York, 1980.
Vol. 329, No. I, pp. E-91, Printed m Great Britain
1992
91