Environ. Sci. Technol. 1990,2 4 , 920-924
Aspects of Fluid Flow through Small Flaws in Membrane Liners John C. Walton'
Geosciences Unit, Idaho National Engineering Laboratory, P.O. Box 1625, Idaho Falls, Idaho 83415 Budhi Sagar
Southwest Research Institute, 6220 Culebra Road, San Antonio, Texas 78228 H Fluid flow through flaws (punctures, seams, or tears) in geomembranes used as flow barriers in the liners of landfills and surface impoundments is examined. Performance of a geomembrane deteriorates fairly rapidly with the first few flaws, much greater than their small area would suggest. As the membrane is penetrated with a greater number of flaws, the degradation asymptotes toward a proportionality with total flaw area. The rate of flow through the flawed membrane is frequently directly proportional to the permeability of the adjacent porous media. Geomembranes that are not used in conjunction with adjacent low-permeability porous materials are unlikely to give satisfactory performance in the event that flaws occur. A better understanding of design aspects that tend to minimize the impacts of flaws in geomembranes can lead to improved performance of landfills and surface impoundments.
?*(K?H) = 0 (1) Appropriate boundary conditions for this problem are as follows: a fixed head a t the hole mouth, a fixed head a t distance, and no flow a t intact membrane portions. Volumetric flow rate a t any point is given by v = -K?H (2) Total flow through the flaw is obtained by integrating eq 2 over the opening surface. Q = S,S(-K?H).ii
(3)
dA
We express this in an axisymmetric cylindrical coordinate system with the z direction normal to the flaw as Q =
J*Y -K a H 2 ~ RdR
(4)
or in a Cartesian coordinate system as
Introduction Low-permeability layers form an integral part of many schemes for isolation and disposal of municipal, hazardous, and/or radioactive wastes in landfills and for storage of sewage, mine tailings, and industrial wastes in surface impoundments. In many cases a barrier layer consists of a low-permeability material (e.g., clay, concrete) with an impermeable geomembrane applied to the surface to further reduce flow rates. Increasingly, many designs for land disposal systems employ geomembranes alone as a flow barrier-saving the additional expense of compacted clay layers. During construction of the facilities and through time the geomembranes will develop punctures, tears, leaks at seams, or other flaws that allow percolation of fluids to occur. In order to design more effective liners it is important that some of the major components influencing performance of flawed geomembranes be evaluated. The goal of this work is to examine the impact of multiple small flaws in geomembranes on percolation of fluids through barrier layers (e.g., landfill liners). The work is parametric in nature, emphasizing the aspects of barrier layer design that may be most important to performance. In particular, the practice of relying exclusively on a geomembrane to form the flow barrier layer is examined. A schematic of the flow system considered in the mathematical analysis is given in Figure 1. The geomembrane is assumed to be affixed to a thicker porous material of low permeability (e.g., a compacted clay layer). The low-permeability material, in conjunction with the geomembrane, is presumed to constitute the barrier to fluid flow. The low-permeability or barrier layer is presumed to lie within a matrix of higher permeability material such as soil or waste where a fixed head is assumed. The mathematical analysis assumes saturated, steady flow, through a series of equally spaced identical flaws. Although these conditions will not correspond exactly to actual field conditions, they are adequate to give much useful parametric information concerning the behavior of these systems. The governing equation for steady-state percolation through flaws is 920
Environ. Sci. Technol., Vol. 24, No. 6, 1990
In the cylindrical coordinate system Q represents the total discharge from the flaw (units of L3/r). In the Cartesian coordinate system Q is the discharge per unit length of the flaw (in the infinite dimension) (units of L 2 / T ) . The equations can be put into dimensionless form by defining the following variables:
8 = K\E(Ho - H,)
8 = k ( H o - H,)
cylindrical Cartesian
The cylindrical and Cartesian coordinate system equations in terms of dimensionless variables are, respectively q = - 2 n l '$r dr q =
(7)
-21'2
dx
The dimensionless forms of eq 1 in cylindrical and Cartesian coordinates are
"( + $(k g ) = &(k g ) + $(k:)
rr krs)
0
=0
The dimensionless format assists with generalization of the results.
0013-936X/90/0924-0920$02.50/0
0 1990 American Chemical Society
fluid flow
Fixed Head
(M)
Geommbrane
Relatively Permeable Material
Figure 1. Schematic of flow system. The barrier layer, which includes a clay layer with adjacent geomembrane, is presumed to lie sandwiched between layers of relatively high permeability. Flow occurs through flaws in the geomembrane.
The governing equation for fluid flow (eq 1)is identical with the diffusion equation. Chambre' et al. ( I ) obtained an analytical solution to the diffusion equation for ellipsoidal-shaped openings. Their solution assumes a porous, homogeneous, and isotropic medium, a semiinfinite domain, and a membrane of infinitesimal thickness. For the special case of circular openings, the analytical solution gives a dimensionless flow rate of q = 4.0. Aidun et al. (2) obtained an analytical solution for circular openings with a finite membrane thickness. While the analytical solutions are useful for single flaws, to estimate real world performance it is desirable to estimate release rate from a series of flaws. Generalization of the isolated hole solution to multiple perforations has been attempted by Chambre' et al. (1)by assuming that the solutions for individual apertures can be summed. However, this methodology is limited to situations with widely spaced holes.
Numerical Solution For Thin Membranes Numerical methods can be used to estimate flow rates for any system geometry, flaw size, and flaw spacing. Through the use of dimensionless variables, a relatively small number of numerical simulations can be used to describe a wide variety of flow situations. The initial calculations assume a set of uniformly spaced, identicaI, circular flaws in a geomembrane of infinitesimal thickness. These could be caused by punctures in a thin geomembrane. A finite difference flow and transport code, PORFLO (3), was used to obtain a numerical solution in terms of dimensionless variables. Because of symmetry, the presence of multiple holes of the same dimensions results in no flow boundaries at locations halfway between holes. Thus we need only simulate a single opening with an appropriate placement of the no-flux boundary. If circular holes are closely packed, the no-flow boundaries take on a honeycomb (hexagonal)shape on the membrane surface. To facilitate the solution, the honeycomb is approximated by a circle so that the solution to eq 1can be obtained in a cylindrical coordinate system (Figure 2). The impact of hole spacing is obtained by varying the location to the no-flow boundary. Figure 3 illustrates the impact of flaw spacing and thickness of the barrier layer on flow rate from a single flaw when multiple equally spaced flaws are present. The results are given in terms of dimensionless flaw spacing (ro = Ro/'k) and dimensionless distance to a fixed-head boundary (z, = Z,/\k). The distance to a fixed-head boundary (zo) corresponds to the thickness of the barrier layer. Reduction of the flaw spacing (i.e., larger number of flaws per unit area) reduces the flow rate from each flaw. Accounting for interference between adjacent apertures greatly reduces the projected flow rate. When zo is large,
Fixed ;Head (K)
7 Figure 2. Illustration of simulation geometry.
-
4
al
m
= 3
B
ii u)
I
F0 2 .u)
C
al
.E
n
1
0
Flaw Spacing (diameters) Figure 3. Dimensionless flow rate for multiple circular flaws ( q ) as a function of dimensionless hole spacing ( r o = R,/\k) and dlmensionless thickness of barrier layer (z, = Z,/\k).
I
I
14
4 r
I
i
summation
numerical solution 1 2
one dimensional solution
.......................................................... 1
2
5
10
20
50
......................... 100
200
~
........ 1 0
500
io00
Flaw Spacing (diameters) Figure 4. Comparison of numerical solution for flow from multiple circular flaws with summation of the analytical solution and with the assumption of onedimensional flow. The thickness of the barrier is assumed to be z o = Zo/\k = 1000.
the numerical solution asymptotes to 4.0 as flaw spacing is increased, consistent with the analytical solution. Figure 4 compares the numerical solution for flow through multiple flaws with summation of the analytical solution (1) and with the assumption of simple one-dimensional flow through flaws (4)-two simplifying assumptions that have previously been applied to this type of simulation. For the purpose of the calculation, the thickness of the attached low-permeability layer was assumed to be zo = 10o0. Note that the approximationshave only a limited range of validity. When the flaws are very Environ. Sci. Technol., Vol. 24, No. 6, 1990
921
relatively permeable layer
W
geomembrane
11~ H
z
L
1 \
20 0
1E 4 5
0.0001
0.001
0.01
0.1
1
i H3
Area of Flaws I Total Area
Figure 5. Dimensionless flow ratio (f,) for multiple circular flaws as
a function of proportion of membrane that is flawed and dimensionless thickness of barrier layer.
widely spaced (low proportion of surface flawed), summation of the analytical solution is valid. Similarly, when the flaws are tightly spaced (i.e., most of the membrane is gone), the assumption of one-dimensional flow through the system becomes valid. A useful graphic is to compare the flow from a combined membranellow-permeabilitymaterial with that from the low-permeability material with no membrane (Le., one where the entire surface is exposed). The steady-state flow rate from a single thickness of low-permeability material is
hole or’ flaw
compacted clay layer
relatively permeable layer
H3
Figure 6. Schematic of thick geomembrane simulation geometry. H, - H, is the head drop entering (or exiting) the hole while H, - H , is the head drop in the stem of the hole. W is the hole thickness and \k is the hole radius.
of an analogy with electrical current in resistors connected in series. Since the flow must be the same in the hole and the stem
Rearrangement of the equations to eliminate H2 gives
If the permeability inside the “stem” of the hole (K,) is equivalent to that in the porous media (K2)(i.e., hole filled with porous material) eq 15 becomes or in dimensionless form q’=
mo2 20
+
Division of flow from the low-permeability material flawed membrane by the flow from the low-permeability material alone gives in terms of dimensionless parameters
Where f is the dimensionless flow ratio. This allows characterization of the system in terms of dimensionless geometric variables and dimensionless flow. The dimensionless flow ratio can be used as input to landfill evaluation computer codes such as the HELP model (5). The dimensionless flow ratio, as a function of the proportion of the membrane that is flawed, is illustrated in Figure 5. A dimensionless graph such as Figure 5 can be interpreted in a number of ways. In actual applications of this graph, the thickness of the barrier layer will be fixed. In this situation, large values of zo correspond to smaller flaws. On an area basis, small flaws (large zo) give greater flow rates than large flaws. For the smallest flaw size illustrated (i.e., for zo = 5000 radii), membrane performance is almost completely compromised when less than 0.1% of the surface is perforated.
Thick Geomembranes The above calculations assume the geomembrane is of infinitesimal thickness. If the membrane is of finite thickness ( W), the flow rate is further reduced by resistance to flow when it passes through the stem of the hole (Figure 6). The impact of finite thickness can be approximated either with numerical modeling or with the use 922
Environ. Sci. Technol., Voi. 24, No. 6, 1990
Where w = W/\k is the dimensionless thickness of the membrane, q is the dimensionless flow rate for the infinitesimal thickness case flaw, and qt is the dimensionless flow rate for the finite thickness flaw. Comparison with numerical simulation suggests that the error in this approximation is less than 5%. This accuracy is considered adequate for illustration purposes. Division of flow from the low-permeability material + flawed membrane by the flow from the low-permeability material alone gives a dimensionless flow rate:
Note that in calculating the flow from the material without membrane eq 12 is modified to include the membrane thickness (i.e., assuming the same total thickness in both cases). In field applications the thickness of the low-permeability layer and the membrane are fixed. In this situation the ratio between zo and w is constant, but the magnitude of zo and w depend upon the flaw size, large values being indicative of small flaws. A measure of the importance of membrane thickness, relative to medium thickness, can be obtained by evaluation of zo/w. In Figure 7, a comparison is made between three scenarios: zo/w = QJ,100, and 10. Note that as the membrane becomes thicker (i.e., as zo/w decreases), smaller flaws no longer give significantly greater flow rate, on an area basis, than large flaws. As zo/w decreases the curves approach each other, indicating equivalent flow rates on an area basis. This means that as the size of the flaw becomes small relative to the
al
3 0.5
0.5
E 0.4
0.4
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0.3
K VJ
.-
v)
C
E
0.2
0.2
0.1
0.1
E
0
0 1w
10
1wO
Dimensionless Slit Spacing ” o.ooo1
0.01
1
o.ooo1
0.01
1
Area of Flaws / Total Area Figure 7. Dimensionless flow ratio (t2) as impacted by thickness of membrane. The three sections of the graph correspond to z o / w = 0 3 , 100, and 10. Small flaws (large 1,)are more significantly impacted by resistance through the neck of the flaw than are large flaws. When the membrane is thick (Le., when z o / wis small), small and large flaws no longer give significantly different behavior on an area basis.
Figure 9. Dimensionless flow rate per unit length (9) for multlple linear flaws as a function of dimensionless spacing (x, = X,/\k) and dlmensloniess thickness of barrier layer (z, = Z,/\k). __ 1
0.8
width = 2 \I/
0.6
0.4
0.2
0 7E-05
o.ooo1
0.001
0.01
0.1
1
Area of Flaws / Total Area Flgure 10. Comparison of dlmensbnless flow ratio from linear circular (TI) flaws. Figure 8. Schematic diagram of simulation geometry for series of linear flaws. The simulations are performed by assuming symmetry In the X direction. The flaw has a width of 2\k and the distance between flaws is 2X,.
thickness of the membrane, flow becomes independent of flaw size.
Simulation of Linear Flaws Splitting, scraping, tearing, and inadequate sealing of seams are alternate failure modes for the geomembranes that result in linear features (i.e., slits). In this section, steady-state two-dimensional (x-z) flow from a slit is simulated. Multiple, equally spaced and sized slits are simulated by using a no-flow boundary condition at locations midway between slits. Flow rate is given for a unit length segment of an infinitely long slit of width 2\k (Figure 8). The resulting dimensionless flow rates are illustrated in Figure 9. The ratio of slit flow to that for a low-permeabilitylayer without a membrane can be obtained as
5; =
420 LAO
(c3)
Figure 10 compares the flow ratio of linear and circular flaws (si). On an area basis linear flaws give a greater flow rate. Flow from a slit in a geomembrane of finite thickness can be approximated by the method discussed earlier for circular flaws. The equation obtained is
(b)and
If the permeability inside the “stem” of the slit (K,) is equivalent to that in the porous media (Kz) (e.g., hole filled with porous material) the overall flow rate is approximated by 1 8 (20) 4t =
n - + - = K(H12
H3)
4
These equations, in combination with the dimensionless flows given in the above figure, allow simulation of flow from a variety of situations. The flow ratio is given by
The implications of membrane thickness for linear flaws (eq 21) are illustrated in Figure 11. Figure 11has the same format and interpretation as Figure 7 and leads to the same conclusions. As the membrane becomes thicker, smaller flaws no longer give significantly greater flow rate, on ah area basis, than large flaws. Comparison of Figures 7 and 11suggests that both the shape and size of the flaw become unimportant as the geomembrane thickness increases. Figures 7 and 11 (eqs 17 and 21) and the interpretation of them are only valid for the case where the permeability in the stem of the hole is the same as the permeability of the adjacent potous medium (i-e.,where the flaws become filled with the same material). Plastic porous materials such as clay soils subject to overburden stress are likely to migrate to fill any flaws in the adjacent geomembrane, lending credence to the simplifying assumption for many Environ. Sci. Technol., Vol. 24, No. 6, 1990 923
Acknowledgments 1 (thinmembrane) 1 7, (thick membrane) 1 ~
We thank an anonymous reviewer whose extensive comments greatly improved the work and manuscript. Glossary
A
H
HO H* h K
KO k
ri om1
001
1
om1
001
1
om1
001
1
Area of Flaws / Total Area Figure 11. Dimensionless flow ratio (l,)from linear flaws as impacted by thickness of membrane. The three sections of the graph correspond to z O / w= a,100, and 10. When the membrane is thick (Le., when zo/w is small), different sizes and shapes of flaws do not exhibff
significantly different behavior on an area basis.
systems. Specific situations where the Kl = K , assumption may not hold should be evaluated with eqs 15 and 19.
Conclusion Important factors influencing flow rates through barrier layers in landfill or surface impoundment liners including use of (flawed) geomembranes have been examined. The situations examined cover steady flow through multiple, equally spaced circular and linear flaws. While not exhaustive, the simulations provide significant insight into the performance of geomembranes as flow barriers. Performance of a geomembrane deteriorates fairly rapidly with the first few flaws, much greater than their small area would suggest. As the geomembrane is penetrated with a greater number of flaws, the degradation asymptotes toward a proportionality with total flaw area. The rate of flow through the flawed membrane is, in many situations, directly proportional to the permeability of the surrounding porous media. Thus, a geomembrane that is used exclusively as a barrier layer and not is associated with a compacted clay layer is unlikely to give satisfactory performance in the event that flaws occur. The impact of flaws on performance is greatly minimized if the membrane is adjacent to a porous material of low permeability such as a compacted clay layer. Given the presence of the clay layer, the flow barrier is most effective and robust if the geomembrane is relatively thick and/or if the clay layer as a whole is thick. This has important implications for design of waste disposal facilities. While thin barrier layers can have very impressive performance characteristics in the intact state, their performance deteriorates very rapidly with the presence of flaws in the geomembrane. Conversely, while thick barrier layers are more expensive, and perform only marginally better when intact, their performance is much more robust and likely to be significantly better in the field than a thinner barrier layer composed of the same materials.
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Environ. Sci. Technol., Vol. 24, No. 6, 1990
Q Q' 4
r R
RO V
W
X XO X
2 20
z
r.
9
surface of flaw (L2) hydraulic head ( L ) hydraulic head at flaw mouth ( L ) hydraulic head at distance; usually taken at a distance 2, normal to the flaw ( L ) dimensionless head, (H- H,)/(Ho - H,) hydraulic connectivity (L/)'2 reference hydraulic conductivity ( L / 7') dimensionless hydraulic conductivity, K / K o unit normal vector total flow rate out of flaw in the cylindrical coordinate system (L3/r ) or equals flow rate per unit length of flaw in Cartesian coordinate system (L2Il-7 flow ihr'ough barrier layer without membrane (L3/7') dimensionless flow rate through flaw R/\k, dimensionless radial distance radial distance coordinate ( L ) half spacing between circular flaws Darcy velocity ( L I T ) membrane thickness, W / 9 lateral distance ( L ) half of lateral distance between linear flaws ( L ) X/9,dimensionless lateral distance distance normal to flaw ( L ) thickness of porous media attached to membrane (L) Z/\k, dimensionless normal distance flow rate with/ without membrane characteristic dimension (hole radius or slit half width) ( L )
L i t e r a t u r e Cited (1) Chambre', P. L.; Lee, W. W.-L.; Kim, C. L.; Pigford, T. H.
Steady-State and Transient Radionuclide Transport Through Penetrations in Nuclear Waste Containers. LBL-21806; Lawrence Berkeley Laboratory and Dept. of Nuclear Engineering U. of California, Berkeley, CA, 1986. (2) Aidun, C. K. S.; Bloom, S. G.; Raines, G. E. Radionuclide Transport Through Perforations in Nuclear Waste Containers. Mater. Res. SOC.Symp. Proc. 1988, 112, 261. (3) Runchal, A.; Sagar, B.; Baca, R. G.; Kline, N. W. PORFLO-A Continuum Model for Fluid Flow, Heat Transfer, and Mass Transport in Porous Media: Model Theory, Numerical Methods, and Computational Tests. Rockwell Hanford Operations, RHO-BW-CR-150-P,1985. (4) Priyantha, W. J.; Brown, K. W.; Thomas, J. C.; Lytton, R. L. Leakage Rates through Flaws in Membrane Liners. J. Enuiron. Eng. 1988, 114, p 1401. (5) Schroeder,P. R.; Gibson, A. C.; Smolen, M. D. The Hydrologic Evaluation of Landfill Performance (HELP) Model; EPA/530-SW-84-010;U.S. Environmental Protec-
tion Agency: Washington, DC, 1984; Vol. 11. Received for review October 30,1989. Accepted February 8,1990. Work partially supported by the U S . Nuclear Regulatory Commission, Officeof Research, under DOE Contract No. DEAC07- 761001570.
