Finite element method (FEM) update

Geonet anisotropy and its effect on the planar flow in drainage geocomposites.

By Dhani Narejo

Anisotropy is the characteristic of a material having different values for a property when measured in different directions. Materials can be anisotropic with respect to their physical, mechanical, electrical, hydraulic, and many other types of properties. Anisotropy results from solids being arranged in a regular pattern as opposed to a random structure.

Most geosynthetics have different properties in machine and cross-machine direction but nowhere is this difference more pronounced than for the transmissivity of geonets. More than a decade ago, Sieracke and Maxon (2001)¹ illustrated, based on laboratory test data, that the cross-direction transmissivity of some geonets can be as little as 20% of the machine-direction value.

Since then, other geonets have been introduced in the marketplace with their anisotropy as low as zero, no cross-directional flow. Geonet anisotropy is often shrouded in mystery because manufacturers do not publish cross-direction transmissivity values and design methods cannot account for the cross-directional flow.

The increasing popularity of finite element seepage analysis—as one can surmise by visiting any of the reputable civil engineering journals—is making it possible to understand planar anisotropy in geonets and account for it in simple as well as complex site problems.

Drainage geocomposites are manufactured as large panels ranging in width and length from 1.5–4.6m and 15–90m, respectively. The thickness of most drainage geocomposites is generally less than 15mm with the flow, for all practical purposes, being planar in nature. Liquid percolates down the overlying mass, enters the geonet and flows down the slope almost immediately due to a very high hydraulic conductivity of these materials compared to the underlying geomembrane and overlying soil or waste mass.

For a simple slope, such as a geocomposite installed on a peripheral berm of a landfill cell, or within a dam embankment, or behind a retaining wall, the flow takes place in one direction (i.e., from the crest to the toe of the slope). In such cases, the design procedures for selection of drainage geocomposites use the hydraulic conductivity in the machine direction as the materials are installed with the machine direction oriented down the slope. For short slopes, panels are sometimes installed in the cross-direction but the design and testing is then also performed in the cross-direction.

Many cases of field conditions exist where the flow is multi-directional and it is necessary to account for both the machine and the cross-machine direction hydraulic conductivity of drainage geocomposites. Examples of such applications involve three-dimensional (3-D) slopes and include roadways, green roofs, sports fields, landfill cells, and heap leach pads.

A 3-D slope is shown in Figure 1 consisting of, from bottom up: geomembrane, drainage geocomposite, and cover soil. Using letters m and c to represent machine and cross-direction of a drainage geocomposite, the slope has lengths of L_m and L_c that tilt at an angle β_m and β_c with a geocomposite hydraulic conductivity of k_m and k_c, respectively.

Liquid can enter the slope only from the top, such as by percolation or from a rainfall, and the field drains on both the downstream sides. Any finite element software can be used for a seepage analysis of this problem as long as it can handle 3-D domains. For this paper, Hydrus 2D/3D software was used to perform a steady state seepage analysis for a range of conditions. (Refer to Narejo 2013a, 2013b² for additional information on the materials and methodology used in this paper.)

A 3-D steady state analysis is equivalent to the current state of practice of using Darcy’s law for the design and selection of drainage geocomposites for uni-directional flow.

Figure 2 shows the liquid supply rate for a constant maximum pressure head within drainage geocomposites of different values of anisotropy divided by the liquid supply rate for an isotropic material (i.e., k_c = k_m). For the geometry considered for Figure 2 (β_c = β_m and L_c = L_m) the figure shows that the effect of anisotropy becomes significant only when the cross-machine direction slope length is less than the machine direction length. For L_c/L_m of less than 0.5, the cross-machine direction hydraulic conductivity controls the planar flow rate. Between L_c/L_m of 0.5 and 1.5, both machine and cross-direction hydraulic conductivities are important. For L_c/L_m of greater than 1.5, cross-machine hydraulic conductivity has a negligible effect on the planar flow in a drainage geocomposite.

Figure 2 was based on cases where both machine and cross-machine directions have the same slope angle and slope length. In practice, the length and slope angle along machine and cross-direction vary significantly. As an example, consider a lined field with a length of 20m sloping at 20% and a width of 5m sloping at 5% where three different structures of geonets are being considered for use. These geonets all have the same thickness of 5mm and the same machine direction hydraulic conductivity of 82800mm/hour.

The anisotropy values (i.e., cross-machine to machine direction hydraulic conductivity) for these three geonets are 1.0, 0.7, and 0.2. A 3-D finite element seepage analysis shows that a liquid supply rate of 6.4mm/hour would result in a maximum pressure head of 5mm, 16mm, and 62mm for these geocomposites, as illustrated in the top row of Figure 3, if panels were installed with cross-direction parallel to the shorter side. Therefore, the thickness of the geonets with anisotropy of 0.7 and 0.2 must be increased if pressure head is to be kept within the geocomposite.

Another option would be to consider installing the geocomposites with the machine direction parallel to the shorter side even though the slope is steeper along the longer side. In this case, pressure heads for a liquid supply rate of 6.4mm/hour are 5mm, 8mm, and 11mm as shown in the bottom row of Figure 3. Design engineers and installers need this type of information to be able to select a geonet from the many different types available and develop a panel layout for the specific type of product under consideration.

Anisotropy is an important property of geonets and should be considered in design calculations for slopes involving multi-directional flow. The effect of anisotropy on pressure head depends on many factors that must be considered during a project-specific seepage analysis. Such a seepage analysis can be performed with finite element methods with minimal expense and time.

While this article considered only a steady state analysis and one type of boundary conditions, the site-specific effect of anisotropy can be very different from such simple cases. Many types of geonets structures are available for design engineers to utilize in their projects and the hydraulic performance of these materials has increased tremendously over last 10 years. However, it is no longer necessary to perform calculations based on machine direction transmissivity for projects involving multi-directional flow.

Dhani Narejo, Ph.D., P.E., is president of Narejo Inc., an independent consulting firm based in Conroe, Texas. He is a member of Geosynthetics magazine’s Editorial Advisory Committee.

References

²Narejo, D. (2013a), “Finite Element Mesh Generation for Drainage Geocomposites,” Geosynthetics 2013 conference, April 1–4, Long Beach, Calif.

²Narejo, D. (2013b), “Finite Element Analysis Experiments on Landfill Cover with Geosynthetic Drainage Layer,” Geotextiles and Geomembranes, Vol. 38 (2013), pp. 68–72.

¹Sieracke, M., and Maxon, T. (2001), “Common sense design with geosynthetic material,” Geotechnical Fabrics Report (GFR), Vol. 19, No. 8, Oct./Nov. 2001, pp. 22–23.