By performing a sensitivity analysis that evaluates the impact of the pipe and base slopes on the airspace and construction material quantities, potential savings could be in the range of several tens of thousands of dollars. The mathematical model may also reduce the cost of construction and increase the value gained from additional airspace for landfill owners.
Ali Khatami, Ph.D., P.E.
Landfill designers deal with many technical parameters for the design of various components of landfills, including foundation settlement, slope stability, settlement of waste, leachate collection system hydraulics, maximum leachate head above the liner and airspace, to name a few. Airspace is the space within the permitted boundary of the landfill-lined areas, base grades of disposal cells and final grades of the landfill. Since the disposal airspace of a landfill is considered the actual asset of a landfill owner, which gets consumed with every cubic yard of waste disposed in the landfill, it would make sense that the designer would try to optimize its design to achieve the largest airspace for the landfill.
During the course of the design of a new landfill or an expansion of an existing landfill, the designer selects certain parameters, through various means, for the design of the landfill boundary, base grades and final grades. These parameters are normally discussed with the facility owner and decisions are made. Among a few of these parameters that may be left to the engineer to select are the leachate collection pipe slope (pipe slope) and base slope, where they are used to develop grades at the bottom of the landfill. Traditionally, a herringbone design has been used by designers to establish necessary slopes for the gravity flow of leachate toward a collection point (sump) at the lowest point of a disposal cell. Procedures followed for the selection of the pipe and base slopes may vary from one engineer to another. Occasionally, minimum allowable values are required for the pipe and base slopes in accordance with applicable regulations, which adds another level of complication in the process of selecting the design parameters.
In many cases, a sensitivity analysis to optimize the pipe and base slopes is not performed. Part of the reason that sensitivity analysis is not carried out is the cost of performing the analysis. Sensitivity analysis to optimize the pipe and base slopes begins with developing various scenarios of landfill base grades, followed by performing volume calculations for various combinations of the scenarios with different slopes. The sensitivity analysis evaluates the impact of the pipe and base slopes on the airspace and construction material quantities. The task of developing several scenarios requires a significant effort and may involve many hours of work for each scenario; as a result, the cost of the design project increases significantly, especially if the design includes many disposal cells.
The formulation discussed in this article can potentially cut back on the cost of a sensitivity analysis in a dramatic way because the formulation provides the means to calculate the volume change (i.e., airspace and construction material quantities) among each of two scenarios simply by plugging in a few numbers in an analytical formula. The remainder of this article provides a description of the model, mathematical formulation and two numerical examples illustrating the ease of calculations in using the formula.
Model
The model used for this article is a rectangular cell with grades in accordance with the herringbone concept. Figure 1 shows a typical rectangular base of a disposal cell without showing berms on the four sides of the base. The location of the leachate collection pipe is at the trough of the cell base area. To simplify the model, it was assumed that the pipe is not placed inside a trench. The base has two distinct portions, one on either side of the leachate collection pipe. In each portion, two distinct slopes define the grades, namely the pipe slope and the base slope. The basic assumption for this model is that the pipe slope and the base slope remain fixed throughout the entire area of each portion of the cell, but the base slope may vary from one portion to another portion of the cell with the pipe slope being the common element between the two portions (Figure 2).
The formulation discussed below is for one portion of the base only. Based on the model described above, the lengths of the cell portions are similar, and the pipe slope remains unchanged from one portion to the other. If the two portions are similar in the base slope and width, the result of the calculations can be multiplied by two to get the result for the entire cell. Alternatively, if the two portions have different base slopes and/or widths, each portion must be analyzed separately and the results must be added to get the final result for the entire cell. The model considers two sets of grades (surfaces) for analysis: 1) the original surface designed or suggested by the engineer for the cell prior to the sensitivity analysis, and 2) the trial surface by a slight variation in the pipe slope and/or the base slope for the sensitivity analysis. The formulation discussed below calculates the volume between the original surface and the trial surface without the need to actually generate a grading plan for the trial surface by graphics software (such as AutoCAD) and performing three dimensional volume calculations between the two surfaces.
Formulation
The volume between the original surface and the trial surface may be calculated using differential calculus. An elemental volume presented in Figure 3 is used to define the mathematical relationship between the