Groundwater infiltration through cracked sewer pipes has caused significant economic losses. This paper presents a three-dimensional analytical expression for calculating the steady-state groundwater infiltration rate into sewer systems. As an extension of the previously developed two-dimensional model by the author, this new model can be used to simulate the infiltration through an orifice defect. The derived analytical expression has been validated with experimental results. The new model incorporates all the related resistances during the infiltration process, including the soil head loss and the orifice loss. The soil head loss is assessed with Ergun equation, which involves an inertial loss term in addition to the viscous loss. This has significantly extended the application range of the new expression. A new OS number (the ratio of orifice loss to soil loss) expression is also presented. The order analysis of the OS number expression has demonstrated that in most real cases, the head loss through the soil layer dominates the whole infiltration process.
INTRODUCTION
As sewer pipes age, a number of defects or cracks emerge on the pipe wall due to various factors (Davies et al. 2001; Kuliczkowska 2016). These defects or cracks may serve as the entrance for groundwater infiltration into the sewer pipe when the groundwater table is greatly raised above the pipeline during the rainy season. As a common problem existing in sewer systems worldwide, groundwater infiltration has caused great economic losses. This part of ‘unwanted water’ (Wittenberg & Aksoy 2010) from infiltration increases the cost of wastewater treatment and the energy consumption and operating costs of pumping stations. Therefore, in the past few decades, significant attention has been drawn on the infiltration control and management (De Benedittis & Bertrand-Krajewski 2005).
Many methods have been proposed for the catchment or sub-catchment scale groundwater infiltration estimation, including the traditional water balance method (Brombach et al. 2002; Weiss et al. 2002), the pollutant loads analysis method (Kracht & Gujer 2005), the tracer method (Kracht et al. 2007; Prigiobbe & Giulianelli 2009) and numerical modelling (Aksoy & Wittenberg 2011; Karpf & Krebs 2013; Thorndahl et al. 2016). The pollutant loads analysis and the traditional method are relatively expensive and their precisions are difficult to ensure. The tracer method has good precision, but it can only be applicable to urban catchments, where the suitable isotopic separation between drinking water and potential infiltration sources exists. Although numerical models have been reported with good accuracy, the computational costs for long term modelling will be prohibitive. On the other hand, municipal agencies worldwide are conducting structural assessment on their sewer systems by using advanced inspection technologies (Wirahadikusumah et al. 1998; Costello et al. 2007). Therefore, a method connecting the routine expenditure on inspection with the infiltration estimation would be attractive and beneficial. Guo et al. (2013a) proposed a two-dimensional model for calculating the infiltration rate of the individual sewer pipe segment with a line defect on the pipe wall. The method aimed to connect the routine expenditure on sewer inspection with infiltration estimation to save the budget allocation for the sub-catchments scale infiltration investigation.
As an extensional work of the previous model, this study presents a three-dimension infiltration model. Therefore, the following reasonable assumptions used in the previous study are also kept in this study: the groundwater table above the embedded sewer pipe is horizontal and the surrounding soil is homogeneous and isotropic. Experiments were conducted to verify the derived analytical model. The computed results showed a good agreement with experimental data. From the parametric analysis, the OS number, which is defined as the ratio of orifice loss to soil loss, has been demonstrated to be less than unity in most real cases; this indicates the head loss through the soil dominate the infiltration process.
METHODS
Flow field and governing equation
Schematic of a cracked sewer pipe and flow condition: (a) real flow field; (b) transferred flow field.
Schematic of assumed groundwater flow field.
Analytical derivation
RESULTS AND DISCUSSION
Experimental verification
Schematic of experiment setup: (a) plan view; (b) section A-A; (c) section B-B.
Particle size distribution of experimental sand.
Before the infiltration experiment, tests with no soil media (only water) were conducted. By fitting the results to Equation (10), the integrated orifice loss coefficient k is found to be 2.8. It should be noted that the tests were conducted with the wire covering over the orifice. Therefore, the possible effect from the wire has been incorporated into the measured value of k.
In order to obtain the particle's shape effect on the infiltration flow and avoid the direct measurement of a great deal of individual particle's shape factor, constant–head tests were conducted according to the standard soil test method. The hydraulic conductivity K of the sand material is measured to be 1.13 × 10−3 m/s. Using Equation (4), the values of A, B, and φpdp are calculated to be 885.47 s/m, 6,256.81 m2/s2 and 0.39 mm, respectively. Table 1 summarizes all the related experimental parameters.
Parameters’ values based on experimental results
| dpφp (mm) | ɛ | t (mm) | D0 (mm) | k | K (m/s) | A (s/m) | B (m2/s2) |
|---|---|---|---|---|---|---|---|
| 0.39 | 0.36 | 5 | 10 | 2.8 | 0.00113 | 885.47 | 6,256.81 |
| dpφp (mm) | ɛ | t (mm) | D0 (mm) | k | K (m/s) | A (s/m) | B (m2/s2) |
|---|---|---|---|---|---|---|---|
| 0.39 | 0.36 | 5 | 10 | 2.8 | 0.00113 | 885.47 | 6,256.81 |
Comparison of the results by analytical expressions and experimental data.
Parametric analysis
Walski et al. (2006) proposed an OS number that is the ratio of the orifice head loss and soil head loss to model the leakage from water distribution pipes, and concluded that the OS number would be larger than 1 in most cases. The OS and Reynolds numbers in this study were carefully measured, as shown in Table 2.
OS and Reynolds number based on experimental results
| Water table (mm) | 230 | 250 | 270 | 290 | 310 | 330 |
|---|---|---|---|---|---|---|
| OS number = Δho/Δhs | 0.0054 | 0.006 | 0.0062 | 0.0068 | 0.0071 | 0.0076 |
| Re = ρwVdp/μ | 44 | 49 | 51 | 56 | 60 | 64 |
| Water table (mm) | 230 | 250 | 270 | 290 | 310 | 330 |
|---|---|---|---|---|---|---|
| OS number = Δho/Δhs | 0.0054 | 0.006 | 0.0062 | 0.0068 | 0.0071 | 0.0076 |
| Re = ρwVdp/μ | 44 | 49 | 51 | 56 | 60 | 64 |
As shown in Table 2, the OS numbers were very small and greatly less than unity. The disagreement between the two studies was mainly due to the Reynolds number. Walski et al. (2006) used the Darcy equation to assess the head loss through the soil, however, this is only valid for Re < 10 when the groundwater flow is laminar. The Reynolds number in the vicinity of the orifice in this study was larger than 40 in all cases, as indicated in Table 2. The disadvantages of Walski et al. (2006) analysis have also been noticed by Van Zyl & Clayton (2007) and Collins & Boxall (2013).
where Vo represents the velocity at the orifice and is defined to differ from the specific velocity. The square of the term (gD0)0.5 represents a characteristic velocity at the orifice, then we can deduce that gD0 or (gD0)0.5 should be of the same order with Vo. The hydraulic conductivity K of the backfill sandy soil is usually 10−2–10−5m/s (Terzaghi et al. 1996), then A = 1/K will be of the order of 102–105. The denominator term in Equation (13) should be of the order of 102 or larger, while the numerator can be easily assessed according to the Equation (10) and should be generally less than 102. Therefore, the OS number would be less than 1, and we can conclude that in most cases, the head loss through the soil dominates the infiltration process.
CONCLUSIONS
Based on some reasonable assumptions, including a homogeneous and isotropic aquifer, this paper presents an approximate model for calculating groundwater infiltration rate into sewer systems through orifice or hole damages. Although the damage is assumed to be on top of the sewer pipe in deriving the expression, the proposed model is still applicable if the orifice locates on other places. In order to validate the proposed model, experiments were conducted and a good agreement between predicted results and experimental results was found.
The derived analytical expression combines the head loss through the orifice and the soil. By introducing Ergun equation instead of the Darcy equation to assess the resistance from the soil layer to the groundwater flow, the application range of the new expression has been significantly extended. A new OS number expression is also presented. The order analysis of the OS number has cleared the fact that the head loss through the soil layer dominates the whole infiltration process in most real cases. At present, advanced inspection technologies capture not only the internal structural condition of the sewer pipe, such as the defect type, size, but also the external groundwater table, soil type, etc. Data collected by these technologies provide the probabilities for using the proposed model in the future.
ACKNOWLEDGEMENTS
This work was financially supported by the China Postdoctoral Science Foundation (Grant No. 2013M541781) and by the Jiaxing Water Special Program (project Number: 2011ZX07301-004)