Accurate Assessment of Flow Reduction Due to Biofouling in Open Channels

1. Introduction

Biofilm development in water supply and drainage systems is relevant for the design, management, and maintenance of urban, agricultural, or industrial hydraulic infrastructures [1]. The presence of this biological growth in pipes and open channels adversely affects hydraulic performance by increasing friction and energy losses, thereby increasing operation, maintenance, and environmental costs [2]. Despite its importance, incorporating biofilm effects into the design, maintenance, and operation decision-making processes of hydraulic infrastructures remains challenging due to its complex and dynamic nature [3].

This paper presents a method for the accurate quantification of flow reduction caused by biofouling in open channels. The approach is based on Manning’s equation [4] combined with a probabilistic formulation of the input quantities. Emphasis is placed on Manning’s roughness coefficient, and to its probabilistic representation using a beta distribution. The parameters of this distribution are derived from established reference values [5,6] accounting for channel type, construction material, surface condition, and the presence of biofilm.

The propagation of measurement uncertainty propagation from the input quantities to the flow rate is carried out using the Monte Carlo method [7,8]. The results are crucial for conformity assessment [9] of open channels with biofilm, based on the adoption of an appropriate decision rule and the definition of a tolerance upper limit for flow reduction. The proposed approach provides a robust tool to support risk-informed decision-making [10,11], particularly in maintenance planning and interventions scheduling. This tool can be used by infrastructure managers to determine cleaning (see Figure 1), and rehabilitation requirements with a quantifiable level of confidence, thereby reducing the risk associated with premature or delayed interventions and the corresponding economic costs.

The proposed approach is demonstrated through a case study of a sub-system of the Alqueva agricultural irrigation network, the Álamos—Loureiro trapezoidal concrete open channel, located in southern Portugal, designed for a maximum flow rate of 37 m³·s^-1 [12].

2. Flow Measurement in Open Channels

In the context of open channels flow, Chézy’s equation is one of the earliest empirical formulations (proposed in seventeenth century) for calculating the cross-sectional average flow velocity, v, being given by

v = C · √(R_h · S),

(1)

where C is Chézy’s roughness coefficient, R_h is the hydraulic radius, and S is the channel slope. An alternative formulation was later proposed by Manning in the late of the nineteenth century [13], given by

v = (1/n) · R_h^2/3 · S^1/2,

(2)

where n is the Manning’s roughness coefficient. This empirical formula supports one of the most widely used methods for flow rate measurement calculation in open channels, given by

q_v = v · A = (1/n) · R_h^2/3 · S^1/2 · A,

(3)

where q_v is the volumetric flow rate and A is the cross-sectional flow area. In this context, the hydraulic radius is defined as

R_h = A / P_w,

(4)

where P_w the channel’s wet perimeter. Manning’s equation can be applied to a wide range of hydraulic engineering systems, including rivers, streams, floodplains, open channels, sewers and partially filled pipes, under steady, uniform free surface conditions [14]. The input quantities are primarily related to the channel geometry, flow depth and resistance to flow, the latter being represented by Manning’s roughness coefficient. For this parameter, reference values are available in literature [e.g., 5], typically provided as minimum, expected and maximum values for different types of channels (both artificial and natural), as well as for varying surface texture conditions. In artificial channels, these values depend on construction material and surface finish. Table 1 presents typical reference values of Manning’s roughness coefficient for concrete open channels [5].

The development of biofilm increases the uncertainty associated with Manning’s roughness coefficient, because of the complex and non-linear nature of this process [2,3]. This is related to: (i) viscoelastic and flow-dependent behavior, whereby, higher flow velocity can induce more compact biofilm and regular detachment of biofilm patches, decreasing the flow resistance; (ii) spatial heterogeneity, resulting from irregular and non-uniform layers of biofilm in terms of thickness, density and structure in different regions of the same channel; (iii) temporal variability, as biofilm is a metabolically active material undergoing cycles of growth, maturation, sloughing and restructuring.

3. Case Study

The case study illustrates the application of the proposed approach and concerns to a sub-system of the Alqueva agricultural irrigation network, namely the Álamos-Loureiro channel, located in southern Portugal (see Figure 2). The general design specifications of this open channel [12] are presented in Table 2.

The channel is designed for a maximum flow rate of 37 m³·s^-1, corresponding to a flow depth of 3.70 m. Considering the total channel height of 4.50 m, a freeboard of 0.80 m was adopted to reduce the risk of overtopping caused by flow fluctuations associated with transient flow conditions and high intensity precipitation events.

The construction process of the channel (see Figure 3) allows to classify the surface condition as trowel finish, during the initial stage of hydraulic operation.

For a trapezoidal channel (see Figure 2.b), the following equations are used to determine the top width of the flow cross-section, B, and the corresponding area, A, and the wet perimeter, P_w:

B = b + 2 · h · tan (π / 4)

(5)

A = 0.5 · (B + b) · h;

(6)

P_w= b + 2 · h / cos (π / 4).

(7)

4. Measurement Uncertainty Evaluation

4.1. Functional Diagram and Probabilistic Formulation of the Input Quantities

Figure 4 presents the functional diagram related to the flow rate measurement in the studied open channel.

According to Figure 4, five input quantities are identified: Manning’s roughness coefficient, n; the flow depth, h; the channel bed width, b; and the channel slope horizontal and vertical dimensions, H and V, respectively. The probabilistic formulation of these quantities is presented in the following sub-sections.

4.1.1. Dimensional and Geometric Quantities

The probabilistic formulation presented in Table 3 is based on the rounding adopted in the definition of design values. A uniform distribution is assumed for each quantity, with a half-width equal to half of the corresponding rounding value.

For the slope vertical dimension and the bed width values, a rounding of 0.01 m was considered, whereas a rounding of 1 m was adopted for the horizontal dimension. These rounding values are higher than the target measurement uncertainties specified in the construction quality control procedures. In the case of the flow depth, in situ measurements with ultrasonic sensors are also associated with a target measurement uncertainty lower than the adopted rounding value of 0.01 m.

4.1.2. Manning’s Roughness Coefficient

For the probabilistic formulation of Manning’s roughness coefficient, a beta distribution is adopted as an appropriate representation of the dispersion of values associated to this input quantity. This choice is supported by the availability of minimum, expected and maximum reference values [5] to clean (biofilm-free) condition. The beta distribution is defined by two parameters, p₁ and p₂, allowing different, smooth and flexible probability distributions within a bounded interval defined by the minimum and maximum values.

This distribution avoids generating values outside the known minimum and maximum reference values, respectively, while allowing asymmetric probability distributions, namely, when the expected value is different from the average value between minimum and maximum values. The influence of biofilm development can be incorporated through appropriate parameterization of the distribution, allowing a realistic probabilistic formulation of a complex, spatial and time-varying, and non-linear phenomenon.

In this study, the beta distribution parameterization is based on minimum, expected and maximum reference values of Manning’s roughness coefficient. The procedure involves the calculation of the corresponding normalized average, variance estimation (by the PERT method) and normalization. Table 4 presents p₁ and p₂ for the reference values shown in Table 1. Figure 5 and Figure 6 illustrate the corresponding beta distribution of values associated with Manning’s roughness coefficients.

In this study, three levels of biofilm effect were considered—reduced, moderate and severe—based on reported flow reduction due to biofouling [1]. The corresponding minimum, expected and maximum of Manning’s roughness coefficient values are presented in Table 5, together with the associated parameterization of the beta distribution.

In the quantification of the reduced biofilm effect, a clean concrete channel with trowel finish was considered.

4.2. Propagation of Measurement Uncertainty

The Monte Carlo Method (MCM) was used to propagate measurement uncertainties from the input quantities to the output quantity (flow rate), as illustrated in Figure 4. The selection of this numerical method was motivated by the probabilistic modelling of the Manning’s roughness coefficient using a beta distribution, which allows dealing with asymmetrical dispersion of values, as is the case, when minimum, expected and maximum reference values are available. The MCM provides accurate and convergent results when the uncertainty propagation involves non-linear mathematical models, such as Manning’s equation (see expression 3). The influence of asymmetrical input distributions can also be evaluated in the dispersion of output values.

In this context, the main guidelines of the GUM—Guide to the expression of Uncertainty in Measurement / Supplement 1 [8] were followed. A total of 10⁶ simulations were executed for each input quantity, ensuring convergence of the flow rate (output quantity) distribution, with a computational accuracy level below 0.1 m³·s^-1.

A dedicated computational routine was developed in a MATLAB environment for this purpose, using the Mersenne-Twister pseudo-random number generator [15]. The same routine was used to conduct a sensitivity analysis of the input quantities, with the aim of identifying the main contributions for the measurement uncertainty of the flow rate. This analysis was carried out by introducing identical relative increase in the uncertainty of each input quantity and observing the corresponding normalized increase in the uncertainty of the output quantity.

5. Results

5.1. Intermediate Quantities

Table 6 presents the results of the numerical simulation obtained using the MCM method for the intermediate quantities associated with the flow rate (output quantity), considering a reduced biofilm effect (clean surface condition).

The relative 95% expanded uncertainty of Manning’s roughness coefficient at this confidence level is, approximately, 10%. Complementary results were obtained for the flow cross-sectional average velocity (by equation 2), namely, an average of 1.38 m·s^-1, a mode of 1.37 m·s^-1 and a (absolute) 95% expanded measurement uncertainty equal to 0.14 m·s^-1, equivalent to a 10% of the mentioned estimate.

5.2. Output Quantity

Table 7 presents the measurement estimates and uncertainties of the flow rate, considering the intermediate results (in Table 6), for the studied trapezoidal concrete open channel under reduced biofilm conditions (clean surface).

Figure 7. Numerical dispersion of values of the output quantity for a flow depth equal to 3.70 m.

Figure 7. Numerical dispersion of values of the output quantity for a flow depth equal to 3.70 m.

Additional simulations carried out for flow depths between 0.5 m up to overtopping limit of 4.5 m, revealed a constant relative 95% expanded measurement of 10%, for flow rate values between 1 m³·s^-1 and 50 m³·s^-1. Within this range, Manning’s roughness coefficient was identified as the dominant contributor (close to 99%) to the overall measurement uncertainty, while the contributions of the remaining input quantities were found to be negligible. For lower flow depths close to 0.1 m, the relative 95% expanded measurement uncertainty of the flow rate increases up to 14%, reflecting the additional influence of flow depth on the total uncertainty.

Figure 8 illustrates the relationship between Manning’s roughness coefficient and the flow rate for the studied channel, considering the maximum operation flow depth (3.70 m) and different concrete surface textures (see Table 1). Appendix A presents analogous relationships for the same open channel geometry considering different construction materials (cement, wood, hybrid materials, gravel, brick, masonry and stone), based on Manning’s roughness coefficient reported in [5].

As expected, an increase in Manning’s roughness coefficient results in a reduction in the estimated flow rate, while the relative 95% expanded measurement uncertainty varies between 7% and 12% depending on the type of concrete surface textures.

Table 8 presents the results obtained for the flow rate, considering different levels of biofilm development (as shown in Table 5).

As expected, the increase of the biofilm effect magnitude, from a reduced to a severe level, corresponds to a flow rate decrease in terms of expected value and absolute measurement uncertainty.

5.3. Flow Reduction and Conformity Assessment

The results presented in Table 8 enable the quantification of the impact of moderate and severe biofilm conditions relative to the reduced effect (clean channel) condition. The measurement estimates and associated uncertainties reported in Table 9, were obtained by propagating the flow rate results from Table 8 using the MCM.

Conformity assessment is crucial to ensure compliance of the hydraulic infrastructure performance with the requirements defined by management entities, thereby supporting confidence in risk-informed decision-making with significant impacts on operational, maintenance and environmental costs.

In the presented case study, the implementation of a conformity assessment procedure [9,10,11] is supported by:

(i)

a tolerance upper limit, defining the boundary between acceptable and unacceptable flow reduction due to presence of biofilm;

(ii)

flow reduction measurement results, including both estimates and associated uncertainties, as presented in Table 9;

(iii)

the selection of a decision rule, corresponding to an objective criterion for comparing the measured flow reduction with the tolerance upper limit specified by the hydraulic management entity.

When considering decisions regarding cleaning and rehabilitation of the channel due to biofouling, two types of incorrect decisions can occur: premature or delayed interventions, associated with probabilities α and β, respectively. Both cases imply costs for the management entity. Following the general guidelines mentioned in [9], a 2 × 2 decision matrix P can be defined, where the diagonal elements, (1–α) and (1–β), represent the probabilities of correct decisions, while the off-diagonal elements, α and β, correspond to the probabilities of incorrect decisions. These are associated with type I and type II errors, respectively: a type I error occurs when the channel is incorrectly classified as non-compliant (unnecessary intervention), while a type II error occurs when the channel is incorrectly classified as compliant (required intervention is not carried out).

Infrastructure managers can prioritize their risk analysis by focusing on either type I or II probabilities and their associated consequences. In the case studied, a type I error leads to the risk of unnecessary costs related to the operational shutdown, loss of revenue, and cleaning and rehabilitation activities. Conversely, a type II error implies a risk of non-compliance with service commitments (e.g., inability to deliver the contracted flow rate), and an increased risk of failure due to overtopping, driven by flow fluctuations under transient flows or intense precipitation events.

Based on the geometrical representations shown in Figure 9 and assuming the applicability of the Central Limit Theorem [16] to the flow reduction results (Table 9), a Gaussian probability function, Φ, can be assumed for the measurement uncertainty, u(Δq_V). The following decision rule can be adopted, focusing on the control of type I error:

(a) the biofouled channel is considered conform, if the hypothesis H₀: P(Δq_V ≤ T_u) ≥ (1—α) is true;

(b) the biofouled channel is considered non-conform, if the hypothesis H₀: P(Δq_V ≤ T_u) < (1—α) is false.

In this case, the probability of conformity is given by (8)

P_c = P (Δq_V ≤ T_u) = Φ [ (T_u − Δq_V) / (u(Δq_V) ],

(8)

where P_c is the conformance probability, Δq_V is the flow reduction estimate and T_u is the tolerance upper limit.

Table 10 presents the results of the conformity assessment for the two previously analysed biofilm-induced flow reduction scenarios (see Table 9), considering a 5% type I error probability and a tolerance upper limit of 33% (indicated by the vertical line in Figure 9).

It should be noted that, in the case of severe biofilm effect, if a simple acceptance / rejection decision rule was applied, without accounting for the measurement uncertainty in the flow reduction, the channel would be classified as conform, since the measurement estimate (31.3%) is below the tolerance upper limit (33%). In that case, the probability of a type I error (false acceptance) would increase to 50%, reducing confidence in the intervention decision.