## Abstract

Brazilian semi-arid productive techniques use rainwater harvesting systems, which are a sustainable water management practice with a low environmental impact that have been adopted as an alternative to meet water demand worldwide. The aim of this study was to find an optimal sizing methodology for rainwater harvesting systems using local parameters allied to the lowest system cost. The analyses were based on a system supplying 95% of a 250 L/d demand for a goat herd in Feira de Santana, Bahia, Brazil. The area available for system implementation did not limit the analysis. The only limiting parameters were the quantity and quality of water required for the herd. The results indicated that, among the combinations of catchment area and water tank volume capable of meeting the defined demand, there is an optimal set, with minimum cost. This was a catchment covering 220 m² with a 21.1 m³ water tank, equivalent to a 0.62 demand fraction (FD). The variance influence of meeting service efficiency and demand, in the system's implementation and performance, was also analyzed.

## INTRODUCTION

The average annual precipitation in Brazil's semi-arid region is 750 mm, and the average annual evapotranspiration 2,500 mm (Montenegro & Montenegro 2012). In addition to irregular precipitation distribution through the year, the region is characterized by shallow soils with low infiltration capacity and scarce groundwater resources. These conditions lead to high water deficits, limiting the agricultural and livestock productivity of the region (Alves et al. 2016).

Goat husbandry stands out as an effective choice for Brazil's northeastern semi-arid region. Goat farming does not need large areas and, unlike other species, goats have low water loss, making them resistant to the high-temperature variance and water stress typical of the region (Kaliber et al. 2016).

Brazil's Northeast Region is home to approximately 9.1 million goats (Figure 1), about 93.0% of the Brazilian herd (IBGE 2016). However, the limited capacity of water support, associated with low water quality, affects goat farming negatively in the region (Mdletshe et al. 2017).

Figure 1

Feira de Santana's location in Brazil's Northeast Region, with emphasis on the country's main goat farming areas.

Figure 1

Feira de Santana's location in Brazil's Northeast Region, with emphasis on the country's main goat farming areas.

Worm infections are a serious sanitary problem in goat farming, and directly related to water supply quality. Infection affects the welfare of the animals and increases their stress levels, compromising production efficiency and quality (Palhares & Guidoni 2012; Mad-Ali et al. 2018).

In this type of scenario, rainwater harvesting and storage systems (RWHSs) are technically feasible alternatives. They are already used for both domestic purposes and animal production (Ayantunde et al. 2018; Vargas et al. 2019).

In the Northeast Region, the ‘One Land Two Waters Program’ (Programa Uma Terra Duas Águas – P1 + 2) has been the basis for implementing one RWHS each for domestic, farming and livestock use in each rural property (ASA 2019).

International experience has shown successful use of RWHS in agricultural production (Rockström & Falkenmark 2015; Yosef & Asmamaw 2015) and animal production (Ayantunde et al. 2018; Londra et al. 2018). In Brazil, rainwater is used in poultry, pig and cattle husbandry (Palhares & Guidoni 2012; de Proença et al. 2015; Feitosa et al. 2018).

Despite the above, adequate system design methodologies have not been adopted, limiting the systems' potential. Suitable use of relevant parameters – for example, catchment area, and average annual precipitation and distribution – individualize the results for each region (Ghisi 2010; Araujo & Cohim 2016). Studies have shown that using uniform water tank volumes and/or catchment areas decreases system efficiency (Cohim & Orrico 2015; Gomes 2017).

An approach is needed that relates demand-meeting efficiency with an optimal RWHS, including tank capacity and catchment area in an integrated system approach, and allowing the greatest return on investment in the region's water infrastructure.

In this study, an optimal design method for RWHSs for goat husbandry is proposed in relation to the northeast region's semi-arid conditions, with a case study in Feira de Santana Bahia.

## MATERIALS AND METHODS

### Study area

Feira de Santana is in the central-north of Bahia, Brazil, between the Recôncavo Basin and the semi-arid region (Figure 1). The city's rainfall regime is well defined, with 43% precipitating from April to July. Precipitation data were extracted from the INMET platform (2018), for the Feira de Santana Station, code 83221, for the period 1 January 1998 to 31 December 2017 (20-year period). The annual average was 664 mm.

### Rainwater harvesting systems

The system design was based on the P1 + 2 model, which uses a surface covered with concrete slabs, known as boardwalks, as a catchment area. This is a low cost and widely used technology in Brazil's semi-arid region because of the One Land Two Waters Program (P1 + 2). The system has plate-type cisterns produced using concrete slabs and built in situ by a voluntary workforce from the region. The catchment area is constructed to incline at 2%. The system also includes a decanter for solids removal, improving the quality of the stored water. Figure 2 shows a typical layout (floor plan and section) of a catchment x cistern system.

Figure 2

Layout of the boardwalk and cistern system (floor plan and section).

Figure 2

Layout of the boardwalk and cistern system (floor plan and section).

### Calculation criteria

To calculate the daily demand, a herd of 50 goats was considered – the average size in the region (IBGE 2012), (Table 1) with average water consumption per goat of 5 L/d. Total water consumption was thus 250 L/d.

Table 1

1–45–910–1920–4950–99100–199200–499>500
Semi-arid Brazil 3.7% 8.1% 16.9% 31.9% 18.5% 10.4% 8.2% 2.3%
Semi-arid Bahia 1.9% 5.5% 14.8% 33.1% 21.1% 11.2% 9.4% 3.0%
1–45–910–1920–4950–99100–199200–499>500
Semi-arid Brazil 3.7% 8.1% 16.9% 31.9% 18.5% 10.4% 8.2% 2.3%
Semi-arid Bahia 1.9% 5.5% 14.8% 33.1% 21.1% 11.2% 9.4% 3.0%
System behavior was modeled on the basis of serial water balances at daily intervals (Fewkes 2000) – Equations (1)–(4):
(1)
(2)
(3)
where Q(t) (L) is the volume of water flowing in interval t (day), P(t) precipitation (mm), A the catchment area (m²), and C the flow coefficient, which is taken as 0.8. The coefficient θ varies between 0 and 1. When θ = 0, it describes the condition where consumption of daily demand is complete before the rainwater runoff in the time interval is added to the tank, when θ = 1 it was made afterwards. In this study, θ was taken as 0.6. Q(t) (L) is the water demand in time interval t, V(t) the water volume stored in the same interval (L), R the cistern's reserve capacity (L), and Y(t) (L) the yield of what is stored in interval t.
Equation (4) was used to verify the service efficiency of the RWHS in meeting the demand:
(4)
where and correspond to the sum of the rainwater volume used and the demand, respectively, during the period concerned.

The tank volumes and related catchment areas were calculated assuming an EA – Equation (4) – equal to 95%. EA reflects the overall system performance for the simulated period. Thus, system performance for each year of the series was analyzed, taking into account the rainfall anomaly index (RAI) and the precipitation concentration degree (PCD).

The RAI characterizes dry and rainy year severity on the basis of annual precipitation volume (Pereira et al. 2017), and is calculated for each year using Equations (5) and (6).

For positive anomalies:
(5)
For negative anomalies:
(6)
where, N is the annual precipitation (mm), (mm) the historic series annual average, (mm) the average of the seven lowest annual precipitations in the series, and M the average of the seven highest annual precipitations.
To calculate PCD, the year, based on the trigonometric circle, is divided into 12 parts (months) at an angle of 30°. Ri corresponds to the annual precipitation, and Rxi and Ryi correspond to the annual precipitations from the x- and y- axes, respectively. rij is the monthly precipitation, in which i and j represent the year and month, respectively. PCD was calculated using Equations (7)–(10) (Li et al. 2011) and varies from 0 to 1. Values close to 0 represent well-distributed rainfall through the year, while values close to 1 indicate the concentration of rains in a shorter period (Araujo & Cohim 2016).
(7)
(8)
(9)
(10)

The model was tested for the highest catchment area costs and a catchment area already available, and finally, for demand values that could supply herds of between 30 and 500 goats.

Some analyses were based on the dimensionless standard parameter FD (demand fraction), depending on the quantity and variety of parameters used for the simulation. FD expresses the relationship between annual demand, D (L), and the product of the catchment area, A (m²), and average annual precipitation, P (mm) (Fewkes 2000) – Equation (11).
(11)

### Costs

The systems' costs were calculated on the basis of the materials and workforce required for construction, etc. Operating and maintenance costs were not considered.

Budgets were made for cisterns with capacities of between 5 and 100 m³, allowing interpolation to obtain the costs of cisterns of any capacity in that interval. For the catchment area, the average unit cost was budgeted, allowing calculation of the total cost (TC) of a catchment area.

## RESULTS AND DISCUSSION

The cost curves obtained for the catchment system components are shown in Figure 3.

Figure 3

Cost of constructing cisterns (a), and catchment areas (b).

Figure 3

Cost of constructing cisterns (a), and catchment areas (b).

Curves for five capture areas of different sizes – 163, 180, 200, 250 and 300 m² – are shown in Figure 4. Tank capacities above 45,000 L were omitted to make the figure clearer. Figure 4 shows that, for EA = 95%, the 250 L/d demand could only be met satisfactorily with a catchment area of about or exceeding 180 m2. As can be seen, there is a volume that satisfies the demand with the desired efficiency, for each catchment area, and for a catchment area of 163 m², the tank volume is over 90,000 L (Table 2), which makes RWHS deployment uneconomic. It is noted that, for the same service (EA) and demand levels, larger catchment areas need relatively smaller cisterns.

Table 2

Comparison of RWHS cost to meet demand of 250 L/d, with various combinations of catchment area and cistern capacity

Catchment area (m²)Cistern volume (m³)Catchment area costCistern costTotal cost
163 94.9 R$4,080.00 R$ 11,460.00 R$15,540.00 180 38.8 R$ 4,500.00 R$7,432.09 R$ 11,932.09
200 25.5 R$5,000.00 R$ 6,067.66 R$11,067.66 250 17.0 R$ 6,250.00 R$4,987.97 R$ 11,237.97
300 14.6 R$7,500.00 R$ 4,634.25 R$12,134.25 Catchment area (m²)Cistern volume (m³)Catchment area costCistern costTotal cost 163 94.9 R$ 4,080.00 R$11,460.00 R$ 15,540.00
180 38.8 R$4,500.00 R$ 7,432.09 R$11,932.09 200 25.5 R$ 5,000.00 R$6,067.66 R$ 11,067.66
250 17.0 R$6,250.00 R$ 4,987.97 R$11,237.97 300 14.6 R$ 7,500.00 R$4,634.25 R$ 12,134.25

Note – R$1 is roughly US$ 0.25.

Figure 4

Cistern volume × service efficiency for different catchment areas.

Figure 4

Cistern volume × service efficiency for different catchment areas.

Technically, all catchment area and cistern combinations have the same service efficiency, which is characteristic of a mathematical indeterminacy. However, as Table 2 shows, the TC is different for each combination.

TC is the sum of two plots whose tendencies are inverse. As the catchment area increases, its cost increases while that of the cistern decreases. Thus there must be a minimum TC value, which is the optimum sought – see Table 2.

The TC curve for a 95% EA demand (Figure 5) was obtained by applying the cost equations (Figure 3) to a range of catchment areas and their respective volumes. The equation for the TC curve provided a good fit to the data points, with R² = 0.99. The minimum value is obtained by equating the equation's first derivative to zero and solving it for variable A (catchment area). For 250 L/d demand and 95% EA, the optimal solution is A = 220 m² and V = 21.1 m³, at TC = R$11,029.25 (Figure 5). Figure 5 System component costs and total cost for D = 250 L/d and EA = 95%. Figure 5 System component costs and total cost for D = 250 L/d and EA = 95%. As there is considerable variation in the catchment area materials and surfaces – for example, ceramics, metal and paving – the sensitivity of the model to their cost variations was evaluated. The results are presented in Figure 6. Figure 6 System cost variation with catchment area cost. Figure 6 System cost variation with catchment area cost. Increasing the catchment area cost leads to a smaller optimal area, which is reflected in the total system cost. The area, volume, and cost values are shown in Table 3. Table 3 Comparison of system cost sensitivity to catchment area cost Area cost/m²Optimal area/m²Volume/m³Cost R$ 25.00 220 21.1 R$11,001.61 R$ 50.00 199 25.6 R$16,210.22 R$ 75.00 189 29.0 R$21,057.70 R$ 100.00 184 33.8 R$25,727.03 Area cost/m²Optimal area/m²Volume/m³Cost R$ 25.00 220 21.1 R$11,001.61 R$ 50.00 199 25.6 R$16,210.22 R$ 75.00 189 29.0 R$21,057.70 R$ 100.00 184 33.8 R$25,727.03 If a usable catchment area already exists, the related costs decrease but the optimal area does not change (Figure 7). This is because the catchment area cost curve behavior maintains the same angular coefficient with a smaller intercept value as the area available increases. Figure 7 System cost vs. variation in existing catchment area. Figure 7 System cost vs. variation in existing catchment area. Table 4 shows the results of a year-by-year analysis of the system's reliability for the criterion EA = 95%. The data indicate that in 17 of the 20 years in the study, reliability is equal to or exceeds 90%, thus justifying use of the EA = 95% criterion adopted. Table 4 Precipitation, reliability, RAI and PCD in the period 1998 to 2017. YearP (mm/a)Reliability (d)ReliabilityPCDIntensity class (RAI) 2013 372 236 65% 0.26 Extremely dry 2012 375 237 65% 0.25 2017 425 275 75% 0.26 2009 582 331 91% 0.39 Dry 1998 586 327 90% 0.39 2011 604 365 100% 0.11 2001 619 365 100% 0.15 2002 644 365 100% 0.15 2016 670 330 90% 0.39 Wet 2015 670 365 100% 0.18 1999 689 365 100% 0.25 2014 714 365 100% 0.07 2006 716 345 95% 0.35 2007 766 335 92% 0.38 Very wet 2004 774 353 97% 0.35 2005 783 365 100% 0.14 2003 785 365 100% 0.29 2008 792 365 100% 0.13 2010 850 365 100% 0.27 Extremely wet 2000 864 365 100% 0.04 YearP (mm/a)Reliability (d)ReliabilityPCDIntensity class (RAI) 2013 372 236 65% 0.26 Extremely dry 2012 375 237 65% 0.25 2017 425 275 75% 0.26 2009 582 331 91% 0.39 Dry 1998 586 327 90% 0.39 2011 604 365 100% 0.11 2001 619 365 100% 0.15 2002 644 365 100% 0.15 2016 670 330 90% 0.39 Wet 2015 670 365 100% 0.18 1999 689 365 100% 0.25 2014 714 365 100% 0.07 2006 716 345 95% 0.35 2007 766 335 92% 0.38 Very wet 2004 774 353 97% 0.35 2005 783 365 100% 0.14 2003 785 365 100% 0.29 2008 792 365 100% 0.13 2010 850 365 100% 0.27 Extremely wet 2000 864 365 100% 0.04 The table is organized in order of increasing depth of precipitation in the year. On the basis of the RAI and local standards, there was a predominance of wet periods in the series during the period studied. The three years in which EA was below 90% were precisely those classified as extremely dry. The results are not as consistent, however, for the other years. For example, the years classified as ‘dry’ between 2001 and 2011 have reliability of 100%, while those between 2004 and 2007, classified as very wet, do not achieve that level – that is, maximum reliability. This is explained by the PCD, which is influenced by precipitation distribution through the year (Li et al. 2011). It is noted that EA is 100% in years with below-average precipitation – for example, 2011 with 604 mm – while in 2007, with 766 mm, the EA achieved is 92%. As shown in Table 4, in all years with PCD exceeding 0.3 the EA reached 100%, which proves the PCD's importance. That, in turn, confirms that basing the RWHS design solely on the guaranteed annual precipitation yields inadequate results. For example, Zhu et al. (2015) recommend that RWHS design for animal watering is based on rainfall equal to or exceeding 70%, corresponding approximately to a 1.5 year return period. In order to generalize application of the methodology, the sensitivity of FDoptimal to other demand values were analyzed. Demands from 150 to 2,500 L/d· were tested, corresponding to the demands of 90% of the goat herds in Brazil's northeast Region – see FDoptimal values (in Table 5, column 3. The variation is sufficiently small (SD = 0.01), however, in view of the inherent uncertainties in the rainfall data, to justify use of the average FD value = 0.67. Adopting this value to calculate the area and tank volume, TC values are obtained that differ by no more than 3% from the specific values of each demand (Table 5). Table 5 Comparison between system costs for FDoptimal and FD average Demand (L/d)FD optimalCost (FD Optimal R$)Cost (FD Average R$)Variation 150 0.60 R$ 7.566.78  R$7.747.58 2% 250 0.62 R$ 10.915.52  R$11.110.32 2% 350 0.64 R$ 14.018.85  R$14.181.39 1% 450 0.66 R$ 16.951.31  R$17.138.95 1% 550 0.67 R$ 19.752.36  R$19.973.19 1% 650 0.68 R$ 22.504.25  R$22.763.68 1% 1,000 0.70 R$ 31.645.70  R$32.138.90 2% 1,500 0.72 R$ 43.946.00  R$44.883.98 2% 2,000 0.73 R$ 55.759.00  R$57.025.61 2% 2,500 0.74 R$ 67.178.00  R$69.328.84 3% Demand (L/d)FD optimalCost (FD Optimal R$)Cost (FD Average R$)Variation 150 0.60 R$ 7.566.78  R$7.747.58 2% 250 0.62 R$ 10.915.52  R$11.110.32 2% 350 0.64 R$ 14.018.85  R$14.181.39 1% 450 0.66 R$ 16.951.31  R$17.138.95 1% 550 0.67 R$ 19.752.36  R$19.973.19 1% 650 0.68 R$ 22.504.25  R$22.763.68 1% 1,000 0.70 R$ 31.645.70  R$32.138.90 2% 1,500 0.72 R$ 43.946.00  R$44.883.98 2% 2,000 0.73 R$ 55.759.00  R$57.025.61 2% 2,500 0.74 R$ 67.178.00  R\$ 69.328.84 3%

## CONCLUSIONS

Definition of the catchment area and tank capacity to meet a given demand and service efficiency is a mathematically indeterminate problem. The solution can be obtained by minimizing the TC of the rainwater harvesting system, which comprises the sum of the catchment area and cistern costs.

As several combinations of catchment area and cistern volume can meet the required service efficiency, this methodology has shown that the installation cost is an adequate system choice criterion. For any given combination of demand level and service efficiency, the higher the catchment area's unit cost, the smaller the area yielding the lowest TC.

The availability of a suitable catchment area does not change the value of FDoptimal, although it does reduce the total system cost.

A 95% service efficiency in animal watering was adequate in Feira de Santana, where 100% was achieved in 11 of the 20 years and more than 90% in 17 of them.

The year by year performance of the system, measured by service efficiency, is influenced by the RAI and the PCD.

An increase in demand of more than 1,600% causes an increase of less than 25% in the value of FDoptimal. Use of the average FDoptimal for demand range tested in the RWHS design in Feira de Santana yields a maximum difference of 3% in total system cost compared to that calculated with the actual FDoptimal.

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