1 Introduction

Irrigation in general involves taking water from natural or artificial sources and supplying it to command areas where crops are grown. Different scales and types of irrigation schemes are being practiced throughout the world. In some regions, small reservoirs or tanks are built to provide common artificial water sources for rural communities. Tank irrigation systems in Monsoon Asia have a long history, and discussions on better management of them are still ongoing. Arumugam and Mohan (1997) proposed a supply-side decision support system for the release strategy from Veeranam Lake, which is a large historic irrigation tank in South India with a capacity of 26 million m3. Unami et al. (2005) investigated the optimal water management strategy for a system of three tanks irrigating paddy fields of 40 ha in Japan. Srivastava et al. (2009) examined a rainwater recycling system consisting of six tanks and five open wells for the full irrigation of the command area of 23 ha in Eastern India. Guerra et al. (1990) analyzed the hydrological processes in terraced rice fields with farm reservoirs extending in central Luzon, Philippines. A farm reservoir of their study with a capacity of 2,000 m3 can support dry season rice irrigation for an area of 0.4 ha. Panigrahi et al. (2007) carried out experimental studies on a rainfed rice-mustard cropping system consisting of a small on-farm reservoir with a capacity of 61 m3 and a farm area of 800 m2. It was revealed that such a system is an economically feasible option for small-scale supplemental irrigation. Smallholder irrigation schemes are also being developed in semi-arid regions of Sub-Saharan Africa (SSA), where erratic rainfall and high evaporation are serious constraints on agricultural production. Micro-dams, dugouts, sub-surface runoff harvesting tanks, and rooftop rainwater harvesting systems are all different rainwater harvesting practices found in the Makanya catchment of rural Tanzania (Pachpute et al. 2009). Here, the smallholder farmers have locally established robust water allocation arrangements among the micro-dams interconnected with furrows (Mul et al. 2011). Makurira et al. (2007) analyzed the water balance of the Manoo micro-dam system, which is part of the Makanya catchment. The storage capacity of the micro-dams in the Makanya catchment ranges from 200 to 1,600 m3. Ngigi et al. (2005 found that small on-farm reservoirs with capacities of 30–100 m3 are adequate for supplemental irrigation to produce cabbage in the Laikipia district of Kenya, when they are combined with plot-scale low-head drip irrigation systems, which were introduced in the late 1990s. However, Kulecho and Weatherhead (2005) observed that the drip irrigation system was often discontinued in different parts of eastern Kenya due to lack of maintenance, irrelevant cultural background, and unreliable water supply. The Guinea savanna agro-ecological zone of West Africa is drawing research attention recently because of the extreme contrast between rainy and dry seasons (ILRI, 1993). Thousands of micro-dams are distributed over the upper and central parts of the Volta Basin (Liebe et al. 2005; Liebe et al. 2009; Leemhuis et al. 2009). The command areas of Tono and Dorongo irrigation schemes in the Upper East Region of Ghana are 2,500 and 10 ha, respectively. Mdemu et al. (2009) noticed that the smaller Dorongo scheme, which alone experienced drought, had better water productivity than that in the larger Tono scheme. Faulkner et al. (2008) made comparison between another set of two small reservoir irrigation schemes, namely, Tanga and Weega schemes in the Upper East Region of Ghana, where water demand in the irrigated crop fields was estimated to be in the range of 3.7–5.9 mm/day on weekly basis. In both schemes, the reservoirs’ capacities are almost the same with command areas of 1.6 and 6.0 ha in the Tanga and the Weega schemes, respectively. This variance resulted in different irrigation water depths actually supplied during the experimental period, which were in the ranges of 22.0–37.9 and 8.9–14.4 mm/day in the Tanga and the Weega schemes, respectively. The above two comparative studies of the irrigation schemes in Ghana attributed the reason for better performance of the schemes with scarce water availability to efficient water management, which might reduce some of the water losses as Carter et al. (1999) considered.

From the above-mentioned literature review, it can be inferred that more research effort should be directed toward seeking the maximum performance of the minimum sized irrigation schemes, though the operation of micro-dams is not yet clearly understood in conjunction with small-scale subsistence farming managed at the community level in semi-arid SSA. In this paper, a comprehensive description of an optimal water management strategy for a micro-dam existing in the Northern Region of Ghana, having a capacity of 14,748 m3 and fully irrigating a potential command area of 840 m2, is outlined. The large dam capacity relative to the small command area is in fact quite reasonable under the assumption that no inflow to the reservoir can be expected during the whole period of dry season irrigation from mid-November to mid-March, which may be 120 days. The water level of the reservoir monotonically decreases in the dry season, while it is multiply utilized for domestic use, livestock watering, and bricks molding, as commonly practiced in SSA (Senzanje et al. 2008). Importance of the micro-dam as an aquatic habitat is secondary (Unami et al. 2012). Therefore, water management should primarily aim neither to empty the reservoir nor dry up the command area during the irrigation period, regardless of economic performance. A minimum stochastic model is developed to represent the physical processes in the micro-dam irrigation scheme to be optimally managed. The storage volume of the reservoir, the volume of readily available water contained in the soil of the command area, and the intake flow rate from the reservoir for irrigation to the command area are all considered as stochastic processes. A rigorous stochastic control approach that solves the Hamilton–Jacobi–Bellman (HJB) equation is employed. This method has an advantage over the conventional scenario-based optimization with a simulation method that do not take all the events that may occur into account (Srivastava 1996) or with an inexact programming method of predetermining the bounds of uncertainties (Lu et al. 2009).

The analytical or numerical solutions of the HJB equations are being studied mostly in the field of financial engineering where economic indices such as discounted utilities are to be maximized (Morimoto and Kawaguchi 2002; Chaumont et al. 2006; Baten and Miah 2007; Yiu et al. 2010). However, the concept presented here does not explicitly involve the economic aspect of the micro-dam irrigation schemes, but the determination of the optimal water management strategy is mathematically reduced to a stochastic control problem maximizing the expectation of the first exit time from a spatio-temporal domain where the stochastic processes should stay within. A computational procedure based on the finite element method is implemented to give a comprehensive description of the optimal water management strategy as well as the values of expected first exit time.

In the next Sect. 2, the physical details of the micro-dam irrigation scheme as well as the surrounding environment are described. In Sect. 3, the stochastic model is developed to formulate the stochastic control problem. In Sect. 4, being the main objective of this paper, it is demonstrated that the stochastic control problem can be numerically solved to determine the optimal intake flow rate based on the knowledge of current states of the storage volume of the reservoir and the soil moisture in the command area. Section 5 presents the conclusions.

2 Description of the micro-dam irrigation scheme

Unami et al. (2009) have monitored the hydrology of a Ghanaian inland valley containing six micro-dams (Dam 0 through Dam 5), which collect ephemeral surface flow after torrential rains and seepage flow of subsurface water from the ground during the rainy season ahead of the dry season. Figure 1 is a map showing the positions of the six micro-dams in relief of the inland valley depicted using the SRTM digital elevation data (Farr et al. 2007). Groundwater resources in the inland valley are not reliable, since shallow hand dug wells (HDWs) often dry up and deeper tube wells produce brackish water (Unami et al., 2012). Out of the six micro-dams, the most upstream one (Dam 5), located at coordinates 09°28′15″N 001°03′21″W and having a catchment area of 42 ha, is chosen as the micro-dam irrigating the potential command area during the dry season. In the catchment, seepage flows in the hydromorphic farmlands contribute to runoff significantly (Unami and Alam, in press). Figure 2 shows a satellite image of the micro-dam irrigation scheme, whose frame is delineated in Fig. 1. There is no perennial natural water source in the nearby area, but Dam 5 provides two adjacent communities with year-round water source. Dam 5 is equipped with a concrete spillway of 14.7 m wide and underground pipes connected to HDWs, which are installed out of the frame of Fig. 2. Some shallow tube wells (STWs) have also been constructed in the vicinity of Dam 5. Locations of the two STWs are shown in Fig. 2. Water from the HDWs and the STWs are primarily used for domestic purposes in the communities. The site of Dam 5 is fenced with barbed wire to prevent grazing animals from entering, but this arrangement is not working perfectly due to poor maintenance.

Fig. 1
figure 1

Six micro-dams in relief of the inland valley and location of the irrigation scheme

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Fig. 2
figure 2

Satellite image of the micro-dam irrigation scheme

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Figure 3 shows the rainfall intensity and water level of Dam 5 observed every 15 min during the period from September 1, 2007 through to March 24, 2011. A drop of the water level occurring every dry season is linear irrespective of the stages, suggesting that evaporation from the water surface is the dominant source of water losses in Dam 5. Filling up time of the reservoir during the rainy seasons significantly varies from year to year. The rising trend of the full water level over the years is due to siltation occurring at the upstream side of the reservoir, where the concrete spillway is installed. Here, the full supply water level (FSL) is set as EL168.0m, which is the crest level of the concrete spillway, while the dead-storage level (DSL) is assumed to be EL166.4m. As marked on the time axis of Fig. 3, four dry periods (DP07/08 through DP10/11) during which the water level of Dam 5 monotonically decreases are extracted from the whole observation period. The arithmetic mean (AM) and the standard deviation (SD) of the water level decrease in 24 h are calculated for each dry period and summarized in Table 1 together with its starting date, ending date, and duration.

Fig. 3
figure 3

Observed time series data of rainfall intensity and water level of Dam 5

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Table 1 Estimation of water loss during dry periods in Dam 5
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Using information obtained from water level data, satellite images and field surveys, the water surface area A of Dam 5 is related to the water depth h down to the DSL as

$$ A = A(h) = - 926.07h^{2} + 12510h $$
(1)

implying that

$$ x = x(h) = - 308.69h^{3} + 6255.0h^{2} $$
(2)

where x is the storage volume. The best fit curve for the relation between x and A is

$$ A = A(x) = A_{\max } \left( \frac{x}{V} \right)^{0.47} $$
(3)

where A max = A(h max), V = x(h max), and h max are the water surface area, the storage volume, and the water depth, respectively at FSL. These values are calculated as A max = 17,645 m2, V = 14,748 m3, and h max = 1.6 m, respectively.

There is no evidence that water from Dam 5 has been used for dry season irrigation, which is also not common in the area (Yiridoe et al. 2006). One of the motivations for developing the micro-dam irrigation scheme with Dam 5 is that farmers from a nearby community carried out small-scale gardening using the water from another of the six micro-dams, particularly, Dam 1 located at coordinates 09°28′36″N 001°00′50″W, during the dry season of early 2010. An exceptional torrential rain of 110 mm for one night on February 12th recovered the water levels of the reservoirs and might trigger off the farmers’ dry season small-scale gardening. Dam 1 is located 4.8 km downstream from Dam 5 and is rather a simple dugout intended for livestock watering, equipped with an intake pipe to draw water from the reservoir to a cattle trough. The catchment area of Dam 1 is 1,226 ha, including those of Dam 2 through Dam 5. At the time of the small-scale gardening, water was diverted from the cattle trough to basins with beds for the cultivation of Hibiscus calyphyllus, which provides vegetable leaves for human consumption as well as fiber materials. The dimensions of Dam 1 are as large as h max = 1.8 m, A max = 91,733 m2, and V = 79,320 m3, while the total area of the command is 350 m2.

The area immediately downstream side of Dam 5, which is triangular in shape as shown in Fig. 2, is proposed as the potential command area of the micro-dam irrigation scheme. One side of the triangular area is the dam embankment and the other two are planted with trees, so as to control entry of animals. The total area is a = 840 m2, and the land surface slope is less than 1% around an elevation of EL166.35m. Located in the hydromorphic valley bottom, the soil of the area is moderately permeable Dystric Planosols, which is normally used for rice cultivation (CERSGIS 2005). However, rainfed farming system for upland crops such as maize, yam, and groundnut is currently practiced in the area, since Dam 5 bypasses surface water flows which originally covered the valley bottoms during the rainy seasons.

Top soil in the depth between 0.0 and 0.2 m is considered as the target layer for irrigation. In this layer of D = 0.2 m thickness, the soil texture is classified as loam (sand 50%, silt 40%, clay 10%). The soil moisture and matric head data were simultaneously collected at a site close to the command area as shown in Fig. 4. Then, the wilting point θ w , the field capacity θ c , and the saturation θ s in terms of volumetric water content are estimated as 0.05, 0.30 at most, and 0.52, respectively. It is assumed that θ max, the upper limit of tolerable volumetric water content, is equal to 0.35, being a value between θ c and θ s . Based on the estimation and the observation by Faulkner et al. (2008), a reasonably large value of 20 mm/day is assumed for E c , the average water consumption rate. However, there are different small-holder farmers’ practices to conserve soil moisture in the semi-arid SSA (Chiroma et al. 2006), making it difficult to determine the actual water consumption rate, which may include evapotranspiration varying with different growth stages, deep percolation, conveyance losses, and so on. The volumetric water content θ that can be observed may differ from the actual one. The model proposed here comprehensively represents these uncertainties in terms of the stochastic processes, where volatility is the sole model parameter prescribing the magnitude of fluctuations.

Fig. 4
figure 4

Soil moisture and matric head data collected at the site of a hydromorphic valley bottom

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Dam 5 is not equipped with water intake facilities available for the command area. Using solar photovoltaic (PV) powered electric pumps is a feasible option for lifting water from the reservoir, since solar PV electrification is developing in the rural areas of Ghana (Obeng and Evers 2010). A PV pumping system unit consisting of a submerged electric direct current (DC) pump and a solar panel without a rechargeable battery is commercially available, and it can operate during daylight hours from 09:00 to 15:00 with the expected solar radiation at the site. Every PV pumping system unit is assumed to be equipped with a 12 mm diameter hose with a Manning’s roughness coefficient of 0.01, crossing over the dam embankment, in order to convey the irrigation water from the DSL of the reservoir to the command area. Then, the operating flow rate Q P (L/min) of a single PV pumping system unit is related to the storage volume x as

$$ Q_{P} = Q_{P} (x) = 8.680 + 1.243\left( \frac{x}{V} \right)^{0.53} . $$
(4)

3 Stochastic model and optimal control

The dynamics of the micro-dam irrigation scheme is modeled as a system of two stochastic differential equations

$$ \left\{ {\begin{array}{*{20}l} {{\text{d}}X_{t} = \left( {Q(X_{t} ) - u} \right){\text{d}}t} \\ {{\text{d}}Y_{t} = \left( {u - \bar{q}(Y_{t} )} \right){\text{d}}t + \sigma (Y_{t} ){\text{d}}B} \\ \end{array} } \right. $$
(5)

where X t is the storage volume of the reservoir at the time t, Y t is the volume of readily available water contained in the soil of the command area at the time t, Q is the uncontrolled water balance flowing into the reservoir, u is the intake flow rate from the reservoir for irrigation in the command area, \( \bar{q} \) is the average volumetric water consumption rate in the command area, σ is the volatility of soil moisture in the command area, and B is the one-dimensional Brownian motion. The presence of the stochastic term σ(Y t )dB involves including all uncertainties. The farmers that manage the micro-dam irrigation scheme are assumed to know the current states of X t and Y t so as to give feed-back to u. A temporal domain (0, T) is considered as the maximum irrigation period, whose duration is equal to T. The worst case considered is the drying up of either the reservoir or the command area during the course of the irrigation period, bringing the irrigation scheme to ruin. Reaching θ max of the volumetric water content is not tolerated. Exceeding h max of the water depth of the reservoir is unfavorable, though it does not happen almost surely. Thus, a spatial x–y domain of the stochastic processes (X t , Y t ) is prescribed as

$$ \Upomega_{x} \times \Upomega_{y} = \left( {0,\;V} \right) \times \left( {y_{\min } ,y_{\max } } \right) $$
(6)

where y min = aDθ w , and y max = aDθ max. Then, the stochastic control problem is mathematically formulated as maximization of the cost function

$$ J^{u} \left( {s,x,y} \right) = E^{s,x,y} \left[ {g\left( {\hat{T},X_{{\hat{T}}} ,Y_{{\hat{T}}} } \right)} \right] $$
(7)

where \( \hat{T} \) is the first exit time of (X t , Y t ) from the spatio-temporal domain (0, T) × Ω x  × Ω y , and g is the function evaluating the severity of the ruin. Setting

$$ g\left( {\hat{T},X_{{\hat{T}}} ,Y_{{\hat{T}}} } \right) = g\left( {\hat{T}} \right) = - \frac{{T - \hat{T}}}{T} $$
(8)

is appropriate in the sense that it is continuous on the boundary of (0, T) × Ω x  × Ω y , monotone increasing with respect to \( \hat{T} \), and vanishing at \( \hat{T} = T \). The intake flow rate u is constrained in the set U of admissible controls, which is the closed interval [u min, u max]. An optimal control u * is u = u(s, x, y) that achieves the maximum of the cost function J u(s, x, y), satisfying u min ≤ u ≤ u max. The expectation of the first exit time \( \hat{T}^{*} \) under the optimal control u * is inversely given by

$$ E^{s,x,y} \left[ {\hat{T}^{*} } \right] = T\left( {1 - J^{{u^{*} }} \left( {s,x,y} \right)} \right). $$
(9)

According to Øksendal (2007), the maximum Φ of J u(s, x, y) and the optimal control u * are governed by the HJB equation

$$ \frac{\partial \Upphi }{\partial s} + \left( {Q - u^{*} } \right)\frac{\partial \Upphi }{\partial x} + \left( {u^{*} - \bar{q}} \right)\frac{\partial \Upphi }{\partial y} + \frac{{\sigma^{2} }}{2}\frac{{\partial^{2} \Upphi }}{{\partial y^{2} }} = 0 $$
(10)

with the boundary condition

$$ \Upphi (s,x,y) = g(s,x,y). $$
(11)

The HJB equation (Eq. 10) is rewritten as

$$ \frac{\partial \Upphi }{\partial s} + Q(x)\frac{\partial \Upphi }{\partial x} - \bar{q}\frac{\partial \Upphi }{\partial y} + \frac{{\sigma^{2} }}{2}\frac{{\partial^{2} \Upphi }}{{\partial y^{2} }} - zu^{*} = 0 $$
(12)

where z is the multiplier

$$ z = \frac{\partial \Upphi }{\partial x} - \frac{\partial \Upphi }{\partial y} $$
(13)

from which the optimal control u * is determined as

$$ u^{*} = \left\{ {\begin{array}{*{20}c} {u_{\max } } & {\left( {z < 0} \right)} \\ {\text{any}} & {\left( {z = 0} \right)} \\ {u_{\min } } & {\left( {z > 0} \right)} \\ \end{array} } \right. $$
(14)

so as to maximize the left hand side of (Eq. 10).

4 Computational method and application

As a consequence of the discussions in the preceding sections, the model parameters are determined as summarized in Table 2, where y s is the volume of water contained in the soil of the command area when saturated. However, the volatility σ is assumed to be a positive convex function in Ω y with a constant parameter C v in the form

Table 2 Key parameters used for the optimal control of the micro-dam irrigation scheme
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$$ \sigma = \sigma (y) = C_{v} \frac{{y_{s} - y}}{{y_{s} - y_{\min } }}y, $$
(15)

so that σ vanishes when the soil moisture is at the wilting point or when the soil is saturated. In designing a control system, specifying the value of C v implies how much fluctuation is presumed in the soil moisture.

The bounds of the set of admissible controls become

$$ u_{\min } = 0 $$
(16)

and

$$u_{{\max}} = u_{{\max}} (s,x) = \left\{ {\begin{array}{ll} {N_{P} Q_{P} (x)} & {\left( {9 < \text{mod} (24s,24) < 15} \right)} \\ 0 & {\left( {{\text{Otherwise}}} \right)} \\\end{array} } \right.$$
(17)

where N P is the number of the PV pumping system units. Here, N P is chosen as eight (8), so that averaged u max cover \( \bar{q} \) .

The finite element method is employed for numerically solving the partial differential equation Eq. 10 to obtain Φ and u * as functions of s, x, and y. The same upwind discretization scheme as in Unami et al. (2010) is applied to the weak form of Eq. 10, which is written as

$$ \int\limits_{{\Upomega_{y} }} {\left( {w \, \frac{\partial\Upphi }{\partial s} + w(Q - u^{*} )\frac{\partial \Upphi}{\partial x} + w\left( {u^{*} - \bar{q}} \right)\frac{\partial\Upphi }{\partial y} - \frac{1}{2}\frac{{\partial (w\sigma^{2})}}{\partial y}\frac{\partial \Upphi }{\partial y}}\right){\text{d}}y = 0} $$
(18)

for any weight w in H 10 (Ω y ), the space of functions having certain regularity properties in Ω y and vanishing at the boundary of Ω y . The spatial domain Ω x  × Ω y is discretized into rectangular sub-domains of equal size Δx by Δy, which are generically represented as (iΔx, (i + 1)Δx) × (kΔy, (k + 1)Δy) for integers i and k. The unknown Φ is attributed to each node (iΔx, kΔy) and is denoted by Φ i,k . Then, there are four estimates z ±± for the multiplier z in each subdomain;

$$ z_{ - - } = \frac{{\Upphi_{i + 1,k} - \Upphi_{i,k} }}{\Updelta x} - \frac{{\Upphi_{i,k + 1} - \Upphi_{i,k} }}{\Updelta y}, $$
(19)
$$ z_{ - + } = \frac{{\Upphi_{i + 1,k + 1} - \Upphi_{i,k + 1} }}{\Updelta x} - \frac{{\Upphi_{i,k + 1} - \Upphi_{i,k} }}{\Updelta y}, $$
(20)
$$ z_{ + - } = \frac{{\Upphi_{i + 1,k} - \Upphi_{i,k} }}{\Updelta x} - \frac{{\Upphi_{i + 1,k + 1} - \Upphi_{i + 1,k} }}{\Updelta y}, $$
(21)

and

$$ z_{ + + } = \frac{{\Upphi_{i + 1,k + 1} - \Upphi_{i,k + 1} }}{\Updelta x} - \frac{{\Upphi_{i + 1,k + 1} - \Upphi_{i + 1,k} }}{\Updelta y}. $$
(22)

As a computational rule applied to each sub-domain, the rigor of Eq. 14 is relaxed as

$$ u^{*} = \frac{\mu }{4}u_{\max } $$
(23)

where μ is the number of negative estimates z ±±. Practically, the number n P of operating pumps may be 0, 2, 4, 6, or 8. Finally, imposing the boundary condition Eq. 11 and employing a fully implicit scheme in the temporally backward direction with an increment Δs, a system of linear equations is composed for updating the values of Φ i,k and is solved with the Gauss–Seidel method. However, a predictor–corrector scheme is further applied so that the optimal control is determined not with the future values of Φ i,k but with the current ones.

Specifying three different values of C v as 0.20, 0.30, and 0.40, the computational procedure is implemented. The computational grid sizes are \( \Updelta x = \frac{V}{150} \), \( \Updelta y = \frac{{y_{\max } - y_{\min } }}{150} \), and \( \Updelta s = \frac{1}{1,440} \) day (1 min). The results at noon are shown in Figs. 5, 6 and 7 in terms of n P under the optimal water management strategy and \( E^{s,x,y} \left[ {\hat{T}^{*} } \right] \), with halving temporal intervals (68, 34, and 17 days) so that both the first and the last days are included. The diagrams of n P on the x–y plane outline the irrigation principle, which is based on supply-side strategy when \( \left| {\frac{{\partial n_{P} }}{\partial x}} \right| \gg \left| {\frac{{\partial n_{P} }}{\partial y}} \right| \) and demand-side strategy when \( \left| {\frac{{\partial n_{P} }}{\partial x}} \right| \ll \left| {\frac{{\partial n_{P} }}{\partial y}} \right| \). Stopping all the pumps when the reservoir is near empty is an obvious supply-side strategy, as is computationally reproduced, but otherwise demand-side strategy is predominant in all cases of C v -values. Switching θ, which is defined as the maximum volumetric water content where all the pumps should operate, varies with s and x when C v  = 0.20 but is almost constant at 20–25% when C v  = 0.30 and 0.40. Gray zones with n P  = 2, 4, or 6 appear where the expected first exit time is almost constant in the x–y plane, as Eq. 13 indicates. Minor numerical oscillations in appearance of the gray zones are seen when the reservoir is near full.

Fig. 5
figure 5

Number of operating pumps (left column) and expected first exit time (right column) for s = 12:00 h of the 1st, the 69th, the 103th, and the 120th days with C v  = 0.20

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Fig. 6
figure 6

Number of operating pumps (left column) and expected first exit time (right column) for s = 12:00 h of the 1st, the 69th, the 103th, and the 120th days with C v  = 0.30

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Fig. 7
figure 7

Number of operating pumps (left column) and expected first exit time (right column) for s = 12:00 h of the 1st, the 69th, the 103th, and the 120th days with C v  = 0.40

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The numerical solutions of Φ are totally oscillation free. The contour graphs of \( E^{s,x,y} \left[ {\hat{T}^{*} } \right] \) on the x–y plane quantitatively represent the performance of the optimal water management strategy. The minimum of \( E^{s,x,y} \left[ {\hat{T}^{*} } \right] \) is achieved on the boundary of Ω x  × Ω y for every s. The sub-domain of Ω x  × Ω y where the value of \( E^{s,x,y} \left[ {\hat{T}^{*} } \right] \) is large appears near the boundary x = V with moderate y, and then it expands to the whole Ω x  × Ω y as the time s approaches to T. A critical time s c , which is defined as the minimum s where there exists (x, y) such that the expected first exit time \( E^{s,x,y} \left[ {\hat{T}^{*} } \right] \) is greater than 119 days, is on the 38th, 39th, and 96th day for C v  = 0.20, 0.30, and 0.40, respectively. In other words, the micro-dam irrigation scheme can be expected to last for about 80 days if C v  = 0.20 or 0.30, provided that the reservoir is full and the soil moisture of the command area is moderate at the initial stage. While, the micro-dam irrigation scheme becomes short of water within 24 days if the value of C v is larger as 0.40.

For the sake of comparison, another computational run for C v  = 0.20, the least severe case, is implemented with u * = u max, implying that the pumps are always operated at full capacity. The results are shown in Fig. 8. The critical time s c is on the 116th day. Unlike the controlled cases, the value of \( E^{s,x,y} \left[ {\hat{T}^{*} } \right] \) is almost uniform in the whole Ω x  × Ω y at every stage of time.

Fig. 8
figure 8

Number of operating pumps (left column) and expected first exit time (right column) for s = 12:00 h of the 1st, the 69th, the 103th, and the 120th days with C v  = 0.20, not using the optimal control

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The computational procedure is implemented for different values of the command area a as well, though detailed results are not presented here. When a = 420 m2, half of the original value, the critical time s c is on the 33th, 34th, and 90th day for C v  = 0.20, 0.30, and 0.40, respectively. When a = 210 m2, one quarter of the original value, the critical time s c is on the 31th, 32th, and 90th day for C v  = 0.20, 0.30, and 0.40, respectively. Qualitative behavior of the numerical solutions of u * and Φ for these values of a is almost the same as that for the original a = 840 m2, except that switching θ shifts to the dry side and the gray zones increase as the value of a becomes small. In practice, soil moisture should be controlled on the dry side if the command area is small, but it has the least effect of prolonging the irrigation period. When a = 1,680 m2, the double of the original value, the critical time s c is on the 110th, 112th, and 115th day for C v  = 0.20, 0.30, and 0.40, respectively. In that case there is substantially no possibility of successful dry season farming in the enlarged micro-dam irrigation scheme.

The results obtained above indicate that the rigorous stochastic control approach with the minimum model leads to simple optimal management strategies, which is consistent with common conventional practices of small-scale subsistence farming. This methodology is advantageous over the conventional scenario-based optimization by not requiring many ambiguous model parameters as well as offering feasible solutions at the community level.

5 Conclusions

A stochastic control approach is applied to the optimal water management of an irrigation scheme consisting of a reservoir, a command area, and water intake facilities. Only the first exit time is considered as the performance index, targeting at small-scale subsistence farming where profit is not a primary concern. The computational procedure that solves the HJB equation for the stochastic processes provides comprehensive information on the optimal control. Demonstrations are given for the micro-dam irrigation scheme planned at a site within the Guinea savanna agro-ecological zone of West Africa. The model parameters are estimated by different observed data, with specified values of volatility. No inflow is expected into the reservoir and the evapotranspiration is extremely high during the dry seasons, when the irrigation scheme is operational. This is a very severe condition observed here, which is not encountered in neither the Monsoon Asia nor in East Africa. Though the capacity of the micro-dam is seemingly large relative to the small command area, the computational results indicated that the irrigation scheme cannot last long enough for the cultivation of dry season crops when the volatility is too large. When the volatility is small enough, the initial water stored in the reservoir can support full irrigation for about 80 days under the optimal water management strategy, which is predominantly based on the demand-side principle. The feed-back rule is so simple that water should be taken from the reservoir when the command area requires water, and it is considered feasible at the community level. These facts explain the actual farmers’ practice of small-scale gardening in the smaller command area using the water from the larger reservoir. However, introduction of water intake facilities such as the PV pumping system unit will be a constraint to realizing the potential of the irrigation scheme.