Maximizing on-farm groundwater recharge with surface reservoir releases: a planning approach and case study in California, USA

Abstract

A hydro-economic approach for planning on-farm managed aquifer recharge is developed and demonstrated for two contiguous sub-basins in California’s Central Valley, USA. The amount and timing of water potentially available for recharge is based on a reoperation study for a nearby surface-water reservoir. Privately owned cropland is intermittently used for recharge with payments to landowners that compensate for perceived risks to crop health and productivity. Using all cropland in the study area would have recharged approximately 4.8 km³ (3,900 thousand acre-feet) over the 20-year analysis period. Limits to recharge effectiveness are expected from (1) temporal variability in recharge water availability, (2) variations in infiltration rate and few high-infiltration recharge sites in the study area, and (3) recharged water escaping from the study area groundwater system to surface water and adjacent sub-basins. Depending on crop tolerance to ponding depth, these limitations might be reduced by (1) raising berm heights on higher-infiltration-rate croplands and (2) creating dedicated recharge facilities over high-infiltration-rate sites.

Résumé

Une approche hydro-économique visant à planifier la recharge d’un aquifère gérée en plein champ a été conçue et appliquée au niveau de deux sous-bassins contigus dans la Vallée Centrale en Californie, Etats-Unis d’Amérique. La quantité et le moment où l’eau est potentiellement disponible pour la recharge sont définis à partir d’une étude de reconfiguration d’un réservoir d’eau de surface situé à proximité. Les terres cultivées sur des propriétés privées sont utilisées par intermittence pour la recharge, avec une indemnisation des propriétaires pour compenser les risques identifiés concernant l’état des cultures et leur productivité. En ayant recours à la totalité des terres cultivées de la zone d’étude permettrait une recharge d’environ 4.8 km³ (1,560 milliers d’hectares) au cours de la période d’analyse sur 20 ans. Des limites à l’efficacité de la recharge sont attendues (1) de la variabilité temporelle de la disponibilité en eau de recharge, (2) des variations du taux d’infiltration et du faible nombre de sites à forte recharge dans l’aire d’étude, et (3) des fuites de l’eau rechargée hors du système hydrogéologique vers les eaux de surface et les sous-bassins adjacents. En tenant compte de la tolérance des cultures à la hauteur d’inondation, ces limitations peuvent être réduites (1) en augmentant la hauteur des digues dans les terres cultivées à fort taux d’infiltration et (2) en créant des infrastructures de recharge dédiées pour les sites à fort taux d’infiltration.

Resumen

Se desarrolló y demostró un enfoque hidroeconómico para planificar la recarga de acuíferos administrados en fincas de dos subcuencas contiguas en el Valle Central de California, EEUU. La cantidad y el tiempo del agua potencialmente disponible para la recarga se basan en un estudio de reoperación para un embalse de agua superficial cercano. Las tierras de cultivo de propiedad privada se utilizan de forma intermitente para la recarga con pagos a los propietarios de la tierra que compensan los riesgos percibidos para la salud y la productividad de los cultivos. El uso de todas las tierras de cultivo en el área de estudio se habría recargado aproximadamente 4.8 km³ (3,900 mil acres-pie) durante el período de análisis de 20 años. Se esperan límites para la efectividad de la recarga a partir de (1) variabilidad temporal en la disponibilidad de agua de recarga, (2) variaciones en la tasa de infiltración y pocos sitios de recarga de alta infiltración en el área de estudio, y (3) agua recargada que se escapa del sistema de agua subterránea del área de estudio al agua superficial y subcuencas adyacentes. Dependiendo de la tolerancia del cultivo a la profundidad del embalse, estas limitaciones podrían reducirse en (1) elevando las alturas de los terraplenes en las tierras de cultivo con mayor índice de infiltración y (2) creando instalaciones de recarga dedicadas en los sitios con alto índice de infiltración.

摘要

在美国加州中央谷的两个相邻的子盆地,开发并论证了一种规划农田管理含水层补给的水文经济方法。可供补充的水量和时间是根据附近地表水水库的再运行研究确定的。私人拥有的农田被间歇性地用于向土地所有者支付补偿,以补偿作物健康和生产力可能面临的风险。在20年的分析期间,使用研究地区的所有耕地将补充约4.8 km3(390万英亩-英尺)。研究区补水有效性的限制包括:(1)补水有效性的时间变异性;(2)研究区入渗速率的变化和高入渗补水点较少;(3)从研究区地下水系统向地表水及邻近亚盆地补水。根据作物对积水深度的耐受性,这些限制可以通过以下措施来降低:(1)在入渗率较高的农田上提高护堤高度;(2)在入渗率较高的土地上建立专用的补给设施。

Resumo

Uma abordagem hidro-econômica para o planejamento da recarga de aquíferos gerenciados na fazenda é desenvolvida e demonstrada para duas subacias contíguas no Vale Central da Califórnia, EUA. A quantidade e o tempo da água potencialmente disponível para recarga é baseado em um estudo de reoperação para um reservatório de água de superfície próximo. As terras de propriedade privada são utilizadas de forma intermitente para recarregar com pagamentos a proprietários de terras que compensam os riscos percebidos para a saúde e a produtividade das culturas. O uso de todas as terras cultiváveis ​​na área de estudo teria recarregado aproximadamente 4.8 km³ (3,900 mil acres) ao longo do período de análise de 20 anos. Os limites para a efetividade da recarga são esperados de (1) variabilidade temporal na disponibilidade de água de recarga, (2) variações na taxa de infiltração e poucos locais de recarga de alta infiltração na área de estudo, e (3) água recarregada escapando do sistema de águas subterrâneas da área de estudo para águas superficiais e subacias adjacentes. Dependendo da tolerância da cultura à profundidade das represas, essas limitações podem ser reduzidas por (1) elevação das alturas de barragens em terras de alta taxa de infiltração e (2) criação de instalações de recarga dedicadas em locais de alta taxa de infiltração.

Appendix

Details regarding formulation of the linear programming model are presented in the following sections.

Ponded water drainage

The upper bound for recharge water applied is specified as Eqs. (4, 16 and 28) and based on a requirement that water ponded on the field not overtop an assumed perimeter berm of height HB. A series of steps are taken to develop an expression for the maximum allowable recharge volume.

An ordinary differential equation and initial condition for water mass balance in a recharge pond during filling is formulated and solved:

$$ A\ \mathrm{d}h/\mathrm{d}t=Q\hbox{--} A\ \left(I/{H}_0\right)\kern0.5em h $$

(33)

$$ h(0)=0 $$

(34)

$$ h=\left[\left(Q\ {H}_0\right)/\left(I\ A\right)\right]\ \left[1\hbox{--} {\mathrm{e}}^{-\left(\mathrm{I}/{H}_0\right)t}\right] $$

(35)

where:

A :

is the ponding area

h :

is the ponding depth

t :

is time

Q :

is the rate of inflow

I :

is the reference rate ponded water infiltrates the subsurface

H ₀ :

is ponding depth associated with I

The quantity (I/H₀) in Eq. (33) normalizes the infiltration rate by the ponding depth used to estimate the quantities summarized in Fig. 7 (Maples et al. 2017; S. Maples, Hydrologic Sciences Graduate Group, University of California Davis, unpublished manuscript, 2018) and allows scaling by h to simulate variation in infiltration rate with ponding depth. Evaporation is not considered in the pond mass balance because recharge operations are considered during the winter when evaporative losses are expected to be small.

Filling the pond at a constant rate until the maximum ponding depth is reached at a specified time is represented by substituting h = HB, Q = Q_max and t = T into Eq. (35). Rearrangement yields:

$$ {Q}_{\mathrm{max}}=\left(I\ A\ \mathrm{HB}/{H}_0\right)/\left[1\hbox{--} {\mathrm{e}}^{-\left(I\ T/{H}_0\right)}\right] $$

(36)

An ordinary differential equation for water mass balance in a recharge pond during draining is formulated and solved for the general condition:

$$ \mathrm{d}h/\mathrm{d}t=\hbox{--} \left(I/{H}_0\right)\ h $$

(37)

$$ h={\mathrm{e}}^{-\left(I/{H}_0\right)t+K} $$

(38)

Substituting t = t – T into Eq. (38) so that Eqs. (35) and (38) initiate at the same time and rearranging yields:

$$ h=K\ {\mathrm{e}}^{-\left(I/H0\right)\ \left(t\hbox{--} T\right)} $$

(39)

Equating Eqs. (35) and (39) at time t = T, solving for K and substituting into Eq. (39) yields:

$$ h=\left[\left(Q\ {H}_0\right)/\left(I\ A\right)\right]\ \left[1\hbox{--} {\mathrm{e}}^{-\left(I/{H}_0\right)T}\right]\ {\mathrm{e}}^{-\left(I/{H}_0\right)\ \left(t\hbox{--} T\right)} $$

(40)

Substituting Eq. (36) for Q and rearranging yields an expression for filling to time T and then draining thereafter:

$$ h=\mathrm{HB}\ {\mathrm{e}}^{-\left(I/{H}_0\right)\ \left(t\hbox{--} T\right)} $$

(41)

Assume that the pond must be filled and drained within 1 month to allow operational flexibility such that the land could be used for purposes other than recharge during the following month. A 1-month filling and draining cycle is represented by introducing a terminal boundary condition for Eq. (41): h(1) = ɛ HB, where ɛ is a small increment. Solving for T yields:

$$ T=1+\left({H}_0/I\right)\ \ln \left(\varepsilon \right) $$

(42)

Substituting Eq. (42) into Eq. (41) and the result into Eq. (36) yields an expression for Q_max:

$$ {Q}_{\mathrm{max}}=\left(\mathrm{HB} AI/{H}_0\right)/\left[1-{e}^{-\left(I/{H}_0+\ln \left\{\varepsilon \right\}\right)}\right] $$

(43)

Multiplying this expression for Q_max by the Eq. (42) for T results in an expression for the maximum recharge volume that can be added to a pond in a single month:

$$ {\mathrm{RV}}_{\mathrm{max}}=\left[\left(\mathrm{HB}\;A\right)\left(I/{H}_0+\ln \left\{\varepsilon \right\}\right)\right]/\left[1-{e}^{-\left(I/{H}_0+\ln \left\{\varepsilon \right\}\right)}\right] $$

(44)

Equation (44) is based on an infiltration rate derived for water ponded on the deeper geologic materials. Because a lower hydraulic conductivity soil overlays the geology, the expression is scaled by a factor that accounts for the effective vertical hydraulic conductivity of the layered porous medium:

$$ {K}_{\mathrm{scale}}={K}_{\mathrm{eff}}/{K}_{\mathrm{geol}} $$

(45)

$$ {K}_{\mathrm{eff}}=\left({b}_{\mathrm{soil}}+{b}_{\mathrm{geol}}\right)/\left[\left({b}_{\mathrm{soil}}/{K}_{\mathrm{soil}}\right)+\left({b}_{\mathrm{geol}}/{K}_{\mathrm{geol}}\right)\right] $$

(46)

where:

K _eff :

is the effective vertical hydraulic conductivity calculated as the harmonic mean of the conductivities of the soil and geologic layers

K _geol :

is the averaged vertical hydraulic conductivity of the deeper geologic materials (Maples et al. 2017; S. Maples, Hydrologic Sciences Graduate Group, University of California Davis, unpublished manuscript, 2018)

K _soil :

is the vertical hydraulic conductivity of the soil (taken as 3 × 10⁻² ft/day, or 10⁻⁵ cm/s, based on Brush et al. 2013)

b _geol :

is the thickness of the unsaturated zone in the geologic materials (Maples et al. 2017; S. Maples, Hydrologic Sciences Graduate Group, University of California Davis, unpublished manuscript, 2018)

b _soil :

is the thickness of the soil layer (taken as 1 ft or 0.3 m)

Applying the scaling factor to Eq. (44) yields the general expression used for Eq. (4).

$$ {\mathrm{RV}}_{\mathrm{max}}=\left[\left({K}_{\mathrm{scale}}\ \mathrm{HB}\ A\right)\left(I/{H}_0+\ln \left\{\varepsilon \right\}\right)\right]/\left[1-{\mathrm{e}}^{-\left(I/{H}_0+\ln \left\{\varepsilon \right\}\right)}\right] $$

(47)

Dividing Eq. (47) by A yields a general expression for the maximum recharge depth that can be added to a pond in a single month. This is used for Eqs. (16) and (28).

$$ {D}_{\mathrm{max}}=\left[\left({K}_{\mathrm{scale}}\ \mathrm{HB}\right)\left(I/{H}_0+\ln \left\{\varepsilon \right\}\right)\right]/\left[1-{\mathrm{e}}^{-\left(I/{H}_0+\ln \left\{\varepsilon \right\}\right)}\right] $$

(48)

The formulation is somewhat sensitive to the value chosen for ɛ with smaller values reducing the upper bound. Using a value of 0.01 appeared reasonable for this analysis. Finally, the assumed 1-month filling and drainage cycle could be adjusted by extending the approach described here to simulate pulsed flooding for crop root health (Dahlke et al. 2018).

Groundwater elevation calculation

The upper bound on groundwater elevation is specified as Eqs. (6), (18 and (30) based on ground surface elevation and an assumed required freeboard to avoid waterlogging of soil. This consideration can be important for down-flow parts of basin where recharge might not be applied but water levels may rise as a result of recharge water redistribution by means of groundwater flow (Niswonger et al. 2017). The groundwater elevation itself is based on a linearized representation of groundwater head response to addition of water to the system at a particular location and time (Reilly et al. 1987; Gorelick et al. 1993; Ahlfeld and Mulligan 2000). The representation is most accurate for confined systems but works well for unconfined conditions when the head change in response to the addition of water is small relative to the saturated thickness, as is the case for this work.

The groundwater simulation model used in this work (coarse-grid version of C2VSim; Brush et al. 2013) was manipulated to generate the background groundwater heads (H) as well as the mounding responses (M) for the control locations. The background heads were based on running the original model. Information for M was generated through a series of steps: (1) altering the model by stripping out all unmanaged hydrologic stresses, (2) making a suite of runs with the altered model separately simulating a managed stress for each potential recharge location using a unit recharge volume (RVu) in the first time step of the model, (3) running the altered model once with no managed stresses and (4) calculating the differences in heads at control locations between the runs from steps 2 and 3. The resulting information for M is a set of vectors containing transient mounding responses at each control location for each potential recharge location. The vectors are then arranged in tableaus as described by Gorelick et al. (1993) to create a matrix M for each control location.

The information developed for M is used as a groundwater elevation simulator that represents increases in elevation over time as a linear combination of responses to monthly recharge volumes. The responses (1) are produced by recharge events simulated for single time steps in any model element within the study area and any time step over the planning horizon, (2) scale with the magnitude of recharge volume and (3) can be summed to simulate combinations of recharge events over space and time.

Reformulation of Lagrange multiplier for berm height

A generalized form of constraint Eq. (28) is as follows:

$$ D\le \left[\left({K}_{\mathrm{scale}}\mathrm{HB}\right)\left(I/{H}_0+\ln \left\{\varepsilon \right\}\right)\right]/\left[1-{e}^{-\left(I/{H}_0+\ln \left\{\varepsilon \right\}\right)}\right] $$

(49)

When this constraint is binding in the linear programming solution, the Lagrange multiplier will be non-zero and indicate the change in the optimal value of the objective function for an increase of 1 in the right-hand side (RHS). If HB in Eq. (49) were increased by 1, the RHS would increase by [K_scale (I/H₀ + ln{ɛ})]/[1 – e^{–(It/H0 + ln{ɛ})}]. Multiplying the Lagrange multiplier value from Eq. (49) by this quantity converts the original linear programming result, Lagrange multiplier for Eq. (49), into a Lagrange multiplier for HB. Summing the converted Lagrange multipliers for each model element over all time steps in the planning horizon provides a location-specific estimate for total increase in recharge over the planning horizon for a unit increase in berm height.