The performance of time series models is assessed using synthetic head series simulated with a numerical model that solves Richards' equation for variably saturated flow. Heads were simulated in a homogeneous unconfined aquifer between two parallel canals; measured daily precipitation and potential evaporation are specified at the land surface and root water uptake is simulated. The head response to a precipitation event is nonlinear and depends on the saturation degree and rainfall before and after the precipitation event while evaporation reduction occurs during summers. Synthetic series were generated for 27 years and three different soil types; the unsaturated zone thickness varies between 0 and >5 m. The synthetic head series were simulated with a linear and nonlinear time series model. Performance of a linear time series model with four parameters, using a scaled Gamma response, gave R² values ranging from 0.67 to 0.96. The nonlinear time series model with five parameters simulates recharge using a root zone reservoir after which the head response to recharge is simulated with a scaled Gamma response function. The nonlinear time series model was able to simulate all synthetic head series very well with R² values above 0.9 for almost all models. The head response of the nonlinear model to a single precipitation event compares well to the response of the variably saturated groundwater model. The provided scripts may be used to simulate synthetic head series for other climates or for systems with additional complexity to assess the performance of other data-driven models.

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Time series analysis is a data-driven approach to analyze time series of heads measured in an observation well. Time series models are commonly much simpler and give much better fits than regular groundwater models. Time series analysis with response functions gives insight into why heads vary, while such insight is difficult to gain with black box models out of the artificial intelligence world. An important application is to quantify the contributions to the head variation of different stresses on the aquifer, such as rainfall and evaporation, pumping, and surface water levels. Time series analysis may be applied to answer many groundwater questions without the need for a regular groundwater model, such as what is the drawdown caused by a pumping station? Or, how long will it take before groundwater levels recover after a period of drought? Even when a regular groundwater model is needed to solve a groundwater problem, time series analysis can be of great value. It can be used to clean up the data, identify the major stresses on the aquifer, determine the most important processes that affect flow in the aquifer, and give an indication of the fit that can be expected. In addition, it can be used to determine calibration targets for steady-state models, and it can provide several alternative calibration methods for transient models. In summary, the overarching message of this paper is that it would be wise to do time series analysis for any application that uses measured groundwater heads.@en

Time series analysis is an increasingly popular method to analyze heads measured in an observation well. Common applications include the quantification of the effect of different stresses (rainfall, pumping, etc.), and the detection of trends and outliers. Pastas is a new and open source Python package for the analysis of hydrogeological time series. The objective of Pastas is twofold: to provide a scientific framework to develop and test new methods, and to provide a reliable ready-to-use software tool for groundwater practitioners. Transfer function noise modeling is applied using predefined response functions. For example, the head response to rainfall is simulated through the convolution of measured rainfall with a Gamma response function. Pastas models are created and analyzed through scripts, ensuring reproducibility and providing a transparent report of the entire modeling process. A Pastas model can be constructed in seven simple steps: import Pastas, read the time series, create a model, specify the stresses and the types of response functions, estimate the model parameters, visualize output, and analyze the results. These seven steps, including the corresponding Python code, are applied to investigate how rainfall and reference evaporation can explain measured heads in an observation well in Kingstown, Rhode Island, USA. The second example demonstrates the use of scripts to analyze a large number of observation wells in batch to estimate the extent of the drawdown caused by a well field in the Netherlands. Pastas is free and open source software available under the MIT-license at http://github.com/pastas/pastas.

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An approach is presented to determine groundwater flow in unconsolidated aquifers with a heat pulse response test using a heating cable and a fiber-optic cable. The cables are installed together using direct push so that the cables are in direct contact with the aquifer. The temperature response is measured for multiple days along the fiber-optic cable with Distributed Temperature Sensing (DTS). The new approach fits a two-dimensional analytical solution to the temperature measurements, so that the specific discharge can be estimated without knowledge of the position of the fiber-optic cable relative to the heating cable. Two case studies are presented. The first case study is at a managed aquifer recharge system where fiber-optic cables are inserted 15 m deep at various locations to test the fitting procedure. Similar and relatively large specific discharges are found at the different locations with little vertical variation (0.4–0.6 m/day). The second case study is at a polder, where the water level is maintained 2 m below the surrounding lakes, resulting in significant groundwater flow. The heating and fiber-optic cables are inserted to a depth of 45 m. The specific discharge varies 0.07–0.1 m/day and is significantly larger in a thin layer at 30-m depth. It is shown with numerical experiments that the estimated specific discharge is smoother than in reality due to vertical conduction, but the peak specific discharge is estimated correctly for layers thicker than ∼1.5 m.

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An approach is presented to determine the seasonal variations in travel time in a bank filtration system using a passive heat tracer test. The temperature in the aquifer varies seasonally because of temperature variations of the infiltrating surface water and at the soil surface. Temperature was measured with distributed temperature sensing along fiber optic cables that were inserted vertically into the aquifer with direct push equipment. The approach was applied to a bank filtration system consisting of a sequence of alternating, elongated recharge basins and rows of recovery wells. A SEAWAT model was developed to simulate coupled flow and heat transport. The model of a two-dimensional vertical cross section is able to simulate the temperature of the water at the well and the measured vertical temperature profiles reasonably well. MODPATH was used to compute flowpaths and the travel time distribution. At the study site, temporal variation of the pumping discharge was the dominant factor influencing the travel time distribution. For an equivalent system with a constant pumping rate, variations in the travel time distribution are caused by variations in the temperature-dependent viscosity. As a result, travel times increase in the winter, when a larger fraction of the water travels through the warmer, lower part of the aquifer, and decrease in the summer, when the upper part of the aquifer is warmer.

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An approach is presented to determine the specific discharge in an unconsolidated aquifer with a heat pulse response experiment using a heating cable and a fiber optic cable. The cables are installed using direct-push so that the cables are in direct contact with the aquifer and the disturbance of the aquifer due to installation is small. The temperature increase and decrease are measured for multiple days along the fiber optic cable with a Distributed Temperature Sensing system (DTS). A two-dimensional analytical solution is fitted to each of the measurement points along the fiber optic cable and the specific discharge is estimated, resulting in a profile of the specific discharge over depth. With the suggested approach it is not needed to specify the distance between the heating cable and the fiber optic cable. Two case studies are presented. The first case study contains a setup with one heating cable and multiple fiber optic cables inserted at various distances. Similar specific discharges are measured independent of the distance between the heating cable and the fiber optic cable, thereby showing that the approach gives consistent results. The second case study focusses on the ability to measure vertical variation in the specific discharge and the effect of neglecting vertical conduction. A heating cable and a fiber optic cable are installed as a pair up to a depth of 47 m. A thin layer was found at 30 m depth where groundwater moves 2.5 times faster. Variations in specific discharge of such magnitude result in vertical heat exchange. Numerical simulations are used to quantify the effect of vertical heat exchange on the estimated specific discharge. @en

Heads vary as a result of a variety of stresses on the aquifer, including rainfall, evaporation, pumping, and variations in surface water levels. One of the common objectives of groundwater models is to untangle measured groundwater head variations into contributions of these different stresses, for example to determine if lower heads are caused by a reduction in rainfall, an increase in pumping, or the construction of a new drainage system. Time series analysis with response functions is a relatively new groundwater modeling technique to model groundwater dynamics at observation wells. The method is fully data-driven. It requires measured time series of the head in an observation well and of the stresses on the aquifer. No other knowledge, including knowledge of aquifer parameters, is required. Two of the main difficulties in application of this method are the choice of the response function and the approach to simulate the movement of recharge through the unsaturated zone. Response functions are often based on analytic solutions to simple groundwater flow problems. Movement of recharge through the unsaturated zone is modeled numerically with a reservoir-type model. In this presentation, examples will be given of untangling head responses using the newly developed Pastas software. Pastas is free and open-source software, and is available from https://github.com/pastas/pastas. @en

An analytic approach is presented for the simulation of variations in the groundwater level due to temporal variations of recharge in surficial aquifers. Such variations, called groundwater dynamics, are computed through convolution of the response function due to an impulse of recharge with a measured time series of recharge. It is proposed to approximate the impulse response function with an exponential function of time which has two parameters that are functions of space only. These parameters are computed by setting the zeroth and first temporal moments of the approximate impulse response function equal to the corresponding moments of the true impulse response function. The zeroth and first moments are modeled with the analytic element method. The zeroth moment may be modeled with existing analytic elements, while new analytic elements are derived for the modeling of the first moment. Moment matching may be applied in the same fashion with other approximate impulse response functions. It is shown that the proposed approach gives accurate results for a circular island through comparison with an exact solution; both a step recharge function and a measured series of 10 years of recharge were used. The presented approach is specifically useful for modeling groundwater dynamics in aquifers with shallow groundwater tables as is demonstrated in a practical application. The analytic element method is a gridless method that allows for the precise placement of ditches and streams that regulate groundwater levels in such aquifers; heads may be computed analytically at any point and at any time. The presented approach may be extended to simulate the effect of other transient stresses (such as fluctuating surface water levels or pumping rates), and to simulate transient effects in multi-aquifer systems.

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