Improving global-scale model representations of near-surface soil moisture
and groundwater hydrology is important for accurately simulating terrestrial
processes and predicting climate change effects on water resources. Most
existing land surface models, including the default E3SM Land Model (ELMv0),
which we modify here, routinely employ different formulations for water
transport in the vadose and phreatic zones. Clark et al. (2015) identified a
variably saturated Richards equation flow model as an important capability
for improving simulation of coupled soil moisture and shallow groundwater
dynamics. In this work, we developed the Variably Saturated Flow Model (VSFM)
in ELMv1 to unify the treatment of soil hydrologic processes in the
unsaturated and saturated zones. VSFM was tested on three benchmark problems
and results were evaluated against observations and an existing benchmark
model (PFLOTRAN). The ELMv1-VSFM's subsurface drainage parameter,

Groundwater, which accounts for 30 % of freshwater reserves globally, is a
vital human water resource. It is estimated that groundwater provides
20–30 % of global freshwater withdrawals (Petra, 2009; Zektser and
Evertt, 2004), and that irrigation accounts for

Local environmental conditions modulate the impact of rainfall changes on groundwater resources. For example, high-intensity precipitation in humid areas may lead to a decrease in groundwater recharge (due to higher surface runoff), while arid regions are expected to see gains in groundwater storage (as infiltrating water quickly travels deep into the ground before it can be lost to the atmosphere; Kundzewicz and Doli, 2009). Although global climate models predict changes in precipitation over the next century (Marvel et al., 2017), few global models that participated in the recent Coupled Model Intercomparison Project (CMIP5; Taylor et al., 2012) were able to represent global groundwater dynamics accurately (e.g., Swenson and Lawrence, 2014)

Modeling studies have also investigated impacts, at watershed to global
scales, on future groundwater resources associated with land-use (LU) and
land-cover (LC) change (Dams et al., 2008) and ground water
pumping (Ferguson and Maxwell, 2012; Leng et al., 2015).
Dams et al. (2008) predicted that LU changes would result
in a small mean decrease in subsurface recharge and large spatial and
temporal variability in groundwater depth for the Kleine Nete basin in
Belgium. Ferguson and Maxwell (2012) concluded that groundwater-fed
irrigation impacts on water exchanges with the atmosphere and groundwater
resources can be comparable to those from a 2.5

Groundwater models are critical for developing understanding of groundwater systems and predicting impacts of climate (Green et al., 2011). Kollet and Maxwell (2008) identified critical zones, i.e., regions within the watershed with water table depths between 1 and 5 m, where the influence of groundwater dynamics was largest on surface energy budgets. Numerical studies have demonstrated impacts of groundwater dynamics on several key Earth system processes, including soil moisture (Chen and Hu, 2004; Liang et al., 2003; Salvucci and Entekhabi, 1995; Yeh and Eltahir, 2005), runoff generation (Levine and Salvucci, 1999; Maxwell and Miller, 2005; Salvucci and Entekhabi, 1995; Shen et al., 2013), surface energy budgets (Alkhaier et al., 2012; Niu et al., 2017; Rihani et al., 2010; Soylu et al., 2011), land–atmosphere interactions (Anyah et al., 2008; Jiang et al., 2009; Leung et al., 2011; Yuan et al., 2008), vegetation dynamics (Banks et al., 2011; Chen et al., 2010), and soil biogeochemistry (Lohse et al., 2009; Pacific et al., 2011).

Recognizing the importance of groundwater systems on terrestrial processes,
groundwater models of varying complexity have been implemented in land
surface models (LSMs) in recent years. Groundwater models in current LSMs
can be classified into four categories based on their governing equations.
Type-1 models assume a quasi-steady state equilibrium of the soil moisture
profile above the water table (Hilberts et al., 2005; Koster et
al., 2000; Walko et al., 2000). Type-2 models use a

The Energy, Exascale, Earth System Model (E3SM) is a new Earth system modeling project sponsored by the US Department of Energy (E3SM Project, 2018). The E3SM model started from the Community Earth System Model (CESM) version 1_3_beta10 (Oleson et al., 2013). Specifically, the initial version (v0) of the E3SM Land Model (ELM) was based off the Community Land Model's (CLM's) tag 4_5_71. ELMv0 uses a Type-2 subsurface hydrology model based on Zeng and Decker (2009). In this work, we developed in ELMv1 the Type-4 Variably Saturated Flow Model (VSFM) to provide a unified treatment of soil hydrologic processes within the unsaturated and saturated zones. The VSFM formulation is based on the isothermal, single phase, variably saturated (RICHARDS) flow model within PFLOTRAN (Hammond and Lichtner, 2010). While PFLOTRAN is a massively parallel, three-dimensional subsurface model, the VSFM is a serial, one-dimensional model that is appropriate for climate-scale applications.

This paper is organized into several sections: (1) a brief review of the ELMv0 subsurface hydrology model; (2) an overview of the VSFM formulation integrated in ELMv1; (3) an application of the new model formulation to three benchmark problems; (4) development of a subsurface drainage parameterization necessary to predict global water table depths (WTDs) comparable to recently released observationally constrained estimates; (5) comparison of ELMv1 global simulations with the default subsurface hydrology model and VSFM against multiple observations using the International Land Model Benchmarking package (ILAMB; Hoffman et al., 2017); and (6) a summary of major findings.

Water flow in the unsaturated zone is often described by the

Flow in the saturated zone is modeled as an unconfined aquifer below the soil column based on the work of Niu et al. (2007). Exchange of water between the soil column and unconfined aquifer depends on the location of the water table. When the water table is below the last hydrologically active soil layer in the column, a recharge flux from the last soil layer replenishes the unconfined aquifer. A zero-flux boundary condition is applied to the last hydrologically active soil layer when the water table is within the soil column. The unconfined aquifer is drained by a flux computed based on the SIMTOP scheme of Niu et al. (2007) with modifications to account for frozen soils (Oleson et al., 2013).

In the VSFM formulation integrated in ELMv1, we use the mass conservative
form of the variably saturated subsurface flow equation
(Farthing et al., 2003; Hammond and Lichtner,
2010; Kees and Miller, 2002):

In order to close the system, a constitutive relationship is used to express
saturation and relative permeability as a function of soil matric pressure.
Analytic water retention curves (WRCs) are used to model effective
saturation (

The sink of water due to transpiration from a given plant functional type
(PFT) is vertically distributed over the soil column based on area and root
fractions of the PFT. The top soil layer has an additional flux associated
with balance of infiltration and soil evaporation. The subsurface drainage
flux is applied proportionally to all soil layers below the water table.
Details on the computation of water sinks are given in
Oleson et al. (2013). Unlike the default
subsurface hydrology model, the VSFM is applied over the full soil depth (in
the default model, 15 soil layers). The VSFM model replaces both the

We use a cell-centered finite volume discretization to decompose the spatial
domain,

The discretized form of the left-hand-side term and first term on the right-hand side of Eq. (14) are approximated as

We perform temporal integration of Eq. (

Rearranging terms of Eq. (

In this work, we find the solution to the nonlinear system of equations
given by Eq. (

Soil properties and discretization used in the three test problems described in Sect. 2.3.

We tested the VSFM with three idealized 1-dimensional test problems. First,
the widely studied problem for the 1-D Richards equation of infiltration in dry
soil by Celia et al. (1990) was used. The problem setup consists
of a 1.0 m long soil column with a uniform initial pressure of

Second, we simulated transient one-dimensional vertical infiltration in a
two-layered soil system as described in Srivastava and Yeh (1991).
The domain consisted of a 2 m tall soil column divided equally into two soil
types. Except for soil intrinsic permeability, all other soil properties of
the two soil types are the same. The bottom soil is 10 times less permeable
than the top (Table 1). Unlike Srivastava and Yeh (1991), who used
exponential functions of soil liquid pressure to compute hydraulic
conductivity and soil saturation, we used Mualem (1976) and
van Genuchten (1980) constitutive relationships. Since our choice
of constitutive relationships for this setup resulted in absence of an
analytical solution, we compared VSFM simulations against PFLOTRAN results.
The domain was discretized in 200 control volumes of equal soil thickness.
Two scenarios, wetting and drying, were modeled to test the robustness of
the VSFM solver robustness. Initial conditions for each scenario included a
time invariant boundary condition of 0 m (

Third, we compare VSFM and PFLOTRAN predictions for soil under variably
saturated conditions. The 1-dimensional 1 m deep soil column was discretized
in 100 equal thickness control volumes. A hydrostatic initial condition was
applied such that the water table is 0.5 m below the soil surface. A time
invariant flux of

We performed global simulations with ELMv1-VSFM at a spatial resolution of
1.9

For evaluation and calibration, we used the Fan et al. (2013)
global

In the VSFM formulation, the dominant control on long-term
ground water depth is the
subsurface drainage flux,

With the optimal

Comparison of VSFM simulated pressure profile
(blue line) against data (red square) reported in
Celia et al. (1990) at time

For the 1D Richards equation infiltration in dry soil comparison, we evaluated the solutions at 24 h against those published by Celia et al. (1990; Fig. 1). The VSFM solver accurately represented the sharp wetting front over time, where soil hydraulic properties change dramatically due to nonlinearity in the soil water retention curve.

For the model evaluation of infiltration and drying in layered soil, the results of the VSFM and PFLOTRAN are essentially identical. In both models and scenarios, the higher permeability top soil responds rapidly to changes in the top boundary condition and the wetting and drying fronts progressively travel through the less permeable soil layer until soil liquid pressure in the entire column reaches a new steady state by about 100 h (Fig. 2).

Transient liquid pressure simulated for a
two-layer soil system by VSFM (solid line) and
PFLOTRAN (square) for

We also evaluated the VSFM predicted water table dynamics against PFLOTRAN predictions from an initial condition of saturated soil below 0.5 m depth. The simulated water table rises to 0.3 m depth by 1 day and reaches the surface by 2 days, and the VSFM and PFLOTRAN predictions are essentially identical (Fig. 3). Soil properties, spatial discretization, and time step used for the three single-column problems are summarized in Table 1 These three evaluation simulations demonstrate the VSFM accurately represents soil moisture dynamics under conditions relevant to ESM-scale prediction.

Transient

The simulated nonlinear WTD–

Global estimate of

For 79 % of the global grid cells, the ensemble range of simulated WTD
spanned the F2013 dataset. The optimal value of

Bias, root mean square error (RMSE), and correlation (

The ELMv1-VSFM predictions are much closer to the F2013 dataset (Fig. 6a)
using optimal globally distributed

Globally averaged WTD in ELMv1-VSFM simulations with default

Seasonal monthly mean soil moisture differences for
top 10 cm between ELMv1-VSFM simulations with optimal
and default

ILAMB benchmark scores for latent heat flux
(LH), sensible heat flux 640 (SH), terrestrial water
storage anomaly (TWSA), and surface runoff.
The calculation of ILAMB metrics and scores
are described at

The ILAMB package (Hoffman et al., 2017) provides a
comprehensive evaluation of predictions of carbon cycle states and fluxes,
hydrology, surface energy budgets, and functional relationships by
comparison to a wide range of observations. We used ILAMB to evaluate the
hydrologic and surface energy budget predictions from the new ELMv1-VSFM
model (Table 3). Optimal

Finally, we evaluated the computational costs of implementing VSFM in ELM
and compared them to the default model. We performed 5-year-long simulations
for default and VSFM using 96, 192, 384, 768, and 1536 cores on the Edison
supercomputer at the National Energy Research Scientific Computing Center.
Using an optimal processor layout, we found that ELMv1-VSFM is

The significant improvement in WTD prediction using optimal

In this study, we assumed a spatially homogeneous depth to bedrock (DTB) of
150 m. Recently, Brunke et al. (2016) incorporated a
global

Lateral water redistribution impacts soil moisture dynamics (Bernhardt et al., 2012), biogeochemical processes in the root zone (Grant et al., 2015), distribution of vegetation structure (Hwang et al., 2012), and land–atmosphere interactions (Chen and Kumar, 2001; Rihani et al., 2010). The ELMv1-VSFM developed in this study does not include lateral water redistribution between soil columns and only simulates vertical water transport. Lateral subsurface processes can be included in LSMs via a range of numerical discretization approaches of varying complexity, e.g., adding lateral water as source/sink terms in the 1-D model, implementing an operator split approach to solve vertical and lateral processes in a noniterative approach (Ji et al., 2017), or solving a fully coupled 3-D model (Bisht et al., 2017, 2018; Kollet and Maxwell, 2008). Additionally, lateral transport of water can be implemented in LSMs at a subgrid level (Milly et al., 2014) or grid cell level (Miguez-Macho et al., 2007). The current implementation of VSFM is such that each processor solves the variably saturated Richards equation for all independent soil columns as one single problem. Thus, extension of VSFM to solve the tightly coupled 3-D Richards equation on each processor locally while accounting for lateral transport of water within grid cells and among grid cells is straightforward. The current VSFM implementation can also be easily extended to account for subsurface transport of water among grid cells that are distributed across multiple processors by modeling lateral flow as source/sink terms in the 1-D model. Tradeoffs between approaches to represent lateral processes and computational costs need to be carefully studied before developing quasi- or fully three-dimensional LSMs (Clark et al., 2015).

Transport of water across multiple components of the soil-plant-atmosphere continuum (SPAC) has been identified as a critical process in understanding the impact of climate warming on the global carbon cycle (McDowell and Allen, 2015). Several SPAC models have been developed by the ecohydrology community and applied to study site level processes (Amenu and Kumar, 2008; Bohrer et al., 2005; Manoli et al., 2014; Sperry et al., 1998), yet implementation of SPAC models in global LSMs is limited (Clark et al., 2015). Similarly, current generation LSMs routinely ignore advective heat transport within the subsurface, which has been shown to be important in high-latitude environments by multiple field and modeling studies (Bense et al., 2012; Frampton et al., 2011; Grant et al., 2017; Kane et al., 2001). The use of PETSc's DMComposite in VSFM provides flexibility for solving a tightly coupled multi-component problem (e.g., transport of water through the soil-plant-atmosphere continuum) and multi-physics problem (e.g., fully coupled conservation of mass and energy equations in the subsurface). DMComposite allows for an easy assembly of a tightly coupled multi-physics problem from individual physics formulations (Brown et al., 2012).

Starting from the climate-scale land model ELMv0, we incorporated a unified
physics formulation to represent soil moisture and groundwater dynamics that
are solved using PETSc. Application of VSFM to three benchmarks problems
demonstrated its robustness to simulate subsurface hydrologic processes in
coupled unsaturated and saturated zones. Ensemble global simulations at
1.9

An optimal

The Brooks and Corey (1964) water retention curve of Eq. (10)
has a discontinuous derivative at

In practice, setting

The Brooks–Corey water rendition curve for estimating
liquid saturation,

Let

Given a breakpoint

As shown in Fig. A1, the two reductions differ mainly in that setting

Both reductions require solving a nonlinear expression, either
Eqs. (A3)
or (A4), for

The residual equation for the VSFM formulation at

The density at the interface of control volume,

The first term on the right-hand side of Eq. (27) is computed as the product of
distance-weighted harmonic average of intrinsic permeability,

By substituting Eqs. (28), (29), and (30) in Eq. (27), we obtain

The discretized equations of VSFM leads to a system of nonlinear equations
given by

The derivative of the accumulation term in

The derivative of flux between

Lastly, the derivative of the Darcy velocity between the

VSFM uses a two-stage check to determine an acceptable numerical solution

Stage-1: at any temporal integration stage, the model attempts to solve the
set of nonlinear equations given by Eq. (

Stage-2: after a numerical solution for the nonlinear problem is obtained, a
mass balance error is calculated as the difference between input and output
fluxes and change in mass over the integration time step. If the mass balance
error exceeds 10–5 kg m

The stand-alone VSFM code is available at

The supplement related to this article is available online at:

GB developed and integrated VSFM in ELM v1.0. GB and WJR designed the study and prepared the manuscript. GEH provided a template for VSFM development. DML implemented smooth approximations of water retention curves in VSFM.

The authors declare that they have no conflict of interest.

This research was supported by the Director, Office of Science, Office of Biological and Environmental Research of the US Department of Energy under contract no. DE-AC02-05CH11231 as part of the Energy Exascale Earth System Model (E3SM) programs. Edited by: Jatin Kala Reviewed by: Minki Hong and two anonymous referees