Water flow in the unsaturated zone is often described by the θ-based Richards equation: ∂θ∂t=-∇⋅q-Q, where θ (m3 of water m-3 of soil) is the volumetric soil water content, t (s) is time, q (m s-1) is the Darcy water flux, and Q (m3 of water m-3 of soil s-1) is a soil moisture sink term. The Darcy flux, q, is given by q=-K∇(ψ+z), where K (m s-1) is the hydraulic conductivity, z (m) is height above some datum in the soil column and ψ (m) is the soil matric potential. The hydraulic conductivity and soil matric potential are modeled as non-linear functions of volumetric soil moisture following Clapp and Hornberger (1978): K=ΘiceKsatθθsat2B+3,ψ=ψsatθθsat-B, where Ksat (m s-1) is saturated hydraulic conductivity, ψsat (m) is saturated soil matric potential, B is a linear function of percentage clay and organic content (Oleson et al., 2013), and Θice is the ice impedance factor (Swenson et al., 2012). ELMv0 uses the modified form of the Richards equation of Zeng and Decker (2009) that computes Darcy flux as q=-K∇(ψ+z-C), where C is a constant hydraulic potential above the water table, z∇, given as C=ψE+z=ψsatθE(z)θsat-B+z=ψsat+z∇, where ψE (m) is the equilibrium soil matric potential, z (m) is height above a reference datum, θE (m3 m-3) is volumetric soil water content at equilibrium soil matric potential, and z∇ (m) is the height of the water table above a reference datum. ELMv0 uses a cell-centered finite volume spatial discretization and backward Euler implicit time integration. By default, ELMv0's vertical discretization of a soil column yields 15 soil layers of exponentially varying soil thicknesses that reach a depth of 42.1 m Only the first 10 soil layers (or top 3.8 m of each soil column) are hydrologically active, while thermal processes are resolved for all 15 soil layers. The nonlinear Darcy flux is linearized using Taylor series expansion and the resulting tri-diagonal system of equations is solved by LU factorization.

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): ∂(ϕswρ)∂t=-∇⋅(ρq)-Q, where ϕ (m3 m-3) is the soil porosity, sw (-) is saturation, ρ (kg m-3) is water density, q (m s-1) is the Darcy velocity, and Q (kg m-3 s-1) is a water sink. We restrict our model formulation to a one-dimensional system and the flow velocity is defined by Darcy's law: q=-kkrμ∇(P+ρgz), where k (m2) is intrinsic permeability, kr (-) is relative permeability, μ (Pa s) is viscosity of water, P (Pa) is pressure, g (m s-2) is the acceleration due to gravity, and z (m) is elevation above some datum in the soil column.

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 (se) se=sw-sr1-sr, where sw is saturation and sr is residual saturation. We have implemented Brooks and Corey (1964; Eq. 10) and van Genuchten (1980; Eq. 11) WRCs: se=-PcPc0-λifPc<Pc0,1ifPc≥Pc0,se=1+αPcn-mifPc<0,1ifPc≥0, where Pc (Pa) is the capillary pressure, Pc0 (Pa) is the air entry pressure, α (Pa-1) is inverse of the air entry pressure, λ (-) is the exponent in the Brooks and Corey parameterization, and n (-) and m (-) are the van Genuchten parameters. The capillary pressure is computed as Pc=P-Pref where Pref is Pc0 for Brooks and Corey WRC and typically the atmospheric pressure (=101 325 Pa) is used for the van Genuchten WRC. In addition, a smooth approximation of Eqs. (10) and (11) was developed to facilitate convergence of the nonlinear solver (Appendix A). Relative soil permeability was modeled using the Mualem (1976) formulation: κrse=se0.51-1-se1/mmifP<Pref,1ifP≥Pref, where m=1-1/n. Lastly, we used an EoS for water density, ρ, that is a nonlinear function of liquid pressure, P, and liquid temperature, T, given by Tanaka et al. (2001): ρP,T=1+k0+k1T+k2T2P-Prefa51-T+a12T+a2a3T+a4, where k0=50.74×10-11(Pa-1),k1=-0.326×10-11(Pa-1C-1),k2=0.00416×10-11(Pa-1C2),a1=-3.983035(C),a2=301.797(C),a3=522558.9(C-2),a4=69.34881(C),a5=999.974950(kgm-3).

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 θ-based Richards equation and the unconfined aquifer of the default model and uses a zero-flux lower boundary condition. In the VSFM model, water table depth is diagnosed based on the vertical soil liquid pressure profile. Like the default model, drainage flux is computed based on the modified SIMTOP approach and is vertically distributed over the soil layers below the water table.

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 -10.0 m (=3535.5 Pa). The time invariant boundary conditions applied at the top and bottom of the soil column are -0.75 m (=93989.1 Pa) and -10.0 m (=3535.5 Pa), respectively. The soil properties for this test are given in Table 1. A vertical discretization of 0.01 m is used in this simulation.

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 (=1.01325×105 Pa) for the lowest control volume and a constant flux of 0.9 and 0.1 cm h-1 at the soil surface for wetting and drying scenarios, respectively.

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 2.5×10-5 m s-1 is applied at the surface, while the lowest control volume has a boundary condition corresponding to the initial pressure value at the lowest soil layer. The soil properties used in this test are the same as those used in the first evaluation.

We performed global simulations with ELMv1-VSFM at a spatial resolution of 1.9∘ (latitude) × 2.5∘ (longitude) with a 30 (min) time step for 200 years, including a 180-year spin-up and the last 20 years for analysis. The simulations were driven by CRUNCEP meteorological forcing from 1991 to 2010 (Piao et al., 2012) and configured to use prescribed satellite phenology.

For evaluation and calibration, we used the Fan et al. (2013) global ∼1 km horizontal resolution WTD dataset (hereafter F2013 dataset), which is based on a combination of observations and hydrologic modeling. We aggregated the dataset to the ELMv1-VSFM spatial resolution. ELM-VSFM's default vertical soil discretization uses 15 soil layers to a depth of ∼42 m, with an exponentially varying soil thickness. However, ∼13 % of F2013 land grid cells have a water table deeper than 42 m. We therefore modified ELMv1-VSFM to extend the soil column to a depth of 150 m with 59 soil layers; the first nine soil layer thicknesses were the same as described in Oleson et al. (2013) and the remaining layers (10–59) were set to a thickness of 3 m.

In the VSFM formulation, the dominant control on long-term ground water depth is the subsurface drainage flux, qd (kg m-2 s-1), which is calculated based on water table depth, z∇ (m), Niu et al. (2005): qd=qd,maxexp-fdz∇, where qd,max (kg m-2 s-1) is the maximum drainage flux that depends on grid cell slope and fd (m-1) is an empirically derived parameter. The subsurface drainage flux formulation of Niu et al. (2005) is similar to the TOPMODEL formulation (Beven and Kirkby, 1979) and assumes the water table is parallel to the soil surface. While Sivapalan et al. (1987) derived qd,max as a function of lateral hydraulic anisotropy, hydraulic conductivity, topographic index, and decay factor controlling vertical saturated hydraulic conductivity, Niu et al. (2005) defined qd,max as a single calibration parameter. ELMv0 uses fd=2.5 m-1 as a global constant and estimates maximum drainage flux when WTD is at the surface as qd,max=10sin⁡β kg m-2 s-1, where β (radians) is the mean grid cell topographic slope. Of the two parameters, fd and qd,max, available for model calibration, we choose to calibrate fd because the uncertainty analysis by Hou et al. (2012) identified it as the most significant hydrologic parameter in CLM4. To improve on the fd parameter values, we performed an ensemble of global simulations with fd values of 0.1, 0.2, 0.5, 1.0, 2.5, 5.0, 10.0, and 20 m-1. Each ensemble simulation was run for 200 years to ensure an equilibrium solution, and the last 20 years were used for analysis. A nonlinear functional relationship between fd and “WTD” was developed for each grid cell and then the F2013 dataset was used to estimate an optimal fd for each grid cell.

With the optimal fd values, we ran a ELM-VSFM simulation using the protocol described above. We then used the ILAMB package to evaluate the ELMv1-VSFM predictions of surface energy budget, terrestrial water storage anomalies (TWSAs), and river discharge (Collier et al., 2018; Hoffman et al., 2017). ILAMB evaluates model prediction bias, RMSE, and seasonal and diurnal phasing against multiple observations of energy, water, and carbon cycles at in-situ, regional, and global scales. Since ELM-VSFM simulations in this study did not include an active carbon cycle, we used the following ILAMB benchmarks for water and energy cycles: (i) latent and surface energy fluxes using site-level measurements from FLUXNET (Lasslop et al., 2010) and globally from FLUXNET-MTE (Jung et al., 2009); (ii) TWSA from the Gravity Recovery And Climate Experiment (GRACE) observations (Kim et al., 2009); and (iii) stream flow for the 50 largest global river basins (Dai and Trenberth, 2002). We applied ILAMB benchmarks for ELMv1-VSFM simulations with default and calibrated fd to ensure improvements in WTD predictions did not degrade model skill for other processes.

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).

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.

The simulated nonlinear WTD–fd relationship is a result of the subsurface drainage parameterization flux given by Eq. (20; Fig. 4a and b). For 0.1≤fd≤1, the slope of the WTD–fd relationship for all grid cells is log–log linear with a slope of -1.0±0.1. The log–log linear relationship breaks down for fd>1, where the drainage flux becomes much smaller than infiltration and evapotranspiration (Fig. 4c and d). Thus, at larger fd, the steady state z∇ becomes independent of fd and is determined by the balance of infiltration and evapotranspiration.

For 79 % of the global grid cells, the ensemble range of simulated WTD spanned the F2013 dataset. The optimal value of fd for each of these grid cells was obtained by linear interpolation in the log–log space (e.g., Fig. 4a). For the remaining 21 % of grid cells where the shallowest simulated WTD across the range of fd was deeper than that in the F2013 dataset, the optimal fd value was chosen as the one that resulted in the lowest absolute WTD error (e.g., Fig. 4b). At large fd values, the drainage flux has negligible effects on WTD, yet simulated WTD is not sufficiently shallow to match the F2013 observations, which indicates that either evapotranspiration is too large or infiltration is too small. There was no difference in the mean percentage of sand and clay content between grids cells with and without an optimal fd value. The optimal fd has a global average of 1.60 m-1±2.68 m-1 and 72 % of global grid cells have an optimal fd value lower than the global average (Fig. 5).

The ELMv1-VSFM predictions are much closer to the F2013 dataset (Fig. 6a) using optimal globally distributed fd values (Fig. 6c) compared to the default fd value (Fig. 6b). The significant reduction in WTD bias (model – observation) is mostly due to improvement in the model's ability to accurately predict deep WTD using optimal fd values. In the simulation using optimal globally distributed fd values, all grid cells with WTD bias >3.7 m were those for which an optimal fd was not found. The mean global bias, RMSE, and R2 values improved in the new ELMv1-VSFM compared to the default model (Table 2). The 79 % of global grid cells for which an optimal fd value was estimated had significantly better water table prediction with a bias, RMSE, and R2 of -0.04, 0.67, and 0.99 m, respectively, as compared to the remaining 21 % of global grid cells that had a bias, RMSE, and R2 of -9.82, 18.08, and 0.31 m, respectively. The simulated annual WTD range, which we define to be the difference between maximum and minimum WTD in 1 year, has a spatial mean and standard deviation of 0.32 and 0.58 m, respectively, using optimal fd values (Fig. 7a). The annual WTD range decreased by 0.24 m for the 79 % of the grid cells for which an optimal fd value was estimated (Fig. 7b).

Globally averaged WTD in ELMv1-VSFM simulations with default fd and optimal fd values were 10.5 and 20.1 m, respectively. Accurate prediction of deep WTD in the simulation with optimal fd caused very small differences in near-surface soil moisture (Fig. 8). The 79 % of grid cells with an optimal fd value had deeper globally averaged WTDs than when using the default fd value (24.3 vs. 8.6 m). For these 79 % of grid cells, the WTD was originally deep enough to not impact near-surface conditions (Kollet and Maxwell, 2008); therefore, further lowering of WTD led to negligible changes in near-surface hydrological conditions.

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 fd values had inconsequential impacts on simulated surface energy fluxes at site-level and global scales. Optimal fd values led to improvement in prediction of deep WTD (with a mean value of 24.3 m) for grid cells that had an average WTD of 8.7 m in the simulation using default fd values. Thus, negligible differences in surface energy fluxes between the two simulations are consistent with the findings of Kollet and Maxwell (2008), who identified decoupling of groundwater dynamics and surface processes at a WTD of ∼10 m. There were slight changes in the bias and RMSE for predicted TWSA, but the ILAMB score remained unchanged. The TWSA amplitude is lower for the simulation with optimal fd values, consistent with the associated decrease in annual WTD range. ELM's skill in simulating runoff for the 50 largest global watersheds remained unchanged. Two additional 10-year-long simulations were performed to investigate the sensitivity of VSFM simulated WTD to spatial and temporal discretization. Results show that simulated WTD is insensitive to temporal discretization, and has small sensitivity to vertical spatial resolution. See the Supplement for details regarding the setup and analysis of results from the two additional simulations.

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 ∼30 % more expensive than the default ELMv1 model (the Supplement, Fig. S1). We note that the relative computational cost of the land model in a fully coupled global model simulation is generally very low. Dennis et al. (2012) reported computational cost of the land model to be less than 1 % in ultrahigh-resolution CESM simulations. We therefore believe that the additional benefits associated with the VSFM formulation are well justified by this modest increase in computational cost. In particular, VSFM allows for a greater variety of mesh configurations and boundary conditions, and can accurately simulate WTD for the ∼13 % of global grid cells that have a water table deeper than 42 (m) (Fan et al., 2013).

The significant improvement in WTD prediction using optimal fd values demonstrates VSFM's capabilities to model hydrologic processes using a unified physics formulation for unsaturated–saturated zones. However, several caveats remain due to uncertainties in model structure, model parameterizations, and climate forcing data.

In this study, we assumed a spatially homogeneous depth to bedrock (DTB) of 150 m. Recently, Brunke et al. (2016) incorporated a global ∼1 km dataset of soil thickness and sedimentary deposits (Pelletier et al., 2016) in CLM4.5 to study the impacts of soil thickness spatial heterogeneity on simulated hydrological and thermal processes. While inclusion of heterogeneous DTB in CLM4.5 added more realism to the simulation setup, no significant changes in simulated hydrologic and energy fluxes were reported by Brunke et al. (2016). Presently, work is ongoing in the E3SM project to include variable DTB within ELM and future simulations will examine the impact of those changes on VSFM's prediction of WTD. Our use of the “satellite phenology” mode, which prescribes transient leaf area index profiles for each PFT in the grid cell, ignored the likely influence of water cycle dynamics and nutrient constraints on the C cycle (Ghimire et al., 2016; Zhu et al., 2016). Estimation of soil hydraulic properties based on soil texture data is critical for accurate LSM predictions (Gutmann and Small, 2005) and this study does not account for uncertainty in soil hydraulic properties.

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).