Accurately capturing the complex soil-water and groundwater interactions is vital for describing the coupling between subsurface–surface–atmospheric systems in regional-scale models. The nonlinearity of Richards' equation (RE) for water flow, however, introduces numerical complexity to large unsaturated–saturated modeling systems. An alternative is to use quasi-3-D methods with a feedback coupling scheme to practically join sub-models with different properties, such as governing equations, numerical scales, and dimensionalities. In this work, to reduce the nonlinearity in the coupling system, two different forms of RE are switched according to the soil-water content at each numerical node. A rigorous multi-scale water balance analysis is carried out at the phreatic interface to link the soil-water and groundwater models at separated spatial and temporal scales. For problems with dynamic groundwater flow, the nontrivial coupling errors introduced by the saturated lateral fluxes are minimized with a moving-boundary approach. It is shown that the developed iterative feedback coupling scheme results in significant error reduction and is numerically efficient for capturing drastic flow interactions at the water table, especially with dynamic local groundwater flow. The coupling scheme is developed into a new HYDRUS package for MODFLOW, which is applicable to regional-scale problems.

Numerical modeling of the soil-water and groundwater interactions has to deal with flow components and governing equations at different scales. This adds significant complexity to model development and calibration. Unsaturated soil-water and saturated groundwater flows with similar properties are usually integrated into a whole modeling system. Although physically consistent and numerically rigorous, methods involving the 3-D Richards' equation (RE; Richards, 1931) tend to be computationally expensive and numerically unstable due to the large nonlinearity and the demand for dense discretization (Kumar et al., 2009; Maxwell and Miller, 2005; Panday and Huyakorn, 2004; Thoms et al., 2006; Zha et al., 2013a), especially for problems with multi-scale properties. In this work, parsimonious approaches, which appear in different governing equations and coupling schemes, are developed for modeling the soil-water and groundwater interactions at regional scales.

Simplifying the soil-water flow details into upper flux boundaries has been widely used to simulate large-scale saturated flow dynamics, such as the MODFLOW package and its variants (Langevin et al., 2017; Leake and Claar, 1999; McDonald and Harbaugh, 1988; Niswonger et al., 2011; Panday et al., 2013; Zeng et al., 2017). At the local scale, in contrast, the unsaturated flow processes are usually approximated with reasonable simplifications and assumptions in RE (Bailey et al., 2013; Liu et al., 2016; Paulus et al., 2013; Šimůnek et al., 2009; van Dam et al., 2008; Yakirevich et al., 1998; Zha et al., 2013b).

The original RE, also known as the

For regional problems, the vadose zone is usually conceptualized into paralleled soil columns without lateral connections. The resulting quasi-3-D coupling scheme (Kuznetsov et al., 2012; Seo et al., 2007; Xu et al., 2012; Zhu et al., 2012) significantly reduces the dimensionality and complexity. According to how the messages are transferred across the coupling interface, the quasi-3-D methods are categorized into (1) the fully coupling scheme, which simultaneously builds the nodal hydraulic connections of sub-models at both sides and implicitly solves the assembled matrices; (2) the one-way coupling scheme, which delivers the soil-water model solutions onto the groundwater model without feedback mechanism; and (3) the feedback (or two-way) coupling scheme, which explicitly exchanges the head and/or flux solutions in vicinity of the interface nodes.

The fully coupling scheme (Gunduz and Aral, 2005; Zhu et al., 2012) is numerically rigorous but tends to increase the computational burden under practical conditions. For example, the potentially conditional diagonal dominance causes non-convergence for the iterative solvers (Edwards, 1996). Owing to high nonlinearity in the soil-water sub-models, the assembled matrices can only be solved with unified small time steps, which adds to the computational expense. The one-way coupling scheme, as adopted by the UZF1 package for MODFLOW (Grygoruk et al., 2014; Niswonger et al., 2006) as well as the free drainage mode in SWAP package for MODFLOW (Xu et al., 2012), assumes that the water-table depth has a minor influence on flow interactions at the phreatic interface and is thus problem specific.

The feedback coupling methods, in contrast, are widely used (Kuznetsov et al., 2012; Seo et al., 2007; Shen and Phanikumar, 2010; Stoppelenburg et al., 2005; Xie et al., 2012; Xu et al., 2012) as a compromise of numerical accuracy and computational cost. In a feedback coupling scheme, the soil-water and groundwater sub-models can be built with governing equations and numerical schemes at different scales. For flow processes with multi-scale components, such as boundary geometries, parameter heterogeneities, and hydrologic stresses, the scale-separation strategy can be implemented easily. Although the feedback coupling method is numerically more rigorous than a one-way coupling method, and tends to reduce the inconsistency of head and/or flux interfacial boundaries, some concerns arise.

The first concern is the numerical efficiency of the iterative and non-iterative feedback coupling methods. The non-iterative approach (Twarakavi et al., 2008; Xu et al., 2012) usually leads to significant error accumulation when dealing with a dynamically fluctuating water table, especially with large time-step sizes. The iterative methods, in contrast (Kuznetsov et al., 2012; Stoppelenburg et al., 2005; Xie et al., 2012), by converging the head and/or flux solutions at the coupling interface, are numerically rigorous but computationally expensive, especially when solving the coupled sub-models with a unified time-stepping scheme (Kuznetsov et al., 2012). Good balance between cost and effect is needed to maintain practical utility of the iterative feedback coupling scheme.

The second concern lies in the scale-mismatching problem. For groundwater models (Harbaugh et al., 2005; Langevin et al., 2017; Lin et al., 2010; McDonald and Harbaugh, 1988), the specific yield at the phreatic surface is usually represented by a simple large-scale parameter, while for soil-water models (Niswonger et al., 2006; Šimůnek et al., 2009; Thoms et al., 2006), the small-scale phreatic water release is influenced by the water-table depth and the unsaturated soil moisture profile (Dettmann and Bechtold, 2016; Nachabe, 2002). Delivering small-scale solutions of the soil-water models onto the large-scale interfacial boundary of the groundwater model, as well as maintaining the global mass balance, usually introduces significant nonlinearity to the entire coupling system (Stoppelenburg et al., 2005). Conditioned by this, the mismatch of numerical scales in the coupled sub-models causes significant coupling errors and instability. A multi-scale water balance analysis at the phreatic surface helps to relieve such difficulties.

The third concern is the nontrivial lateral fluxes between the saturated regions controlled by the vertical soil columns, which are usually not considered in previous study (Seo et al., 2007; Xu et al., 2012). Though rigorous water balance analysis is conducted to address such inadequacy (Shen and Phanikumar, 2010), the lateral fluxes solved with a 2-D groundwater model usually require additional efforts to build water budget equations in each subdivision represented by the soil columns. A moving-boundary strategy helps to avoid the saturated lateral flow in the groundwater body.

In this work, the

In this paper, the governing equations at different scales, the multi-scale water balance analysis at the phreatic surface, and the iterative feedback coupling scheme for solving the whole system, are presented in Sect. 2. Synthetic numerical experiments are described in Sect. 3. Numerical performance of the developed model is investigated in Sect. 4. Conclusions are drawn in Sect. 5.

To address the aforementioned first concern, governing equations for subsurface flow are given at different levels of complexity (Sect. 2.1), numerical solutions of these equations are presented (Sect. 2.2), and nonlinearity in the soil-water sub-models is reduced by a generalized switching scheme that chooses appropriate forms of RE according to the hydraulic conditions at each numerical node (Sect. 2.3). Then, an iterative feedback coupling scheme is developed to solve the soil-water and groundwater models at independent scales (Sect. 2.4). As for the second concern, a multi-scale water balance analysis is conducted to deal with the scale-mismatching problem at the phreatic surface (Sect. 2.5). To cope with the third concern, a moving Dirichlet boundary at the groundwater table is assigned to the soil-water sub-models (see Appendix Sect. A1); the Neumann upper boundary for the saturated model is provided in Sect. A2, and the relaxed iterative feedback process is presented in Sect. A3.

The mass conservation equation for unsaturated–saturated flow is given by

The governing equation for the saturated zone (Eq. 3) is spatially and
temporally approximated in the same form as the MODFLOW 2005 model (Harbaugh
et al., 2005; Langevin et al., 2017). Celia's modification (Celia et al.,
1990; Šimůnek et al., 2009) is applied to the

The spatial discretization of Eq. (4), as well as the water balance analysis
for each node, is based on the nodal flux in element

When

Hereinafter,

Due to lower nonlinearity of hydraulic diffusivity (

For element

Note that the equation switching method takes full advantage of the

The Dirichlet and Neumann boundaries are iteratively transferred across the phreatic interface. The groundwater head solution serves as the head-specified lower boundary of the soil columns, while the unsaturated solution is converted into the flux-specified upper boundary of the groundwater model. Due to moderate variation of the groundwater flow, the predicted water-table solution is usually adopted in advance as the Dirichlet lower boundary of the fine-scale soil-water flow models (Seo et al., 2007; Shen and Phanikumar, 2010; Xu et al., 2012), which in sequence then provides the Neumann upper boundary for successively solving the coarse-scale groundwater flow model. Section A1 provides the method for a moving Dirichlet lower boundary, while Sect. A2 presents the Neumann upper boundary for the 3-D groundwater model. In Sect. A3, the relaxed iterative feedback coupling scheme is used to solve the unsaturated–saturated sub-models at two sides of the coupling interface.

Schematic of the space- and time-splitting strategy for coupling
models at two independent scales. For a groundwater model, spatial
discretization is expected to be large (

Coupling models at different scales requires consistency in their spatial and
temporal scales at the interface (Downer and Ogden, 2004; Rybak et al.,
2015). A space- and time-splitting strategy (see Fig. 1) is adopted to
separate sub-models at different scales. That is, the soil-water models are
established by

The Dirichlet–Neumann coupling of the soil-water and groundwater
flow models at different scales.

In this section, a range of 1-D, 2-D, 3-D, and regional numerical test cases
are presented. The 1-D tests are benchmarked by the globally refined
solutions from the HYDRUS-1D code (Šimůnek et al., 2009). The 2-D and
3-D “truth” solutions are obtained from the fully 3-D variably saturated
flow (VSF; Thoms et al., 2006) model. At the regional
scale, a synthetic case study suggested by Twarakavi et al. (2008) is
reproduced. The codes are run on a personal computer with a 16 GB RAM and
3.6 GHz Intel Core (i3-4160). A maximal number of feedback iterations is set
at 20. Soil parameters for the van Genuchten model (van Genuchten, 1980) are
given in Table 1. The root-mean-square error (RMSE) of the solution

Soil parameters used in the test cases.

The 1-D case is used to investigate the benefit brought by switching
Richards' equation in the unsaturated zone. A soil column is initialized with
a hydrostatic water-table depth of 800 cm. That is,

Miller et al.'s problem (Miller et al., 1998) is reproduced in scenario 1. A
dry sandy soil column (see soil no. 1 in Table 1) experiences a large
constant flux infiltration at the soil surface of

In scenario 2, Hills et al.'s problem (Hills et al., 1989) is considered.
The soils no. 2 and no. 3 from Table 1 are alternatively layered with a
thickness of 20 cm within the first 80 cm of depth. Below 80 cm (

Rapidly changing atmospheric upper boundary conditions for scenario 2, case 1.

The coupled unsaturated model is discretized into a fine grid with

Schematic of the cross section for test case 2.
Two pumping wells with screens of

A 2-D case is analyzed with sharp groundwater flow (see Fig. 4). To minimize
the unsaturated lateral flow, the soil surface is set with a non-flux boundary.
The bottom and lateral boundaries are also non-flux. Two pumping stresses are
applied to the cross-sectional field with

Different number of subzones partitioned for the quasi-3-D simulations in case 3. The vadose zone is partitioned into 16, 12, 9, 5, and 3 subzones.

Case 3 is used to investigate the efficiency and applicability of a quasi-3-D
coupling model in comparison of the fully 3-D approaches. A phreatic aquifer
with

The precipitation, evaporation, and pumping rates in 12 stress periods.

A hypothetical test case from literature (Niswonger
et al., 2006; Prudic et al., 2004; Twarakavi et al., 2008) for a large-scale
simulation is reproduced here. The overall alluvial basin is divided into
uniform grids with

Input of the synthetic regional problem including

The numerical difficulty in a coupled unsaturated–saturated flow system
originates from the nonlinearity of the soil-water models, heterogeneity of
the parameters, and the variability of the hydrologic stresses
(Krabbenhøft, 2007; Zha et al.,
2017). In our work, the overall complexity of an iteratively coupled
quasi-3-D model could be lowered by (1) taking full advantage of the

Two scenarios in case 1 were selected to address the first point. Sudden
infiltration into a dry sandy soil and the rapidly altering atmospheric upper
boundaries were tested to illustrate the importance of applying a
switching-form RE for lowering the nonlinearity in the soil-water models. To
evaluate the benefits brought by a switching-form RE, the numerical stability
was first considered, as shown in Fig. 7. The coupled model in our work was
tested with

The time-step sizes through the simulation of

Comparison of soil moisture content at

Reducing the complexity of a coupling system can also be attained by
smoothing the exchanged information in space and time. As suggested by
Stoppelenburg et al. (2005), a time-varying specific yield calculated by the
small-scale soil-water models,

The traditional non-iterative feedback coupling methods cannot maintain sound mass balance near the phreatic surface, especially for problems with drastic flow interactions.

One reason is that, to launch a new step of a sub-model at either side of the
phreatic interface, the non-iterative feedback methods usually employed a
predicted interfacial boundary without correction, which inevitably
introduced coupling errors. In traditional non-iterative methods
(Seo et al., 2007; Xu
et al., 2012), such shortcomings could be alleviated by refining the macro-time-step size (

Water table changing with time for different macro-time-step sizes
(

The other factor contributing to the coupling errors in the traditional
method lies in neglecting the saturated lateral flux between adjacent soil
columns (Seo et al., 2007; Stoppelenburg et al., 2005; Xu et al., 2012). In
practical applications, the fluxes in and out of the saturated parts of the
soil columns differ, which adds to the complexity of the coupling scheme.
Although a strict water balance equation is established (Shen and Phanikumar,
2010), the concern centers on the spatial scale-mismatching problem. That is,
when the coarse-grid groundwater flow solutions are converted into the
vertically distributed fine-scale source and sink terms for the soil columns,
an extra downscaling approach is needed to ensure their accuracy. Here we
carried out a multi-scale water balance analysis above the phreatic surface.
The fine-scale saturated lateral flows were thus excluded from Eq. (10). The
benefits of the moving-boundary approach can be seen in case 2, which
produces significant saturated lateral flux. We carried out a comparative
analysis against the traditional stationary-boundary methods (Seo et al.,
2007; Xu et al., 2012). The 2-D solution of VSF was taken as the truth.
Figure 10 presents the effectiveness of the moving-boundary method. Five
stationary soil columns with three different lengths (

Comparison of RMSE of

In coupling two complicated modeling systems, a common agreement has been reached that; feedback coupling, either iteratively (Markstrom et al., 2008; Mehl and Hill, 2013; Stoppelenburg et al., 2005; Xie et al., 2012) or non-iteratively (Seo et al., 2007; Shen and Phanikumar, 2010; Xu et al., 2012), is numerically more rigorous than the one-way coupling scheme. The main difference between the above two methods lies in the ability to conserve mass within a single step for back-and-forth information exchange. In an iterative method, the head and/or flux boundaries are iteratively exchanged. There is a cost–benefit tradeoff for obtaining higher numerical efficiency.

During the late stages of the recharge in scenario 1 of case 1, the
groundwater table rises quickly, which increases the burden on the coupling
scheme. In our work, feedback iteration was conducted to eliminate the
coupling error within the back-and-forth boundary exchange. To investigate
how the feedback iteration influences the numerical accuracy as well as
computational cost, solutions were compared with different closure criteria,
instead of different maximal numbers of feedback iterations. For this
purpose, scenario 1 in case 1 is tested with a range of closure criteria
indicated by closure

When the wetting front approached the phreatic surface (at

The feedback coupling schemes, either iteratively or non-iteratively, increase the degree of freedom of the users to manage the sub-models with different governing equations, numerical algorithms, and the heterogeneities in parameters and variabilities in hydrologic stresses. For practical purposes, a significant concern is how to efficiently handle the complicated and scale-disparate systems.

For problems with rapid changes in groundwater flows, as in case 2, the hydraulic gradient at the phreatic surface was large. Using a single soil column for such a complex situation introduced significant coupling errors at the water table, (see Fig. 12a). Although more subzones portioned means higher accuracy for the coupling method, five or more soil columns were adequate enough in approaching the truth. Furthermore, for the saturated nodes deep in the aquifer, such differences in coupling errors were of minor influence (see Fig. 12b).

Comparison of

In case 3, a simple pumped and irrigated region was simulated with different numbers of soil columns. A range of tests with total numbers of 16, 12, 9, 5, and 3 soil columns were carried out to obtain a cost–benefit curve, shown in Fig. 13c. When partitioning the vadose zone into more than 12 soil columns, there was a slight reduction in solution errors (RMSE) and a significant increase in the computational cost caused by solving more 1-D soil-water models. Although parallelled computation could further reduce the numerical cost, representing the vadose zone with three sequentially calculated soil-water models achieved acceptable accuracy, as presented in Fig. 13a and b. The computational cost for obtaining the fully 3-D solution with VSF was 15.561 s, which was more than 11 times larger than an iterative feedback coupling method with soil-water models sequentially solved. Problems in more complicated real-world situations can thus be simplified to achieve higher numerical efficiency.

Prudic et al.'s problem was originally designed to validate a streamflow routing package (Prudic et al., 2004). Stressed by soil surface infiltration, pumping wells, and the general head boundary, the synthetic case was used to evaluate several unsaturated flow packages for MODFLOW (Twarakavi et al., 2008). Based on their studies, in case 4, we compared the developed iterative feedback coupling method with HPM. The hydraulic conductivity, as well as its heterogeneity, was forced to be consistent within the saturated and unsaturated zones, which is different from the case in Twarakavi et al. (2008). Figure 14a gives the contours for the final phreatic head solutions, indicating a good match of the phreatic surface with the HYDRUS package. Figure 14b–e present the absolute head difference of the method developed here and the HYDRUS package at the end of stress periods 3, 6, 9, and 12. The dark color blocks indicated the largest difference in the head solution. According to Fig. 6d, the saturated grid cells controlled by the soil columns of no. 3, no. 9, no. 10, no. 15, and no. 19 were suffering from the largest deviation, although with the same horizontal partitioning of the unsaturated zone. The strict iteratively two-way coupling contributes to such an accuracy improvement.

For unsaturated–saturated flow situations, the vadose zone flow is important. Figure 15 presents the water content profiles at subzones 1, 3, 5, 7, and 9 as examples. The solution obtained from the developed model matched well with the original HPM solution. For practical purposes, the manually controlled stress periods for the unsaturated sub-models are no longer constrained. In our method, the soil-water models run at disparate numerical scales, which makes it possible to handle daily or hourly observed information rather than a stress period lasting 2 or more days in traditional coupled models.

Comparison of water content profiles obtained from the HYDRUS
package for MODFLOW (Seo et al., 2007) and the developed iterative feedback
coupling method. Subzones 1, 3, 5, 7, and 9 are shown as an example. (

Fully 3-D numerical models are available but are numerically expensive for simulating the regional unsaturated–saturated flow. The quasi-3-D method presented here, in contrast, with horizontally disconnected adjacent unsaturated nodes, significantly reduces the dimensionality and complexity of the problem. Such a simplification brings about computational cost saving and flexibility for better manipulation of the sub-models. However, the nonlinearity in the soil-water retention curve, as well as the variability in realistic boundary stresses of the vadose and saturated zones, usually results in the scale-mismatching problems when attempting numerical coupling. In this work, the soil-water and groundwater models were coupled with an iterative feedback (two-way) coupling scheme. Three concerns about the multi-scale water balance at the phreatic interface are addressed using a range of numerical cases in multiple dimensionalities. We conclude with the following:

A new HYDRUS package for MODFLOW was developed by switching the

Stringent multi-scale water balance analysis at the water table was conducted to handle scale-mismatching problems and to smooth the information delivered back and forth across the interface. In our work, the errors originating from inadequate phreatic boundary predictions were reduced firstly by a linear extrapolation method and then by an iterative feedback. Compared with the traditional stepwise extension method, the linear extrapolation significantly reduced the coupling errors caused by scale mismatching. For problems with severe soil-water and groundwater interactions, the coupling errors were significantly reduced by using an iterative feedback coupling scheme. The multi-scale water balance analysis mathematically maintained numerical stabilities in the sub-models at disparate scales.

When a moving phreatic boundary was assigned to the soil columns during the phreatic water balance analysis, it avoided the coupling errors originating from the saturated lateral fluxes. In practical applications for regional problems, the fluxes into and out of the saturated parts of the soil columns differed, which added to the complexity and phreatic water balance error of the coupling scheme. With a moving Dirichlet lower boundary, the saturated regions of the soil-water models were removed. The coupling error was significantly reduced for problems with major groundwater flow. Extra cost saving was achieved by minimizing the matrix sizes of the soil-water models.

Future investigation will focus on regional solute transport modeling based on the developed coupling scheme. Surface flow models, as well as the crop models, which appear to be less nonlinear than the subsurface models, will be coupled in an object-oriented modeling system. The RS- and GIS-based data class can then be used to handle more complicated large-scale problems.

All the data and codes used in this study can be requested by email to the corresponding author Yuanyuan Zha at zhayuan87@gmail.com.

The bottom node of a soil column is adaptively located at the phreatic
surface, which makes it an area-averaged moving Dirichlet boundary:

To simulate the multi-scale flow process within a macro-time step

Flowchart of the relaxed iterative feedback coupling
scheme. The relaxation is conducted at the interfacial Dirichlet–Neumann
boundaries during the feedback iterations (except for the time

The moving Dirichlet boundary introduces the need for water balance of a
moving balancing domain above the water table (see Fig. 2b), which is bounded
by a specific elevation above the phreatic surface,

Assuming that the activated top layer in a three-dimensional groundwater model
is conceptualized into a phreatic aquifer, the governing equation for this
layer is given by

The relaxed feedback iteration method (Funaro et al., 1988; Mehl and Hill,
2013) is used to improve the convergence of head and/or flux at the phreatic
surface (see Fig. A1). The Dirichlet lower boundary head for the soil
columns,

JZ, YZ, and JY developed the new package for soil-water movement based on a switching Richards' equation; JZ and YZ developed the coupling methods for efficiently joining the sub-models. JZ, YZ, JY, and LS made non-negligible efforts in preparing the paper.

The authors declare that they have no conflict of interest.

This work was funded by the Chinese National Natural Science Foundation (no. 51479143 and 51609173). The authors would thank Ian White and Wenzhi Zeng for their laborious revisions and helpful suggestions for the paper as well as the editorial team for the HESS Journal. Edited by: Bob Su Reviewed by: two anonymous referees