These authors contributed equally to this work.

Fluid and melt transport in the solid mantle can be modeled as a two-phase flow in which the liquid flow is resisted by the compaction of the viscously deforming solid mantle. Given the wide impact of liquid transport on the geodynamical and geochemical evolution of the Earth, the so-called “compaction equations” are increasingly being incorporated into geodynamical modeling studies. When implementing these equations, it is common to use a regularization technique to handle the porosity singularity in the dry mantle. Moreover, it is also common to enforce a positive porosity (liquid fraction) to avoid unphysical negative values of porosity. However, the effects of this “capped” porosity on the liquid flow and mass conservation have not been quantitatively evaluated. Here, we investigate these effects using a series of 1- and 2-dimensional numerical models implemented using the commercial finite-element package COMSOL Multiphysics^{®}. The results of benchmarking experiments against a semi-analytical solution for 1- and 2-D solitary waves illustrate the successful implementation of the compaction equations. We show that the solutions are accurate when the element size is smaller than half of the compaction length. Furthermore, in time-evolving experiments where the solid is stationary (immobile), we show that the mass balance errors are similarly low for both the capped and uncapped (i.e., allowing negative porosity) experiments. When Couette flow, convective flow, or subduction corner flow of the solid mantle is assumed, the capped porosity leads to overestimations of the mass of liquid in the model domain and the mass flux of liquid across the model boundaries, resulting in intrinsic errors in mass conservation even if a high mesh resolution is used. Despite the errors in mass balance, however, the distributions of the positive porosity and peaks (largest positive liquid fractions) in both the uncapped and capped experiments are similar. Hence, the capping of porosity in the compaction equations can be reasonably used to assess the main pathways and first-order distribution of fluids and melts in the mantle.

The fluid and melt within the Earth's mantle, as well as their transport from depth to surface, play a key role in the geodynamical and geochemical evolution of our planet. At depth, the presence of small fluid and melt fractions (up to 1 %–10 %) affects the bulk physical properties of mantle rocks (Mei et al., 2002; Zimmerman and Kohlstedt, 2004; Dohmen and Schmeling, 2021). This effect partly influences the vigor of mantle convection (e.g., Ogawa and Nakamura, 1998) and potentially assists the localization of deformations (Holtzman et al., 2003; Zimmerman and Kohlstedt, 2004; Katz et al., 2006) and may thus be a key ingredient for the functioning of plate tectonics. At the depths at which they are generated and through their journey to the surface, fluid and melt extract incompatible and fluid-mobile elements from the mantle rocks, thereby controlling planetary differentiation and contributing to the growth of the continental crust (Gerya and Meilick, 2011; Jagoutz and Kelemen, 2015). The ascent and eruption of magmas lead to the formation of volcanoes over and between tectonic plates, linking the evolution of the solid Earth to the evolution of the atmosphere (Lopez et al., 2023).

Because of the wide impact that fluid and melt have on the Earth system, it is crucial to constrain their migration pathways and spatial distribution. These can be inferred on the basis of geophysical imaging (e.g., magnetotellurics and seismic tomography). However, such methods lead to interpretations that are often non-unique given the dependence of the observables on multiple factors. Further, they are only present-day static images of a dynamic process. Forward modeling of liquid transport is a tool that can help to quantify the fluid and melt migration and their spatial distribution in the mantle and to study the coupled fluid/melt-mantle dynamics at a geodynamic scale (from 1 to 100s of kilometers).

One of the pioneering studies on liquid (aqueous fluid and melt) transport in the solid Earth was that of McKenzie (1984) (see also Scott and Stevenson, 1984; Fowler, 1985), who derived a two-phase flow theory based on continuum mechanics for a liquid in a viscously deforming porous solid matrix (mantle rocks). In this theory, a buoyant liquid phase percolates through the solid phase, where the liquid viscosity is many orders of magnitude lower than that of the permeable mantle matrix. The liquid flow follows Darcy's law but experiences resistance due to the compaction of the solid matrix. Using this theory, liquid flow was evaluated in various geodynamic settings, including mid-ocean ridges (Katz, 2008; Keller and Katz, 2016; Cerpa et al., 2018; Sim et al., 2020; Pusok et al., 2022), subduction zones (Dymkova and Gerya, 2013; Wilson et al., 2014; Cerpa et al., 2017, 2018; Rees Jones et al., 2018; Wang et al., 2019), continental rifts (Schmeling, 2010; Li et al., 2023), and an intraplate context (Keller et al., 2013; Dannberg and Heister, 2016).

The mantle away from the vicinity of the plate boundaries is generally thought to be relatively dry except in the specific regions where the presence of volatiles and melts has been suggested, e.g., in the shallow asthenospheric mantle (Chantel et al., 2016; Cerpa et al., 2019; Debayle et al., 2020) and near the 410 and 660 km discontinuities (Bercovici and Karato, 2003). However, in the application of the two-phase flow equations to the mantle, the near-zero porosity limit leads to a singularity, and it is therefore difficult to handle numerically (Arbogast et al., 2017; Dannberg et al., 2019). Thus, the equations are commonly regularized by imposing a small porosity across the entire model domain (e.g., Wilson et al., 2014; Cerpa et al., 2017). Along with an assumption of small porosity, it is also necessary to “cap” the porosity field to avoid the development of negative porosity values, which naturally arises from the governing equations. However, despite the widespread use of a capped porosity in numerical models, its impacts on the liquid flow and mass conservation have not been quantitatively evaluated.

In the present study, we investigate the effect of regularization with the capped porosity on the liquid flow and mass conservation in the two-phase flow model for the Earth's mantle using the commercial finite-element package COMSOL Multiphysics^{®} (COMSOL hereafter). COMSOL was used previously in the context of mantle convection and successfully benchmarked (e.g., Lee, 2013; Yu and Lee, 2018; Trim et al., 2021). For example, it was used to study the liquid transport in the mantle wedges of subduction zones when considering a simplified porous flow model that did not incorporate the effect of matrix compaction (e.g., Wada and Behn, 2015; Lee et al., 2021; Lee and Kim, 2021). It was also used to investigate compaction-driven segregation of porosity in shear bands (Butler, 2017). Here, we implement the governing equations that account for the compaction of the mantle matrix in COMSOL and validate the implementation by benchmarking the model solution against a semi-analytical solution for 1- and 2-dimensional (1- and 2-D) solitary waves. We then evaluate the effects of a capped porosity on liquid flow and mass conservation by comparing the mass balance between the capped and uncapped experiments using four different flow fields for the solid matrix: stagnant, Couette flow, convective flow, and subduction corner flow. One of the advantages of COMSOL is that it has the potential to perform coupling between different physics. Thus, the results of the present study can provide the basis for future applications of COMSOL for coupling the two-phase flow equations with other solid Earth processes, such as chemical reactions and heat transfer by liquids.

We follow the reformulation of the physics of two-phase flow in the mantle in which only the solid-state mantle flow (solid flow) influences the porous flow (i.e., one-way coupling) under the small-porosity approximation (e.g., Spiegelman, 1993; Katz et al., 2007; Katz, 2022). Such a formulation has been described in detail for previous studies which provided the derivation of the non-dimensionalized governing equations for the solid and liquid flow (Wilson et al., 2014; Cerpa et al., 2017). Below, we briefly describe the equations.

The governing equations for solid flow are the non-dimensionalized incompressible Stokes equations and the heat equation in a non-dimensionalized form (Wilson et al., 2014; Cerpa et al., 2017; Lee et al., 2021):

In what follows, we neglect the effect of the gradients of dynamic pressure term on the fluid flow since they are expected to be negligibly small compared to those of the compaction pressure in most of the regions of the convective mantle which we focus on (see discussion in Cerpa et al., 2017). Given this assumption, the non-dimensionalized governing equations for liquid flow are

The reference compaction length

The non-dimensional bulk viscosity

Given the relationships shown in Eqs. (8) and (9), Eqs. (4) and (5) become singular if

For the Stokes equations (Eqs. 1 and 2), we use the Creeping Flow (CF) module in COMSOL with quadratic and linear elements for the velocity and pressure, respectively. For the heat equation (Eq. 3), we use the Heat Transfer in Fluids (HT) module with quadratic and continuous Galerkin finite elements. The standard stabilization methods for the streamline and crosswind diffusions are used for both the CF and HT modules. To solve the time-dependent equations (Eqs. 1–3), we use the generalized-alpha method, adopting second-order accuracy of time integration with the direct, fully coupled PARDISO solver.

Equation (4) is solved using the Transport of Diluted Species (TDS) module with the stabilization method for the streamline diffusion. Equation (5) is solved using the TDS module for a benchmarking experiment against a semi-analytical solution for 1- and 2-D solitary waves (Sect. 3), and the generalized Coefficient Form of PDE (CFPDE) module is used for all the other experiments (Sect. 4) because the CFPDE module is more flexible with the boundary conditions that can be applied (e.g., the Weak Contribution option). We applied the CFPDE module to solve Eq. (5) to test its consistency with the TDS module, and the porosity differences were found to be smaller than 10

Simpson and Spiegelman (2011) derived a semi-analytical solution for a solitary wave which travels in the direction opposite to gravity (upward) at a fixed speed (

The governing equations for the solitary waves are as shown in Eqs. (4) and (5) with

The semi-analytic solution for a solitary wave defined for a particular choice of the triplet (

To quantify the growing error in the solitary wave with time, we calculate the phase shift and phase error of the wave relative to the semi-analytical solution along the vertical line that passes through the center of the model domain (e.g., Simpson and Spiegelman, 2011). The phase shift is estimated by tracking the location of the peak porosity value of the solitary wave relative to the central node (at a distance of 0.5) by fitting a second-order polynomial to the values at the central node and the nodes above and below it. The calculation of the phase error consists of two steps. First, the calculated porosity values at the nodes are interpolated using a piecewise cubic spline to obtain the waveform; then, the waveform is migrated back by the phase shift and the phase error is calculated over the nodes as follows:

We benchmark the model solutions using three solitary wave solutions for different choices of the triplet (

We evaluate the effect of the element size on the solitary waves with the triplet (

The 1-D experiment using an element size of

Overall, with the refined initial condition, the absolute net phase shift is relatively small (

Although the benchmark experiment is not designed for a specific spatiotemporal scale of geological interest, it is worth noting that the solution remains accurate up to a dimensional time of 0.05 Myr (the dimensional time using the model parameters shown in Table 1). The time-evolving problems described below consider longer timescales relevant to geological applications.

Model parameters.

Although the benchmarking models described above verify the successful implementation of the compaction equations, the effects of the capped porosity on the liquid flow and mass conservation should be quantitatively evaluated. In the evaluation, we start with the simplest case – where the solid does not flow (a stagnant solid) – and then apply three solid-flow patterns that are applicable to Earth's mantle: Couette flow, convective flow, and subduction corner flow. Although no analytical solution exists for the modeling schemes, the relatively simple flow patterns that are applied in the models allow reasonable quantification of the sensitivity of liquid flow and mass conservation to the use of a capped porosity.

To monitor the accuracy of our computations over time, we evaluate the mass balance of the liquid (e.g., Lee et al., 2021). Since we assume a constant liquid density, this is equivalent to evaluating the volume balance of the liquid. The latter evaluation involves two steps. First, we evaluate the accumulated volume of liquid (

Theoretically, assuming that the time-integration scheme is accurate, the sum

Firstly, we consider a 2-D time-evolving problem with a prescribed porosity at the bottom boundary of a square domain (the height and width are 50, equivalent to 50 km in dimensional units) of a stagnant (immobile) porous solid (Fig. 3a). We solve Eqs. (4) and (5) with

Model boundary conditions used for the 2-D models of two-phase flow with four different solid flow patterns:

The Dirichlet liquid boundary condition at the bottom boundary is specified using a Gaussian function:

where

A constant time step of 0.02 (2000 years) is used to satisfy the Courant criterion. The liquid flow is calculated for a model time of 300 (30 Myr). We use square elements of size

In both the capped and uncapped experiments, the solitary waves ascend vertically from the bottom boundary. With time, the waves tend to become vertically elongated, eventually merging into a channel with periodic highs and lows of porosity (Fig. 4a and b). In the uncapped experiment, negative porosity values down to

In the capped experiment, the accumulated volume of liquid (

The evolution of the relative volume-balance error (Fig. 4f) illustrates the influence of the enforced positive porosity in the capped experiment. The enforcement results in an overestimation of the net accumulated volume of liquid until a model time of

Here, we consider the evolution of a liquid flow through a Couette flow of a porous solid from left to right (maximum solid velocity: 3, equal to 3 cm yr

Due to the Couette flow, the solitary waves originating from the bottom boundary are diverted rightwards in both the capped and uncapped experiments (Fig. 5a and b). With time, the solitary waves tend to form channels displaying periodic highs and lows of porosity in both experiments, leading to very similar peak volume outfluxes of liquid at the right boundary (Fig. 5c). However, the uncapped experiment additionally develops two negative-porosity channels under the positive-porosity channel, yielding negative-volume outfluxes of liquid at the right boundary (at

The time evolutions of the integrated volume outfluxes of liquid at the right boundary (

The enforced positive porosity in the capped experiment leads to a larger net accumulated volume of liquid compared to that in the uncapped experiment, in which the accumulated volume is counterbalanced by negative porosity. Thus,

Here, we consider the evolution of the liquid flow through a convective porous solid using the same model domain, boundary conditions, and methods shown in Sect. 4.1.1 (Fig. 3c). We apply free slip to all four boundaries to solve the solid flow. To solve the heat equation, the top and bottom temperatures are fixed at 0 and 1 (0 and 1000

Due to the clockwise solid convection, the solitary waves ascending from the bottom boundary are diverted leftwards and rightwards in the lower and upper model domains, respectively, in both the capped and uncapped experiments (Fig. 6a and b). Most of the ascending liquid leaves the domain through the top boundary, but a fraction of it continues to be entrained in the convective solid and remains within the domain. The entrained liquid then merges with newly ascending liquid from the bottom boundary. With time, the waves tend to form a quasi-steady-state channel in the capped experiment, which leads to a quasi-steady-state integrated volume outflux of liquid (which slowly increases) through the top boundary (

In both the capped and uncapped experiments, the evolution of

As in Sects. 4.1 and 4.2, the enforced positive porosity in the capped experiment results in an overestimation of

Lastly, we consider the evolution of the liquid flow through a subduction corner flow in which the solid-state flow in the corner wedge is kinematically driven by the subducting slab. The height and width of the model are 50 and 52.8 (equal to 50 and 52.8 km), respectively (Fig. 3d). The subducting slab has a dip of 45

To solve the solid flow, free-slip and open-boundary conditions are prescribed for the top and right boundaries of the mantle wedge, respectively (Fig. 3d). To reach steady-state corner flow in the mantle wedge, Eqs. (1) and (2) without the buoyancy term in Eq. (2) are solved for a modeling period of 500 (50 Myr) (e.g., Yu and Lee, 2018).

To solve the liquid flow, free-outflux and zero-influx boundary conditions are prescribed for all the boundaries except for the base of the uppermost slab layer, which is prescribed as having zero porosity (Fig. 3e). A zero gradient of compaction pressure is prescribed for all the boundaries. A Gaussian source term (the same function as in Eq. 16) for

The general trends for the liquid flow in the wedge are similar in both the capped and uncapped experiments, both of which tend toward stable dynamics after a model time of 50. Due to the downdip solid flow, the solitary waves originating from the top layer tend to be diverted slightly rightwards as they ascend through the bottom half of the wedge. In the top half of the wedge, the inward (leftward) corner flow causes leftward advection of the ascending waves before they reach the top boundary (Fig. 7a and b). Although most of the liquid passes upward through the wedge, a fraction of it is entrained by the corner flow and leaves the model domain across the right boundary.

Both the capped and uncapped experiments yield periodic high and low volume fluxes at the top and right boundaries (

Compared to that in the capped experiment, the uncapped experiment shows a lower

To check the sensitivity of our results to the choice of element size, we run additional experiments with square element sizes of

All the stagnant porous solid experiments show similar porosity evolutions regardless of the element size chosen. It is important to note that the uncapped and capped experiments converge towards different minimum errors. A significant decrease in the relative volume-balance error is observed with increasing mesh resolution in the uncapped experiments; the error diminishes to

Relative volume-balance error calculated from capped and uncapped experiments with various element sizes ranging between

All the other experiments (those for Couette flow, convective flow, and subduction corner flow) show similar porosity evolutions regardless of the element size chosen. In all cases, the uncapped experiment always shows better liquid volume conservation (i.e., smaller absolute error values) with the increase in mesh resolution because the outflux at the boundaries is more accurately calculated at increased mesh resolution (Fig. 8b–d). The increase in mesh resolution in the capped experiments does not remove the intrinsic error that results from the overestimated porosity in the model domain and the subsequent overestimation of the outflux at the boundaries, though the outflux at the boundaries is more accurately calculated at increased mesh resolution.

In this study, we first conducted a series of benchmarking experiments against a semi-analytical solution for solitary waves (Simpson and Spiegelman, 2011). Although the specifics of the numerical approach used in the study (nonlinear solvers, finite element order, continuous Galerkin finite elements, etc.) differ from those used in previous studies (e.g., Simpson and Spiegelman, 2011; Wilson et al., 2017; Wang et al., 2019), we obtain a relatively accurate solution when the element size is

Next, we performed time-evolving experiments. Our capped experiments show that the best accuracy for mass conservation occurs in models where the background mantle is stationary (Figs. 4a and 8a) after those models have passed their early stages. In other experiments where the solid flows (Couette flow, convective flow, and subduction corner flow), the error in the mass balance is not negligible owing to the overestimation of porosity induced by the capped porosity and the subsequent overestimation of the mass flux across the model boundaries. Because the overestimation in the capped experiments is intrinsic, increasing the mesh resolution does not significantly reduce the relative volume-balance error (Fig. 8). Thus, the estimated volume of liquid in the model domain and the volume flux of liquid through the model boundaries (e.g., the amount of magma in the mantle and the migration of magma through the overlying lithosphere) should be carefully interpreted when the solid phase is deforming. We thus emphasize that, in future applications, caution should be taken when using the melt and flux estimations. However, the distributions of the positive porosity and peaks (the largest positive liquid fractions) in the model domain and boundaries, respectively, are quite similar in both the capped and uncapped experiments. Hence, it is reasonable to use a capped porosity in the compaction equations for assessing the main fluid pathways and the first-order distribution of fluids and melts in the mantle. Models of liquid transport in the mantle with a capped porosity field may be compared to geophysical imaging (e.g., magnetotellurics and seismic tomography) of the asthenosphere, which illuminates the first-order distribution of fluids and melts in, for example, the sub-arc mantle wedges of subduction zones (Mcgary et al., 2014; Cordell et al., 2019; Bie et al., 2022).

The models were run with the commercial finite-element package COMSOL Multiphysics^{®} (

The data used to generate the figures are shared at

CL and NC conceived the study. CL and NC designed and ran the models, analyzed the results, and wrote the article. DH revised the Python code for the benchmarking experiments and produced the figures. IW wrote the article. All authors discussed the results and their consequences and contributed to the writing of the final article.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We thank the editor Juliane Dannberg, the reviewer Samuel Butler, an anonymous reviewer, and Chenyu Tian for their constructive comments which have improved our manuscript.

This research has been supported by the National Research Foundation of Korea (grant no. 2022R1A2C1004592 for CL and DH) and the National Science Foundation (grants no. EAR-2246804 for IW).

This paper was edited by Juliane Dannberg and reviewed by Samuel Butler and one anonymous referee.

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