Fluid flow through rock occurs in many geological
settings on different scales, at different temperature conditions and with
different flow velocities. Depending on these conditions the fluid will be
in local thermal equilibrium with the host rock or not. To explore the
physical parameters controlling thermal non-equilibrium, the coupled heat
equations for fluid and solid phases are formulated for a fluid migrating
through a resting porous solid by porous flow. By non-dimensionalizing the
equations, two non-dimensional numbers can be identified controlling thermal
non-equilibrium: the Péclet number

Fluid flow through rock occurs in many geological settings on different scales, at different temperature conditions and with different flow velocities. Depending on these conditions the fluid will be in local thermal equilibrium with the host rock or not. On a small scale, e.g., grain scale, usually thermal equilibrium is valid. Examples include melt migration through a porous matrix in the asthenosphere or in crustal magmatic systems at super-solidus temperatures (e.g., McKenzie, 1984), groundwater or geothermal flows in sediments or cracked rocks (e.g., Verruijt, 1982; Furbish, 1997; Woods, 2015), or hydrothermal convection in the oceanic crust (e.g., Davis et al., 1999; Harris and Chapman, 2004; Becker and Davies, 2004). On a somewhat larger scale local thermal equilibrium may not always be reached. Examples of such flows include melt migration in the mantle or crust at temperatures close to or slightly below the solidus where melt may be focused and migrates through systems of veins or channels (Kelemen et al., 1995; Spiegelman et al., 2001). Within the upper oceanic crust water may also migrate through systems of vents or channels (Wilcock and Fisher, 2004). At even larger scales and at sub-solidus conditions, magma rapidly flows through propagating dikes or volcanic conduits (e.g., Lister and Kerr, 1991; Rubin, 1995; Rivalta et al., 2015) and is locally at non-equilibrium with the host rock.

Heat transport associated with most of such flow scenarios is usually described by assuming thermal equilibrium between the fluid and solid under slow flow conditions (e.g., McKenzie, 1984). Alternatively, for more rapid flows such as melts moving in dikes through a cold elastic or visco-elasto-plastic ambient rock, the fluids are assumed to be isothermal (e.g., Maccaferri et al., 2011; Keller et al., 2013). However, on a local scale of channel or dike width, thermal interaction between rising hot magma and cold host rock is important and may lead to effects such as melting of the host rock and freezing of the magma with important consequences for dike propagation and the maximum ascent height (e.g., Bruce and Huppert, 1990; Lister and Kerr, 1991; Rubin, 1995). Clearly, in such rapid fluid flow scenarios melt is not in thermal equilibrium with the ambient rock.

Thus, there exists a transitional regime, which, for example, may be associated with melt focusing into pathways where flow is faster and thermal equilibrium might not be valid anymore. In such a scenario it might be possible that channelized flow of melt might penetrate deeply into sub-solidus ambient rock, and thermal non-equilibrium delays freezing of the ascending melts and promotes initiation of further dike-like pathways. Indeed, for mid-oceanic ridges compositional non-equilibrium has proven to be of great importance for understanding melt migration and transport evolution (Aharonov et al., 1995; Spiegelman et al., 2001). Thus, it appears plausible that in cases of sufficiently rapid fluid flow, e.g., due to channeling or fracturing, thermal non-equilibrium may also become important. Describing this non-equilibrium macroscopically, i.e., on a scale larger than the pores or channels, is the scope of this paper.

While the physics of thermal non-equilibrium in porous flow is well studied in more technical literature (e.g., Schumann, 1929; Spiga and Spiga, 1981; Kuznetsov, 1994; Amiri and Vafai, 1994; Minkowycz et al., 1999; Nield and Bejan, 2006; de Lemos, 2016), so far it has attracted only little attention in the geoscience literature, but see Schmeling et al. (2018) and Roy (2020). The basic approach in all these studies is the decomposition of the heat equation for porous flow into two equations, one for the solid and one for the migrating fluid. The key parameter for thermal non-equilibrium is a heat exchange term between fluid and solid, which appears as a sink in the equation for the fluid and as a source in the equation for the solid. Usually, this heat exchange term is assumed proportional to the local temperature difference between fluid and solid (Minkowycz et al., 1999; Amiri and Vafai, 1994; de Lemos, 2016; Roy, 2020). However, Schmeling et al. (2018) showed that in a more general formulation the heat exchange term depends on the complete thermal history of the moving fluid through the possibly also moving solid. Here we will follow the common assumption and use the local temperature difference formulation. While Schmeling et al. (2018) showed that the magnitude of thermal non-equilibrium essentially depends on the flow velocity, or more precisely on the Péclet number, here we will more generally explore the parameter space.

While thermal non-equilibrium of an arbitrary porous flow system depends on
many parameters, our approach is to reduce the complexity of the system and
systematically explore the non-dimensional parameter space. It will be shown
that only two non-dimensional parameters control thermal non-equilibrium in
porous flow, namely the Péclet number and the porosity. In our simple 1D
model setup with constant flow velocity a third non-dimensional number, the
non-dimensional initial thermal gradient

We start with considering a general two-phase matrix–fluid system with
variable properties and solid and fluid velocities and subsequently apply
simplifications. The two phases are incompressible, and we assume local
thermal non-equilibrium conditions; i.e., the two phases exchange heat. The
equations for conservation of energy of this system are given by de
Lemos (2016), for example. Assuming constant pressure, the conservation of energy of the
fluid phase is given by

Symbols, their definition, and physical units used in this study.

Non-dimensionalization is useful for interpreting models involving a large
number of parameters. It usually helps reduce the number of parameters
and identifies non-dimensional parameters that control the evolution of the
system. We write the two energy conservation equations in a non-dimensional
form, using

Introducing the fluid-filled pore width

We assume that the fluid and solid phases have the same densities and
thermal properties (but relax this assumption later in Sect. 5.1.3):

In the following we consider a homogeneous two-phase matrix–fluid system in
1D with a porosity constant in space and time, i.e.,

The fluid and solid heat conservation equations are solved in a 1D domain of
height

Initial and boundary conditions.

This model setup adds a third non-dimensional scaling parameter to the
system, namely

The equations are solved by a MATLAB (MATLAB R2021b) code using a finite
difference scheme central in space for the conduction terms, upwind for the
advection term, and explicit in time. The spatial resolution is

First, some example numerical results are shown in Fig. 2 to understand the physics and the typical behavior.

Typical model evolution for

Three different models have been run, all with

model 1 –

model 2 –

model 3 –

At each

The absolute maximum temperature difference in space and time does not
depend on boundary conditions (see also Sect. 5.1.2 where the influence of
boundary conditions is discussed) or on the

Comparing the heat fluxes of model 1 (

The maximum temperature difference of a model can be defined as the maximum
value reached in space and time (see Fig. 2f). A series of models have been
carried out for the two different non-dimensional parameters

For all

Maximum fluid–solid temperature differences

In this section a simplified analytical solution for the

We are interested in an analytical solution of Eq. (20) controlling the
non-equilibrium temperature difference

From Eq. (22) the maximum value of the depth-dependent temperature
difference

The analytical solution for

Equation (22) is, however, complicated, and the assessment of the relative
importance of

When

To obtain the limit of Eq. (22) for

Before exploring the full parameter space, we first give a short overview of expected parameter ranges in magmatic systems.

In natural magmatic systems such as mid-ocean ridges,

While the melt fraction does not influence

To get an idea about the expected order of magnitude of the macroscopic
thermal gradient

In Fig. 4 we explore

Main regimes of the maximum fluid–solid temperature
differences

Although the assumptions used to get the analytic solution (Eq. 22) are very
specific, they are reasonable considering the conditions in the models when

Comparison of depth- and time-dependent numerical
solutions with the time-independent analytical solutions for different
parameters

The boundary conditions we chose at the top (

Constant thermal gradient equal to the initial thermal gradient in the solid and fluid phases (Neumann condition). This was the boundary condition used in the models.

The thermal gradient is set to 0 at the top (Neumann condition).

Both fluid and solid temperatures are set to 0 at the top (Dirichlet condition).

Temperature at the top is numerically calculated from the full Eqs. (17) and (18) using one-sided (upwind) positions for the first and second derivatives (open boundary).

Temporal evolution of vertical profiles of

We have also tested an open boundary condition for the fluid and a Robin
boundary condition for the solid imagining a lid on top of our model with a
constant temperature gradient and a fixed surface temperature. Choosing the
surface temperature in such a way that the initial thermal gradient within
our system and within the lid are identical, this boundary condition adds a
new non-dimensional parameter, the lid thickness

In summary, the influence of boundary conditions on fluid and solid
temperature evolution depends mostly on the domain size

As an initial condition we used a linear temperature profile and initial
equilibrium between solid and fluid. A non-linear initial temperature
profile between

Time- and depth-dependent numerical solutions (thin
curves) as in Fig. 5 but for step-function initial conditions:

While for simplicity we used equal physical properties for the fluid and
solid, in many circumstances they might be significantly different. Equal
properties are good approximations for magmatic systems where differences of
density and thermal parameters are small (order of 10 %), whereas porous
flows of water or gases through rocks or other technical settings may be
characterized by larger differences. Allowing for different material
properties adds four new parameters, namely the ratio of diffusivities, the
ratio of densities, the ratio of heat capacities, and a new effective thermal
conductivity

As Eqs. (31) and (32) cannot be merged into one time-independent
ordinary differential equation for

Time- and depth-dependent profiles of the fluid–solid
temperature differences as in Fig. 5.

It is interesting to apply the results for different physical properties to
a geologically relevant setting, namely water flowing through sedimentary
rocks. Given that the high heat capacity of water is about 3 times
larger than that of rock, and the density is almost 3 times less, the
product

Time- and depth-dependent profiles of the fluid–solid
temperature differences as in Fig. 5, but for fluid-to-solid property ratios
typical for water flowing through sedimentary rocks, i.e.,

It is interesting to evaluate the timescales for reaching the maximum
non-equilibrium temperature differences and the steady state. For every
numerical model, we recorded the time needed to reach 90 % of the maximum
temperature differences between fluid and solid,

For evaluating timescales the numerically determined
times of models with various parameters

In regime 1 (high

In regime 2 (low

In regime 3 (low

We now test the possible occurrence of thermal non-equilibrium in natural
magmatic systems based on the suggested controlling non-dimensional
parameters, namely the Péclet number

partially molten regions with interstitial melts sitting at grain corners, grain edges, or grain faces with low (0.0001 %–6 %) melt fractions (see, e.g., the discussion in Schmeling, 2006);

merging melt channel or vein systems with high-porosity (

propagating dikes or other volcanic conduits.

10

10

Interstitial melts are at

Channeling and veining may result in

After transition to diking

To make our scaling laws and timescales for reaching maximum thermal
non-equilibrium more accessible, it is worth writing them in dimensional
form. First, to estimate the Péclet number of a natural system, combining Eqs. (9) and (16) gives

In conclusion, we showed that in magmatic systems characterized by two-phase flows of melts with respect to solid, thermal non-equilibrium between melt and solid may arise and becomes important under certain conditions. The main conclusions are summarized as follows.

From non-dimensionalization of the governing equations, three non-dimensional
numbers can be identified controlling thermal non-equilibrium: the Péclet
number

In regime 1 (high

In regime 2 (low

In regime 3 (low

The timescales for reaching thermal non-equilibrium scale with the
advective timescale in the high-

Applying the results to natural magmatic systems such as mid-ocean ridges
can be done by estimating appropriate orders of

In the studied setup

The derived scaling laws for thermal non-equilibrium are valid for equal solid and fluid properties. Assuming different properties such as for a water–sandstone system results in similar maximum non-equilibrium temperature differences, but in significantly different time evolutions.

The MATLAB code is listed in the Supplement and is available on request.

There are no underlying research data other than the graphs given in the figures. These graphs can be retrieved by the MATLAB programs given in the Supplement.

The supplement related to this article is available online at:

LC formulated the scientific question and the theory of the first version and wrote a first draft including model results. HS did the final writing up, improved the scaling, and reran all models.

The contact author has declared that neither they nor their co-author has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We gratefully acknowledge the excellent reviews by John Rudge and Cian Wilson, who stimulated us into significantly improving the scaling.

This research has been supported by the Deutsche Forschungsgemeinschaft (grant no. 403710316).

This paper was edited by Juliane Dannberg and reviewed by John Rudge, Cian Wilson, and Samuel Butler.