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 Pe describing the fluid velocity and
the porosity ϕ. The equations are solved numerically for the fluid and
solid temperature evolution for a simple 1D model setup with constant flow
velocity. This setup defines a third non-dimensional number, the initial
thermal gradient G, which is the reciprocal of the non-dimensional model height
H. Three stages are observed: a transient stage followed by a stage with
maximum non-equilibrium fluid-to-solid temperature difference, ΔTmax, and a stage approaching the steady state. A simplified
time-independent ordinary differential equation for depth-dependent Tf-Ts is derived and solved analytically. From these
solutions simple scaling laws of the form Tf-Ts=fPe,G,z are derived. Due to scaling they do not
depend explicitly on ϕ anymore. The solutions for ΔTmax and
the scaling laws are in good agreement with the numerical solutions. The
parameter space PeG is systematically explored. Three regimes can be
identified: (1) at high Pe (>1/G) strong thermal non-equilibrium
develops independently of Pe, (2) at low Pe (<1/G) non-equilibrium
decreases proportional to decreasing Pe⋅G, and (3) at low Pe (<1)
and G of the order of 1 the scaling law is ΔTmax≈Pe. The scaling
laws are also given in dimensional form. The dimensional ΔTmax
depends on the initial temperature gradient, the flow velocity, the melt
fraction, the interfacial boundary layer thickness, and the interfacial area
density. The time scales for reaching thermal non-equilibrium scale with the
advective timescale in the high-Pe regime and with the interfacial diffusion
time in the other two low-Pe regimes. Applying the results to natural
magmatic systems such as mid-ocean ridges can be done by estimating
appropriate orders of Pe and G. Plotting such typical ranges in the Pe–G regime
diagram reveals that (a) interstitial melt flow is in thermal equilibrium, (b) melt channeling such as revealed by dunite channels may reach moderate
thermal non-equilibrium with fluid-to-solid temperature differences of up to
several tens of kelvin, and (c) the dike regime is at full thermal
non-equilibrium.
Introduction
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 G, is identified, which is equal to
the reciprocal non-dimensional model height H=1/G. The
non-dimensionalization allows application of the results to arbitrary
magmatic or other systems. The aim is to derive scaling laws that allow an
easy determination of whether thermal equilibrium or non-equilibrium is to
be expected and quantitatively to estimate the maximum temperature
difference between fluid and matrix. The results will be applied, the
magmatic system of a mid-ocean ridge setting.
Governing equations and model setupHeat conservation equations
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
cp,f∂ϕρfTf∂t+∇⋅ϕρfvfTf=∇⋅ϕλf∇Tf-Qfs.
For the definition of all quantities, see Table 1. Equation (1) can be
rearranged into
cp,fTf∂ϕρf∂t+ϕρf∂Tf∂t+Tf∇⋅ϕρfvf+ϕρfvf⋅∇Tf=∇⋅ϕλf∇Tf-Qfs.
Mass conservation for the fluid phase is given by
∂ρfϕ∂t+∇⋅ρfϕvf=0.
Inserting Eq. (3) into Eq. (2), conservation of energy for the fluid phase becomes
cp,fρfϕ∂Tf∂t+vf⋅∇Tf=∇⋅ϕλf∇Tf-Qfs.
In a similar way, the conservation of energy of the solid phase is given by
cp,sρs1-ϕ∂Ts∂t+vs⋅∇Ts=∇⋅1-ϕλs∇Ts+Qfs,
which, assuming that vs=0, is further simplified:
cp,sρs1-ϕ∂Ts∂t=∇⋅1-ϕλs∇Ts+Qfs.
The term Qfs in the fluid and solid heat conservation equations is the
interfacial heat exchange term between the two phases (fluid and solid). In
general, it depends on the local thermal history of the two phases and the
history of the heat exchange (Schmeling et al., 2018). In a simplification
it can be written as a combination of the interfacial area density S with the
dimension [m-1], the interfacial boundary layer thickness δ, the
effective thermal conductivity λeff, and the temperatures of
the two phases:
Qfs=Sλeffδ(Tf-Ts).
In general, the term δ is time dependent. Schmeling et al. (2018),
however, provide evidence that taking an appropriate constant value for
δ (depending on fluid velocity) gives a good approximation of
Qfs and allows for reasonable modeling of temperature evolution with
time. In most of the following parametric study, we use this simplification
for δ by assuming it is constant with time.
Symbols, their definition, and physical units used in this
study.
SymbolDefinitionUnitscp,f,s,0Specific heat at constant pressure for the fluid, solid, or reference, respectivelyJ kg-1 K-1c, csGeometrical constant for fluid pore space or solid phase, respectively. For melt channels or low porosity films c=2 and for tubes c=4 (Eqs. 11, 12)–cthConstant for thermal boundary layer and 2.32 for cooling half space–ds, dfCharacteristic length scale of solid or fluid phase, respectivelymfSubscript used for fluid–gFunction describing part of the ϕ dependence of df and ds (Eq. 35)–GInitial temperature gradient, taken positive for temperature decreasing with height, mostly non-dimensional(T m-1)HHeight of the model, mostly non-dimensional(m)LScaling length used for non-dimensionalization (Eq. 9)mM(z)Function describing the depth dependence of the analytical solution of Tf-Ts for small Pe (Eq. 27)–Pe, PeDPéclet number based on fluid velocity (Eq. 16) or based on Darcy velocity (Eq. 33), respectively–QfsInterfacial heat exchange rate from fluid to solidJ s-1 m-3r1, r2Constants of analytical solution (Eq. 23)–sSubscript used for solid–SInterfacial area density, i.e., interfacial area per volumem-1t, tcharTime, characteristic timescales, respectively. “char” indicates the characteristic time for diffusion or advection over a characteristic length L or H: diffL, diffH, advL, and advHst0Scaling time (Eq. 10)sTf,sTemperature of the fluid or solid, respectivelyKΔT0, ΔTmaxInitial temperature difference between top and bottom used as scaling temperature, and maximum difference between fluid and solid temperature in space and time, respectivelyKvf,sVelocity of the fluid or solid, respectivelym s-1vf0Constant fluid velocity in the model, used for scalingm s-1vDVolumetric flow rate (Darcy velocity) (=ϕvf)m s-1x, y, zCoordinates, distancemα, βFunctions used for analytical solution (Eq. 24)–δInterfacial boundary layer thicknessmκf,s,0Thermal diffusivity of the fluid, solid, or reference, respectivelym2 s-1λf,sThermal conductivity of the fluid or solid, respectivelyW m-1 K-1λeffEffective thermal conductivity at the solid–fluid interfaceW m-1 K-1ϕ,ϕ0Porosity or scaling porosity, respectively–ρf,s,0Density of the fluid, solid, or reference, respectivelykg m-3Scaling and non-dimensionalization
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
T=ΔT0T′,t=t0t′,v=vf0v′,x,y,z=L⋅x′,y′,z′,
where ΔT0 is the macroscopic scaling temperature difference of
the system, i.e., the initial temperature difference between top and bottom;
x,y, and z are distance; vf0 is the scaling fluid velocity; L is the scaling
length
L=ϕ01-ϕ0δS,
with ϕ0 as a scaling porosity; and t0 is the scaling time
based on the diffusion time over the length L,
t0=L2/κ0
(see Table 1 for definitions). Primed quantities are non-dimensional.
Introducing the fluid-filled pore width df and the solid width
ds, which may be the grain size or distance between fluid channels, the
interfacial area density S scales with
S=cϕ0df
for melt channels, tubes, pockets for all melt fractions, and melt films
at small melt fractions, while S scales with
S=cs1-ϕ0ds
for melt channels, films, and suspensions at all melt fractions. Here c is a
geometrical constant of the order of 2 for melt channels, 4 for melt tubes, 6
for melt pockets, and 2 for melt films at small melt fractions. It should be
emphasized that Eqs. (11) and (12) are different ways of calculating the
same S. The geometrical constant cs is of the order of 2 for melt channels
and 6 for melt films or suspensions. Thus, the scaling time and scaling
length can also be written as
t0=1-ϕ0dfδcκ0=ϕ0dsδcsκ0
and
L=1-ϕ0δdfc=ϕ0δdscs.
Equation (9a) shows that L scales both with the geometric mean of df and
δ at small melt fractions and with the geometric mean of ds
and δ at large melt fractions. Thus, L is a natural length scale
associated with thermal equilibrium of fluid-filled pores. The above scaling
laws for S justify using the term ϕ01-ϕ0 in
the scaling length L. It should be noted that we introduce and understand
ds as the average distance between melt-filled pores or channels, which
can be considerably larger than the grain size. Then both δ and
ds and thus L can reach some considerable fraction of the system
dimension.
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):
cp,f=cp,s=cp,0,ρf=ρs=ρ0,κf=κs=λeffcp,0ρ0=κ0.
From Eqs. (4), (6), and (7) we get the non-dimensional energy conservation
equations for the fluid and solid phases, respectively:
14ϕ∂Tf′∂t′+Pevf′⋅∇Tf′=∇⋅ϕ∇Tf′-ϕ01-ϕ0Tf′-Ts′,151-ϕ∂Ts′∂t′=∇⋅1-ϕ∇Ts′+ϕ01-ϕ0Tf′-Ts′.
From these equations we notice that the thermal evolution of the two-phase
system is controlled by two non-dimensional numbers: the scaling porosity
ϕ0 and the Péclet number Pe defined as
Pe=vf0Lκ0.
This number has already proven to be of high significance for determining
whether thermal non-equilibrium is present or not (Schmeling et al., 2018),
and the highest Pe corresponds to the largest temperature difference between
fluid and matrix. In the following we drop the primes, keeping all equations
non-dimensional, if not indicated otherwise.
In the following we consider a homogeneous two-phase matrix–fluid system in
1D with a porosity constant in space and time, i.e., ϕ=ϕ0. We assume a constant fluid velocity which will be expressed
in terms of Pe, thus we choose the non-dimensional velocity
vf=1. This simplifies Eqs. (14) and (15) to
∂Tf∂t+Pe∂Tf∂z=∂2Tf∂z2-1-ϕ0Tf-Ts
and
∂Ts∂t=∂2Ts∂z2+ϕ0Tf-Ts,
respectively. As we are interested in the evolution of the non-equilibrium
temperature difference between the solid and fluid, subtraction of Eq. (18)
from Eq. (17) gives
∂Tf-Ts∂t-∂2Tf-Ts∂z2+Pe∂Tf∂z+Tf-Ts=0,
which is equivalent to
∂Tf-Ts∂t-∂2Tf-Ts∂z2+Pe∂Tf-Ts∂z+Tf-Ts=-Pe∂Ts∂z.
Note that while the temperatures Tf and Ts explicitly depend on two
non-dimensional numbers Pe and ϕ0, the temporal evolution of the
temperature difference Tf-Ts explicitly depends only
on Pe. However, implicitly it is still a function of ϕ0 because
Ts on the right-hand side of Eq. (20) depends on ϕ0 via Eq. (18). Only for cases or stages with Ts independent of ϕ0 as
proposed in Sect. 4 is the temperature difference Tf-Ts a function of only one non-dimensional parameter, Pe, and not
of ϕ0.
Model setup
The fluid and solid heat conservation equations are solved in a 1D domain of
height H. Other geometries could also be easily explored but are not
considered here, since we focus on studying the relative control of the
scaling parameters on thermal non-equilibrium evolution. At time t<0, both solid and fluid are at rest, in equilibrium. Both initial
temperatures decrease linearly from 1 to 0 with z; therefore a constant
temperature gradient of -G=-1/H is present in both phases (see Fig. 1). As a boundary condition both phase temperatures are set to 1
(non-dimensional temperature) at z=0. At z=H a constant thermal gradient
condition ∂T/∂z=-1/H (non-dimensional) is imposed for both
phases. At z=0 the advective flux is fixed by the constant temperature
condition (i.e., it is equal to Peϕ0) while at z=H it evolves
freely with the fluid temperature (i.e., it is given by TfPeϕ0,
all non-dimensional). This top boundary condition needs some justification:
the hyperbolic partial differential Eqs. (17) or (18) require two
well-defined boundary conditions each, Dirichlet (fixed temperature),
Neumann (fixed thermal gradient), Robin (linear combination of Neumann and
Dirichlet), or Cauchy (fixed temperature and thermal gradient). Applying the
Dirichlet condition at the bottom leaves either a Dirichlet, a Neumann, or a
Robin condition to specify for the top. Different combinations of these
boundary conditions can be applied separately for the fluid and the solid.
For example, for the fluid a Neumann condition with zero or a small
temperature gradient may be reasonable, while for the solid one may consider
a Robin boundary condition mimicking a thick conductive lid with an internal
constant temperature gradient and a fixed surface temperature. However,
temporal changes of the temperature at the top (for our system, equal to the bottom of
the lid) would lead to unphysical variations in the constant slope of the
temperature gradient within the imagined lid. This is because the
temperature in the lid can only vary on the diffusive timescale of the lid,
which is much longer than all timescales in our model as long as the lid is
thicker than H. In fact, in an open outflow situation like our system
the evolution of the temperature, the thermal gradient, and the total
(advective plus conductive) heat flux are not known a priori, but depend on the
evolution within the system. In the early stage of model evolution, both the
solid and fluid have a thermal gradient inherited from the initial condition,
which is advected upwards in the fluid. Thus it seems most appropriate to
use the Neumann condition as a boundary condition for both the solid and
fluid. Only at later stages does this boundary condition impose artifacts in the
temperature field close to the top boundary. The limitations of this top
boundary condition are tested and discussed in Sect. 5.1.2.
Initial and boundary conditions.
This model setup adds a third non-dimensional scaling parameter to the
system, namely G=1/H. It defines the initial non-dimensional temperature
gradient or conductive heat flux, positive for a flux directed upwards. To
summarize, the temperatures depend on the non-dimensional parameters Pe, ϕ0, and G.
Numerical scheme
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 dz=0.1 or,
for a few cases in Fig. 3 below, min0.1,H/100 for
H<10. The time step was chosen as dt=14mindz/Pe,dz2, i.e., taking the minimum of the Courant or diffusion criterion.
Tests with higher spatial and temporal resolution have been carried out and
did not change the results visibly. The global heat balance has also been
checked: the maximum relative heat balance error can be defined as δq=qtotz=0-qtotz=H-H∂Tmean∂tqtotz=0+qtotz=H/2max , where
qtot is the total non-dimensional vertical heat flux (conductive and
advective) and Tmean is the mean temperature of the model. δq has an error on the order of 1 (due to the upwind scheme) with respect to the
grid size dz; i.e., it is approximately equal to const⋅dz, where
the constant is of the order of 0.2 (i.e., 2 %) for high Péclet numbers and
drops to 0.1 (1 %) or 0.01 (0.1 %) for Pe≅1 or smaller,
respectively.
Numerical model results
First, some example numerical results are shown in Fig. 2 to understand the
physics and the typical behavior.
Typical model evolution for Pe=1, two different melt fractions ϕ, and two different non-dimensional
temperature gradients G (i.e., heights
H). (a) Model 1 is with
G=0.1 (H=10) and
ϕ=0.1. Red and blue curves show the fluid and solid
temperatures at different non-dimensional times t
as indicated by the legend, respectively. Initial temperatures are almost
identical to the t=0.5 curves. Steady state is reached at about
t=100; the curves of the last two times plot on each
other. (b) Model 2 with G=0.01 (H=100), otherwise as in (a). Steady state
is not fully reached. (c) Conductive (blue and red curves) and advective
(magenta curve) heat fluxes through the solid and fluid, respectively, of
model 1. Line styles indicate the same times as in (a). (d) Same as (c) but for
model 2. (e) Temporal evolution of fluid and solid temperatures,
Tf (red) and
Ts (blue), respectively, at the top
of model 2 with ϕ=0.1 and model 3 with ϕ=0.2. G=0.01
(H=100) for both models. (f) Evolution of fluid–solid temperature difference
(Tf-Ts) at different distances
z in model 2 (ϕ=0.1, solid
curves) and in model 3 (ϕ=0.2, dashed curves). (g) Zoomed-in early temporal evolution of solid and fluid temperatures of models 2 and 3 shown in (e). (h) Zoomed-in early temporal evolution of temperature
difference of models 2 and 3 shown in (f).
Evolution of temperatures and thermal non-equilibrium with time
Three different models have been run, all with Pe=1 and the following other
parameters:
model 1 – G=0.1 (H=10), ϕ=0.1,
model 2 – G=0.01 (H=100),
ϕ=0.1, and
model 3 – G=0.01 (H=100), ϕ=0.2.
Figure 2a
and b show Tf and Ts as functions of z at different times as
indicated for two initial temperature gradients, G=0.1 (H=10) and G=0.01
(H=100). Figure 2c shows the different contributions to the
depth-dependent conductive and advective heat fluxes through the solid and
fluid phases, respectively. Figure 2e shows the evolution of Tf and
Ts with time at the top of the domain, for the same model 2 as in Fig. 2b and for model 3 with a higher melt fraction ϕ=0.2. Figure 2f
shows the evolution of (Tf-Ts) at different distances z of model
2 (ϕ=0.1) and of model 3 (ϕ=0.2). At each depth of the
system, the fluid and solid temperatures, as well as the temperature
difference and the heat fluxes, evolve following three stages.
Stage 1. During this transient stage the fluid temperature
increases faster than the solid temperature (Fig. 2a, b, e, f), and the
temperature difference (Fig. 2f, h) increases. During this stage, the fluid
temperature increases rapidly at first, and then the temperature increase slows
down. The conductive heat fluxes in both solid and fluid decrease rapidly
and more slowly later, while the advective heat flux rapidly increases. As
for the solid temperature, it first increases slowly and then faster and
faster. At t=0, the fluid velocity is suddenly set to non-zero; thus the
fluid temperature starts to deviate from equilibrium and increases due to
these new conditions. If the solid temperature were maintained constant with
time, the fluid temperature would probably reach a steady-state profile,
depending on boundary conditions, fluid velocity, and solid temperature.
While the fluid temperature increases faster than the solid temperature, the
fluid–solid temperature difference, and thus the heat transfer term, increases
too, forcing the solid temperature to progressively increase. At the end of
stage 1 the maximum temperature difference is approached (Fig. 2h). Because
the solid temperature has not risen significantly at that time (at t=4 in
the example) compared to the fluid temperature (Fig. 2g), different melt
fractions do not affect the temperature differences during this stage (Fig. 2h in which all curves merge into one curve). This observation confirms the
expectation from Eq. (20) that the temperature difference does not depend on
melt fraction as long as the solid temperature is independent of ϕ, which
is the case as long as Ts stays close to its initial profile.
Stage 2. The fluid and the solid temperatures increase at similar
rates, constant with time (Fig. 2e), and the temperature difference remains
constant and at a maximum at the top (Fig. 2f). Solid–fluid heat transfer is
at a maximum during this stage. As Ts is no longer constant in
time, different melt fractions lead to different rates of temperature
increase (Fig. 2e) and also to different evolutions of (Tf–Ts) (Fig. 2f solid curves compared to dashed curves). At higher melt fraction the heat
transfer into the solid increases (see last term in Eq. 18), resulting in a
faster increase in the solid temperature whose gradient flattens earlier.
Thus, the end of stage 2 is reached earlier (Fig. 2e).
Stage 3. As the fluid temperature rises closer to the Tf value at the bottom, its increase slows down, and heat transfer, and thus
temperature difference, decreases. In model 1 (Fig. 2a), steady state is
reached while the fluid and solid temperatures are still far from 1. This is
due to the influence of boundary conditions, as the heat transferred from
the fluid phase to the solid phase is compensated for by the solid phase heat
loss at the top of the domain. In model 2 (Fig. 2b), boundary conditions at
z=H are applied farther away from the bottom, therefore allowing for a
higher increase in temperatures when reaching the steady state.
At each z we observe that the temperature difference first increases rapidly
to reach a maximum after a short time (stage 1), hereafter t=4 (Fig. 2h).
The resulting amplitude of the temperature difference is identical at the
different z positions and for both melt fractions. Then it stays constant at
this maximum value (stage 2) and finally decreases (stage 3) (Fig. 2f). The
higher in the model, the longer the temperature difference remains at
maximum. A higher melt fraction accelerates the decrease in (Tf-Ts).
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 z position or on the melt
fraction and therefore looks to be an interesting observable. It could
indeed be useful for getting a first-order estimate of thermal
non-equilibrium conditions and possible temperature differences in a
magmatic system. In the following sections we study how this maximum
temperature difference evolves when varying the parameter Pe.
Comparing the heat fluxes of model 1 (G=0.1) with those of model 2
(G=0.01) shows the importance of heat advection by the fluid phase: in
model 1 (Fig. 2c) the conductive contribution through the solid is of the same
order of magnitude as the advective contribution by the fluid because the
initial temperature gradient G and the porosity φ are the same
(=0.1). In model 2 (Fig. 2d), in spite of the same Péclet number, the
smaller initial temperature gradient (G=0.01) reduced the conductive contribution with
respect to the advective contribution by a factor of about 10, and advection
dominates. Furthermore, the two models demonstrate that the conductive heat
flux contributions may be important with respect to advective and interphase
heat flux for sufficiently large initial thermal gradient (here 0.1), while
it has been neglected in several earlier investigations (e.g., Schumann,
1929; Spiga and Spiga, 1981).
Maximum temperature difference
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 Pe and
G=1/H, and ΔTmax has been determined for each model (Fig. 3).
Some first observations can be made.
For all Pe values, ΔTmax is proportional to Pe (Fig. 3a) as long as ΔTmax is somewhat smaller than the absolutely possible maximum 1, which
is asymptotically approached for high Pe.
ΔTmax is proportional to G, i.e., to the non-dimensional temperature
gradient for G<0.1.
ΔTmax reaches a maximum for large G of the order of 1, i.e., when H reaches
1 or the dimensional H reaches the length scale L.
ΔTmax is essentially independent of ϕ as models with
different ϕ values almost merge in the same points shown in Fig. 3. This has
been verified by running all models of Fig. 3 with melt fractions between
0.1 and 0.9 (not shown).
These observations suggest the existence of several domains in which scaling
laws for ΔTmax could be derived, based on the two scaling
parameters. In the next section, we propose an analytical derivation of
ΔTmax values to obtain scaling laws and confirm the observed
proportionalities.
Maximum fluid–solid temperature differences
Tf-Ts of
numerical models (asterisks) with different parameters, plotted (a) as a
function of the Péclet number Pe for
G=0.1 and ϕ=0.1 and (b) as a function of the initial thermal gradient
G for Pe=1 and ϕ=0.1. The solid lines give the
analytic solutions. The inset in (a) shows the comparison of the analytic
solution Eq. (22) with the different limits derived in Sect. 2.1 and 2.2.
The black curve represents the analytic solution, and the colored straight lines
show the results in the high or low value limits of Eqs. (26) to (30),
respectively. A larger version of the inset is given as Fig. S1 in the
Supplement. The analytical solution for the full parameter range
Pe–G is given in Fig. 4.
Scaling laws derived from analytical solution
In this section a simplified analytical solution for the z-dependent
temperature difference between fluid and solid will be derived. From this
solution the maximum temperature differences ΔTmax can be
obtained, and scaling laws will be derived.
Analytical solution of the governing equations
We are interested in an analytical solution of Eq. (20) controlling the
non-equilibrium temperature difference Tf-Ts. We
simplify the problem by considering the hypothetical case in which Tf-Ts does not change with time and, moreover, in which the
thermal gradient in the solid phase is fixed and linear, with ∂Ts/∂z=-G=-1/H (non-dimensional, with dimensions
G=ΔT0/H). Although different from initial and steady-state stages
described in the 1D models (Sect. 3.1), this hypothetical case is quite
similar to what can be observed at the very beginning of the second stage
described in Sect. 3.1 (see Fig. 2f, h). In this second stage, the
evolution of Tf and Ts was observed to be quite similar indeed.
Besides, at the end of stage 1 (Sect. 3.1), Ts remains close to
initial conditions; therefore a fixed linear gradient of slope -G=-1/H is justified. Since the maximum temperature difference between the two
phases is observed starting from the end of stage 1 and during stage 2
(Sect. 3.2), it does not seem unreasonable to consider this hypothetical
case for deriving the maximum temperature difference. Using these
assumptions, Eq. (20) becomes
∂2Tf-Ts∂z2-Pe∂Tf-Ts∂z-Tf-Ts=-PeG.
While in the general case of Eq. (20) the temperature difference implicitly
depends on ϕ0, i.e., on the three non-dimensional parameters Pe, ϕ0, and G, Eq. (21) no longer depends on ϕ0 because we replaced
∂Tsϕ0/∂z with -G, which is independent of ϕ0. Equation (21) is a second-order ordinary differential equation for Tf-Ts whose solution can be analytically derived as (see
Supplement for details)
Tf-Ts=αer1z+βer2z+PeG,
where r1 and r2 are the roots of the associated equation of Eq. (21):
r1=12Pe-Pe2+4,r2=12Pe+Pe2+4.
The parameters α and β are constrained by the boundary
conditions Tf-Ts=0 at z= 0 and ∂Tf-Ts∂z=0 at z=H:
α=PeGr2r1er1-r2/G-r2,β=PeGr1r2er2-r1/G-r1.
The third term in Eq. (22) is a particular solution for Eq. (21).
Comparison with numerical models
From Eq. (22) the maximum value of the depth-dependent temperature
difference (Tf-Ts) can be determined. It is found that the maximum
is always at z=H. This value will be denoted as ΔTmax and has
been calculated for all parameter combinations used for the numerical
models. In Fig. 3 these analytical solutions are plotted as solid lines
together with the numerical solutions (asterisks). The agreement is very
good. For most cases the differences between the numerical and analytical
solutions are well below 1 %; only when ΔTmax reaches values
of about 0.6 and higher do the differences become >1 %, up to
6 %. This general good agreement is another justification for using the
time-independent Eq. (21) to obtain an analytical solution of an
intrinsically time-dependent process as long as we are interested only in
the maximum value of (Tf-Ts). Other reasons for the observed
differences between the analytical and numerical solutions include numerical
errors when determining the particular times when maximum temperature
differences are reached, especially for the models which are in the regime
close to ΔTmax=1 where the ΔTmax(Pe) curves
become non-linear (Fig. 3a).
Scaling laws for temperature differences at certain parameter limits
The analytical solution for ΔTmax fits very well with our model
results and therefore looks to be ideal for getting a better understanding
of the relative influences of the two controlling parameters Pe and G,
described in Sect. 2.2 and 2.3. The Péclet number is already known to be
of great importance for thermal equilibrium and non-equilibrium conditions.
Inspecting the last term in Eq. (22) we notice that a high Pe and a high
initial thermal gradient should favor higher temperature differences. This
has been demonstrated in Fig. 3.
Equation (22) is, however, complicated, and the assessment of the relative
importance of Pe and G for different possible regimes is limited. In this
section, we study the evolution of Tf-Ts, i.e., also
ΔTmax, in a few limiting cases. This enables us to better
understand the influence of each parameter and to derive some scaling laws
for different regimes.
Limit Pe→0
When Pe tends to 0, we have the condition
Pe≪2.
With this condition Eq. (22) tends to the following limit (see Supplement):
Tf-Ts=PeG1-M,
with
M=cosh(z)+cosh2G-z1+cosh(2/G),
which simplifies for z=H=1/G to
M=1cosh(1/G).
This is the limit for Pe→0. This limit gives predictions for ΔTmax in very good agreement with Eq. (22) for Pe<1 (having
G=0.1) (see inset in Fig. 3a or Fig. S1 in the Supplement). In
the limit G→0 and finite Pe<1/G we get the limit for MM→e-z.
Thus, for both small Pe and small G the temperature difference (Eq. 26) can be
written as
Tf-Ts=PeG1-e-z.
Equation (29) confirms the proportionalities observed in Fig. 3, namely ΔTmax∝Pe (Fig. 3a) and ΔTmax∝G (Fig. 1b).
Limit Pe→∞
To obtain the limit of Eq. (22) for Pe→∞, Eq. (22) can be
linearized with respect to 4/Pe2≪1.
Applying the rule of L'Hôpital, Eq. (22) tends to the following limit:
Tf-Ts=Gz.
For details, see the Supplement. This limit is also the solution of
Eq. (21) when neglecting the diffusive and heat transfer terms. As
demonstrated in the Supplement, this limit predicts ΔTmax values in very good agreement with Eq. (22) for Pe>100 (Fig. 3a, inset).
Exploring the domains for the maximum temperature difference including all limits
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, Pe is expected to evolve
from very low values of the order of 10-5 to 10-3 in partially molten
regions with distributed porous flow to higher values of the order of 1 or larger
at depths where channels have merged and further to very high values of
the order of 105 in dike systems (Schmeling et al., 2018).
While the melt fraction does not influence ΔTmax (cf. Eqs. 22,
30), it does influence the long-term temporal behavior because Ts is ϕ0-dependent (see Eq. 20). Therefore, here follow some words about possible melt
fractions. As melt flow may occur at very small melt fractions (McKenzie,
2000; Landwehr et al., 2001), large ϕ values are not expected in
natural mantle magmatic systems or in dike systems in the crust. Values of
channel volume fraction generally remain below a few percent up to tens of
percent (in dunite channels up to 10 %–20 %, Kelemen et al., 1997).
To get an idea about the expected order of magnitude of the macroscopic
thermal gradient G=1/H of the system, we have to evaluate the scaling length
L used to scale the dimensional H. L scales with the geometric mean of the
channel width df and the interfacial boundary layer thickness δ
(Eqs. 9 with 11). L would evolve non-linearly with the width of melt pathways,
which may increase by several orders of magnitude as 3D grain junctions
eventually merge to 1D dikes. As will be shown in Sect. 5.3 in more detail
the resulting non-dimensional G ranges between an order of 1 and an order of 10-5.
In Fig. 4 we explore ΔTmax variations using the analytical
solution Eq. (22), in which ΔTmax depends on Pe and G. Three main
regimes can be distinguished.
Regime 1. For high Pe values, Tf-Ts tends to the
relationship described in Eq. (30). The temperature difference increases
linearly with distance from the bottom (z=0), reaching ΔTmax=1 at z=H. In the whole region the fluid temperature remains
constant and at a maximum of 1 while the solid temperature increases linearly
with z from 0 to 1. The proportionality of ΔTmax to G disappears
because the maximum value of z is equal to H=1/G.
Regime 2. For Pe≪1, or more precisely, for Pe≪1G
represented by the oblique dashed line in Fig. 4, Tf-Ts varies with distance from the bottom according to 1-e-z and is proportional to Pe and G. This means that large temperature
gradients favor large temperature differences. In this domain, Tf-Ts tends to the relationship presented in Eq. (29).
Regime 3. For a large initial temperature gradient G close to 1 (small H) and
Pe≪1, Tf-Ts tends to the relationship proposed
in Eq. (26). In this domain, Tf-Ts is proportional to
Pe but no more to G because M is a function of G gradually canceling the
proportionality to G, which is visible in regime 2. The depth dependence is
given by 1-Mz, which at G=1 increases
non-linearly from about 0 to 0.4 with increasing z.
Main regimes of the maximum fluid–solid temperature
differences ΔTmax
due to thermal non-equilibrium obtained by the analytical solution (Eq. 22)
in the parameter space of the Péclet number Pe
and temperature gradient G. The asymptotic limits
are indicated by the formulas, and M(z) is given by
Eq. (27) with 1-MG,z increasing non-linearly from about 0 to
about 0.4 with increasing z for
G in the range 0.3 to 3. Regime boundaries are shown as
dashed lines. Typical parameter combinations for magmatic settings such as
interstitial melts or dikes are indicated by the orange rectangles which
extend further to the left, well below log10G of -3. Note that two slices through this
field at G=0.1 and at
Pe=1 have already been shown in Fig. 3.
DiscussionLimitationsComments on the analytic solution
Although the assumptions used to get the analytic solution (Eq. 22) are very
specific, they are reasonable considering the conditions in the models when
ΔTmax is reached, and it fits the numerical results very well.
This is shown in Fig. 5 where for various combinations of Pe and G the
time-dependent temperature differences Tf-Ts are
shown as functions of depth together with the analytical solutions using Eq. (22). For all examples the position of the maximum temperature differences
lies at z=H. A major simplification used in Eq. (21) was time independence.
Obviously, the resulting analytical solutions represent stage 2, which
is quasi steady state in contrast to stage 1 when the temperature difference
builds up and stage 3 when the long-term behavior is approached. We
emphasize that this analytical solution is a very good approximation of the
depth-dependent temporal maximum temperature difference that can be reached
in such porous systems.
Comparison of depth- and time-dependent numerical
solutions with the time-independent analytical solutions for different
parameters Pe and G as
indicated in the panel titles. In each panel the curves show Tf-Ts profiles for progressive times, and the colors are
cyclically varied with time from blue to yellow, starting with blue (bold
curve). The bold red dashed curve shows the analytical solution Eq. (22),
which represents a very good estimate of the depth-dependent temporal
maximum of the temperature difference. In each panel the first five curves are
plotted at time increments of 0.5 (0.025 for Pe=100) and the later curves with 5 (1 for
Pe=100). The total non-dimensional
times of each panel are 100 (500 for G=0.01). Steady state is reached in the models with
G=0.1. The model with G=0.01 would
need t= 10 000 to reach steady state. The melt fraction
was chosen as ϕ0=0.1.
Boundary conditions at top and initial conditions
The boundary conditions we chose at the top (z=H) are suitable for cases
with little temperature evolution (regimes 2 and 3, low Pe) and for early
stages for regime 1 but might be inappropriate for high temperature
increases (high Pe – regime 1) at later stages (see Sect. 4.3.4). In order
to quantify the influence of this choice of boundary conditions on our
results, we compared the evolution of Tf-Ts profiles for three Péclet numbers and two values of G, using four
different boundary conditions at the top (Fig. 6).
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).
Mathematically, the open boundary condition is not a rigorous boundary
condition because both the temperature and temperature gradient
intrinsically depend on the temperature evolution within the model.
Therefore, it cannot be applied to the analytical solution of Sect. 4.1.
Numerically it works well for our system without producing instabilities or
oscillations. Comparing the top and bottom rows of Fig. 6, the constant
temperature gradient condition produces quite similar results as the open
boundary condition for all Pe and G values tested during the first and second
stages of temporal evolution (see Sect. 3.1). The agreement becomes worse
for stage 3 when approaching steady state at large Pe values. Comparing the other two
boundary conditions (second and third rows of Fig. 6) with the constant
gradient condition (top row) shows that the effect of the top boundary
during stages 1 and 2 is still small sufficiently far away from the top. Only
for the small Pe case (left column of Fig. 6) do the zero gradient and zero
temperature conditions strongly affect the upper half of the domain by
diffusion. Yet the maximum temperature difference of the constant gradient
case is nearly reached by the other two boundary conditions further within
the domain, not at the top. The special case of high Pe and small G with zero
temperature boundary condition (third row, fourth column in Fig. 6)
shows a strong build-up of Tf-Ts close to the top when approaching
the steady state. This stems from the large local temperature gradient built
up near the top as a result of transforming the difference in advective heat
in- and output PeTinflux-PeToutflux=Pe into a
high conductive outflux ∂T/∂z at the top.
It is unlikely that such situations occur in natural systems.
Temporal evolution of vertical profiles of Tf-Ts for models with different Péclet numbers and different
initial temperature gradients G. In each panel
the curves show Tf-Ts profiles for progressive times, and the colors are cyclically
varied with time from blue to yellow, starting with blue (bold curve). The
first five curves of the Pe<100
(respectively Pe=100) models were
taken with time increments of 1 (respectively 0.1) and the later curves with 10
(respectively 1). The total time was 100 in all models with
G=0.1 and 500 in the models
with G=0.01. Steady state
is reached in the models with G=0.1. The models with
G=0.01 would need t= 10 000 to reach
steady state. In each row the top boundary conditions are assumed as
indicated at the left.
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 Hlid. Tests show that
lid thicknesses larger than H give results in general agreement with the
constant gradient or open boundary models of Fig. 6. Localized differences
with up to 30 % higher Tf-Ts values occur near
the top, but they disappear for Hlid≥10H. Lid thicknesses smaller
than H force the solid temperature at the top to values closer to 0, while
the fluid temperature remains high. This results in significantly higher
Tf-Ts values than in Fig. 6 (top row) close to the
top. However, typical natural magmatic systems are on the Hlid>H side,
suggesting that our constant gradient boundary condition is a good
approximation.
In summary, the influence of boundary conditions on fluid and solid
temperature evolution depends mostly on the domain size H and on the value of
Pe. The larger these two parameters, the less important the influence of
boundary conditions within almost the whole model domain. If one is
interested in the maximum value of Tf-Ts in space and time, the
tests show that this value can safely be picked at z=H when using the
constant temperature gradient boundary condition.
As an initial condition we used a linear temperature profile and initial
equilibrium between solid and fluid. A non-linear initial temperature
profile between Tf=Ts=1 at the bottom and Tf=Ts=0 at the top
would have spatially varying temperature gradients with sections with
gradients larger than those assumed in our model. As the temperature
gradient strongly influences thermal non-equilibrium (see e.g., Eq. 22, which
explicitly contains the temperature gradient G), the above results are
expected to be different, and a stronger thermal non-equilibrium is expected
in regions with higher gradients. Schmeling et al. (2018) used a step
function with Tf=Ts=1 at z=0 and Tf=Ts=0 at z>0 as the initial condition, i.e., an extremely non-linear profile near z=0.
Assuming this initial temperature profile, Fig. 7 shows the temporal
behavior of the temperature difference for selected parameter combinations,
equal to the parameters used in Fig. 5. The analytical solutions for the
time-independent cases (Eq. 22) are also shown. As expected, at early stages
the temperature differences are significantly larger than given by the
analytical solutions by a factor of 2 or more shortly after the onset of the
evolution. At later stages (stage 2 or 3) the time-dependent solutions
approach or pass through the analytical solutions. Thus, we may state that
the analytical solutions depicted in the regime diagram in Fig. 4 represent
lower bounds of thermal non-equilibrium compared to settings with non-linear
initial temperature profiles.
Time- and depth-dependent numerical solutions (thin
curves) as in Fig. 5 but for step-function initial conditions:
Tf=Ts=1
at z=0 and
Tf=Ts=0
at z>0 at
t=0. The bold dashed red curves are
the time-independent analytical solutions as in Fig. 5. In each panel the
curves show Tf-Ts profiles for progressive times, and the colors are cyclically
varied with time from blue to yellow, starting with blue (bold curve). In
each panel the first five curves (and later curves, respectively) are plotted
at time increments of (a) 0.5 (5), (b) 1 (10), (c) 0.5 (5), and (d) 0.025 (1).
The total non-dimensional times of each panel are 100 (500 for
G=0.01). Steady state is reached in
the models with G=0.1. The model with
G=0.01 would need t= 10 000 to reach
steady state. As porosity ϕ=0.1 is assumed.
Different densities and thermal properties of the two phases
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 λeff for the interface between the two phases
with different properties. To evaluate how many new non-dimensional numbers
are introduced, we non-dimensionalize the equations assuming different
material properties for the two phases. We use the fluid properties as
scaling quantities and assume that they are independent of temperature,
pressure, and depth. We modify the scaling length and Péclet number by
defining L̃=Lλeff′ with λeff′=λeff/λf and Pẽ=Peλeff′. With this scaling, Eqs. (14) and (15) turn into (for
clarity, primes indicate non-dimensional quantities)
ϕ∂Tf′∂t′+Pẽv′⋅∇Tf′=∇⋅(ϕ∇Tf′)-ϕ01-ϕ0(Tf′-Ts′)
and
(1-ϕ)∂Ts′∂t′=κs′ρs′cp,s′∇⋅((1-ϕ)∇Ts′)+ϕ01-ϕ01ρs′cp,s′(Tf′-Ts′).
Inspection of these equations shows that two more non-dimensional numbers
are introduced: the ratio of diffusivities κs′ and the ratio of
the products' density and heat capacity, ρs′cp,s′.
As Eqs. (31) and (32) cannot be merged into one time-independent
ordinary differential equation for Tf-Ts as in
Sect. 4.1, we numerically tested some cases with Pẽ=Pe=1
and λeff′=1 in which the diffusivity ratio and the ratio of
ρs′cp,s′ were varied between 0.1 and 10 (see Fig. 8). The
results show that for the fixed combination of Pe=1 and λeff′=1 the magnitude of thermal non-equilibrium remains on the same
order of magnitude O(0.1) as for equal properties (Fig. 8). However, the
time dependence is significantly affected: for a high ratio of κs′=10 (i.e., the solid is strongly conducting) the solid temperature
profile remains close to the constant initial gradient, and the temperature
difference rapidly converges to a steady state similar to the analytical
solution depicted in Fig. 5a. In contrast, for a low κs′=0.1
the solid temperature departs more strongly from the initial linear
gradient, and the solid–fluid temperature difference slowly drops with
time in the long term. Varying the potential to store heat in the solid,
i.e., ρs′cp,s′, Fig. 8e and f show that a high value slows down
the long-term time-dependent variations, while a small value leads to rapid
long-term temporal variations in Tf-Ts and faster
convergence to the steady state, which is similar to the case with equal properties.
Time- and depth-dependent profiles of the fluid–solid
temperature differences as in Fig. 5. (a) Reference models (as in Fig. 5a)
with Pe=1,
G=0.1, ϕ=0.1, and equal fluid-to-solid properties. (b–f) Profiles as in (a) but with
solid-to-fluid property ratios as indicated in the titles of each panel,
and λeff′=1. The properties in (b) are typical for water in sedimentary
rocks. In every panel but (b) the first five curves were taken with time
increments of 0.5 and the later curves with 5. In (b) the first 5 curves
were taken with time increments of 0.4875 and the later curves with 4.875. The
total time was 100 in all models. Steady state is reached in all models.
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 ρs′cp,s′ is about 0.78, i.e., of an order of 1. However, the
thermal diffusivity of water is significantly smaller than that of rock,
typically by a factor of 16; i.e., κs′ is about 16. We tested a few
cases (Fig. 9) with Péclet numbers and initial thermal gradients G (i.e.,
inverse model heights) (assuming for simplicity λeff′=1) equal
to the cases depicted in Fig. 5. The time-dependent profiles behave
similarly to those in Fig. 5, with very similar maxima of the temperature
differences (red dashed curves in Fig. 5) relevant for stage 2. The only
important difference is that the water–sedimentary rock case more rapidly
approaches the late steady states of stage 3, and these stages are closer to
the maximum red dashed curves. These results suggest that the absolute
values of maximum thermal non-equilibrium temperature differences shown in
the regime diagram Fig. 4 are also applicable to a water–sedimentary rock
system.
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., ρs′cp,s′=0.78,
κs′=16, and
λeff′=1. Pe and G have been
chosen as indicated in the panel titles (as in Fig. 5), and
ϕ=0.1 was assumed. In each panel the curves show
Tf-Ts profiles for progressive times, and the colors are
cyclically varied with time from blue to yellow, starting with blue (bold
curve). The first five curves were taken with time increments of 0.4875 and the
later curves with 4.875. The total time was 100 in all models with
G=0.1 and 200 in the models
with G=0.01. Steady state
is reached in the models with G=0.1. The model with
G=0.01 would need t= 10 000 to reach
steady state.
Timescales
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, t90%, and
the time needed to reach steady state, tsteady. The latter has been
determined as the time at which the maximum difference between Tfz-Ts(z) curves at two subsequent time
steps becomes less than 10-8ΔTmax. These times can be
compared with different timescales that may characterize the evolution of
temperatures in the models. These timescales can be based on advection over
a characteristic distance dchar, giving tadvd=dchar/vf0, or on diffusion over the characteristic
distance, giving tdiffd=dchar2/κ0.
We tested these timescales with the two natural length scales of the
models. The first is the scaling length L (equal to 1 non-dimensional),
representing essentially the geometric mean of the channel width of the
pores, df, and the interfacial boundary layer thickness δ. The
second is the model height H. Grouping the models depending on the regime
they belong to (see Sect. 4.3.4 and Fig. 4), we plotted the recorded
times t90% and tsteady versus the characteristic timescales mentioned above. Good agreement with the characteristic timescales
is indicated by observed times fitting to the dashed x=y lines (Fig. 10).
For evaluating timescales the numerically determined
times of models with various parameters Pe and G representing the three different
regimes 1, 2, and 3 (different symbols) are plotted against characteristic
scaling times. (a) Times for reaching 90 % of the maximum temperature
difference ΔTmax are
plotted against either the advective timescale
tadvH based on model height
H for regime 1 models or against the scaling
time t0 for regime 2 models, or
against the diffusive timescale
tdiffH based on the model height
H. (b) Times for reaching steady states are
plotted against the characteristic diffusive timescales,
tdiffH, based on model height
H for all three regimes. Models close to the dashed
line (y=x) are in best
agreement with the characteristic times. In (a) regime 2 times are taken
dimensionally by multiplying the observed times and the non-dimensional
scaling time t0′=1 by
some arbitrary dimensional times
t0.
In regime 1 (high Pe), t90% is proportional to tadvH
(Fig. 10a, blue circles). In this regime the high value of Pe makes the
fluid temperature increase fast. It reaches its maximum value during the
time under which significant fluid–solid heat transfer builds up and the
solid temperature is still low. This corresponds to the time for traveling
the full distance H. During stages 2 and 3 the solid temperature increases and
the temperature difference decreases before steady state is reached. The
time for reaching steady state (Fig. 10b, circles) varies roughly linearly
with tsteady∝tdiffH. For most cases it is controlled by
diffusion through the solid over distances of the order of H. The case with large
H (circle in Fig. 10b below dashed line) apparently reaches the steady state
earlier, but still later than on a corresponding advective timescale based
on H (not shown). Inspecting this model shows that during stages 2 and 3 the
high Pe number facilitates approaching thermal equilibrium rapidly within large
parts of the model and reducing the effective length scale (and
characteristic timescale) over which non-equilibrium is still present.
In regime 2 (low Pe and G<0.1, i.e., H>10) the time for reaching ΔTmax is controlled by interfacial heat transfer (Fig. 10a, red
asterisks) on the length scale L, resulting in t90% proportional to
t0. The time for reaching steady state is controlled by the diffusion
timescale across the height of the system (Fig. 10b).
In regime 3 (low Pe and high G (small H)), time for reaching ΔTmax
is similar to or shorter than the diffusion time based on the model height H
(Fig. 10a, black crosses). The flattening of the curve indicates that
non-equilibrium is reached faster for some models because Pe reaches the order of 1
and the advective timescale starts to take over. The time for reaching
steady state (Fig. 10b, crosses) varies linearly with tsteady∝tdiffH. Clearly, it is also controlled by diffusion.
Applications to magmatic systems
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 Pe, the initial thermal gradient G
(=1/H), and the melt fraction ϕ. Typical stages of melt flow for
mid-ocean ridges include three stages:
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 (> 10 %–20 %)
channels identified as dunite channels after complete melt
extraction (Kelemen et al., 1997);
propagating dikes or other volcanic conduits.
Let us assume typical overall melt fractions of 1 % to 20 % for stages (b) and (c). Schmeling et al. (2018) discussed possible Péclet numbers for such
systems based on a Darcy-flow-related Péclet number:
PeD=vDdsκ0.
As we preferably use the melt pore dimension df in our scalings (Eqs. 9a
and 10a), we need to relate it to the solid phase dimension ds by using
ds=dfgϕ,g=1-ϕmelt channelsϕ1-ϕmelt tubes.
Using Eqs. (34), (9a), and (16), we arrive at the Péclet number used here.
Pe=PeD1gc1-ϕδdf
Schmeling et al. (2018) reviewed and estimated typical pore or channel
spacings ds of 10-3–10-2 m for stage (a), 0.1 m for early stage (b) increasing to 1–100 m for late stage (b), and 100–300 m for stage (c)
(dikes). Arguing for typical geometries, spreading rates, and melt extraction
rates, Schmeling et al. (2018) estimated the Darcy velocity to range between
10-10 and 10-9 m s-1. With these parameters PeD numbers
for the three stages can be estimated for the three stages as
10-7 to 10-5,
10-5 to 10-4 at depths where channel distances are of the order of 0.1 m
and 10-4 to 0.1 at shallower depths where the channel distances have
increased to the order of 1 to 100 m,
>105 for the dike stage.
To estimate Péclet numbers as defined here (Eq. 35), typical interfacial
thermal boundary layer thicknesses δ are needed. As the thermal
interfacial heat exchange intrinsically is time-dependent, a good estimate is
δ=cthκ0t (in dimensional form), where cth is
a constant for a thermal boundary layer, equal to 2.32 for a cooling half
space (Turcotte and Schubert, 2014). Assuming that the characteristic time can be expressed by the
(dimensional) fluid velocity v0 and system height H, i.e., by t=H/v0=Hϕ/vD, we may express vD in terms of the Péclet number
PeD. With the resulting t and δ we arrive at the following
Péclet number (H and df are dimensional or non-dimensional):
Pe=PeD3/4cthcg-3/4Hdf1/41-ϕ.
For mid-ocean ridge settings we assume H to represent the transition
region between the lithosphere and asthenosphere with a thickness of the
order of 1 to 10 km, and we use Eq. (34) to insert typical df values. They
increase from 10-4 m for interstitial melts (stage a) to 10-3 to
10-1 m for the channeling stage (b) (see Schmeling et al., 2018) to
>10 m for the dike stage (c). The resulting Péclet number (Eq. 36)
is of the order of 10-3 to 0.5 for stage a, 10-2 during
the early stage (b) and 10-2 to 1 during the later stage (b)
appropriate for dunite systems, and 104 to 107 for the dike
stage (c). To estimate typical non-dimensional thermal gradients G′ (or layer
thickness H′), the above estimate for δ and df can be inserted into
the scaling length L (Eq. 9a) to arrive at a non-dimensional G′=1/H′.
G′=Hds-3/4g-1/2ϕ3/4cthcPeD-1/41-ϕ
With the derived estimates for the three stages, G′ is of the order of 10-6
to 10-2.7 for stage (a), 10-4–10-2.5 increasing to 10-4–0.6 for stage (b), and 10-5–10-2 for the dike stage (c). These
resulting stages for Pe and G′ are indicated in the regime diagram (Fig. 4). All three stages extend far into the domain, with G values smaller than
0.001. Thermal non-equilibrium of the three stages can be summarized as follows.
Interstitial melts are at full thermal equilibrium.
Channeling and veining may result in moderate thermal non-equilibrium at sufficiently high thermal
gradients.
After transition to diking full thermal non-equilibrium is predicted.
A similar exercise can be done for continental magmatic systems. We skip
such an explicit evaluation here but note that silicic melt viscosities are
typically higher than those of basaltic melts at mid-ocean ridges. Thus,
Péclet numbers are expected to be smaller, but non-dimensional thermal
gradients (Eq. 37) might be larger, resulting in a downward and rightward
shift of the natural stages indicated in Fig. 4.
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
Pe=vf0κ0ϕ01-ϕ0δS,
indicating that for very small or very large melt fractions Pe becomes very
small. One may use Eqs. (11) or (12) to write Pe also in terms of pore or solid
(grain or channel spacing) dimensions df or ds,
respectively. The scaling laws and characteristic timescales for the three
regimes we found (Fig. 4) are in dimensional form.
Regime 1. For large Pe values the maximum non-equilibrium temperature difference is
simply equal to the imposed temperature difference, ΔTmax=ΔT0, and the characteristic time to reach
maximum non-equilibrium is simply tchar=H/vf0, i.e., the total time
of a fluid particle for passing through the system.
Regimes 2 and 3. For small Péclet numbers (Pe<HSϕ01-ϕ0δ) the maximum temperature
difference scales likeΔTmax=Gvf0ϕ01-ϕ0δκ0S=vf0ϕ01-ϕ0δHκ0SΔT0,and the
characteristic time for reaching this non-equilibrium scales with t0,
i.e.,tchar=ϕ01-ϕ0δκ0S.These relations can easily be used to assess the potential of thermal
non-equilibrium in systems of fluid flow through solids with given
geometrical properties and fluid fractions.
In the above discussion we used the terms moderate thermal non-equilibrium, which we may identify with
ΔTmax′ of a few percent to, say, 30 %, while full thermal non-equilibrium includes
higher values up to 100 %. To translate this into dimensional
ΔTmax values, what are typical ΔT0 values for mid-ocean ridges? In our example we defined Has the
thickness of the transition region between lithosphere and asthenosphere.
Such a transition zone may be defined by the depth region bounded by the
asthenospheric temperature Tasth and, say, 0.8Tasth, i.e.,
ΔT0=0.2Tasth≅200 K. Thus moderate
non-equilibrium may be represented by excess temperatures of the melt with
respect to the solid between, say, 6 and 60 K, while full thermal
non-equilibrium suggests the full ΔT0 range of 60 to 200 K or even higher in the case of dikes extending up into the lithosphere
with temperatures below 0.8Tasth. These typical temperature estimates
may have some implications for whether the solid rock will melt or the melt
will freeze. However, this discussion is beyond the scope of this paper.
Conclusions
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 Pe, the melt porosity ϕ, and the initial non-dimensional
temperature gradient G in the system. The maximum possible non-equilibrium
solid–fluid temperature difference ΔTmax is
controlled only by two non-dimensional numbers: Pe and G. Both numerical and
analytical solutions show that in a Pe–G parameter space three regimes
can be identified.
In regime 1 (high Pe(>1/G)) strong thermal non-equilibrium
develops independently of Pe, and a non-dimensional scaling law
Tf-Ts=Gz has been derived.
In regime 2 (low Pe(<1/G) and low G(<0.3)) non-equilibrium
decreases proportionally to decreasing Pe and G, and the non-dimensional scaling
law reads Tf-Ts=PeG1-e-z.
In regime 3 (low Pe(<1) and G of the order of 1) non-equilibrium scales with
Pe and G and is depth-dependent. The scaling law is
Tf-Ts=PeG1-Mz,
where M(z) depends on G.
Further conclusions include the following.
The timescales for reaching thermal non-equilibrium scale with the
advective timescale in the high-Pe regime and with the interfacial diffusion
time in the other two low-Pe-number regimes.
Applying the results to natural magmatic systems such as mid-ocean ridges
can be done by estimating appropriate orders of Pe and G. Plotting such typical
ranges in the Pe–G regime diagram reveals that (a) interstitial melt flow is in
thermal equilibrium, (b) melt channeling as revealed by dunite channels, for example,
may reach moderate thermal non-equilibrium, and (c) the dike regime is at
full thermal non-equilibrium.
In the studied setup G was constant, leading to conservative estimates of
thermal non-equilibrium. Any other depth-dependent initial temperature
distributions generate higher non-equilibrium than reported here.
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.
While for simplicity the presented approach has been done essentially for
constant model parameters, it can easily be extended to vertically varying
parameters. Thus, tools are provided for evaluating the transition from
thermal equilibrium to non-equilibrium for anastomosing systems (Hart, 1993).
Code availability
The MATLAB code is listed in the Supplement and is available on
request.
Data availability
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: https://doi.org/10.5194/se-13-1045-2022-supplement.
Author contributions
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.
Competing interests
The contact author has declared that neither they nor their co-author has any competing interests.
Disclaimer
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
We gratefully acknowledge the excellent reviews by John Rudge and Cian
Wilson, who stimulated us into significantly improving the scaling.
Financial support
This research has been supported by the Deutsche Forschungsgemeinschaft (grant no. 403710316).
Review statement
This paper was edited by Juliane Dannberg and reviewed by John Rudge, Cian Wilson, and Samuel Butler.
References
Aharonov, E., Whitehead, J. A., Kelemen, P. B., and Spiegelman, M.:
Channeling instability of upwelling melt in the mantle, J. Geophys. Res.,
100, 20433–20450, 1995.
Amiri, A. and Vafai, K.: Analysis of Dispersion Effects and Non-Thermal
Equilibrium, Non-Darcian, Variable Porosity In-compressible Flow Through
Porous Media, Int. J. Heat Mass Tran., 37, 939–954, 1994.
Becker, K. and Davis, E.: On situ determinations of the permeability of the
igneaous oceanic crust, in: Hydrogeology of the Oceanic Lithosphere,
311–336, edited by: Davis, E. and Elderfield, H., CambridgeUniv. Press, ISBN 0 521 81929, 2004.
Bruce, P. M. and Huppert, H. E.: Solidification and melting along dykes by
the laminar flow of basaltic magma, in: Magma transport and
storage, edited by: Ryan, M. P., Wiley, Chichester, 87–101, ISBN 0 471 92766 X, 1990.
Davis, E. E., Chapman, D. S., Wang, K., Villinger, H., Fisher, A. T., Robinson, S. W., Grigel, J., Pribnow, D., Stein, J., and Becker, K.: Regional heat flow variations across the sedimented Juan
de Fuca ridge eastern flank: constraints on lithospheric cooling and lateral
hydrothermal heat transport, J. Geophys. Res., 104, 17675–17688, 1999.de Lemos, M. J. S.: Thermal non-equilibrium in heterogeneous media, Springer
Science+Business Media, Inc., 10.1007/978-3-319-14666-9, 2016.
Furbish, D. J.: Fluid Physics in geology, Oxford University Press, New York,
476 pp., ISBN 0-19507701-6, 1997.
Harris, R. N. and Chapman, D. S.: Deep seated oceanic heat flow, heat deficits
and hydrothermal circulation, in: Hydrogeology of the Oceanic Lithosphere,
311–336, edited by: Davis, E. and Elderfield, H., CambridgeUniv. Press, ISBN 0 521 81929, 2004.
Hart, S. R.: Equilibration during mantle melting: a fractal tree model, P.
Natl. Acad. Sci. USA, 90, 11914–11918, 1993.Kelemen, P. B., Whitehead, J. A., Aharonov, E., and Jordahl, K. A.:
Experiments on flow focusing in soluble porous media, with applications to
melt extraction from the mantle, J. Geophys. Res., 100, 475–496,
10.1029/94JB02544, 1995.
Kelemen, P. B., Hirth, G., Shimizu, N., Spiegelman, M., and Dick, H. J. B.:
A review of melt migration processes in the adiabatically upwelling mantle
beneath oceanic spreading ridges, Philos. T. R. Soc. S.-A, 355, 283–318, 1997.
Keller, T., May, D. A., and Kaus, B. J. P.: Numerical modelling of magma
dynamics coupled to tectonic deformation of lithosphere and crust, Geophys.
J. Int., 195, 1406–1442, 2013.Kuznetsov, A. V.: An investigation of a wave of temperature difference
between solid and fluid phases in a porous packed bed, Int. J. Heat Mass
Tran., 37, 3030–3933, 10.1016/0017-9310(94)90358-1, 1994.
Landwehr, D., Blundy, J., Chamorro-Perez, E. M., Hill, E., and Wood, B.:
U-series disequilibria generated by partial melting of spinel lherzolite,
Earth Planet. Sc. Lett., 188, 329–348, 2001.
Lister, J. R. and Kerr, R. C.: Fluid-mechanical models of crack propagation
and their application to magma transport in dykes, J. Geophys. Res., 96, 10049–10077, 1991.
Maccaferri, F., Bonafede, M., and Rivalta, E.: A quantitative study of the
mechanisms governing dike propagation, dike arrest and sill formation, J.
Volcanol. Geoth. Res., 208, 39–50, 2011.
McKenzie, D.: The generation and compaction of partially molten rock, J.
Petrol., 25, 713–765, 1984.
McKenzie, D.: Constraints on melt generation and transport from U-series
activity ratios, Chem. Geol., 162, 81–94, 2000.
Minkowycz, W. J., Haji-Sheikh, A., and Vafai, K.: On departure from local
thermal equilibrium in porous media due to a rapidly changing heat source:
the Sparrow number, Int. J. Heat Mass Tran., 42, 3373–3385, 1999.Nield, D. A. and Bejan, A.: Convection in Porous Media, 3rd Edn.,
Springer Science+Business Media, Inc., ISBN 978-3-319-84189-2, 2006.
Rivalta, E., Taisne, B., Buger, A. P., and Katz, R. F.: A review of
mechanical models of dike propagation: Schools of thought, results and
future directions, Tectonophysics, 638, 1–42, 2015.Roy, M.: Thermal disequilibrium during melt-transport: Implications for the
evolution of the lithosphere-asthenosphere boundary, arXiv [preprint], arXiv:2009.01496,
2020.
Rubin, A. M.: Propagation of magma-filled cracks, Annu. Rev. Earth
Planet. Sc., 23, 287–336, 1995.Schmeling, H.: A model of episodic melt extraction for plumes, J. Geophys.
Res., 111, B03202, 10.1029/2004JB003423, 2006.
Schmeling, H., Marquart, G., and Grebe, M.: A porous flow approach to model
thermal non-equilibrium applicable to melt migration, Geophys. J. Int., 212,
119–138, 2018.
Schumann, T. E. W.: Heat transfer: A liquid flowing through a porous prism,
J. Frankl. Inst., 208, 405–416, 1929.
Spiegelmann, M., Kelemen, P. B., and Aharonov, E.: Causes and consequences of
flow organization during melt transport: The reaction infiltration
instability in compactible media, J. Geophys. Res., 106, 2061–2077, 2001.Spiga, G. and Spiga, M.: A rigorous solution to a heat transfer two phase
model in porous media and packed beds, Int. J. Heat Mass Tran., 24, 355–364, 10.1016/0017-9310(81)90043-0, 1981.
Turcotte, D. and Schubert, G.: Geodynamics, Cambridge University
Press, Cambridge, ISBN 978-0-521-18623-0, 2014.Verruijt, A.: Theory of Groundwater Flow, The Macmillan Press Ltd., London
and Basingstoke, 141 pp., 10.1007/978-1-349-16769-2, 1982.
Wilcock, W. S. D. and Fisher, A. T.: Geophysical constraints on the
subseafloor environment near Mid-Ocean ridges, 51–74, in: Subseafloor
Biosphere, edited by: Cary, C., Delong, E., Kelley, D., and Wilcock, W. S. D.,
Washington DC, American Geophysical Union, ISBN 0-87590-409-2, 2004.
Woods, A. W.: Flow in Porous Rocks: Energy and Environmental Applications,
Cambridge University Press, Cambridge, 289 pp., ISBN 978-1-107-06585-7, 2015.