HESS Opinions: Linking Darcy's equation to the linear reservoir
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Hydrol. Earth Syst. Sci., 22, 1911–1916, 2018
https://doi.org/10.5194/hess-22-1911-2018
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.
HESS Opinions: Linking Darcy’s equation to the linear reservoir
Hubert H. G. Savenije
Delft University of Technology, Delft, the Netherlands
Correspondence: Hubert H. G. Savenije (h.h.g.savenije@tudelft.nl)
Received: 26 September 2017 – Discussion started: 6 October 2017
Revised: 25 January 2018 – Accepted: 26 February 2018 – Published: 19 March 2018
Abstract. In groundwater hydrology, two simple linear equa- though scaling issues occur in virtually all earth sciences,
tions exist describing the relation between groundwater flow what distinguishes hydrology from related disciplines, such
and the gradient driving it: Darcy’s equation and the linear as hydraulics and atmospheric science, is that hydrology
reservoir. Both equations are empirical and straightforward, seeks to describe water flowing through a landscape that
but work at different scales: Darcy’s equation at the labo- has unknown or difficult-to-observe structural characteris-
ratory scale and the linear reservoir at the watershed scale. tics. Unlike in river hydraulics or atmospheric circulation,
Although at first sight they appear similar, it is not trivial where answers can be found in finer grid 3-D integration of
to upscale Darcy’s equation to the watershed scale without equations describing fluid mechanics, in hydrology this can-
detailed knowledge of the structure or shape of the under- not be done without knowing the properties of the medium
lying aquifers. This paper shows that these two equations, through which the water flows. The subsurface is not only
combined by the water balance, are indeed identical pro- heterogeneous, it is also virtually impossible to observe. We
vided there is equal resistance in space for water entering the may be able to observe its behaviour and maybe its proper-
subsurface network. This implies that groundwater systems ties, but not its exact structure. Groundwater is not a continu-
make use of an efficient drainage network, a mostly invisi- ous homogeneous fluid flowing between well-defined bound-
ble pattern that has evolved over geological timescales. This aries (as in open channel hydraulics), but rather a fluid flow-
drainage network provides equally distributed resistance for ing through a medium with largely unknown properties. In
water to access the system, connecting the active groundwa- other words, the boundary conditions of flow are uncertain
ter body to the stream, much like a leaf is organized to pro- or unknown. As a result, hydrological models need to rely on
vide all stomata access to moisture at equal resistance. As effective, often scale-dependent, parameters, which in most
a result, the timescale of the linear reservoir appears to be cases require calibration to allow an adequate representation
inversely proportional to Darcy’s “conductance”, the propor- of the catchment. These calibration efforts typically lead to
tionality being the product of the porosity and the resistance considerable model uncertainty and, hence, to unreliable pre-
to entering the drainage network. The main question remain- dictions.
ing is which physical law lies behind pattern formation in But fortunately, there is good news as well. The structure
groundwater systems, evolving in a way that resistance to of the medium through which the water flows is not random
drainage is constant in space. But that is a fundamental ques- or arbitrary; it has predictable properties that have emerged
tion that is equally relevant for understanding the hydraulic by the interaction between the fluid and the substrate. Sim-
properties of leaf veins in plants or of blood veins in animals. ilar structures manifest themselves in the veins of vegeta-
tion, in infiltration patterns in the soil, and in drainage net-
works in river basins, emerging at a wide variety of spatial
and temporal scales. Patterns in vegetation or preferential in-
1 Introduction filtration in a soil can appear at relatively short, i.e. human,
timescales, but surface and subsurface drainage patterns, par-
One of the more fundamental questions in hydrology is how ticularly groundwater drainage patterns, evolve at geological
to explain system behaviour manifest at catchment scale timescales. Under the influence of strong gradients, these pat-
from fundamental processes observed at laboratory scale. Al-
Published by Copernicus Publications on behalf of the European Geosciences Union.1912 H. H. G. Savenije: HESS Opinions: Linking Darcy’s equation to the linear reservoir
terns can evolve more quickly, but even in groundwater sys- In general, we see that patterns emerge wherever a liquid
tems with relatively small hydraulic gradients “high perme- flows through a medium, provided there is sufficient gradient
ability features” appear to be present, regulating spring flow to build or erode such patterns. Likewise, such patterns must
(Swanson and Bahr, 2004). be present in the substrate through which groundwater flows,
There is a debate on the physical process causing pattern although these are generally not considered in groundwater
formation. Most scientists agree that it has something to do hydrology. If such patterns were absent, then the groundwa-
with the second law of thermodynamics, but what precisely ter system would be the only natural body without patterns,
drives pattern formation is still debated. Terms in use are which is not very likely.
maximum entropy production, maximum power, minimum This paper is an opinion paper. The author does not pro-
energy expenditure (e.g. Rodriguez-Iturbe et al., 1992, 2011; vide proof of concept. It is purely meant to open up a debate
Kleidon et al., 2013; Zehe et al., 2013; Westhoff et al., 2016), on how the linear drainage of an active groundwater body can
and the “constructal law” (Bejan, 2015). However, this paper be connected to Darcy’s law. The discussion forum of this pa-
is not about the process that creates patterns, but rather on us- per contains an active debate between the author, reviewers
ing the fact that such patterns exist in groundwater systems and commenters that provides more background.
to explore the connection between laboratory and catchment
scale.
2 The linear reservoir
How to connect laboratory scale to system scale?
At catchment scale, the emergent behaviour of the ground-
Dooge (1986) was one of the first to emphasize that hydrol- water system is the linear reservoir. Figure 1 shows a hy-
ogy behaves as a complex system with some form of organi- drograph of the Ourthe Occidentale in the Ardennes, which
zation. Hydrologists have been surprised that in very hetero- on a semi-log paper shows clear linear recession behaviour,
geneous and complex landscapes a relatively simple empiri- overlain by short and fast rainfall responses by rapid sub-
cal law, such as the linear reservoir, can manifest itself. Why surface flow, infiltration excess overland flow, or saturation
is there simplicity in a highly complex and heterogeneous overland flow. The faster processes are generally non-linear,
system such as a catchment? but as the catchment dries out, the fast processes die out,
The analogy with veins in leaves, or in the human body, the recharge to the groundwater system stops, and only the
immediately comes to mind. Watersheds and catchments groundwater depletion remains. Even during depletion, short
look like leaves. In a leaf, due to some organizing princi- runoff events may superimpose the depletion process with-
ple, the stomata, which take CO2 from the air and combine it out additional recharge, in which case the depletion contin-
with water to produce hydrocarbons, require access to a sup- ues following a straight line on semi-logarithmic paper (see
ply network of water and access to a drainage network that Fig. 1).
transports the hydrocarbons to the plant. Such networks are This behaviour is very common in first order streams, and
similar to the arteries and veins in our body where oxygen- even in higher order streams. In water resources management
rich blood enters the cells, and oxygen-poor blood is re- it is well know that recession curves of stream hydrographs
turned. The property of veins and arteries is “obviously” that can be described by exponential functions, which is congru-
all stomata in the leaf, and cells in our body, have “equal” ent with the linear reservoir of groundwater depletion. It fol-
access to water or oxygen-rich blood and can evacuate the lows from the combination of the water balance with the
products and residuals, respectively. Having equal access to linear reservoir concept. During the recession period there
a source or to a drain implies experiencing the same resis- appears to be a disconnect between the root zone system
tance to the hydraulic gradient. If a human cell has too high that interacts with the atmosphere and the groundwater that
a resistance to the pressure exercised by the heart, then it is drains towards the stream network. These two separate “wa-
likely to die off. Likewise, too low resistance could lead to ter worlds” are well described by Brooks et al. (2010) and by
cell failure or erosion. As a result, the network evolves to an McDonnell (2014) and are substantiated by different isotopic
optimal distribution of resistance to the hydraulic gradient. signatures. As a result, we see that during recession only the
In a similar way, drainage networks have developed on the groundwater reservoir is active.
land surface of the Earth. Images from space show a wide va- If during recession, the catchment is only draining from
riety of networks, looking like fractals. Rodriguez-Iturbe and the groundwater stock, then the water balance can be de-
Rinaldo (2001) connected these patterns to minimum energy scribed by
expenditure. Hergarten et al. (2014) used the concept of min-
dSg
imum energy dissipation to explain patterns in groundwater = −Qg ,
drainage. Kleidon et al. (2013), however, showed that such dt
patterns are components of larger Earth system functioning at where Sg [L3 ] is the active groundwater storage and Qg
maximum power, whereby the drainage system indeed func- [L3 T−1 ] is the discharge of groundwater to the stream net-
tions at minimum energy expenditure. work.
Hydrol. Earth Syst. Sci., 22, 1911–1916, 2018 www.hydrol-earth-syst-sci.net/22/1911/2018/H. H. G. Savenije: HESS Opinions: Linking Darcy’s equation to the linear reservoir 1913
Figure 1. During the recession period, the Ourthe has a timescale of 1772 h for groundwater depletion, acting as a linear reservoir. Superim-
posed on the recession we see faster processes with much shorter timescales.
The linear reservoir concept assumes a direct proportional- well at representing hydraulic heads. However, such regional
ity between the active (i.e. dynamic) storage of groundwater groundwater models are generally calibrated solely on water
and the groundwater flowing towards the drainage network: levels (hydraulic head) and seldom on flow velocities, trans-
port of solutes, or flows, leading to equifinality in the deter-
Sg
Qg = , mination of spatially variable k values.
τ Swanson and Bahr (2004) identified preferential flow even
where τ is the timescale of the drainage process, which is in mildly sloping terrain. Therefore it is reasonable to assume
assumed to be constant. Combination with the water balance that under stronger gradients preferential flow becomes more
leads to prominent. In sloping areas, the hypothesis is that the subsur-
face is organized and cannot be assumed to consist of layers
Qg = Q0 exp (−t/τ ) , with relatively homogeneous properties. Under the influence
of a stronger hydraulic gradient, drainage patterns occur in
where Q0 is the discharge at t = 0. So the exponential reces- the substrate more or less following the hydraulic gradient
sion, which we observe at the outfall of natural catchments, along the streamlines. This happens everywhere in nature
is congruent with the linear reservoir concept. But how does where water flows through an erodible or soluble material.
this relate to Darcy’s law, which applies at laboratory scale? An initial disturbance leads to the evolution of a drainage
network that facilitates the transport of water through the
3 Upscaling Darcy’s law erodible material. Initial disturbances can be cracks, sedi-
mentation patterns, animal burrows, former root channels,
Darcy’s law reads as follows: etc. The formation of the network can be by physical ero-
sion and deposition (breaking up, transporting, and settling
dϕ particles) but can also be by chemical activity (minerals go-
v = −k ,
dx ing into solution or precipitating). The latter is the dominant
where v is the discharge per unit area, or filter veloc- process in groundwater flow. The precipitation that enters the
ity [L T−1 ]; k is the conductance [L T−1 ]; ϕ is the hydraulic groundwater system through preferential infiltration (Brooks
head [L]; and x [L] is the distance along the stream line. In et al., 2009; McDonnell, 2014) is low in mineral composition
a drainage network, these streamlines generally form semi- and hence aggressive to the substrate. The minerals that we
circles, perpendicular to the lines of equipotential, draining find in the stream during low flow (when the river is fed by
almost vertically downward from the point of recharge and groundwater) are the erosion products of the drainage net-
subsequently upward when seeping to the open drain (see work being developed. In the mineral composition of the
Fig. 2 for a conceptual sketch). stream we can see pattern formation at work and from the
Henry Darcy (1803–1858) found this relationship under transport of chemicals by the stream we may derive the rate
laboratory conditions, but the law also appears to work fine in at which this happens.
regions with modest slopes, where one or more layers can be In contrast to the physical drainage structures that we can
identified with conductivities representative for the sediment see on the surface (e.g. river networks, seepage zones on
properties of these layers. In such relatively flat areas, upscal- beaches), sub-surface drainage structures are hard to observe.
ing from the laboratory scale to a region with well-defined But they are there. On hillslopes, individual preferential sub-
layer structure appears to work rather well. This is clear from surface flow channels have been observed in trenches, but
the many groundwater models, such as MODFLOW, that do
www.hydrol-earth-syst-sci.net/22/1911/2018/ Hydrol. Earth Syst. Sci., 22, 1911–1916, 20181914 H. H. G. Savenije: HESS Opinions: Linking Darcy’s equation to the linear reservoir
Figure 2. Conceptual sketch of an unconfined phreatic groundwater body draining towards a surface drain. H is the head of the phreatic
water table with respect to the nearest open water.
complete networks are hard to observe without destroying is W . The head difference to the nearest open drain is H .
the entire network. Darcy’s equation then becomes
The hypothesis is that under the ground a drainage sys-
tem evolves that facilitates the transport of water to the sur- H H
v=k = ,
face drainage network in the most efficient manner. As was W rg
demonstrated by Kleidon et al. (2013) an optimal drainage
where rg [T] is the resistance against drainage. This way of
network maximizes the power of the sediment flux, which
expressing the resistance is similar to the aerodynamic resis-
involves maximum dissipation in the part of the catchment
tance and the stomatal resistance of the Penman–Monteith
where erosion takes place and minimum energy expenditure
equation. It is the resistance of the flux to a difference
in the drainage network. This finding is in line with the find-
in head. So, instead of assuming a constant width to the
ings of Rodríguez-Iturbe and Rinaldo (2001, p. 253), who
drainage network, we assume a constant resistance to flow.
found that minimum energy expenditure defines the structure
This is in fact the purpose of veins in systems like leaves or
of surface drainage. Although a surface drainage network has
body tissues, such as lungs or brains or muscles. The veins
2-D characteristics on a planar view, the groundwater system
make sure that the resistance of liquids to reach stomata in
has a clear 3-D drainage structure. The boundary where open
the leaf, or cells in living tissue, is optimal and equal through-
water and groundwater interact also has a complex shape.
out the organ. But also in innate material, where gravity and
This is the boundary where the groundwater seeps out at at-
erosive powers have been at work for millennia, the sys-
mospheric pressure, indicated in Fig. 2 by the dotted blue
tem is evolving towards an equally distributed resistance to
line. This boundary of interaction follows the stream network
drainage, much in line with the minimum expenditure theory
and moves up and down with the water level of the stream.
of Rodriguez-Iturbe and Rinaldo (1997).
To describe this 3-D drainage network conceptually, we can
Building on Darcy’s equation, an infinitesimal area dA of
build on the analogy with a fractal-like (mostly 2-D) struc-
a catchment drains as follows:
ture of a leaf or a river drainage network, but it is not the
same. dQg = vdA.
Fractal networks can be described by width functions that
determine the average distance of a point to the network. Interestingly, this drainage (recharge to the groundwater) is
Let’s call this distance W . Let’s now picture a cross section downward, so that we can assume that dA lies in the hori-
over a catchment with an unconfined phreatic groundwater zontal plane. If we integrate the discharge over the area of
body draining towards an open water drain (see Fig. 2 for the catchment that drains on the outfall, and assuming a con-
a conceptual sketch). At a certain infinitesimal area dA of the stant resistance, we obtain
catchment, the drainage distance to the sub-surface network
Hydrol. Earth Syst. Sci., 22, 1911–1916, 2018 www.hydrol-earth-syst-sci.net/22/1911/2018/H. H. G. Savenije: HESS Opinions: Linking Darcy’s equation to the linear reservoir 1915
Z Z flow at the outfall of catchments, we may be able to deter-
1 HA Sg mine the rate of growth of the drainage network, and – if the
Qg = vdA = H dA = = ,
A rg A rg rg n mineral content of the substrate is known – the origin of the
erosion material. I think it is an interesting venue of research
where n [–] is the average porosity of the active groundwa- to study the expansion of such networks as a function of the
ter body (which is the groundwater body above the drainage mineral composition of the groundwater feeding the stream
level). We see that the areal integral of the head H equals the network, possibly supported by targeted use of unique trac-
volume of saturated substrate above the level of the drain. ers.
Multiplied by the porosity, this volume equals the amount of This paper does not provide an explanation for the fact that
groundwater stored above the drainage level, which equals in recharge systems groundwater drains as a linear reservoir.
the active storage of groundwater Sg . Comparison with the In fact, it raises more fundamental questions: if a catchment
linear reservoir provides the following connection between has exponential recession, congruent with a linear reservoir,
the system timescale τ , the resistance rg , and the average then what causes the resistance to entering the drainage net-
porosity n: work to be constant? What is the process of drainage pattern
W formation? If the sub-surface forms fractal-like structures,
τ= n = rg n. then which formation process lies behind it? The reason why
k
this property evolves over time is still to be investigated, but
As a result, we have been able to connect the timescale of
it is likely that the reason should be sought, in some way or
the linear reservoir to the key properties of Darcy’s equation,
another, in the second law of thermodynamics.
being the average porosity, the conductance and the distance
We know from common practice that in mildly sloping
to the sub-surface drainage structure, or better, to the aver-
areas, groundwater models that spatially integrate Darcy’s
age porosity and the resistance to drainage. This resistance to
equation are quite well capable of simulating piezometric
drainage is assumed to be constant in space, but will evolve
heads. We also know that predicting the transport of pollu-
over time, as the fractal structure expands. However, at a hu-
tants in such systems is much less straightforward, requiring
man timescale, this expansion may be considered to be so
the assumption of dual porosities (which are in fact patterns).
slow that the system can be assumed to be static.
In more strongly sloping areas, such numerical models are
much less efficient at describing groundwater flow. This can,
4 Discussion and conclusion of course, be blamed on the heterogeneity of the substrate,
but one could also ask oneself the question of whether direct
In groundwater flow, connecting the laboratory scale to the application of Darcy’s law is the right approach at this scale.
system scale requires knowledge on the structure, shape, and If under the stronger gradient of a hillslope preferential flow
composition of the medium that connects the recharge in- patterns have developed, then we should take the properties
terface to the drain. Here we have assumed that, much like of these patterns into account. Fortunately, nature is kind and
we see in a homogenous medium, the flow pattern follows helpful. It has provided us with the linear reservoir that we
streamlines perpendicular to the lines of equal head, form- can use as an alternative for a highly complex 3-D numerical
ing semicircle-like streamlines. This implies that flow in the model that has difficulty reflecting the dual porosity of pat-
upper part of the streamlines is essentially vertical and that terns that we cannot observe directly, but of which we can
integration of Darcy’s law over the cross section of a stream see its simple signature: the linear reservoir with exponen-
tube takes place in the horizontal plane, and not in a plain tial recession. Hopefully groundwater modellers are going to
perpendicular to the gradient of the hillslope. make use of that property, particularly in larger scale mod-
The second assumption is that, over time, patterns have elling studies.
evolved along these streamlines by erosion of the substrate.
It is then shown that if the resistance to flow between the
recharge interface and the drainage network is constant over Data availability. No data sets were used in this article.
the area of drainage, the linear reservoir equation follows
from integration. This constant resistance to the hydraulic
gradient is similar to what we see in leaves or body tissue. Competing interests. The author declares that he has no conflict of
What are the evolutionary dynamics of the drainage net- interest.
work? It is likely that the drainage network makes use of
cracks and fissure present in the base rock, but subsequently
expands and develops by minerals going into solution. As
a result, these networks never stop developing, continuously Acknowledgements. The author would like to thank
refining and expanding the fractal structure. In relatively Wouter Berghuijs and Mark Cuthbert and the two re-
young catchments such structures may not be fully devel- viewers Stefan Hergarten and Axel Kleidon for the very
oped. By sampling the chemical contents of springs and base interesting exchange of opinions on the discussion fo-
www.hydrol-earth-syst-sci.net/22/1911/2018/ Hydrol. Earth Syst. Sci., 22, 1911–1916, 20181916 H. H. G. Savenije: HESS Opinions: Linking Darcy’s equation to the linear reservoir
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