Most of the existing models of speleogenesis are limited to situations where flow in all conduits is pressurized. The feedback between the distribution of hydraulic head and growth of new solution conduits determines the geometry of the resulting conduit network. We present a novel modeling approach that allows a transition from pressurized (pipe) flow to a free-surface (open-channel) flow in evolving discrete conduit networks. It calculates flow, solute transport and dissolution enlargement within each time step and steps through time until a stable flow pattern is established. The flow in each time step is calculated by calling the US Environmental Protection Agency Storm Water Management Model (US Environmental Protection Agency, 2014), which efficiently solves the 1-D Saint-Venant equations in a network of conduits. Two basic scenarios are modeled, a low-dip scenario and a high-dip scenario. In the low-dip scenario a slightly inclined plane is populated with a rectangular grid of solution conduits. The recharge is distributed to randomly selected junctions. The results for the pressurized flow regime resemble those of the existing models. When the network becomes vadose, a stable flow pathway develops along a system of conduits that occupy the lowest positions at their inlet junctions. This depends on the initial diameter and inlet position of a conduit, its total incision in a pressurized regime and its alignment relative to the dip of the plane, which plays important role during the vadose entrenchment. In the high-dip scenario a sub-vertical network with recharge on the top and outflow on the side is modeled. It is used to demonstrate the vertical development of karst due to drawdown of the water table, development of invasion vadose caves during vadose flow diversion and to demonstrate the potential importance of deeply penetrating conductive structures.
Karst aquifers are among the most prolific water reservoirs. Due to their heterogeneity and anisotropy, their efficient exploitation and protection face many challenges. The role of solution conduits in karst aquifers has been a topic of numerous studies. Estimates show that conduits carry about 99 % of flow within karst aquifers and present efficient transport pathways for potential pollutants (Worthington, 1999). However, we have only limited insight into karst aquifers; the position of conduit systems is largely unknown, except for the parts accessible for human exploration or encountered directly by drilling or indirectly by geophysical techniques.
Speleogenesis (e.g., the evolution of conduit networks in karst aquifers) has been one of the main topics in karst studies of the last century (Ford and Williams, 2007). Many conceptual models of speleogenesis have been proposed based on field observations (Audra et al., 2007; Ford and Ewers, 1978; Audra and Palmer, 2013; Palmer, 1991) and inference from basic principles of flow. However, to gain insight into the processes governing speleogenesis, different physical models have been built and followed by numerical models that are based on the physical and chemical principles of flow, dissolution and transport.
The main objectives of speleogenetic modeling are to test the conceptual models, to determine and evaluate the role of different geological, hydrological and geochemical factors and to find mechanisms that govern the evolution of conduit networks in karst aquifers. Few examples of direct field application have been published (Epting et al., 2009).
Ewers (1982) applied hardware (physical) models made from plaster of Paris or salt, and discovered several key mechanisms that were later largely confirmed and extended by numerical models. Numerical modeling of single conduit evolution (Dreybrodt, 1990, 1996; Palmer, 1991; Dreybrodt and Gabrovsek, 2000) revealed a feed-back mechanism between flow and dissolution rates and stressed the importance of higher-order dissolution kinetics (Dreybrodt, 1990, 1996; Palmer, 1991; White, 1977) for the evolution of extended conduits. Such kinetics has been proven experimentally for limestone and gypsum (Eisenlohr et al., 1999; Jeschke et al., 2001). More elaborated models of 2-D fractures with statistical aperture fields (Hanna and Rajaram, 1998; Szymczak and Ladd, 2011) showed that nonlinear kinetics is not necessary for the evolution of extended patterns of solution conduits.
The initial stage of speleogenesis is characterized by slow enlargement of proto-conduits, which is accelerated by positive feedback between flow and dissolution rate under constant head conditions. Dissolution widening increases the flow rate along an initial fracture. Then as the flow rate increases, fresh aggressive solution penetrates deeper into the fracture and in turn accelerates widening and flow rates. This feedback mechanism leads to breakthrough, when flow and widening rate increase by several orders of magnitude in a very short time (Dreybrodt, 1990, 1996; Palmer, 1991; Dreybrodt and Gabrovsek, 2000). At breakthrough the initiation stage of conduit development ends and the enlargement stage starts. The time needed to reach breakthrough is termed breakthrough time.
Evolution of 2-D fracture network under pressurized flow. Panels
show aperture widths and dissolution rates at different stage of evolution.
Size of the domain is 1 km
Individual fractures have been assembled into fracture networks in order to model patterns of evolving conduit systems (Lauritzen et al., 1992; Groves and Howard, 1994; Siemers and Dreybrodt, 1998; Kaufmann and Braun, 2000; Liedl et al., 2003). A typical benchmark setting emerged out of the Ewers's hardware models. It includes a plane populated with initial proto-channels (fractures/tubes) with inputs and outputs at different hydraulic heads. These models revealed the competition between different pathways connecting inputs to outputs, as already observed by Ewers (Ewers, 1982; Ford and Williams, 2007) in the physical model.
To review some of these basic mechanisms, a simple scenario is shown in
Fig. 1. It consists of a plane with a rectangular grid of fractures. The
boundary conditions are shown in Fig. 1a: the sides of the network are
marked geographically N, S, E, W (north, south, east, west). No-flow conditions are applied on the N
and S boundaries. Water enters the network at two inputs (In), In1 and In2 at the
W side, initially at constant head
At 0.99
After breakthrough (Fig. 1b), P1 is widened with the maximum dissolution rate along its entire length. It becomes increasingly uniform and so does the hydraulic gradient along it (see Figs. 1b and 2). The gradient builds up between the high head region along still pre-breakthrough (plugged) P2 and post-breakthrough (released) P1, which triggers the growth of conduits connecting P2 to P1. Gray arrows show some principle directions of growth.
Two post-breakthrough scenarios are envisaged:
In Fig. 1c and d, the constant head is kept at both inputs. New
connections between P2 and P1 evolve, while P2 also grows towards the exit.
The network expands along the existing pathways by growth of new bypasses
(some are shown by gray arrows) until all possible flow paths evolve (not
shown). Of course, all catchments have limits and such conditions cannot
last for long. In Fig. 1e and f, the recharge at In1 and In2 is limited to
To summarize: in pressurized flow conditions, the evolution of the network starts with
competition of pathways connecting inputs to outputs and continues with
their integration and expansion until head gradients along un-evolved
pathways are high enough for pathways to breakthrough. The evolution is
controlled by the feedback mechanism between the distribution of hydraulic
head and growth of new conduit pathways. This interplay is affected by many
parameters which reflect local hydrology, geology and geochemistry.
Many other scenarios of early speleogenesis have been modeled to study factors such as the role of geochemical conditions and mixing corrosion, exchange flow between the matrix and conduit network, and the role of insoluble rocks in the evolution of conduits (Dreybrodt et al., 2005). Numerical models have also been used to assess increased leakage at dam sites or other hydraulic structures where unnaturally high hydraulic gradients cause a short breakthrough time (Dreybrodt, 1996; Romanov et al., 2003; Hiller et al., 2011).
In real situations the available recharge cannot sustain pressurized flow within the evolving network, and the conduits undergo a transition from pressurized to free-surface flow conditions. Most accessible cave systems have undergone such a transition.
Though most models have only considered pressurized flow, Annable and Sudicky (1998) and Annable (2003) developed an elaborate model of the evolution of a single partially filled conduit embedded in variably saturated fractured media under laminar flow conditions. The extension of such a model to networks with turbulent flow remains a future challenge.
Here we develop a model that goes beyond the dynamics depicted in Fig. 1 by incorporating the transition to, and further evolution in, a free-surface flow regime.
List of rate constants and other parameters used in this work.
Profile of hydraulic head along pathways P1 (dashed lines) and P2
(full lines) from Fig. 1. Profiles are taken at different time steps, given
in units of breakthrough time (
The vertical evolution of karst has been under debate for more than a century, starting with classical concepts of Katzer, Grund, Davis, Swinnerton, Rhoads and Sinacori and others (Palmer, 2007). The Four State Model of Ford and Ewers (1978) elegantly combines these concepts and relates cave geometry to the density of permeable fissures.
Gabrovšek and Dreybrodt (2001) and Kaufmann (2003) modeled a 2-D vertical cross section of a karst system to explore the evolution of karst aquifers in the dimension of length and depth (sensu lato Ford and Ewers, 1978). They have shown the important role of water table drawdown in speleogenesis. These models considered dissolution in the phreatic part of an aquifer only and partly modeled the formation of drawdown vadose passages (Ford, 1988; Ford and Williams, 2007). Conceptual models have been developed that hypothesize the diversion of vadose water and formation of invasion vadose systems (Ford, 1988;Ford and Williams, 2007; Palmer, 2007; Audra and Palmer, 2013). However, these conceptual models have not been tested by numerical models.
In the following sections we describe how the model is built and present two basic modeling scenarios, each with several representative cases. We focus on the description of new mechanisms of flow pathway selection and discuss the results in view of the existing conceptual models.
Figure 3 shows a conceptual framework for the modeling presented in this
work. We assume a plane populated with conduits with water-soluble walls,
similar to that in Fig. 1. Water enters the conduit network at selected
junctions indicated by arrows in Fig. 3. The direct recharge into a junction
is limited either by the elevation of the land surface (
Conceptual framework. A conduit network with point recharge at
selected locations indicated by arrows. Recharge is limited by the position
of the land surface
The basic workflow of the model follows the same scheme as in the models
cited above (e.g., Dreybrodt et al., 2005) and includes the
following steps:
Define the network of conduits and boundary conditions (water inlets and outlets). Calculate flow in the network. Couple flow, dissolution and transport to calculate dissolution rates
in all conduits. Change the conduit diameter within a time step according to the dissolution
rate and return back to Step 2 or exit the loop when a stable flow pattern is
established or no substantial changes in flow pattern are expected. The flow does not depend on the dissolved load. Timescales for flow, dissolution and transport can be separated from the
timescale for widening, i.e., the evolution goes through a set of stationary
states within which the widening is constant.
We also assume that
We assume that the network has passed the initial (inception) stage of speleogenesis and that turbulent flow has already been established in the network. The reader is referred to the work of Dreybrodt et al. (2005) for early evolution in the laminar flow regime. One-dimensional turbulent flow is considered within all conduits. The flow could be either pressurized or free surface.
The use of a Preissmann slot enables use of the same set of equations for conduits with free-surface flow and conduits with pressurized flow.
Flow in partially filled conduits is described by Saint-Venant equations
(Dingman, 2002), which are based on depth-averaged conservation of
mass and momentum. Several numerical techniques are used to solve them
(Dingman, 2002). Our model invokes an open source package Storm
Water Management Model (abbreviated SWMM from here on), developed primarily
for flow and transport simulation in sewage systems by the US Environmental
Protection Agency (US Environmental Protection Agency, 2014). SWMM solves the set of Saint-Venant
equations to the desired approximation and accuracy using successive
approximations with under-relaxation (Rossman, 2009). Its use for the
simulation of flow in conduit dominated karst systems has been demonstrated
by several authors (Peterson and Wicks, 2006; Gabrovšek and Peric,
2006; Halihan et al., 1998). The pressurized flow is accounted for by
introduction of a fictitious Preissmann slot (Fig. 4) at the top of a
conduit's cross section (Cunge and Wegner, 1964). In this way we
transform a pressurized pipe to an open channel without considerably
changing the hydraulic characteristics and enable use of the same set of
equations for both flow regimes. Friction losses in conduits are calculated
by the Manning equation
SWMM enables easy construction of an arbitrary conduit network and many additional elements, such as reservoirs, catchments, etc., which could be implemented into future upgrades of the models presented here.
Dissolution rates in karst environments are determined by the reaction
kinetics at the rock–water interface (surface controlled dissolution), by
diffusion transport of ionic species between the rock–water boundary and the
bulk solution (transport controlled dissolution), and, in the case of
carbonates, by the rate of CO
In the early evolution of conduit networks, the water in proto-conduits (sub millimeters to few millimeters in size) is close to equilibrium with the mineral being dissolved and dissolution is mostly surface controlled, by higher-order kinetics in case of limestone and gypsum. In turbulent flow conditions, for cases discussed in this work, dissolution in limestone is predominantly surface controlled by first order kinetics, if the input solution has a low saturation ratio. Some issues related to limestone dissolution rates in turbulent flow still remain open; scalloped walls of limestone caves suggest that transport control might play an important role under turbulent flow conditions as well (Covington, 2014).
For these reasons we simplify the dissolution kinetics by assuming a linear
rate law at the rock–water boundary:
Viscous shear stress is related to the friction slope
The ions entering the water increase its saturation state with respect to
the mineral forming the walls, and diminish dissolution rates along the flow
pathways. The increase of concentration within each conduit is described by
a differential equation derived from a mass balance within an infinitesimal
segment of conduit:
Integration of Eq. (11) along a conduit gives the amount of rock dissolved within the conduit. The dissolved load is added to the downstream junction of the conduit and is then treated as a conservative tracer by the pollutant routing code of SWMM.
In most scenarios presented in this work, transport controlled dissolution prevails. Therefore, dissolution rates are dependent on the flow velocity. A case, where the dissolution rates are almost entirely surface controlled, is also presented. Physical constants and parameters used for calculation of flow, dissolution and transport throughout this work, are listed in Table 1.
Growth of a conduit with pressurized flow and a conduit with free-surface flow.
The model structure for the low-dip network.
The model structure for the high-dip scenario.
Dissolution rates are rates of dissolution enlargement
Two basic settings are presented: first a model of a low-dip network is presented as conceptually shown in Fig. 3. This scenario is used to examine the evolution of conduit network in a plan view. In a second scenario, a highly inclined high-dip network is modeled to explore the vertical organization of flow pathways, or evolution of the conduit network in dimension of length and depth (sensu lato Ford and Ewers, 1978).
Figure 6 introduces a model structure for the low-dip network. Circular
conduits with length
Figure 6c shows a side view of the model. The invert elevations increase
from E to W, 1 m per junction. The slope of the W–E conduits is therefore 0.1
and N–S oriented conduits are horizontal. The inlet offset defines how
much a conduit can incise. To keep conduits from bottoming out as they
incise, the inlet offsets,
Each of the junctions on the E boundary are connected to a large conduit
(
In the high-dip model (Fig. 7), the slope of the network (and therefore the conduits) is 0.99 from top to bottom and 0.1 from left to right. We use the expressions vertical for the steep conduits and horizontal for the gradual ones. Water enters on the top side and exits at the seepage face on the right side. The bottom and left boundaries are impermeable. In all junctions, gradual (horizontal) conduits are positioned 1 m above the steep (vertical) conduits, which assures preferential flow along the vertical plane in vadose conditions (see Fig. 7b). Flow along the horizontal conduits is active only when the junction is flooded above their inlets. The outflow is realized as in the low-dip case, with large conduits connecting junctions to outfalls on the right boundary.
We start with a simple scenario where all conduits have the same length (10 m), the same initial diameter (0.005 m) and the same inlet offsets. The network dips from W towards the free-outflow boundary on the E side with the slope 0.1. The model is run for 50 steps of 300 s, in total 15 000 s. The rock used is salt.
Figure 8 presents six snapshots of the network's evolution. Five inputs with
Six snapshots of the evolution of low-dip network with uniform
initial diameters and inlet offsets. Left: flow rates (width) and flow
direction (red
Figure 8a shows the initial situation. All inputs are at the maximal hydraulic heads, and only a small part of available recharge enters the network. High gradient drives fast growth of W–E conduits from In1 and In2 (Fig. 8b and c). Also, pathways heading N and S from In1 and In2 evolve in the pressurized flow regime. To the west of In1 and In2, the development is still slow, as the potential field flattens towards W. On Fig. 8c, the conduits draining In1 and In2 are pressurized and exhibit large flow and widening rates. The gradients from In3 towards the E boundary build up and drive the evolution of pathways from In3 towards the east. When pathways from In1 and In2 are too large to sustain pressurized flow, the hydraulic head in them drops to their topographic height which attracts additional flow from In3. With further time, the evolution progresses upstream. The flow in pathways draining In4 and In5 also increases; it predominantly follows the straight W–E line, although it is also clearly attracted by vadose pathways leading from In3.
Nevertheless, most of the flow from upstream inputs occurs along a direct line of W–E oriented conduits, which evolve most efficiently (Fig. 8c and d). In Fig. 8e, the In3 has become vadose and in a similar manner now attracts flow from In4 and In5. However, the direct line connecting In4 to the boundary takes most of the flow and grows most efficiently. Figure 8f shows the final stable flow configuration. All the inputs drain the available recharge, with the direct pathways between the inputs and the E boundary being the only ones that contain active flow.
A detailed look at Fig. 8 reveals that at any time, looking at the conduits draining a particular node, the highest flow rates are along W–E conduits, which consequently evolve more efficiently than other conduits. The inlet offsets of W–E conduits incise faster than others and eventually the water level at the junction falls below the lower edges of the other conduits, leaving only the W–E conduits active. This is schematically shown in Fig. 9a, where two outlets from a junction are compared; outlet 1 evolves more during the phreatic stage and, therefore, the bottom of the conduit reaches a lower elevation. Consequently, outlet 1 ultimately captures all water during the vadose entrenchment. Several other realizations of this scenario with different recharge rates at the inputs have ended with the same final distribution of active conduits.
Left panel: the geometry of a junction. Right panels (
A network with uniform initial diameters and initial inlet offsets randomly distributed within vertical span of 1 m.
Evolution of a low-dip network with randomly distributed initial diameters.
The time evolution of total discharge from the network in Fig. 11.
Evolution of SW edge of the network in from Fig. 11 before and after transition to free-surface flow.
At this point a short note is needed to explain what is meant by a stable flow configuration. In the case of constant recharge, the configuration is considered to be stable when all junctions are drained by one conduit only, i.e., there are no downstream bifurcations remaining. This is the case in Fig. 8f. In most of the other presented model runs a few outflow bifurcations remain at the last presented time step. These bifurcations would eventually die out if the model was run long enough. We will use the term quasi-stable to describe such situations.
The next step towards less idealized scenarios is to assume that the initial
inlet offsets of conduits are randomly distributed within the range of 1 m.
Figure 10 shows the network when a quasi-stable flow pattern has been established,
which is now more complex than in the previous case. The general evolution
is similar, progressing upstream, but some N and S oriented conduits may
have initial inlets low enough to keep the lowest position until the vadose
transition occurs and they capture all the flow from a junction. This is
schematically illustrated in Fig. 9b. Figure 11 presents the evolution of a
network with initial conduit diameters drawn from a uniform distribution
with a range of 10
Generally, the evolution follows the concepts described in Fig. 8. In the pressurized phase, the selection of efficient pathways depends also on the conduit diameters and the W–E conduits are not necessarily the ones with the highest flow rates.
Figure 12 shows the evolution of total discharge from the network over time.
Initially, most of the available recharge flows over the surface. First In1
and In2 integrate with full recharge, which adds up to 2 m
Another selection mechanism becomes active at the transition to a free-surface flow, which is shown in Fig. 13, where a few snapshots of the SW
part of the network show the evolution of several competing pathways
evolving from input In5. The junctions of interest are marked by 1 to 3 and
enclosed in gray circles at 4800 s. In the pressurized flow regime (4800 s),
the N–S oriented conduits, marked by
When the flow is pressurized, the flow partitioning between two competing
pathways, connecting the same junctions is divided based on the resistance
to flow. Note that conduits
Figure 15 presents a quasi-stable flow and network pattern for the case
identical to the one presented in Fig. 11, but where the plane of the
network is additionally tilted from N to S for 0.3 m per node. The tilting
makes flow towards S preferential to flow towards N, which is clearly seen
in the resulting pattern. The input In4 now joins In3. Since it is near
the boundary, the input In5 has no option to develop towards S, except that
the pathway heading S from the input (conduit
Other scenarios with more complex settings, such as networks with 50
Distribution of flow between two pathways depends on the flow
resistance when the flow is pressurized. The pathway
Quasi-stable state of network with same structure as presented in Fig. 11, but the plane of the network is additionally tilted from N–S, for 0.3 m per node.
Quasi-stable state for the same scenario as in Fig. 11 with dissolution kinetics for limestone.
Evolution of homogenous sub-vertical network. Blue arrows on
Finally, we turn to a network where dissolution rate is predominantly surface
controlled, as is supposed to be the case for limestone. To this end we have
modeled a network, identical to the one in Fig. 11, but with
We now turn to the situation where the network is steep (almost vertical). As this network presents a vertical cross section of karst, we omit the geographical notation and use top, bottom, left and right for the sides of the networks.
Similar models for laminar flow have been presented by Gabrovsek and Dreybrodt (2001) and by Kaufmann (2003). The basic result of these prior models was a continuous drop of the water table due to increased transmissivity of the network and the formation of base level conduits. If a fixed head boundary was applied, competition between a high conductivity zone along the water table and prominent conduits within the phreatic part of the network resulted in a complex pattern of evolved conduits. For many more scenarios of this modeling approach the reader is referred to the book by Dreybrodt et al. (2005)
Figure 17 presents a case where all conduits are 10 m long with initial
diameter of 0.005 m. A maximum possible recharge of 5 L s
Initially (Fig. 17a), a small part of the available recharge enters the network. At the top-right all the recharge is drained directly into the outfall junction (marked by a red circle in Fig. 17a. The flow rates within the conduits are small and dominant along the vertical conduits (top to bottom). Flow along horizontal conduits is small and increases from left to right.
After 600 s (Fig. 17b) the entire network is still pressurized. Horizontal conduits have evolved sufficiently to drain more flow brought in by initially developed vertical conduits. Accordingly, the potential gradient becomes oriented to the right and is the highest close to the boundary. Conduits at the top-right corner experience the fastest growth and capture almost all recharge from the inputs. The flow in the left part of the network is small and the hydraulic potential field is relatively flat there. After 1200 s (Fig. 17c) the top-right corner has become vadose. In this area, the recharge is carried vertically to the water table. The flow rates are the highest along the water table and diminish with distance from it.
However, widening is still substantial below the water table which additionally increases the network permeability and downwards retreat of WT. The process continues until the WT drops to the base level and only vertical recharge conduits and a master conduit at the base continue to grow. The vertical conduits have been widened through the entire evolution; the uppermost for the longest time and they are therefore largest. The diameters decrease from top to bottom. On the other hand, the diameter of horizontal channels increases from left to right, as they evolve only below the water table. Therefore, deeper conduits have more time to evolve.
In the case shown in Fig. 18 we assign a more complex distribution of
initial conduit diameters. The initial diameter (
The initial diameter of the top horizontal line of conduits is 0.1 m.
A recharge of 100 L s
Pathways along the conduits with initially larger diameters evolve efficiently and capture an increasing amount of flow.
At 9000 s (Fig. 18b) about 70 % of the flow is captured by the
junction marked by a blue triangle and denoted by 1 in Fig. 18b. It feeds a
line of vertical conduit that discharges into outflows through horizontal
conduits. Numbers on the conduits in the top-right region denote flow along
the conduits in L s
High-dip network with random initial distribution of conduit
diameters. Flow enters at the top-left edge of the network as pointed by a
blue arrow. Values on
By 24 000 s, the outflow position has retreated to the bottom (Fig. 18c). When the vertical pathway downwards from point 1 becomes vadose, it provides a free-outflow boundary and triggers the development of pathways draining sink points 2 and 3 (Fig. 18b and c), which soon capture all the flow. In Fig. 18c, the flow along the top line has retreated to point 3, and throughout the remainder of the simulation continues to retreat towards the left to points 4 and 5 (Fig. 18d). Ultimately, the flow is captured by the node at point 5 (Fig. 18e). Similarly, the flow migrates from top to bottom, towards the deeper connecting pathways. Figure 18e shows the stable flow situation at 75 000 s, where all the flow follows one single pathway. Downward and leftward progress is slow because some of the conduits to the left are initially small and the permeability is low. In comparison with a uniform network with distributed recharge, the development follows initially prominent pathways, with progressive upstream flow capturing. Soon after a pathway becomes vadose, the flow is overtaken by the evolving pathways to its left.
The progression mechanism, described above, is demonstrated clearly by a final idealized, but telling, example. We assume three vertical conduits (wells) with an initial diameter of 0.2 m, extending completely through the domain in the vertical direction.
These are connected with five evenly spaced horizontal conduits with initial
diameter 0.005 m extending across the domain. All other conduits are
effectively impermeable, with a diameter of 10
Initially (Fig. 19a), all conduits are pressurized. There is almost no gradient left of W3, where evolution is slow or none. High gradients exist between W3 and the outfalls, the highest being along the deepest horizontal conduit, which has the highest flow and evolves most efficiently. As W3 becomes vadose, it presents a free-outflow boundary for the flow from its left and the gradient along the horizontal conduits connecting W2 to W3 builds up. These conduits now experience fast evolution with rates increasing from the top to the bottom (Fig. 19b). The mechanism progresses leftwards: when W2 becomes vadose, W1 connects to it as shown in Fig. 19c. In Fig. 19d, a stable flow condition is shown, where all the flow follows the wells which feed the base level channel.
High-dip network with three prominent conduits (wells), marked by
W1 to W3. A recharge of 100 L s
Sensu lato Palmer (2007) this paper considers the hydrological control of cave patterns, particularly those leading to branchwork cave systems. In the pressurized phase the model gives similar results as the other existing models. This model introduces the selection of flow pathways on a local scale, i.e., at a particular junction, which occurs when a junction becomes vadose. In a long-term perspective, only one outlet conduit drains the node. In nature, down-flow bifurcations are not common in open channels.
In the pressurized phase, the flow out from a junction is distributed to the outlet conduits, according to their resistance to flow and the distribution of hydraulic heads. This also defines the rate of their inlet incision. When a junction becomes vadose, the conduit with the lowest inlet entryelevation has an advantage and is a candidate to take all the flow. However, under vadose conditions the conduit's alignment with respect to the dip of the network becomes important, as higher slope generally invokes higher energy grade, higher flow velocity and faster incision. A conduit that gains advantage in pressurized conditions can be surpassed by a conduit with a higher slope, which has an advantage in free-surface conditions. Once the stable flow pattern is established, the flow follows a system of conduits that all occupy the lowest position in their upstream junctions.
In a homogenous scenario, the evolution is focused to the transitional area between pressurized and free-surface flow, the water table. The flow from the surface is gravitational along the vadose channels down to the water table. There, it is largely focused to the conduits close to the water table. The scenario demonstrates a relatively smooth drawdown of the water table due to increasing permeability in the phreatic zone. The end result is a relatively uniform network with a growing base level conduit. Similar results were obtained by Gabrovšek and Dreybrodt (2001) and by Kaufmann (2003), where only dissolution in the phreatic zone was considered.
The inhomogeneous case demonstrates the evolution of invasion vadose caves based on flow diversion. The drawdown of the phreatic zone is irregular, following fast evolution of prominent pathways and progressive upstream flow capturing. Such a scenario can produce an extended network of steep vadose passages.
Deeply penetrating conductive structures can play an important role as they transfer surface water deep into the massif and redistribute hydraulic gradients. This way fast evolution along deep horizons can be triggered.
The presented model closes some of the open questions, which have not been addressed by the older existing models. The final flow pattern results from all stages of network development, starting with the initial stage, continuing with the growth, integration and expansion under pressurized flow as well as, what is demonstrated by this model, with the final selection of stable flow pathways on a local scale during and after transition to free-surface flow regime.
On the other hand, the model opens new challenges related to evolution of karst aquifers in vadose settings. Further work is needed to improve estimation of dissolution rates and the related role of sediment transport and mechanical erosion. Further steps towards more realistic modeling domain and boundary conditions are also needed. In fact, a single low-dip plane is a scenario which is not common in the nature. A careful step towards 3-D models that simulate speleogenesis, in both phreatic and vadose conditions, is therefore needed. What we mean by careful is the gradual adding of complexity, so that at each new step all mechanisms from previous steps are well understood. The presented model allows such extensions.
At the same time, we have to keep in mind that the modeling results are not stand alone, i.e., they should progress hand in hand with conceptual models based on the field observations.
This work would not have been possible without the support of Slovenian Research Agency, who financed the doctoral work of M. Perne. The work of F. Gabrovšek was funded by the project J2-4093. M. Covington and M. Perne were also supported by the National Science Foundation under grant no. 1226903. We thank to Derek Ford and Steffen Birk for the valuable comments, which helped to improve the manuscript. Edited by: S. Attinger