Reactive transport in porous or fractured media often results in an evolution
of highly conductive flow channels, often referred to as “wormholes”. The
most spectacular wormholes are caves in fractured limestone terrains. Here, a
model of their early evolution is presented. The modeling domain is a
two-dimensional square net consisting of one-dimensional fractures
intersecting each other in a rectangular grid. Fractures have given width
The first numerical models of speleogenesis in terrains of soluble rock considered cave evolution only along one single isolated plane-parallel fracture. First attempts using this concept failed because the linear dissolution rates used caused exponential decline of dissolution widening along the fracture. The cave conduits stopped to grow after penetrating only a few meters into the rock (Dreybrodt, 1996; White and Longyear, 1962).
The experiments of Wigley and Plummer (1976) demonstrated a switch in the dissolution kinetics to a nonlinear regime close to the equilibrium concentration of calcium ions with respect to calcite. Based on these results White (1977) suggested that such a switch reduces the dissolution rates and causes deep penetration of dissolution power into the rock. This allows evolution of caves in geologically reasonable times. Laboratory experiments showed that this concept is valid for limestone (Svensson and Dreybrodt, 1992; Eisenlohr et al., 1999) and also for gypsum (Jeschke et al., 2001). Therefore, it was used in later modeling approaches of speleogenesis (Dreybrodt, 1990; Palmer, 1991), although Dreybrodt et al. (2005a) show that linear kinetics do, in fact, allow the evolution of caves if one considers simultaneously two processes: linear surface kinetics and transport of the dissolved ions by molecular diffusion into the bulk of the water.
In modeling cave genesis, homogeneous fractures with initially even spacing were used as the basic elements in two-dimensional (2-D) models consisting of a net of such fractures (Dreybrodt et al., 2005a). Fracture widening depends only on the distance from the fluid inlet. Therefore, these fractures have been described as one-dimensional (1-D). Using them in speleogenetic models was criticized by Szymczak and Ladd (2011). They showed that homogeneous 1-D fractures exhibit an instability to infinitesimally small perturbations such that the initially evenly propagating dissolution front breaks up into channels, hereafter called “wormholes”. The interaction between these wormholes causes competition, whereby only a few reach the output boundaries, while the others stop growing. This behavior is well known from the evolution of wormholes in porous media (Fredd and Fogler, 1998).
Szymczak and Ladd (2011) questioned the approach used commonly by many model efforts, which uses nets of 1-D fractures (e.g. Dreybrodt et al., 2005a), because the formation of wormholes within individual fractures is not taken into account. This way, the breakthrough time in an individual fracture can be reduced significantly, causing a change in the hydraulic properties of the global fracture network. It is, in principle, possible to meet this criticism by discretizing each single fracture into a 2-D aperture field, to permit wormhole formation – however, at high computational cost.
Alternatively, some models have used circular pipes as basic elements instead of 1-D fractures (Bauer et al., 2005; Kaufmann, 2005). This avoids the formation of wormholes in the 1-D elements of the 2-D net. However, the results of such models are close to those using fractures. This gives confidence that the formation of wormholes in the fracture elements of the 2-D net does not change the general behavior. Therefore, in this paper, we use nets of 1-D fractures to investigate the formation of wormholes in 2-D networks and the interaction between the evolving channels, favoring those that have gained in length compared to their neighbors. In this paper we use the term wormholes because this is common in that context. In our model, caves and wormholes have the same meaning.
Wormhole formation has been in the focus of many researchers from different fields. There are other systems with similar competitive dynamics, where fingers grow. The longer ones screen the shorter ones, thus preventing their growth (Couder et al., 2005) as we find it in this paper.
Budek et al. (2015) investigated the growth of anisotropic viscous fingers in flow of immiscible fluids in a periodic, rectangular network of microfluidic channels. Although the underlying physics is different in both cases and from that in our work, the temporal evolution of viscous fingers is similar as we observe it in the basic case.
A larger class of systems with competitive growth is described in a review article by Krug (1997) dealing with solid state properties of materials generated by molecular beam epitaxy, a topic remote from our system.
Most of the cited work focuses on the mathematical properties of competitive growth. Therefore, they are not perceived by the community of earth science. In this work we take a different, more empirical approach. From the results of model realizations we detect the underlying mechanisms of hydrodynamical flow in the fractures and its interaction with dissolution widening their apertures. We will answer the following questions. Why are breakthrough times reduced even under linear kinetics when a wormhole evolves within a net of water-transmitting fractures? How does the feedback causing breakthrough in a net differ from that active in a 1-D isolated fracture? How do evolving wormholes interact to select the winner and stop the competitors in further growth? How does the instability as described by Szymczak and Ladd (2011) influence the evolution of karst aquifers? We demonstrate that answers can be given by applying the physical mechanisms of flow and dissolution active in a 2-D net of fractures, without using a complex mathematical algorithm.
Here, we describe in short the model suggested by Dreybrodt et al. (2005a). First,
construct a 2-D square net consisting of 1-D fractures
with given width
Modeling domain used in this work is a rectangular grid of fractures
connecting 150
The boundary conditions throughout this work are constant head
In a first step, we calculate the flow in all the fractures of the net. At
each confluence of fractures, we assume complete mixing of the inflowing
solutions. Mass conservation requires
Next, we specify the calcium concentration
By following the order of decreasing heads, we select all nodes where the
concentrations of all the inflowing solutions are known. We assume complete
mixing of these solutions before they are transferred into conduits
transporting the flow away. We repeat this until the input concentrations
for all fractures are determined. From this, the new profiles of the
fractures after a time step
Once the flow rate at the input of a fracture and its input concentration of
calcium are known, dissolution widening of each fracture is calculated by
the following procedure. The widening rate at any point in any fracture is
proportional to the dissolution rate
Discretization and mass conservation in a basic network element; a
1-D fracture. The conservation of mass requires that the change in
concentration
Mass conservation requires that the amount of calcite dissolved from the
walls during the time interval
From this one obtains
For a uniform plane-parallel fracture, integration of Eq. (6) using Eq. (7)
yields
To obtain the temporal evolution of the profile of any fracture Calculate Calculate Calculate the new profile assuming a constant rate in the time
interval
Discretization of the spatial variable
To get a first insight, we start with a homogeneous net with equal aperture
widths
Temporal evolution of a uniform fracture network. Panels in the
upper row show aperture widths as line thickness and relative dissolution
rates (a ratio between the rate in a fracture and the highest dissolution
rate
Figure 3 shows the temporal evolution of the aquifer. The upper panels
illustrate the dissolution rates by a color code shown below and the fracture
aperture widths by a bar code. The distribution of the hydraulic head is
depicted by isolines in the upper panels. The lower panels depict the flow
rates by the thickness of lines shown below and their directions by colors.
Grey means flow downstream (left–right), blue is the direction along a
transverse fracture from the lower boundary to the upper one (flow up), and
red is flow in the opposite direction (flow down). Note that the flow rates
are normalized to
After 1400 years, an almost even front of widening channels has propagated a few meters downstream (Fig. 3a). Flow (Fig. 3e), however, exhibits an uneven distribution. There are domains of transverse flow up (blue) and down (red) originating from the different wormholes.
After 1600 years, due to the instability caused by numerical noise, the front breaks (Fig. 3b and f) and many fractures protrude out from the previously even front. The domains of lateral flow have increased. The green color (Fig. 3e–f) connects nodes at the crests and troughs of the hydraulic potential field and, therefore, marks water divides between the lateral flows originating from the various channels.
After 1800 years (Fig. 3c and g), seven prominent channels have propagated into the net. Transverse flow from the leading channel dominates, whereas lateral flow from the channels staying behind becomes low, as shown by the white regions (Fig. 3g). After 1890 years, only one channel has reached the output boundary, whereas all others have almost stopped growing. Due to the redistribution of the heads, the leading wormhole ejects flow perpendicular to the isolines into the net, as can be seen from the red and yellow regions of dissolution rates close to its tip in Fig. 3h. The shorter losing wormholes are now located in a region with low hydraulic gradients. Therefore, flow in them is reduced.
Temporal evolution of the total flow through the network (solid
brown line), input flow rates, and lengths of wormholes at different positions
(given as a
Figure 4 depicts the temporal evolution of the total flow through the domain
(solid brown line), flow rates into the inputs of the evolving wormholes
(full lines), and their lengths (dashed lines) defined as the distance where
Soon after, the curves separate, where more successful
wormholes (at
In view of the results in Fig. 3, one asks for the detailed mechanisms by which the different wormholes compete for growth and the flow they carry. To this end it would be advantageous to deal with scenarios with only a few competing wormholes.
To this end we use the idea of Upadhyay et al. (2015), who have put seeds into
the entrance region of the modeling domain consisting of areas with increased
fracture aperture width with respect to the apertures widths in the net. This
way the seed triggers wormhole growth from its region. To trigger the
instability of the dissolution front, we insert seeds in the following way.
Using the net of Fig. 3, we select some input point at the left-hand
boundary. From there, we assign a fracture aperture width
Evolution of a seeded network. Same as Fig. 3 but slightly larger
initial apertures by
Figure 5 depicts the evolution of a single wormhole initiated by a seed with
With deeper penetration of the wormhole, the regions of transverse flow
increase and high dissolution rates are active along the entire wormhole
(1455 years; Fig. 5c and g). At 1485 years, shortly before breakthrough, the
vertical fractures (vertical meaning parallel to the
The high transverse flow rates are caused by steep transverse hydraulic
gradients close to the wormhole. This can be envisaged from the lines of
equal heads (isolines) shown in the top panels of Fig. 5. With increasing
distance upstream from the tip of the wormhole, the flow rates decline
because of decreasing hydraulic gradients. At the tip head, gradients in the
horizontal direction rise, and consequently the flow rates increase in this
direction (grey lines). Flow rates drop along the transverse fractures with
greater distance from the wormhole because, at the junctions with horizontal
fractures, flow is partly diverted into these horizontal fractures and then
guided to the output at
Comparison between the case with no seeds (
In Fig. 6, we compare the pressure fields of the evolution in Fig. 3 (no seeds, Fig. 6a) and Fig. 5 (one seed, Fig. 6b) at times where their channels have equal lengths. In Fig. 6c, their head distributions are compared. The black isolines show the head distribution for the scenario without seeds and the red ones with one seed. In the downstream region for heads smaller than 1200 cm, the head distributions become very similar. Therefore, the evolution of the leading wormhole seems to be independent of the presence of other wormholes. On the other hand, the fingers in the non-seeded (homogenous) case (Fig. 6a) would have grown deeper into the domain if the leading channel had not existed.
Breakthrough situation in a network with
To explore this deeper, in Fig. 7 we compare the evolution of a scenario
with three seeds at
In scenario a (Fig. 7a), until 1200 years, all flow rates are equal (Fig. 7c).
Then, the instability causes an advantage for the flow rate through the
input at
To understand the dynamics of the evolution of wormholes, we first investigate in detail the evolution of a single wormhole. Then, we study the competition between two wormholes, which are initiated by identical seeds at various distances from each other. Finally, a scenario with many seeds is shown.
We go back to Fig. 5, the scenario with a single seed
resulting in the creation of a single wormhole. The high transverse flow
rates are caused by steep transverse hydraulic gradients close to the
wormhole. This can be envisaged from the head isolines shown in the top
panels (Fig. 5a–d). With increasing distance upstream from the tip, the flow
rates decline because of decreasing gradients. At the tip head, gradients in
the horizontal direction rise, and consequently, although small, the flow
rates in this direction increase (grey lines). Flow rates drop along the
transverse fractures with lateral distance from the wormhole because, at the
junctions with horizontal fractures, flow is partly diverted into them and
then guided to the output at
Figure 8a illustrates the flow rates in the central fracture along the wormhole as a function of the distance from the input for various times. In the beginning, when the length of the wormhole is small, flow rates along the wormhole are low, and, due to outflow into the vertical fractures, the flow rate declines to a small value at its tip, which is determined by the overall flow resistance of the initial net. With increasing time and length of the wormhole, the vertical outflow increases, allowing rising input flow at the constant head boundary. Close to the tip, due to the flow out into the transverse fractures, flow along the wormhole declines to a value determined by the remaining resistance of the net. This behavior continues until breakthrough of the wormhole.
Figure 8b depicts the transverse flow up and down from the wormhole. In the
beginning, when the wormhole is short, the transverse flow rates increase
steeply by orders of magnitude until they reach a maximum close to its tip
and then they decline rapidly. Note that the sum of the total transverse
outflow along the wormhole and the longitudinal outflow at its tip must be
equal to the inflow at its input. The region where the horizontal flow rates
decline in Fig. 8a marks the region of major transverse outflow. When the
length of the wormhole increases, this region is shifted deeper into the net
and the maximum rate of flow becomes higher, marking the increase in total
transverse outflow in time. Thus, the inflow of fresh solution, aggressive
with respect to limestone, increases with increasing length of the wormhole
supporting further dissolution along its entire length and at its tip. The
vertical fractures carrying the transverse flow from the wormhole into the
upper and lower domain are also subjected to the competition and wormholing,
making some of these fractures more successful than others. This results in
the nonuniform transverse pattern seen in Fig. 7a (wormhole at
Evolution of total flow through the network (full red line), input flow into the seeded wormhole (dashed), and length of the wormhole (blue line).
Figure 9 shows the temporal evolution of the total flow through the domain,
the flow rate into the input of the wormhole, and the length of the
wormhole. In the beginning, flow into the wormhole is low and given by the
resistance of the net. For the first 1000 years, flow remains almost
constant. During this time, the solution front progresses evenly. Then, the
instability causes initiation of the wormhole and flow through it rises.
With increasing length of the wormhole, the resistance between the tip and
the output becomes smaller, and
Profiles of aperture widths (solid lines) and saturation ratios
(
To get more insight, in Fig. 10 we have plotted the aperture widths (solid
lines) and the saturation ratio
The following questions arise. How important is dissolution in the net adjacent to the wormhole? Is its increase in permeability sufficient to create a feedback or is it of only minor influence? To answer these questions, we have investigated a scenario where only the walls along the line of horizontal fractures along the wormhole are soluble and the walls of all remaining fractures in the net are insoluble. Figure 11a depicts a comparison between the input flow rates in scenarios with and without dissolution in the net.
For a net of soluble fractures, there is a long time of low constant flow due to the even solution front propagating slowly downstream. As long as the dissolution front is completely uniform, transverse flow is not possible and enhancement of dissolution triggered by transverse outflow is absent. Each fracture behaves as an isolated fracture. As soon as the instability gives advantage to one fracture it can eject transverse flow and start to grow rapidly until breakthrough is attained.
When the fractures of the net are insoluble, except those of the wormhole, an
even dissolution front cannot be established. The line of soluble fractures
along which the wormhole propagates gains an advantage immediately and
reaches breakthrough in a much shorter time. It is important to note that the
temporal evolutions of the breakthrough curves are almost identical if one
compares them from the time where flow exceeds 1 cm
We therefore postulate that the main mechanism causing progression of the wormhole is an increase in the input flow caused by ejection of transverse flow into the net. In conclusion, the following feedback mechanism seems to be plausible. As soon as one wormhole, for whatever reasons, becomes longer than the neighboring ones, it emits transverse flow that increases its input flow. The resulting enhanced dissolution capacity increases the length from where transverse flow can be emitted, and, consequently, the amount of outflow increases (see Fig. 8) causing growing inflow. It is interesting to note that for a net of soluble fractures the advancing dissolution front retards breakthrough considerably.
In the next step, we study the interaction of two wormholes growing
simultaneously. We construct a net with two seeds, at the various positions
as shown in Fig. 12, which illustrates the temporal evolution of the
aquifer. We start with the two middle panels (Fig. 12b) depicting the
evolution of a domain with two seeds located at
Interaction of two wormholes seeded at distances of 10
Figure 13 shows the dissolution rates and flow rates along the two wormholes
at
Profiles of widening (full lines) and flow rates (dashed lines) for
the case with seeds at
Figure 14 depicts the temporal evolution of the lengths and the input flow rates of the two wormholes for all three scenarios in Fig. 12 until breakthrough. The middle panel (Fig. 14b) depicts the scenario discussed here. In the initial state, the lengths are equal as expected. When the instability becomes active, the lower wormhole grows rapidly, whereas the upper one experiences delayed growth. The same behavior is exhibited by the input flow rates. This pattern is characteristic for the onset of instability in nonlinear systems.
Temporal evolution of flow (full lines) and length (dashed lines) of the interacting wormholes of Fig. 12. Labels corresponds to Fig. 12.
From these findings, one may conclude that the interaction between the
wormholes depends on the distance between them in the
To give further evidence we study a scenario with two seeds as above but
with increased distance between them, which is larger than the distance of
influence. The lowest panels in Fig. 12 show the evolution of a domain with
two seeds at
If, in comparison to Fig. 12b, the distance between the seeds becomes smaller as illustrated in Fig. 12a, one expects increasing dominance of the winning wormhole. At 1400 years, the two competing wormholes exhibit similar lengths in Fig. 12a and b. At later times, however, the losing wormhole stops growth in Fig. 12a, whereas it still gains length in Fig. 12b.
The evolution of flow and length for the three distances is shown in Fig. 14. With increasing distance the time needed to gain advantage for the winner increases until both wormholes become too distant to interact and propagate at the same pace.
From these findings, we conclude that if several wormholes are located inside the region of influence of one another, only one of them will reach breakthrough. This is illustrated in Fig. 15, where 10 seeds are inserted at distances of 30 m. After 1200 years, all seeded wormholes have developed to almost equal lengths. Due to the instability, the wormholes start to grow with differing speeds. At this point, it is not possible to predict the further evolution.
Interaction of set of wormholes seeded at the distance of 15 nodes.
Triangles and numbers denote location and
After 1400 years, three wormholes (at
Evolution of input flow rates (solid lines) and wormhole lengths (dashed lines) for the wormholes marked in Fig. 15.
At the beginning, flow through all fractures is equal. After 1100 years, the
instability causes an advantage for wormholes at
From what we have found so far, the following picture of the evolution of a homogeneous net as described in Fig. 3 arises. Due to the instability of the system to small perturbations (Szymczak and Ladd, 2011), the initially uniform propagation of all equivalent fractures breaks down and some fractures gain advantage. The evolving pattern at that stage is not predictable. After some short time, however, the interaction of the different wormholes determines the head distribution and the flow rates in the entire net. The further evolution proceeds in a deterministic way. Nevertheless, the final patterns are predetermined by the initial instability.
For homogeneous nets with seeds, as in Fig. 14, where an analytical solution predicts equal lengths for all wormholes starting at individual seeds, the final patterns show wormholes only along the directions of the seeds. From this, one can suggest that for heterogeneous nets containing percolating pathways of fractures with aperture widths differing from those in the net, the wormholes should develop favorably along parts of these percolating pathways by a deterministic process where the instability plays no role.
Figure 17 gives an example of a dual network, where a net of prominent
fractures with aperture widths of
Evolution of wormholes in a dual network. The network is a
superposition of the basic network (Fig. 3) and a crude net of prominent
fractures with
After 5 years in the dual network (Fig. 17a and e), small wormholes of equal lengths have developed along all the prominent fractures originating from the input boundary because the prominent fractures act as seeds. This prevents the creation of an even dissolution front along all the inputs without prominent fractures. After 80 years (Fig. 17b and f), wormholes of differing lengths have invaded the domain. The leading one inhibits further growth of all others, as can be seen after 230 years (Fig. 17c and g). After 290 years (Fig. 17d and h), the winning wormhole achieves breakthrough. It is interesting to note that introduction of prominent fractures reduces breakthrough time from 1900 years in the homogeneous net to 290 years (see Fig. 3). There are two reasons for this. First, an even dissolution front cannot evolve because perturbations from one-dimensionality are strong. Second, due to the wider prominent fractures along the pathway and also transverse to the winning channel, input flow is greater than along a pathway with fracture aperture width of 0.02 mm exclusively. Therefore, more calcite can be dissolved and the wormhole proceeds faster to breakthrough. For completeness, Fig. 18 illustrates the flow rates into the evolving wormholes and their lengths.
Another way to break the action of the instability is to select distinct input regions or input points instead of an even head along the entire input boundary. Such cases have been explored in Dreybrodt et al. (2005a).
Evolution of input flow rates (solid lines) and wormhole lengths
(dashed lines) for the wormholes marked in Fig. 17. Numbers on curves denote
the
From our findings so far, we can summarize the evolution of wormholes as follows.
In Fig. 19a, we show the temporal evolution of the inflow into the winning
wormhole for the basic case (see Figs. 3 and 4) in comparison to the
evolution of the same domain with one seed implanted (see Fig. 5). In both
cases we find a long initial period of about 1000 years during the formation
of the even dissolution front with low, almost constant inflow. In both
cases the instability becomes active, but in the seeded domain this happens
about 400 years earlier. After the activation of the instability, a fringe
with several small channels evolves and flow rates rise slightly to a few
tenths of a cubic centimeter per second. In the unseeded case, several
wormholes grow from this fringe with almost uniform rate until one of them
has gained an advantage in its length and from then on develops independently
of its past history. If only one line of fractures connecting the input to
the output boundary is soluble and all other fractures in the net are
insoluble, competition is excluded and the evolution of a dissolution front
is thus not possible, so that the wormhole starts to grow immediately. In
all cases shown in Fig. 19, the evolution of the wormhole is almost
identical from the moment when it has gained an advantage over its competitors.
This is shown in Fig. 19b where we have shifted each curve by its
breakthrough time
Inflow into the wormhole as a function of time
Figure 20, as a further example, illustrates the evolution of a
heterogeneous domain with statistically distributed fracture aperture
widths. These are taken from a lognormal distribution with
Evolution of wormholes in the network with lognormal distribution
of initial aperture widths.
Compared to a single 1-D fracture of the same dimension (300 m
To understand this, we go back to the evolution of the wormhole in a
homogeneous aquifer with one seed. Figure 21a shows, for various times, the
profiles of dissolution rates converted to widening of the fractures in
centimeters per year and the concentration
Profiles of different parameters at given times for the
single-seeded case (network evolution presented in Fig. 5).
With increasing
At each location in the fracture, penetration length increases with time. Therefore, dissolution can penetrate deeper into the fracture and the length of the wormhole increases. With increasing length, the amount of outflow from the wormhole into the net increases, and the flow into the wormhole grows because the effective resistance of the part downstream from its tip is reduced. Therefore, penetration lengths increase and cause deeper penetration of the wormhole into the aquifer. Here, the feedback loop is related to the resistance of the net into which it is embedded.
We have, therefore, explored a scenario where all horizontal fractures
(i.e. parallel to the
Breakthrough time of a network with a central fracture
In our model, the basic element of the 2-D net is treated as a 1-D fracture.
This has been criticized by Szymczak and Ladd (2011). They argue that, due to
the inherent instability, the fracture will not be widened evenly but
wormholes will arise and the breakthrough time will be reduced. Wormholes,
however, are created only in a limited range of
Péclet and Damköhler numbers (Szymczak and
Ladd, 2009). The Péclet number,
To verify this finding we have employed the following approach (Fig. 23). We
consider a fracture that has just been reached by the wormhole (Fig. 23a, b).
It has experienced almost no dissolution so far. We discretize this fracture
into a network of 100 by 200 fractures, each 1 cm long and 1 cm wide with
aperture width of 0.02 cm. (Fig. 23c). Figure 23a shows the 2-D net with the
wormhole and the even dissolution front. A square marks the region enlarged
in Fig. 23b, where the fracture of interest, at the tip of the wormhole, is
marked by the blue arrow. Figure 23c depicts the discretization of this
fracture and the dissolution front that is created in it after some time. The
downstream boundary nodes of the fracture are connected to insoluble
fractures with length
In the first three 1-D fractures an even front develops initially. Due to the instability the front breaks down and wormholes grow there after 1200 years. In contrast, in all the following 1-D fractures downstream, due to the increasing flow after the 2-D-wormhole has arrived there, the Péclet number rises sharply by about 1 order of magnitude within a few tens of years. Therefore these 1-D fractures exhibit an even dissolution front. In our model we have assumed that in each junction of fractures lateral head differences are smoothed in such a way that at the downstream input fractures constant head conditions can be applied. Piotr Szymczak in the interactive comment regarding this work has argued correctly that this assumption is dubious “as the pressure will be highly nonuniform there, with the maximum along the developing wormhole” at the output of the fracture. Due to the computational limitations, however, we have no choice in our approximation. A better approximation might have been to limit the width of the fractures to about one-tenth of the width we use to account for the wormhole formation in the first fractures. The basic behavior of the wormhole formation under these conditions will be similar qualitatively to our findings.
On the other hand under initial boundary conditions of higher head at the input of our 2-D model all fractures may show even compact dissolution fronts due to higher flow through the system. Therefore we have repeated the procedure described above for higher heads imposed onto the two-dimensional net. For heads higher than 27.5 m we find even and compact dissolution fronts in all fractures including the entrance one. We have also repeated the calculation for the basic case (see Fig. 3) with the elevated head of 27.5 m instead of 15 m and found qualitatively similar behavior as for a head of 15 m. From this, one may conclude that wormholing in the 1-D fractures does not change the general behavior of the nets because as pointed out by Piotr Szymczak in his interactive comment “many features of flow-focusing systems are rather generic and independent of the particular model”. Of course our approximation cannot be applied to predict breakthrough times in real systems. The target of our work, however, is to get insight into the processes active during the formations of wormholes.
To reveal the mechanisms governing the evolution of wormholes in fractured limestone aquifers on the scale of several hundred meters, we have used a 2-D fracture network consisting of an array of 1-D fractures with defined hydrodynamic and hydro-chemical properties.
As a basic scenario, we start with a homogeneous network where all the 1-D fractures have identical initial properties. We find that the evolution of such networks proceeds in two steps. In the beginning, an even dissolution front invades slowly into the modeling domain. Dissolution in all fractures in the front is identical. Due to instability inherent in such homogeneous systems, some fractures in the front gain an advantage and dissolution along them penetrates deeper, thus rendering these evolving wormholes slightly longer than their neighbors. Wormholes develop along these advantaged fractures because, due to the differing lengths, the redistribution of hydraulic heads increases flow along these fractures and aggressive water is delivered preferentially from the input, increasing dissolution rates along these fractures. The position of the wormholes cannot be predicted deterministically. Then, the second deterministic step of wormhole formation is triggered. Several competing wormholes invade the aquifer until one of them reaches the output boundary. The other wormholes stop growing.
If one of the input fractures serves as a seed by slightly increasing its
initial aperture width
Wormholes interact with each other. In a scenario with seeds at various distances, one finds a critical distance. If the separation of the wormholes is larger than this critical distance, they grow independently of each other. For smaller distances, interaction is active and the winning wormhole inhibits the growth of the losing one. If many wormholes grow initially, a region of influence can be defined. If two or more growing wormholes are located within this region of influence, only one of them will achieve breakthrough.
For a heterogeneous domain with statistically distributed fracture aperture
widths, taken from a lognormal distribution, there is no appearance of an
even dissolution front. Instead several competing wormholes start to grow
immediately. The initial step of a slowly invading even dissolution front is
prevented since the homogeneity, and its corresponding one-dimensionality, of
the net (i.e. its properties do not depend on the
There are no data associated with the content of this work. All the results are based on the model developed by the authors. The respective codes can be obtained from the authors.
WD initiated the work and wrote the text. FG is the author of the model code. He performed simulations and prepared figures and figure captions. The paper is based on the in-depth discussions of both authors.
The authors declare that they have no conflict of interest.
Franci Gabrovšek acknowledges the financial support from the Slovenian Research Agency (research core funding no. P6-0119). The authors thank Vanessa E. Johnston for careful reading of the paper and for pointing out many errors and inconsistencies.The article processing charges for this open-access publication were covered by the University of Bremen.
This paper was edited by Alberto Guadagnini and reviewed by Piotr Szymczak and Arthur Palmer.