Expected scenarios due to collapsed cavity underneath a dike

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Historical evolution of karst and dikes knowledge

Karst

Karst refers to a distinctive terrain that evolves through dissolution of the bedrock. It is therefore associated primarily with limestone, but also with other carbonate rocks (chalk, dolomite) and other soluble rocks (gypsum, salt) (Worthington and Ford 2009, Waltham et al β005). The origin of the word “karst” can be traced back to several centuries ago in different derivatives of European and Middle Eastern languages, such as the use of “karra” to refer to stone. The region called Kras (between Slovenia and Italy) was the first karstland to receive intensive scientific investigations for its natural characteristics, and hence was viewed as a “classical karstic terrain”. The word, afterward, evolved to karst with the Germans (during the Austro-Hungarian occupation of this country) and typically was used to describe landscape in limestone, dolomite or other soluble material (Huang 2007, Williams 2008).
Typical karstic topography consists of dry streams, sinkholes, caves, enclosed depressions, fluted rock outcrops and large springs (Ford and Williams 1989). Limestone consists mainly of calcite CaCO3, formed by the deposition of organisms (shells and corals) or by (bio) chemical precipitation (for example in lakes). Tectonic processes often fracture sedimentary rocks. This process is often rapid, and water entering these cracks gradually dissolves the rocks creating wider and deeper cracks (Forest Practices Branch 2003). The development of karst is largely determined by the water CO2 content, the geologic processes, the climate factors, the geomorphology and the nature of rocks. Karstic terrains cover around 20% of the whole immersed lands as shown in Figure 1(Ford and Williams 2007 and 1989).
Karstic features form because water that is slightly acidic from absorbing carbon dioxide from the air and the soil dissolves the bedrock and forms pathways and channels inside. These pathways, called “conduits” or “channels”, are like underground plumbing that carries water from the surface to springs located in valleys. Eventually, these conduits are exposed by erosion and, if large enough, become “caves” (Sturman and Spronken-Smith 2001).
Karst lands cover around 25% of the surface of France (see Figure 2). This is the case of most basins of the Seine and Somme, and a significant portion of the other major basins (Loire, Rhone, Garonne, Rhine, Meuse). Nearly a quarter of linear French rivers, 10,000 km, flows on karst bedrock, including 3,000 km in the context of significant karst index including the Val d’Orléans (Gombert et al, 2014).

Cavities and sinkholes

Cavities may appear in rock affected by karstification and cause the subsidence and collapse in upper layers of soils. Sinkhole is considered as a final result when the collapse and propagation of the cavity reaches the surface. According to Waltham et al. (2005), sinkholes can be classified into six types with respect to the mechanisms of the ground failure and the nature of the material that fails and subsides: solution and collapse sinkholes can directly occur in rock, whereas dropout, caprock, suffusion and buried sinkholes occur in the overburden soil layers. Thus, the last three types are named overburden cavities (Figure 3). Gutiérrez et al. (2008) updated the previous classification by dividing the sinkholes into two mains groups and by taking into account evaporitic formations (Figure 4): one of them corresponds to solution sinkholes, generated by differential corrosional lowering of the ground surface where karstic rocks are exposed at the surface or merely soil mantled (bare karst). The other group of sinkholes is subsidence sinkholes, which results from both subsurface dissolution and downward gravitational movement (internal erosion or deformation). These sinkholes, that cause the subsidence of the ground surface, are the most important from a hazard and engineering perspective (Gutiérrez et al. 2008, 2014).
Some sinkholes are shaped like shallow bowls or saucers whereas others have vertical walls; some hold water and form natural ponds (USGS 2007).
In this work, we focus on the cavities in the superficial formations, which are named in several references by “overburden cavities”. Thus, wherever the word “cavity” is found in next paragraphs and chapters, it refers to “overburden cavity”.
Karst problems worldwide create huge annual costs that are increased due to insufficient understanding of karsts by engineers (Filipponi and Jeannin 2008, Parise et al. 2015). The presence of a cavity underneath civil engineering structures raises concerns about the safety and stability of these structures. Figure 5 shows examples of major sinkhole collapses underneath roads and buildings in different cities around the word. Due to this, we can think that a collapse of a hydraulic structure may cause catastrophic, human, financial and environmental losses. Many hydraulic structures (dams, dikes) around the world are built on karstic foundations. For example, the U.S. Army Corps of Engineers (USACE) has numerous dams built on limestone foundations that are susceptible to karstification (Schaefer 2009). Their failure can be induced by the underground cavities.

Dike

This work focuses on one of the hydraulic structures which is called dike, and sometimes called flood defense embankment or river levée (Simm et al., 2012). The river dike (or river levée) is defined as “an embankment whose primary purpose is to furnish flood protection from seasonal high water and which is therefore subject to water loading for periods of only a few days or weeks a year” (USACE β000). The modern word dike or dyke most likely derives from the Dutch word « dijk ». Dyke or dike may refer to: a natural or artificial slope or wall to regulate water levels. A dike is called levee (from the French word “levée”) in American English (Ammerlaan 2007). In this work, the word dike is used.
A flood is defined by the French Ministry of Environment as a « submersion (fast or slow) of an area that can be inhabited; it corresponds to the overflow of water » (Lepetit 2002). Stream and river valleys were preferred sites for human habitation for millions of years, most civilizations on the banks of rivers such as the civilizations of Babylon, Egypt and China because such valleys provide drinking water, livestock watering, and irrigation. Soils in river valleys are also among the most fertile that can be found because they are replenished by annual or more or less frequent flooding (Kusky, 2008). For all that reasons, it was important to provide protection for cities and towns by construction dikes that protect people and property from the river in the flood season, and also in order to increase the height level water for the purpose of navigation.
For the US Army Corps of Engineers (USACE), « all the soil is suitable for the construction of dikes » except the very moist, fine grain or very organic soils. Selection of the type of material is generally based on the availability and proximity to the project area. Maximum slopes for dikes are 1V of 2H, where V and H refer to vertical and horizontal dimensions respectively. Dikes with non-ideal materials such as sand, dams are required to have much shallower side slopes (until 1V to 5H) to prevent damage caused by the infiltration and action of flooding waves (USACE 2000, 2006).
Faced with the risk of flooding, people have always tried to protect themselves against overflowing rivers and streams. They tried to « tame » the rivers to live closer to their surroundings. This protection has been made possible by the gradual construction of dikes.
The construction of dikes was built there more than 3,000 years in Egypt, where a levee system was built along the left bank of the Nile, over more than 1000 km. Civilizations of Mesopotamia and ancient China also built large dike systems (Wei 2012). Like those of the Loire, the oldest dikes still operating today and elevated over time in France date back to the middle ages (Serre 2005, Castanet 2008). There are 8,000 km of dikes that have been identified by the state to this day in France, 5,600 km of them more than a meter tall (CEPRI, 2008).

Literature review

Mechanism of karst and cavities collapse

In fact, there is no obvious mechanical explaining the karst collapse, but there are several theories to explain the collapse of cavities, and certain conditions must be available in order to collapse occurs.
The collapse of cavities can take different forms: subsidence, sinkhole, etc. The sinkhole is a brutal failure of the ground above a cavity. The cavities can collapse due to natural and/or human factors. In general, the materials change over time and that the cavities have a natural tendency to fill themselves. There are classically two theories to explain the mechanisms of a collapse (Chan 1995, Tharp 1999, Salvati et al. 2002, Keqiang et al. 2004, Zhao et al. 2011).
The first theory is internal erosion, which occurs when the groundwater table is lowered due to natural (drought) or human (pumping) factors; the groundwater velocity increases and the hydraulic gradient becomes steeper. Thus, the groundwater outwashes and erodes the soil cover (overburden) to increase the cavity dimensions at the interface between the bedrock and the soil cover. There are two possible scenarios, depending on the following conditions (Waltham and Lu 2007; Parise 2008):
– If the soil cover is thick enough and has enough strength (high cohesion), a natural balanced arc will be formed in the soil cavity and the cavity will not collapse, if there are no other inducing factors.
– If the soil cover is thin and has poor strength (low cohesion), the soil cavity will continue to enlarge until a collapse occurs (cf. Figure 6-a).
The second theory is the vacuum suction erosion theory, which involves a confined karstic aquifer. When groundwater suddenly falls by large amplitude and the water table drops below the floor of the soil cover, the groundwater will change from confined to unconfined, and a relative vacuum of low air pressure will occur between the water table and the floor of the soil cover as shown in Figure 1.6-b.
According to Chan (1995) the sinkhole will not be formed if the soil next to the slot (karstic cave) does not fall into the cave. Hence, to evaluate the potential sinkhole formation, the first step is to examine the stability of the layer of soil immediately above rock head. From theories of arching (Terzaghi 1943, Tharp 1999), the thickness of this layer is very approximately the width of the slot (Figure 7).
Figure 6 Classic two theories to explain the mechanisms of a brutal collapse of cavities. (a) Theory of potential erosion. (b) Vacuum suction erosion theory (after Waltham and Lu 2007; Parise 2008).
A mechanism of sinkhole formation has been described by Beck and Barry (1984). Piping and collapse of the residual soil above a cavity often result in the formation, upward propagation, and eventual collapse of the void (Figure 8). The state of stress and resulting shear strength of the surrounding will govern the stability of the cavity. Under some combinations of cavity diameter and vertical stress, circumferential stresses may be sufficient to maintain soil stability. One can assess the stability by comparing the vertical stress due to the weight and the shear strength developed on the failure surface (Drumm et al. 1990).

Formation and stability of cavities

The most available research is descriptive rather than analytical and regarding sinkhole prediction is closely related to particular sites or geology. According to Zhou and Beck (2008), formation of a cavity is probably preceded by formation of a void at the soil/rock contact by transporting of soil into an opening in the bedrock. Openings in solvable limestone are commonly 10–40 cm in diameter (White et al., 1984). There is commonly a zone of very soft soil just above the rock (Pazuniak 1989, Iqbal 1995, Sowers 1996, Wilson and Beck 1988). This soft zone has also been interpreted as soil that has sloughed into and filled previous soil voids (Sowers 1975, Wilson and Beck 1988). Tensile failure of this layer and perhaps overlying stronger soil will lead to formation of a no-tension cylindrical or dome in the soil (Hodek et al. 1984).
The traditional approach to predict cavity collapse in soil, as opposed to the problem of locating the cavities themselves, has been to use physical modeling and analytical techniques. The stability of soils over cavities was investigated through centrifuge modeling by Craig (1990), Abdulla and Goodings (1996). These centrifuge experiments used idealized cavity configurations, where collapse resulted from the overburden weight alone. Craig (1990) examined the stability of a cylindrical cavity opened up under a two-layered clay sample using two sets of tests. In the first set of tests, overburden weight was gradually increased by increasing the speed of the centrifuge, until the clay layers failed into a preformed cylindrical cavity. In the second set of tests, the centrifuge ran at constant speed while sand was extracted from a void beneath the clay layers. Craig found that the assumption of a simple cylindrical rigid-block failure in the clay was adequate for both sets of tests, providing the ratio of effective overburden depth against cavity diameter was less than unity.
In a similar study, Abdulla and Goodings (1996) investigated the stability of a cemented layer of sand overlying a cylindrical cavity, with and without overburden. They modeled a soil resulting from groundwater extraction in arid regions. The main finding of this study was that the cemented sand layer failed along steeply inclined planes forming a truncated conical section. In thicker cemented layers, however, a compression dome formed with a height of 25–30% of the cavity diameter.
Zhou (1997) demonstrated cavities of 0.2 m in diameter can cause sinkhole collapse in over 15 m thick overburden and the collapse could reach the surface regardless of the thickness of the soil, as long as the karst conduit was not blocked by the collapsed soil. The collapse process ceased every time when the fracture was filled with the soil, and restarted after the filled soil was removed.
Yang and Drumm (1999) analysed stress-strain and indicated the collapse or stability of the cavity is controlled by the relationship between the diameter of the cavity and the thickness of overburden when the cohesion remains constant. For a given value of soil cohesion, the lower bound of the soil cavity diameter and the overburden thickness was described by a power function that is independent of the soil friction angle in cohesive soil. They proposed the following equation to estimate a sort of critical depth (H) for a value of soil cohesion equals to 25 kPa:
H and D are expressed in meters. Where H and D are the height of overburden and diameter of the cavity, respectively. They considered the stable zone is bounded by the line defined by Eq. 1.1. Thus, the height of overburden must be more than H in Eq. 1.1 to be stable.
Zhao et al. (2011) consider that the karst collapse in China depends on several factors: purity of the limestone, degree of karstification, topography, geomorphology, geological structures, covering characteristics and hydrodynamic conditions. They demonstrated that the major cause of most karst collapses in southern China is the rapid change of hydro-climatic conditions which may cause:
-A drastic lowering of the groundwater level, reinforcing the mechanisms of suffosion and / or reducing the load in karst; these would be the main causes of karst collapse in the dry season. In the initial state (Figure 9) and if G + Fh + P ≤ Fa + Fb, the covering layers of the cave are stable. When the groundwater level declined below the bedrock (Fb = 0), if G + Fh + P ≤ Fa, the covering layers of the cave remain stable; where G is the dead weight of the overlying strata, Fh is dynamic hydraulic force, P is the total atmospheric pressure and additional dead load on the surface, Fa is the total resistance force of collapse, and Fb is buoyancy force of the soil. However, if G + Fh + P>Fa, ground fissures and collapse are bound to occur;
-During rain, a sudden increase in saturation in the soil that covers the karst layer, which accelerates the suffosion by increasing the load of the covering 30 to 40% (Figure 10) and a reduction in the cohesion and angle of internal friction of the covering material;
-The extensively distributed red clay is composed of hydrophilic minerals, such as montmorillonite, illite, and kaoline. These minerals are easily disintegrated by groundwater especially in case of repeated cycles of high and low water that create repeated cycles of saturation – desaturation.
-The negative pressure could easily form when groundwater fluctuates during rainfall alternation thus enlarging the soil cavity. This vacuum suction effect would greatly facilitate collapse.
Figure 9 3D Schematic to demonstrate the mechanical condition of a karst collapse. 1 Overlying strata; 2 Limestone; 3 Soil cavity; 4 Karstic cave; 5 Debris; 6 Compression stress and shear stress on the soil arch; 7 Groundwater level; 8 The total atmospheric pressure and additional dead load on the surface; 9 The dead weight of the overlying strata; 10 The total resistance force of collapse; it contains the cohesive strength and interfacial friction of the soil, etc.; 11 Buoyancy force of the soil; 12 Dynamic hydraulic force (according to Zhao et al. 2011).
Figure 10 Mechanical conditions of a limit equilibrium arch of a soil cave. (a) The mechanical structure of a soil cave can be simplified as a limit equilibrium arch, where F the dynamic hydraulic force; G the deadweight of the cover layer; and P the atmospheric pressure. (b) The bending moment diagram of the soil arch. (c) The shearing force diagram of the soil arch (Zhao et al. 2011).
Hyatt and Jacobs (1996) examined the distribution and morphology of 312 sinkholes suddenly appeared in the karstic Dougherty Plain at Albany, Georgia (USA) that resulted of flooding of the Flint River in July 1994. They also evaluated the mode of formation, characterized early stages of the evolution of sinkhole form, and estimated the lowering of the ground surface associated with the development of new sinkholes. They found that 88% of sinkholes occurred inside the limits of flooding, especially in areas of sandy overburden. They presented a descriptive model of the sinkhole form (Figure 11). Sinkholes initiated with the collapse and suffosion of saturated soil arches.
The impact of water on sinkhole formation was analysed by Anikeev (1999) and Sharp (1997, 2003) based on hydrofracturing theory (Figure 12). Anikeev (1999) proposed a simple hydrofracturing criterion that is controlled by the ratio of soil cohesion to the loss of buoyant with considering the case when there is comparatively impervious clay between the bedrock and the main overburden. He found that rapid drawdown in the overburden will induce cracking arches in the clay. A 3 m of drawdown in water level could cause hydrofracturing in soil with cohesion of 25 kPa. While Sharp’s numerical analysis indicate that hydrofracturing was unlikely to occur under steady state pore pressure. Transient pore pressure is a more probable cause of failure. In general, the increased load above the soil accompanied with the loss of buoyancy support caused by a rapid drawdown of water table is initially shifted to the pore water. The effect occurs at the perimeter of the soil cavity, where it is manifested in high pore pressures and consequently high pore pressure gradients. If pore pressure exceeds the sum of the radial compressive stress and tensile strength of the soil, hydrofracturing will occur and the cavity will be enlarged. The new free face that results will exhibit the same high gradient. This will result in a progressive failure that may proceed rapidly, accounting for the locally sudden appearance of sinkholes. The critical factor in this example is the rapidity of drawdown. If drawdown were slow enough to allow consolidation without a significant pore pressure transient, sinkhole formation may be slowed down.

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Table of contents :

Chapter 1:Introduction and literature review
1.1 Historical evolution of karst and dikes knowledge
1.1.1 Karst
1.1.2 Cavities and sinkholes
1.1.3 Dikes
1.2 Literature review
1.2.1 Mechanism of karst and cavities collapse
1.2.2 Formation and stability of cavities
1.2.3 Stability of dikes
1.3 Geo-risk methodology
1.4 References
Chapter 2: The case study (dikes of Val d’τrléans)
2.1 The dikes of Val d’τrléans
2.2 The Val d’τrléans site
2.2.1 Site presentation
2.2.2 Geological context
2.2.3 Hydrogeological context
2.3 Sinkholes and subsidence (historical information)
2.4 Mechanical properties of soils
2.5 Summary
2.6 References
Chapter 3: Effect of the dike upon the stability of the cavity
3.1 Introduction
3.2 Analytical approach of cavity collapse
3.3 Dike effect upon the stability of cavity
3.4 Calculation of the dike effect
3.5 Results analysis and discussion
3.6 Summary
3.7 References
Chapter 4: Effect of cavity upon the stability of the dike slope
4.1 Introduction
4.2 Method and objective
4.3 Slope stability analysis
4.3.1 The Ordinary method of slices
4.3.2 Verification of TALREN results
4.3.3 Initial safety factor of the dike of Val d’τrléans (without cavity)
4.4 Modified method with a cavity underneath the dike
4.5 Application to the Val d’τrléans dike: safety factor of slope stability
4.6 Summary
4.7 References
Chapter 5: Numerical modelling (2D and 3D)
5.1 Objective and method
5.2 General description of the dike of Val d’τrléans model in CESAR-LCPC
5.3 Slope stability calculation by the c-phi reduction method
5.3.1 The theoretical bases of the method
5.3.2 Slope stability calculation for the dike without cavities
5.3.3 Slope stability calculation for the dike with cavities
5.4 Calculation by standard method
5.4.1 Effect of the cavity on the slope stability
5.4.2 Effect of the dike upon the cavity stability
a-Safety factor criterion
b-Vertical displacement criterion
c-Plastic strain criterion
5.5.1 Cavity underneath the centre of the dike
5.5.2 Cavity underneath the dike slope
5.5.3 Effect of the direction of the cavity
5.6.1 Cavity stability and collapse shape
5.6.2 Cavity effect upon the dike slope stability
5.7 Summary
5.8 References
Chapter 6:Expected scenarios due to collapsed cavity underneath a dike
6.1 Introduction
6.2 Scenario of isolated karstic caves
6.2.1 Scenario for one cohesionless alluvium layer
6.2.2 Scenario for one cohesive alluvium layer
6.2.3 Scenario for two alluvium layers
6.3 Scenario for cave connected to a karstic conduit
6.3.1 Scenario for one cohesionless alluvium layer
6.3.2 Scenario for one cohesive alluvium layer
6.3.3 Scenario for two alluvium layers
6.4 Water pressure effect and the final form of the cavity collapse
6.4.1 Resistance force against the water flow Ff
6.4.2 water flow force F
6.4.3 Application to the Val d’Orléans case
6.5 Summary
6.6 References
Chapter 7: General conclusion and perspectives
7.1 General conclusion
7.2 Perspectives
7.1 Conclusion générale
7.2 Perspectives

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