A Percolation Approach to Soil Consolidation
Postgrado de Ingeniería, Universidad Autónoma de Querétaro
Postgrado de Ingeniería, Universidad Autónoma de Querétaro
Instituto de Física Aplicada y Tecnología Avanzada
Recent research trends about fluid flow are being focused to percolation theory as a statistical tool. In this work, the percolation approach is used for explaining the soil behavior in consolidation phenomenon, one of the many phenomena encountered in soil mechanics.
Keywords: Soils percolation, soils consolidation percolation, soils consolidation, percolation, Therzaghi’s consolidation theory, secondary consolidation.
Percolation theory was introduced by Broadbent and Hammersley in 1957. Essentially, they adopted the particles extension concept of a hypothetical fluid through a random medium. Generally speaking, the hypothetical fluid extension through a disordered medium involves some random elements, but the implicit mechanisms maybe simplified as two different types. In the first type, randomness is placed on the fluid, i.e., the particles behave as a diffusion process. Regarding the second type, randomness is attributed to the medium i.e., the medium controls the particles trajectories. The latter was the new situation considered by Broadbent and Hammersley, and the process was called percolation, the name being apparent from the way coffee flows through a percolator (Isichenko 1992). Moreover, percolation is a non-universal phenomenon of broad scope (private communication: Castaño V.M. Notas del curso de Fractales, UAQ; México 1999) that also required its own terminology.
In this work, the percolation approach is used for explaining the behavior phenomenon of soil consolidation, one of the many phenomena encountered in soil mechanics.
MATERIALS AND METHODS
Percolation general formulation involves elementary geometrical objects such as spheres, bounds, sites, etc., that are placed either in a net of finite dimension or a continuum. Net models usually have an irregular geometry and capillary segments of different sizes and shapes, distributed in turn, on an irregular net. One can distinguish one-, two-, or three-dimensional nets. In any case, in a net, the capillary (bonds in current technology) connecting two adjacent nodes (sites) can have a uniform diameter and be surrounded by a number of capillary segments with different diameters.
Chatzis and Dullien (Dullien 1992) studied several two-dimensional nets with 20 to 40 porous-segments wide and depth ranges from 15 to 80 segments. The non-wet phase can be studied inside the net through one of the faces; the two perpendicular sides to the penetration face is supposed to be non-permeable and the fourth face was open.
Initially, it was assumed an arbitrary distribution of the porous diameters. Then they were assigned to the different net bonds in a random fashion, similarly as Fat did (1959). Net bonds are numbered sequentially from 1 to N, where N is the total number of net bonds. This procedure can be implemented in a computer where such vector can be generated by pseudo-random numbers. The accumulative probability distribution function, pk, gives the accumulative number of porous of sized j = k in the net, that is,
where Nj is the total number of porous in the net characterized by a porous size j, for j = 1, 2, ..., n, j=1 being the greatest and j=n the smallest; k (k=1,2,…n) is the smallest porous being traversed at some time while the penetration process.
As a first step in the penetration process, only the largest porous (k=1) and the ones accessible through the net penetration face are penetrated. Then, the second largest ones (k=2) are penetrated, and so forth, until some value of the accumulative porous size distribution, corresponding to a porous of intermediate size. Percolation is denoted then to the particular point in which the fluid penetration reaches the net opposite face. Before this, the penetrated porous are unable of transporting the fluid between the injection and the output face of the sample.
In percolation, there exists a cumulative probability, pcr, called the critical probability. It has been found that the net size of approximately 40 x 40 porous segment is the minimum for representing a true value of pcr with probability of ±0.01. The probability of critical bonds percolation (b), pcr(b) especifies the minimum fraction of net bonds required to be open and, thus, the fluid can penetrate from one net side to the other. In other words, pcr is the percolation threshold and it corresponds to the minimum concentration at which an infinite cluster traverses the space. The theory main keypoint is that for each net there exists a critical probability pcr, 0 < pcr < 1, at which an infinite cluster appears definitely (Dullien 1992).
Two sites are connected if there exists at least a path between them that contains all of its bonds occupied. A set of connected (occupied) sites, communicated via the rule of the closest neighbor, surrounded by void bonds is called a cluster. If the net is very large and if pk is sufficiently small, the size of any connected cluster is small. However, if pk is close to 1, the net must be almost completely connected (Newman 1981).
The simplest problem statement is given as: Given a periodic net inside an space of given dimension, and the probability pk for each net site being occupied (1-pk will be the probability being void), ¿What is the resulting clusters distribution for a given size and other geometrical parameters? Together with this theory of site percolation, the idea of bond percolation can be introduced. In this, the clusters are connected through conductive loops. The loops are conductive with probability pk, and the interrupted ones have probability 1-pk (Dullien 1992).
The consolidation of a clay deposit can be divided in two parts. First, the compression due to the hydrodynamical delay caused by gravitational water, known as Terzaghi’s consolidation theory, this takes into account only the delay of the elastoplastic deformation. The second one, known as secondary consolidation, can be represented as a compression phenomenological law due to viscous effects (Zeevaert 1973).
According to the consolidation theory there are two pressures, namely, the porous pressure and the effective pressure. For any time, the sum of these two pressures remains constant (Juárez and Rico 1974). At the beginning, the porous pressure is the only one that is present. After applying a load, (p, in the water rises a pressure, u, in excess of the hydrostatic such that (p=u. At a later instant, the effective pressure increases whereas the porous pressure decreases. Finally at an infinite time, (p is equal to the effective pressure since the porous pressure vanishes.
Based on Terzaghi’s consolidation theory and regarding the developing stresses for a given load, at t=0, there exists a percolation due to porous pressure, and at t=infinite(, there exists percolation due to the effective pressure. Thus, there exists a critical time, t=tcr, at which there is an intersection of the first one with the latter.
This percolation approach to soil consolidation is better explained in Fig. 1. In that figure, for a first pressure, P1, applied to the soil at t=0, all the pressure is taken by the water, i.e. porous pressure. At this time the percolation is 1. At a later instant, t=t, the porous pressure percolation diminishes, there exits water flow and the effective pressures rise with a cumulative probability pk, very likely less than the threshold value. At t=infinite(, the cumulative probability of the porous pressure vanishes and the cumulative probability of the effective pressure takes a value less, greater than or equal to the percolation threshold.
Figure 1. Behavior analogy of pressures development in soils consolidation via percolation
For a second pressure P2 applied to the soil, all the exerted pressure will be taken by the water again, porous pressure. However, since there is less water, due to the first load, the percolation value is less than 1. At a later time, the percolation value of the porous pressure decreases because of the water exit. On the other hand, effective pressures appear in lesser times than in the first load with a cumulative probability. Finally, at t=infinite, the cumulative probability value of the porous pressure vanishes and the corresponding to effective pressures takes a value greater than in the above load.
For t=0, and greater soil pressures, these will be taken for the remaining water, Thus, the porous pressure cumulative probability will decrease for each pressure increase. At later times, the cumulative probability of the porous pressure decreases whereas the percolation of the effective pressure rises as the pressure increases, and sooner because of the constant diminution of the water soil content. Finally, for t=infinite, the porous pressure cumulative probability vanishes and the percolation of the effective pressure goes to 1, as soil pressure increases.
Based on the above, it could be thought of that for each pressure increment, an effective pressure percolation threshold can be found. However, this idea is discarded because for small loads, the most likely is that only part of the soil skeleton, of given thickness, is working without really existing a threshold. On the other hand, for heavier loads, is likely that the effective pressures had already percolated, that is, the threshold had already appeared. Thus, the percolation threshold value, appears at the intersection of both (porous and effective pressures).
By joining the corresponding values of pressure and time for all load increments, one gets the plot shown in Figure 2. In that figure, at the beginning, the percolation due to the porous pressure has a value of 1, line AB. Then, the porous pressure decreases until it vanishes, and the effective pressure rises as time and pressure increase. From Fig. 2, the intersection of both values of cumulative probability would be equivalent to the percolation threshold of both pressures.
Figure 2. Percolation condition in soils consolidation.
According to soil consolidation fundamentals, this point would represent the starting of the secondary consolidation and, on the other hand, the last moment of percolation due to porous pressure. This means that when the secondary consolidation starts, there are already percolation of some part of the soil particles. Thus, as secondary consolidation goes on, the next behavior of the effective pressure would correspond to form an infinite cluster in the soil with a total percolation of 1, line CD. Ideally, at this moment, the soil consolidation would stop, in agreement with the hypothesis that the soil is incompressible (Juárez and Rico 1974).
CONCLUSIONS AND DISCUSSIONS
In this work, an analogy based on percolation theory has been applied to the explanation of Terzaghi’s soils consolidation theory. The proposed analogy is: as soil exerted pressure are applied, there exists a stresses percolation threshold at the instant in which a defined path rises (stress path) from the load application point to the desired depth. In agreement with pressure and time increments distributions during the consolidation processes, it can be said that when a first pressure increment is applied, at t=0, there exists a total percolation (value of 1) due to porous pressure. At t=t, this decreases until it vanishes, whereas the effective pressure rises until reaches the value 1 (total percolation in t=infinity) as pressure increases. The intersection of both cumulative probability paths (porous and effective pressure) at different pressure and time increments in a consolidation test would be equivalent to find the percolation threshold with a value less than 1. This, in turn, can be interpreted, according to soil consolidation, on the one hand, as the starting of secondary consolidation and, on the other hand, as the last instant where porous pressure percolation exists due to pressure increase.
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