Powerful Software to Simulate Soil Consolidation Problems with Prefabricated Vertical Drains

water

Article Powerful Software to Simulate Soil Consolidation Problems with Prefabricated Vertical Drains

Gonzalo García-Ros 1,*, Iván Alhama 1 and Manuel Cánovas 2 ID 1 Civil Engineering Department, Technical University of Cartagena, Paseo Alfonso XIII 52, 30203 Cartagena, Spain; [email protected] 2 Metallurgical and Mining Engineering Department, Universidad Católica del Norte, Av. Angamos 0610, Antofagasta (Región de Antofagasta), Chile; [email protected] * Correspondence: [email protected]

Received: 2 February 2018; Accepted: 23 February 2018; Published: 27 February 2018

Abstract: The present work describes the program Simulation of Consolidation with Vertical Drains (SICOMED_2018), a tool for the solution of consolidation processes in heterogeneous soils, with totally or partially penetrating prefabricated vertical drains (PVD) and considering both the effects of the smear zone, generated when introducing the drain into the ground, and the limitation in the discharge capacity of the drain. In order to provide a completely free program, the code Next-Generation Simulation Program with Integrated Circuit Emphasis (Ngspice) has been used as a numerical tool while the Matrix Laboratory (MATLAB) code was used to program and create an interface with the user through interactive screens. In this way, SICOMED_2018 is presented as an easy-to-use and intuitive program, with a simple graphical interface that allows the user to enter all the soil properties and geometry of the problem without having to resort to a complex software package that requires programming. Illustrative applications describe both the versatility of the program and the reliability of its numerical solutions.

Keywords: consolidation software; vertical drains; heterogeneous soils; smear; drain resistance

1. Introduction The use of specific software for the understanding of complex physical phenomena involved in engineering problems has become an indispensable tool in modern teaching, mainly due to the high costs involved in laboratory tests. These programs, in spite of the effort and knowledge—especially of programming and calculation—that their development require, are generally designed by the university departments themselves as an interdisciplinary work among several areas of knowledge. The basic objective is to create an interactive computer-to-user communication program that, with minimal help, enables the learning objectives to be met in a shorter time than would be necessary with laboratory tests. In this sense, the design of these programs—avoiding the concept of the ‘black box’—allows the user to access the choice of a large number of parameters associated with programming and design of the model, with the purpose of determining the potential influence of them in numerical results. The ‘network simulation’ research group of the UPCT (Technical University of Cartagena) has significant experience in software development, in the fields of mechanical engineering and coupled flow and transport processes, such as: (i) Program for Designing Simple Fins (PRODASIM) and Program for Heat Conduction (PROCCA-09), for the assembling of transient heat transfer in fins [1]; (ii) Flow and Transport Simulator (FATSIM-A), for the simulation of flow and transport processes [2]; and (iii) Coupled Ordinary Differential Equations by Network Simulation (CODENS_13), for the simulation of mechanical models governed by coupled ordinary differential equations [3]. All of them are available and have been used as tools for numerous research works [4,5], but require a license to

Water 2018, 10, 242; doi:10.3390/w10030242 www.mdpi.com/journal/water Water 2018, 10, 242 2 of 20 use the Personal Computer Simulation Program with Integrated Circuit Emphasis (Pspice), which is responsible for performing the numerical calculations. In this work we present the free program SICOMED_2018 [6], a code for the simulation of transient problems of soil consolidation with the use of prefabricated vertical drains to accelerate the settlement process, easy to use but with numerous applications within the field of civil engineering. The possibility of selecting not only the value of any of the physical parameters involved, with heterogeneous soils of up to three layers, but also those associated with the grid size and the numerical computation parameters allows the user to address specific consolidation scenarios—determining the separate influence of each of the parameters involved—as well as to solve optimization problems such as the relationship between the consolidation time and the depth of the drain. On the other hand, the powerful graphic output environment of the program allows us to visualize and interpret results in a direct and simple way. For these reasons, SICOMED_2018 is presented as a complete software package, with novel contributions, that joins other specific programs for learning the theory and practice of soil consolidation such as Consolidation Testing and Multimedia (CTM) [7]. The performance of SICOMED_2018, which can be acquired through UPCT (Technical University of Cartagena), requires the installation of the free code Ngspice [8]. The graphical communication environment has been developed with MATLAB (under a license that allows its distribution), a very widespread code at a scientific and professional level that has highly developed tools to create user interfaces, while the powerful computational algorithms of Ngspice provide the nearly exact solution of the network model of the consolidation process, whose design is based on the analogy between electric and physical quantities. The design of the numerical model follows the rules of the network simulation method [9], a tool widely known in the scientific world for the implementation of network models. Each term in the equation is an electric current that is balanced with the other terms in a common node of the volume element whose voltage is the solution of the potential quantity (pressure) of the governing equation. A mesh size of the order of 50 volume elements, in one-dimensional (1D) problems, reduces errors with respect to analytical solutions to below 1% [10], which is acceptable for this type of engineering problem. Illustrative applications are presented to demonstrate the efficiency and reliability of the program.

2. Fundamentals of Soil Consolidation with Prefabricated Vertical Drains and Network Model Prefabricated vertical drains (PVD) are a system of soil drainage commonly used in civil engineering in problems of soil consolidation in which the goal is to accelerate final settlement. When a fully saturated clay soil is subjected to the action of loads on its surface, it begins to experience a decrease in volume or settlement over a generally long period, in a transitory process commonly known as consolidation. This volume reduction is due to the gradual expulsion of part of the water contained in the voids between the soil particles, through one or more draining boundaries (generally across the ground surface), as the excess pore pressure initially generated by the applied load is transmitted to the soil, increasing the effective pressure on it. In real scenarios, with ground of considerable thickness and low permeability, the main problem facing the civil engineer is the long consolidation times needed to reach final settlement (on the order of years). For this reason, the installation of PVD is a good solution to reduce the time of consolidation of a soil, as we are significantly shortening the drainage length of the water. Assuming the hypotheses postulated by Terzaghi [11]—(i) both water and soil particles are assumed incompressible; (ii) the ground self-weight is neglected; (iii) the excess pore pressure (u) is caused by the application of the external load (q), constant over time; (iv) the fluid movement in the porous medium is assumed to obey the law of Darcy; (v) the soil skeleton does not creep under the action of a constant effective pressure; (vi) the change in soil thickness in relation to its total volume (hypothesis of 1 + e constant) is considered negligible; (vii) there is uniform applied load, so that the soil deformation only occurs in the vertical direction; (viii) the geotechnical parameters of the soil, included in the consolidation coefficient cv, remain constant; and (ix) the soil rests on an impermeable Water 2018, 10, 242 3 of 20 Water 2018, 10, x FOR PEER REVIEW 3 of 20 stratum—so that that water water drains drains only only by by the the ground ground surface surface (water (water flow flow only only in in the the vertical vertical direction), direction), thethe governing equation equation and and boundary boundary conditions conditions for for the the 1D 1D linear soil consolidation problem problem are given by ∂u ∂2u ∂u= c ∂u (1) ∂t =cv ∂z2 (1) ∂t ∂z u(z=0,t) = 0 Surface drainage condition (2) u(,) = 0 Surface drainage condition (2) ∂u ∂u = 0 Impermeable bottom edge (3) ∂z (z=H,t=) 0 Impermeable bottom edge (3) ∂z(,) u(z,t=0) = uo = q Initial condition. (4) u(,) =u = q Initial condition. (4) The analytical solution for a single layer of soil can be found in several works [12,13], providing solutionsThe analytical for the local solution degree for of settlementa single layer (U) of for soil any can depth be found and instant in several of time, works as well[12,13], as the providing average solutionsdegree of for settlement the local( degreeU) at any of settlement time. (U) for any depth and instant of time, as well as the average degreeThe of installationsettlement of(U PVD,) at any made time. of geosynthetic material, allowing drainage through them, involves the transformationThe installation of of the PVD, 1D problem made of into geosynthet a three-dimensionalic material, (3D)allowing problem, drainage since thethrough direction them, or involvesflow of water the transformation occurs in any ofof the the three 1D problem spatial directions into a three-dimensional (x, y, z). By adopting (3D)a problem, rectangular since layout the direction(Figure1), or which flow allowsof water us tooccurs obtain in volumeany of elementsthe three ofspatial rectangular directions geometry, (x, y, z). the By mathematical adopting a rectangularmodel for the layout typical (Figure section 1), (Figure which2 allows) is governed us to obtain by the volume following elements equations: of rectangular geometry, the mathematical model for the typical section (Figure 2) is governed by the following equations: 2 2 2 ∂u ∂ u ∂ u ∂ u ∂u= c ∂ u + c ∂ u + c ∂ u Governing equation (5) =cv,z 2 +cv,x 2+c v,y 2 Governing equation (5) ∂t∂t , ∂∂zz , ∂x∂x , ∂y∂y

u(ux,y,z(,,,=0,t))==uu(0(,,,≤x≤c,y=0,0≤z≤d,t) =) = 00 Soil surface and and drainage drainage area area o off the PVD the PVD (6) (6) ∂u∂u == 00 Impermeable Impermeable edges (7) (7) ∂n∂n rest of the boundaries

u(,,,x,y,z,t=0))=u= uo InitialInitial condition, condition, (8) (8) where ∂u represents represents thethe partialpartial derivativederivative inin aa directiondirection normalnormal toto thethe boundaryboundary surface. surface. ∂n

RECTANGULAR DISTRIBUTION LONGITUDINAL AND CROSS SECTIONS

d z+ 2b H 2b

Figure 1.1. 3D3D rectangular rectangular consolidation consolidation problem problem with with prefabricated prefabricated vertical vertical drains. drains. Rectangular Rectangular layout. layout.Physical Physical scheme scheme (left) and (left section) and section of the model of the (modelright). (right).

Assuming isotropy in the (x, y) horizontal directions (a common feature of soil behavior), Equation (5) can be simplified to

=c +c + . (9) , ,

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Assuming isotropy in the (x, y) horizontal directions (a common feature of soil behavior), Equation (5) can be simplified to

∂u ∂2u ∂2u ∂2u = c + c + . (9) Water 2018, 10, x FOR PEER REVIEW ∂t v,z ∂z2 v,h ∂x2 ∂y2 4 of 20

AtAt this this point, point, it it is is important important to to note note that, that, as as in in the the linear linear theory theory of of Terzaghi Terzaghi [11], [11], the the consolidation consolidation coefficientscoefficients c cv,zv,z andand c cv,hv,h ofof the the governing governing Equation Equation (9) (9) have have been been assumed assumed constant. constant. As As is is well well known, known, thisthis is is not not entirely entirely true, true, since since in practice the soil properties (k(kvv, k kh,, e e and and a avv)) that that are are grouped grouped by by these these () () kv(1+e) kh(1+e) coefficientscoefficients ( (c,v,z == , c, c, == ) are) are not not constant, constant, but but they they va varyry during during the the consolidation consolidation γwav v,h γwav process,process, a a phenomenon phenomenon that that began began to tobe beanalyzed analyzed many many years years ago ago by Davis by Davis and andRaymond Raymond [14] and [14] manyand many other other authors authors of the of thetime. time. At Atpresent present there there are are numerous numerous publications publications and and research research that that incorporateincorporate the the non-linearity non-linearity of of these these coefficients coefficients in the in the mathematical mathematical models, models, although although it is also it is true also that,true for that, the for usual the usual practical practical values values of the of surface the surface applied applied load (q), load the (q), changes the changes in properties in properties that give that risegive the rise consolidation the consolidation coefficients coefficients (cv,z and (cv,z candv,h) are cv,h small) are small(and the (and variations the variations in these in theseproperties properties may evenmay evencompensate compensate for each for eachother), other), so that so that the the final final result result is is practically practically constant constant consolidation 0 0 coefficients.coefficients. In In effect, effect, the the non-linearity non-linearity of of these these coefficients coefficients grows grows the the higher higher the the σ′σf/fσ′/oσ quotient,o quotient, a ratioa ratio that, that, though though it theoretically it theoretically can can reach reach any any valu value,e, in inpractice practice very very rarely rarely exceeds exceeds the the value value of of2 (or,2 (or, at atmost, most, 2.5); 2.5); a avalue value below below which which the the differe differencence between between the the solutions solutions provided provided by by the the linear linear andand non-linear non-linear models models is is practically practically insignificant insignificant [15]. [15]. In In this this way, way, the the smearing smearing effects effects produced produced by by thethe introduction introduction of of the the vertical vertical drain drain are are reflected reflected in in the the mathematical mathematical model model in in a a very very simple simple way, way, since,since, in in practice, practice, the the soil soil property property that that is is affected affected by by the the smear, smear, for for the the purposes purposes of of the the consolidation consolidation coefficient,coefficient, is is the the hydraulic hydraulic conductivity conductivity [16–18]. [16–18]. So, So, and and for for the the area area of of soil soil that that is is considered considered to to be be affectedaffected by by the the smear smear effects, the hydrauli hydraulicc conductivity values of thethe smearsmear zonezone (k(kv,sv,s and kh,s)) must must simplysimply be be assigned. assigned.

RECTANGULAR DISTRIBUTION SYMMETRY CONDITIONS

Figure 2. Symmetry conditions of the rectangular layout. Figure 2. Symmetry conditions of the rectangular layout.

The set of Equations (6)–(9) has a complex analytical solution (from the point of view of the managementThe set offor Equations the civil engineer), (6)–(9) has so a complexit is presumed analytical much solution more (fromappropriate the point to solve of view them of theby numericalmanagement methods. for the civil engineer), so it is presumed much more appropriate to solve them by numerical methods. 2.1. Network Model Network Model The network method is a technique for the study and numerical solution of any physical The network method is a technique for the study and numerical solution of any physical processes that can be defined by a mathematical model. Starting from the latter, the procedure processes that can be deﬁned by a mathematical model. Starting from the latter, the procedure consists of two stages: (i) elaborate a “network model” or electrical circuit equivalent to the process, consists of two stages: (i) elaborate a “network model” or electrical circuit equivalent to the process, and (ii) numerically simulate the model by means of an adequate electrical circuits resolution code. and (ii) numerically simulate the model by means of an adequate electrical circuits resolution code. Its applications are multiple; González-Fernández [9] uses the network method in solving transport Its applications are multiple; González-Fernández [9] uses the network method in solving transport problems through membranes, heat transfer and electrochemical systems. In recent years it has been successfully applied in other engineering fields such as elastic waves [19], material resistance [20], corrosion [21], magneto-hydrodynamics [22], flow and transport [4], tribology [23], mechanical problems and chaos [24]. On the other hand, among the future research lines in which the network method can be used we find: seepage [25], real-life case models in hydrologic engineering [26–29], and atmospheric dispersion of pollutants [30,31], some of them already in the development phase. The formal equivalence between the network model and the physical process is that both are governed by the same equations (discretized in space, but retaining time as a continuous variable),

Water 2018, 10, 242 5 of 20 problems through membranes, heat transfer and electrochemical systems. In recent years it has been successfully applied in other engineering ﬁelds such as elastic waves [19], material resistance [20], corrosion [21], magneto-hydrodynamics [22], ﬂow and transport [4], tribology [23], mechanical problems and chaos [24]. On the other hand, among the future research lines in which the network

Watermethod 2018 can, 10, x be FOR used PEER we REVIEW find: seepage [25], real-life case models in hydrologic engineering [265– of29 20], and atmospheric dispersion of pollutants [30,31], some of them already in the development phase. establishingThe formal a correspondence equivalence between between the the network dependen modelt variables and the of the physical problem, process pressure is that and both water are flowgoverned in the by physical the same model, equations and the (discretized electrical variables in space, of but the retaining circuit, timevoltages, as a and continuous currents. variable), That is, bothestablishing models aare correspondence governed by the between same theequations dependent referred variables to a volume of the problem, (or cell) element pressure and and by water the sameflow indiscrete the physical equations model, for andthe boundary the electrical conditio variablesns. Consequently, of the circuit, voltages,the errors and from currents. the simulation That is, areboth only models attributable are governed to the geometric by the same mesh equations size, so that referred for an to acceptable a volume number (or cell) of element cells (of and the by order the ofsame 50 in discrete 1D scenarios, equations 4000 for for the 3D boundary problems) conditions. errors are Consequently, much less than the 1% errors in linear from problems the simulation [10]. The are powerfulonly attributable computation to the codes geometric for the mesh numerical size, so thatresolu fortion an of acceptable circuits provide number the of cellsexact (of solution the order of theseof 50 inthanks 1D scenarios, to an optimal 4000 selection for 3D problems) of the calculat errorsion are time much steps, less imposed than 1% by inthe linear Ngspice problems code or [10 by]. Thethe user. powerful computation codes for the numerical resolution of circuits provide the exact solution of theseIn thanks terms toof an the optimal first spatial selection derivative, of the calculationEquation (9) time is written steps, imposed in the form by the Ngspice code or by the user. In terms of=c the, first spatial − derivative, +c, Equation (9)− is written +c in, the form − , (10) nh i h i o nh i h i o h i h i where z∂u and= z∂ denote∂u the− cell∂u output+ and∂ input∂u locations− ∂u in the+ z-direction,∂ ∂u and− the∂u same for x ∂t cv,z ∂z ∂z + ∂z − cv,h ∂x ∂x + ∂x − cv,h ∂y ∂y + ∂y − , (10) and x in the x-directionz and for zy and y in thex y-direction.x y y whereDividingz+ and zthe− denote medium the into cell outputNx × N andy × N inputz volume locations elements in the of z-direction, size Δx × andΔy × the Δz, same and forusingx+ andthe xnomenclature− in the x-direction of Figure and 3, for the y +aboveand yequation− in the can y-direction. be written in the form Dividing the medium into N × N × N volume elements of size ∆x × ∆y × ∆z, and using the ∂u 1 u −u x u −uy z 1 u −u u −u nomenclature=c of Figure 3, the above − equation  can + be c written in the form −  ∂t , z z z , x x x 2 2 2 2 (11) ∂u = 1 ui+∆z−ui − ui−ui−∆z + 1 ui+∆x−ui − ui−ui−∆x ∂t cv,z ∆z 1∆z u −u∆z u −ucv,h∆x ∆x ∆x +c 2 2−− 2 2 , y y (11) y 1 ui+∆y−ui ui−ui−∆y +cv,h ∆2y ∆y − 2 ∆y 2 2 or or " # " #         − − ∂u ui+∆z−ui ui−ui−∆z ui+∆x−ui ui−ui−∆x ui+∆y ui ui ui−∆y =  − + − + − . (12) ∂t = (∆z)2 − (∆z)2+  (∆x)−2   (∆+x)2   (−∆y)2   .( ∆y)2  () () () () () () (12) 2cv,z 2cv,z 2cv,h 2cv,h 2cv,h 2cv,h , , , , , ,

z+ y+

x d

ui+ z d

ui+ yd z

u d d i+ x ui- x ui

ui- yd d

ui- zd y

d x

Figure 3. Nomenclature of the element cell of the 3D rectangular consolidation model.

Each of the terms in the above equation is a flow: ∂u u −u u −u u −u j = ,j=  , j =  ,j=  , ∂t  (z)  (z)  (x) 2c 2c 2c , , , (13) u −u u −u u −u j = ,j = , j =  (x)  (y)  (y) 2c, 2c, 2c, so that Equation (12) can be written as a flow balance:

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Each of the terms in the above equation is a ﬂow:

− − − − j = ∂u , j = ui+∆z ui , j = ui ui−∆z , j = ui+∆x ui , j = ui ui−∆x , C ∂t R+∆z (∆z)2 R−∆z (∆z)2 R+∆x (∆x)2 R−∆x (∆x)2 2cv,z 2cv,z 2cv,h 2cv,h u −u u −u (13) j = i+∆y i , j = i i−∆y R+∆y (∆y)2 R−∆y (∆y)2 2cv,h 2cv,h so that Equation (12) can be written as a ﬂow balance:

jC = jR+∆z − jR−∆z + jR+∆x − jR−∆x + jR+∆y − jR−∆y. (14)

In the electric analogy, by establishing associations between variables so that the excess pore pressure (u) is equivalent to the electrical voltage (V) while the temporal (ﬁrst derivative) and spatial changes (second derivative) of u are two electric currents (J), we have:

ui+∆z − ui j + = 2 , (15) R ∆z (∆z) 2cv,z which is equivalent to Ohm’s law, which relates the current (IR) in a resistance with the voltage (VR) at its ends (terminals): V I = R . (16) R R Thus, the implementation of Equation (15) is carried out by the resistance between the central node and the upper edge of the cell (Figure3). The value of this resistance is

(∆z)2 Ri+∆z = . (17) 2cv,z

By the same reasoning, the term

ui − ui−∆z j − = 2 (18) R ∆z (∆z) 2cv,z is implemented by a new resistor of the same value as Equation (17) between the central node and the lower edge of the cell. Finally, the term ∂u j = (19) C ∂t is equivalent to the equation that relates the electric current at a capacitor (IC) to the voltage or potential difference at its ends (VC): ∂V I = C C . (20) C ∂t Thus, the implementation of this term of Equation (12) is carried out by a capacitor C of value unity connected between the central node and an outer node (which will be the common reference node for all cells). In short, the values of the electrical devices that implements the model of the volume element, whose network is depicted in Figure4, are

(∆z)2 (∆x)2 (∆y)2 C = 1, Ri+∆z = Ri−∆z = ,Ri+∆x = Ri−∆x = ,Ri+∆y = Ri−∆y = . (21) 2cv,z 2cv,h 2cv,h

Notice that, physically, an increase in the consolidation coefficient decreases the value of the resistances and makes the consolidation process faster, as expected. Each cell of the domain is connected to the adjacent cells by means of ideal electrical contacts. As regards the boundaries, a second-type Neumann (homogeneous) condition is adopted at the Water 2018, 10, 242 7 of 20 impermeable edges, which is simply implemented by a resistance of infinite value, while a first-type Dirichlet condition (constant pressure) is assumed at the draining boundaries (soil surface and drainage area of the PVD), which is implemented by a constant voltage source of zero value. Finally, the drain discharge capacity (qw) limitation caused by the unit gradient of pressure at the drain is modeled by fixing the drain permeability (hydraulic conductivity) to the required value. Note that this solution allows us to increase the flow of water through the drain as the water moves Watertowards 2018, the10, x exit FOR border PEER REVIEW of the drain. 7 of 20

d x

ui+ zd

ui+ yd

Ri+ zd

Ri+ yd d d

u Ri- d ui+ x d i- x x u

z i

d d Ri+ x

Ri- y

d u d Ri- z

i- y Ci

u d i- z d y

Figure 4. Network model of the element cell (3D rectangular consolidation). Figure 4. Network model of the element cell (3D rectangular consolidation). 3. The Simulation Program 3. The Simulation Program The objective of this program (SICOMED_2018) is the simulation of the 3D consolidation The objective of this program (SICOMED_2018) is the simulation of the 3D consolidation problem problem with partially or totally penetrating PVD, in heterogeneous soils formed by one, two, or with partially or totally penetrating PVD, in heterogeneous soils formed by one, two, or three layers and three layers and taking into account the effects of the smear zone and the discharge capacity of the taking into account the effects of the smear zone and the discharge capacity of the drain. It must be an drain. It must be an easy-to-use software for the user but able to provide powerful numerical easy-to-use software for the user but able to provide powerful numerical calculations. This requirement calculations. This requirement entails the creation of a practical and simple graphical interface, where entails the creation of a practical and simple graphical interface, where data entry and choice of the data entry and choice of the different options is performed in an orderly, intuitive, and guided way. different options is performed in an orderly, intuitive, and guided way. Also, the program should Also, the program should include a complete guide [32] that allows the user to understand the include a complete guide [32] that allows the user to understand the successive steps in data entry, successive steps in data entry, simulation and representation of results. simulation and representation of results. With regard to the simulation possibilities, the user can use SICOMED_2018 both to design and With regard to the simulation possibilities, the user can use SICOMED_2018 both to design and to optimize the installation of PVD in the ground, since the program allows us to know, in a simple to optimize the installation of PVD in the ground, since the program allows us to know, in a simple way, the great variety of results that will allow engineers to make the decisions that most suit their way, the great variety of results that will allow engineers to make the decisions that most suit their needs: evolution of settlements on the soil surface, excess pore pressure at any point of the ground, needs: evolution of settlements on the soil surface, excess pore pressure at any point of the ground, and average degree of settlement, among others. and average degree of settlement, among others. Figure 5 shows a simplified scheme of the SICOMED_2018 flowchart. 3.1. The Input Data and Network Design 3.1. The Input Data and Network Design Data entry is carried out through the following three windows that appear successively as they Data entry is carried out through the following three windows that appear successively as they are completed (Figures6–8). The start-up screen (Figure6), allows the entry of the problem geometry are completed (Figures 6–8). The start-up screen (Figure 6), allows the entry of the problem geometry (values a, b, and c in Figure2), the definition of the smear zone, and the desired grid size in plant. In the (values a, b, and c in Figure 2), the definition of the smear zone, and the desired grid size in plant. In second window (Figure7), the geometry of the strata (thickness and vertical grid size), parameters the second window (Figure 7), the geometry of the strata (thickness and vertical grid size), parameters to define the consolidation coefficients (both the undisturbed and the smear zone), the depth of the to define the consolidation coefficients (both the undisturbed and the smear zone), the depth of the PVD, and its discharge capacity are introduced. Finally, in the third screen (Figure8), parameters PVD, and its discharge capacity are introduced. Finally, in the third screen (Figure 8), parameters related to the simulation such as the value of the uniform applied load, initial and final calculation related to the simulation such as the value of the uniform applied load, initial and final calculation times, the maximum time step between each iteration, and a relative tolerance parameter used by times, the maximum time step between each iteration, and a relative tolerance parameter used by Ngspice to achieve the convergence are introduced. As a unit of measure of time, the year (y) has been Ngspice to achieve the convergence are introduced. As a unit of measure of time, the year (y) has been chosen, a more appropriate magnitude than the second(s) for this type of problem, while for the other magnitudes the units of the international system have been maintained.

Water 2018, 10, 242 8 of 20 chosen, a more appropriate magnitude than the second(s) for this type of problem, while for the other magnitudes the units of the international system have been maintained. Water Figure2018, 105, x shows FOR PEER a simpliﬁed REVIEW scheme of the SICOMED_2018 ﬂowchart. 8 of 20 Water 2018, 10, x FOR PEER REVIEW 8 of 20

Figure 5. Operating scheme of SICOMED_2018. Figure 5. Operating scheme of SICOMED_2018.

Figure 6. SICOMED_2018 start-up screen. Figure 6. SICOMED_2018 start-up screen. Figure 6. SICOMED_2018 start-up screen.

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Water 2018, 10, x FOR PEER REVIEW 9 of 20

Figure 7. Second data entry screen. Figure 7. Second data entry screen. Figure 7. Second data entry screen.

Figure 8. Third data entry screen. Figure 8. Third data entry screen. Figure 8. Third data entry screen.

Water 2018, 10, x FOR PEER REVIEW 10 of 20 Water 2018, 10, 242 10 of 20 3.2. Simulation and Output Data To increase the flexibility and effectiveness of the program, the user can avoid entering data by 3.2. Simulationworking with and another Output file Data previously saved. For this, SICOMED_2018 includes the options of saving and loading data. Once all the data have been entered or loaded the simulation can start. To do this, To increase the flexibility and effectiveness of the program, the user can avoid entering data by SICOMED_2018 creates the network model file in a specific source code that is run in Ngspice. At the workingend with of the another simulation file the previously user can access saved. the For resu this,lts given SICOMED_2018 in a graphic way. includes SICOMED_2018 the options offers of saving and loadingup to six data. possibilities Once all theof results data have representation been entered (Figure orloaded 9), conveniently the simulation arranged can so start. that To the do this, SICOMED_2018geotechnical creates engineer the can network have all modelthe necess fileary in information a specific source in a simple, code precise that is form: run in Ngspice. At the end of the simulation the user can access the results given in a graphic way. SICOMED_2018 offers up 1. Excess pore pressure in a given column of the soil to six possibilities2. Excess pore of resultspressure representation in a given point (Figureof the soil9), conveniently arranged so that the geotechnical engineer3. canAverage have degree all the of necessary settlement information in a simple, precise form:

1. Excess4. Local pore settlements pressure in in a a given given column column of the of thesoil soil 5. Total settlement in a given point of the surface 2. Excess6. Surface pore settlements pressure in animation. a given point of the soil 3. Average degree of settlement Some of these representations are shown in the illustrative applications of the following section, 4. Localwhich settlements have been inselected a given with column the aim of the of soilpresenting all the potentialities and utilities of 5. TotalSICOMED_2018, settlement such in a givenas: (i) pointanalysis of of the classic surface consolidation problems; (ii) real scenarios of PVD 6. Surfacelayout optimization; settlements animation.(iii) inclusion of the smearing and drain discharge capacity effects; and (iv) verification of results.

Figure 9. Representation selection screen. Figure 9. Representation selection screen. 4. Applications Some of these representations are shown in the illustrative applications of the following section,4.1. which First Scenario: have beenConsolidation selected of a with One-Layer the aim Soil with of presenting PVD all the potentialities and utilities of SICOMED_2018,This application such as: refers (i) analysis to a fixed of classicone-layered consolidation consolidation problems; scenario (ii) in realwhich scenarios the PVD oflayout PVD is layout optimization;already imposed. (iii) inclusion The geometry of the smearingof the problem and and drain the dischargesoil properties capacity are shown effects; in Table and 1 (iv) (note verification that of results.the PVD thickness, td, is not given, as this parameter does not have an influence when the smear and drain discharge capacity effects are not considered, as in this case). The final time of the simulation 4. Applicationshas been set at four years, while the chosen time step is 0.01 years. Finally, a 20 × 16 × 20 grid has been chosen, enough to guarantee precision in the results, with relative errors below 1%. 4.1. First Scenario: Consolidation of a One-Layer Soil with PVD Table 1. Geometry of the problem and soil properties. This application refers to a fixed one-layered consolidation scenario in which the PVD layout is Thickness (H) 3 m already imposed. The geometry of the problem and the soil properties are shown in Table1 (note that Separation between PVDs (2a) 2 m the PVD thickness, td, is notSeparation given, between as this parameterrows of PVDs does (2b) not have 1.6 an m influence when the smear and drain discharge capacity effects are notPVD considered, width (2c) as in this case). The 0.1 m final time of the simulation has been set at four years, whileDepth the chosen of penetration time step of PVD is 0.01 (d) years. Finally, 3 m a 20 × 16 × 20 grid has been chosen, enough to guarantee precisionApplied in theload results, (q) with relative 30 errors kN/m below2 1%. Initial void ratio (eo) 1.23 Table 1. Geometry of the problem and soil properties.

Thickness (H) 3 m Separation between PVDs (2a) 2 m Separation between rows of PVDs (2b) 1.6 m PVD width (2c) 0.1 m Depth of penetration of PVD (d) 3 m Applied load (q) 30 kN/m2 Initial void ratio (eo) 1.23 Water 2018, 10, 242 11 of 20

Table 1. Cont.

Water 2018, 10, x FOR PEER REVIEW 2 11 of 20 Compressibility coefficient (av) 0.0023 m /kN Vertical hydraulic conductivity (kv) 0.0095 m/year Water 2018, 10, x FORHorizontal PEER REVIEW hydraulicCompressibility conductivity coefficient (kh) (av) 0.0023 0.0259 m2/kN m/year 11 of 20 2 Vertical consolidation coefficient (cv,z) 0.94 m /year Vertical hydraulic conductivity (kv) 0.0095 m/year Horizontal consolidation coefficient (c ) 2.56 m2/year Compressibility coefficientv,h (av) 0.0023 m2/kN Horizontal hydraulic conductivity (kh) 0.0259 m/year Vertical hydraulic conductivity (kv) 0.0095 m/year Vertical consolidation coefficient (cv,z) 0.94 m2/year Horizontal hydraulic conductivity (kh) 0.0259 m/year At this point it is importantHorizontal to consolidation note that SICOMED_2018 coefficient (cv,h) has2.56 the optionm2/year to directly enter the values Vertical consolidation coefficient (cv,z) 0.94 m2/year of the consolidation coefficients (cv,z and cv,h), allowing only the analysis of the problem in terms of the Horizontal consolidation coefficient (cv,h) 2.56 m2/year duration ofAt the this process point it (optionsis important 1, 2, to and note 3 that in Figure SICOMED_20189), or provide has the the option values to of directly the soil enter parameters the values of the consolidation coefficients (cv,z and cv,h), allowing only the analysis of the problem in (av, eo, kv andAt this kh )point from it whichis important these to coefficients note that SI areCOMED_2018 deduced—Equation has the option (22)—allowing to directly enter this the second terms of the duration of the process (options 1, 2, and 3 in Figure 9), or provide the values of the soil optionvalues to perform of the anconsolidation analysis of coefficients the problem (cv,z inand terms cv,h), allowing of settlement only the (options analysis 4, of 5, the and problem 6 in Figure in 9): parameters (av, eo, kv and kh) from which these coefficients are deduced—Equation (22)—allowing terms of the duration of the process (options 1, 2, and 3 in Figure 9), or provide the values of the soil this second option to perform an analysis of the problem in terms of settlement (options 4, 5, and 6 in parameters (av, eo, kv and kh) from whichkv(1 these+ e) coefficientskh are(1 +deduced—Equatione) (22)—allowing Figure 9): cv,z = , cv,h = . (22) this second option to perform an analysisw ofa vthe problem in termswav of settlement (options 4, 5, and 6 in () () Figure 9): c, = ,c, = . (22) Once the simulation is run, with a computation time of 10 min on an Intel Core i7-6700 CPU () () c, = ,c, = . (22) 4.00 GHzOnce computer, the simulation the user is run, can with access a computation the results time (Figure of 10 min9). on The an Intel excess Core pore i7-6700 pressure CPU 4.00 can be GHz computer, the user can access the results (Figure 9). The excess pore pressure can be represented representedOnce for allthe thesimulation cells of is one run, or with two a soilcomputation columns, time given of 10 to min the on program an Intel Core by their i7-6700 (X, CPU Y) coordinates 4.00 for all the cells of one or two soil columns, given to the program by their (X, Y) coordinates (Figure (FigureGHz 10). computer, As can bethe seen,user can the access dissipation the results of (Figure the excess 9). The pore excess pressure pore pressure (Figure can 11be )represented occurs faster in 10). As can be seen, the dissipation of the excess pore pressure (Figure 11) occurs faster in those points those pointsfor all the of thecells soil of one closer or two to thesoil columns, draining given edges, to the soil program surface, by and their PVD. (X, Y) The coordinates program (Figure alsoallows of the soil closer to the draining edges, soil surface, and PVD. The program also allows (option 2) the (option10). 2) theAs can representation be seen, the dissipation of the evolution of the excess of this pore unknown pressure (Figure in one, 11) two, occurs or threefaster in locations, those points indicated representation of the evolution of this unknown in one, two, or three locations, indicated to the of the soil closer to the draining edges, soil surface, and PVD. The program also allows (option 2) the to the SICOMED_2018 by byits its(X, (X,Y, Z) Y, coordinates. Z) coordinates. representation of the evolution of this unknown in one, two, or three locations, indicated to the SICOMED_2018 by its (X, Y, Z) coordinates.

Figure 10. Selection of soil columns to be represented and time interval. Figure 10. Selection of soil columns to be represented and time interval. Figure 10. Selection of soil columns to be represented and time interval.

Figure 11. Excess pore pressure evolution in two soil columns (Nz = 20).

Figure 11. Excess pore pressure evolution in two soil columns (Nz = 20). Figure 11. Excess pore pressure evolution in two soil columns (Nz = 20).

Water 2018, 10, 242 12 of 20

Water 2018, 10, x FOR PEER REVIEW 12 of 20 The evolution of the average degree of settlement (the degree of consolidation averaged over all Water The2018, evolution10, x FOR PEER of the REVIEW average degree of settlement (the degree of consolidation averaged over12 of all 20 soil columns) is shown in Figure 12. Thus, it can be observed that 90% of the consolidation is reached soilWater columns) 2018, 10, x isFOR shown PEER REVIEWin Figure 12. Thus, it can be observed that 90% of the consolidation is reached12 of 20 after 1.30after years. The1.30 evolution years. of the average degree of settlement (the degree of consolidation averaged over all soil columns)The evolution is shown of the in Figureaverage 12. degree Thus, of it settlementcan be observed (the degree that 90% of consolidation of the consolidation averaged is reached over all aftersoil columns) 1.30 years. is shown in Figure 12. Thus, it can be observed that 90% of the consolidation is reached after 1.30 years.

Figure 12. Average degree of settlement evolution. Figure 12. Average degree of settlement evolution.

Options 4–6 of Figure Figure9 are also 12. Average available degree (Figures of settlement 13–15) since evolution. parameters av, eo, kv, and kh are Optionsgiven. Note 4–6 that of Figurethe soil9 columns areFigure also closer12. available Average to the degree (Figures PVD willof settlement 13 reach–15 )the since evolution. final parameters settlement before a v, e (Figureo, kv, and 14), kh are given.although NoteOptions that the thevalue4–6 soilof ofFigure columnsthis unknown 9 are closeralso is available th toe thesame PVD(Figures for all will the 13–15) reach columns: since the 0.0927 ﬁnalparameters settlementm. av, eo, k beforev, and k (Figureh are 14), given. Note that the soil columns closer to the PVD will reach the final settlement before (Figure 14), although theOptions value 4–6 of of this Figure unknown 9 are also is theavailable same (Figures for all the 13–15) columns: since parameters 0.0927 m. av, eo, kv, and kh are althoughgiven. Note the that value the of soil this columns unknown closer is th toe same the PV forD allwill the reach columns: the final 0.0927 settlement m. before (Figure 14), although the value of this unknown is the same for all the columns: 0.0927 m.

Figure 13. Evolution of the local settlement in each of the cells of a soil column (Nz = 20).

Figure 13. Evolution of the local settlement in each of the cells of a soil column (Nz = 20). Figure 13. Evolution of the local settlement in each of the cells of a soil column (Nz = 20). Figure 13. Evolution of the local settlement in each of the cells of a soil column (Nz = 20).

Figure 14. Surface settlement evolution of two soil columns.

Figure 14. Surface settlement evolution of two soil columns. Figure 14. Surface settlement evolution of two soil columns. Water 2018, 10, 242 13 of 20 Water 2018, 10, x FOR PEER REVIEW 13 of 20

Figure 15. Screenshot of the settlement evolution animation on the ground surface. Figure 15. Screenshot of the settlement evolution animation on the ground surface. The last representation option (Figure 15) is a video animation of the settlement evolution of the wholeThe soil last surface. representation As representation option (Figure options, 15 )the is auser video can animation choose the of time the settlementinterval to evolutiondisplay and of the the numberwhole soil of surface.frames per As second representation of the an options,imation the (for user faster can or choose slower the display). time interval to display and the number of frames per second of the animation (for faster or slower display). 4.2. Second Scenario: Optimization of PVD Layout 4.2. Second Scenario: Optimization of PVD Layout In this application we analyze and accelerate the process of consolidation of a three-layered soil In this application we analyze and accelerate the process of consolidation of a three-layered soil (whose properties are given in Table 2) on which an embankment is going to be built, transmitting to (whose properties are given in Table2) on which an embankment is going to be built, transmitting to the ground a uniform load of 60 kN/m2. Using SICOMED_2018, different layouts (depth of the ground a uniform load of 60 kN/m2. Using SICOMED_2018, different layouts (depth of penetration penetration and separation of PVDs) will be simulated (with different final and step times, depending and separation of PVDs) will be simulated (with different final and step times, depending on the case), on the case), looking for the optimal solution from the point of view of consolidation time. A 10 × 10 looking for the optimal solution from the point of view of consolidation time. A 10 × 10 × 30 grid has × 30 grid has been chosen (0.2 m high in each cell). been chosen (0.2 m high in each cell). Table 2. Thickness and properties of the three strata. Table 2. Thickness and properties of the three strata. S1 (Upper S2 (Intermediate S3 (Lower Soil Property S (Upper S (Intermediate S (Lower Soil Property Stratum)1 2 Stratum) 3Stratum) Stratum) Stratum) Stratum) Thickness (m) H 1 3 2 Thickness (m) H 1 3 2 InitialInitial void void ratio ratio eo eo 1.51.5 0.9 0.9 0.7 0.7 Compressibility coefficient (m2/kN) a 0.0075 0.0028 0.00125 Compressibility coefficient (m2/kN) v av 0.0075 0.0028 0.00125 Vertical hydraulic conductivity (m/year) kv 0.007 0.015 0.006 v VerticalHorizontal hydraulic hydraulic conductivity conductivity (m/year) (m/year) kh k 0.0220.007 0.04 0.015 0.01 0.006 2 HorizontalVertical consolidation hydraulic coefficient conductivity (m /year) (m/year)cv,z kh 0.240.022 1.04 0.04 0.83 0.01 2 Horizontal consolidation coefficient (m /year) cv,h 0.75 2.77 1.39 Vertical consolidation coefficient (m2/year) cv,z 0.24 1.04 0.83 Horizontal consolidation coefficient (m2/year) cv,h 0.75 2.77 1.39 To address this problem it is necessary to set a definition for the consolidation time (or characteristic time,To to [address33]), as wellthis asproblem to fix ait target is necessary value of to this set time a definition to be achieved for the with consolidation the use of PVDs. time (or So, characteristicwe define the time, characteristic to [33]), timeas well as theas to time fix thata target takes value thesoil of this to reach time anto averagebe achieved degree with of settlementthe use of PVDs.(U) of 90%.So, we In addition,define the PVDs characteristic must reduce time this as timethe ti tome a valuethat takes below the two soil years. to reach In this an way,average we proceeddegree ofto settlement perform the (U simulation) of 90%. In of addition, the problem PVDs in must the most reduce unfavorable this time to scenario, a value that below is, whentwo years. there In are this no way,PVDs we installed proceed in to the perform ground. the For simulation this case of (Figure the pr oblem16), 90% in the of the most average unfavorable degree scenario, of settlement that is, is whenachieved there after are moreno PVDs than installed 30 years, in a valuethe ground. too high For for this work case in (Figure civil engineering. 16), 90% of the average degree of settlement is achieved after more than 30 years, a value too high for work in civil engineering.

Water 2018, 10, 242 14 of 20 Water 2018, 10, x FOR PEER REVIEW 14 of 20 Water 2018, 10, x FOR PEER REVIEW 14 of 20

Figure 16. Average degree of settlement for the case with no PVDs. Figure 16. Figure 16.Average Average degreedegree of settlement for for the the case case with with no no PVDs. PVDs. Figure 17 shows the local settlement evolution of an entire soil column, which is different for eachFigure Figurestratum. 17 17 shows The shows simulation the the local local settlement also settlement provides evolution evolution a final surface of of an an entire settlemententire soil soil column, valuecolumn, of which 0.53which m. is differentis different for for each stratum.each stratum. The simulation The simulation also provides also provides a ﬁnal a surfacefinal surface settlement settlement value value of 0.53 of 0.53 m. m.

Figure 17. Local settlements evolution for the case with no PVDs (Nz = 30). Figure 17. Local settlements evolution for the case with no PVDs (Nz = 30). Figure 17. Local settlements evolution for the case with no PVDs (Nz = 30). In order to reduce the consolidation time, PVDs separated every 2 m (both at x and y spatial directions)In order are to installed. reduce the The consolidation width of the time,PVD PVDsis 0.1 m,separated a typical every commercial 2 m (both value at xfor and this y typespatial of In order to reduce the consolidation time, PVDs separated every 2 m (both at x and y spatial directions)drain of geosynthetic are installed. origin. The widthAs depth of the of penetratPVD is 0.1ion m, we a begintypical with commercial d = 1 m (partiallyvalue for thispenetrating type of directions) are installed. The width of the PVD is 0.1 m, a typical commercial value for this type of drain ofto geosyntheticthe first layer). origin. As the As reduction depth of ofpenetrat to is notion enough, we begin we with increase d = 1the m penetration(partially penetrating up to 4 m drain of geosynthetic origin. As depth of penetration we begin with d = 1 m (partially penetrating drain(partially to the penetrating first layer). drain As the to reductionthe intermediate of to is notstratum) enough, and we 6 increasem (totally the penetrating penetration drain). up to 4The m drain to the ﬁrst layer). As the reduction of to is not enough, we increase the penetration up to 4 m (partiallyresults for penetrating the consolidation drain timeto the (t ointermediate) are shown in stratum) Table 3. and 6 m (totally penetrating drain). The (partiallyresults for penetrating the consolidation drain to thetime intermediate (to) are shown stratum) in Table and 3. 6 m (totally penetrating drain). The results for the consolidationTable 3. time Values (to of) are to as shown a function in Tableof PVD3. penetration (a = 1 m; b = 1 m; c = 0.05 m). Table 3. Values of to as a function of PVD penetration (a = 1 m; b = 1 m; c = 0.05 m). Table 3. Values of to as aDepth function of of PVD PVD (m) penetration to (Years) (a = 1 m; b = 1 m; c = 0.05 m). DepthWithout of PVD drain (m) to (Years) 32 Depth of PVDWithout (m)1 drain 3227 to (Years) Without drain14 4.527 32 1 27 46 4.53 46 3 4.5 6 3

Water 2018, 10, x FOR PEER REVIEW 15 of 20 Water 2018, 10, 242 15 of 20 Since none of the studied options fits the two-year objective, it is necessary to bring PVDs closer. Thus, for the cases d = 6 m and d = 4 m, a set of simulations in which both the distance between PVDs and Sincethe separation none of the between studied rows options of ﬁtsPVDs the are two-year reduced objective, will be it simulated is necessary with to bring SICOMED_2018. PVDs closer. Thus,Characteristic for the casestimes dfor = each 6 m andlayout d =are 4 m,summarized a set of simulations in Table 4. in which both the distance between PVDs and the separation between rows of PVDs are reduced will be simulated with SICOMED_2018.

CharacteristicTable times 4. Values for each of t layouto as a function are summarized of PVD penetration in Table 4and. surface layout (c = 0.05 m).

Table 4. Values of to as aDepth function of of PVD PVD penetration (m) to (Years) and surface layout (c = 0.05 m). Without drain 32 Depth of1 PVD (a = (m)b = 1 m) 27 to (Years) Without4 (a drain = b = 1 m) 4.5 32 1 (a = b = 1 m) 27 4 (a = b6 =(a 1 = m) b = 1 m) 3 4.5 6 (a =6 b (a = 1= m)b = 0.9 m) 2.4 3 6 (a = b6 =(a 0.9 = b m) = 0.8 m) 1.9 2.4 6 (a = b = 0.8 m) 1.9 4 (a = b4 =(a 0.8 = b m) = 0.8 m) 2.5 2.5 4 (a = b4 =(a 0.7 = b m) = 0.7 m) 2.0 2.0 4 (a = b4 =(a 0.65 = b m)= 0.65 m) 1.8 1.8

In viewview of of these these results, results, two two alternatives alternatives emerge emer as age solution as a solution to the problem: to the (i)problem: totally penetrating (i) totally drainspenetrating every drains 1.6 m every (2 linear 1.6 metersm (2 linear of PVD meters per of square PVD meterper square of soil meter surface), of soil which surface), provides which a characteristicprovides a characteristic time of 1.9 time years; of 1.9 and years; (ii) partially and (ii) partially penetrating penetrating drains (4 drains m) every (4 m) 1.4 every m (1.75 1.4 m linear (1.75 meterslinear meters of PVD of per PVD square per square meter meter of soil of surface), soil surface), which which provides provides a characteristic a characteristic time of time 2.0 years.of 2.0 Figuresyears. Figures 18 and 18 19 and illustrate 19 illustrate the last the alternative last alternat showing,ive showing, respectively, respectively, the local the settlementslocal settlements and total and surfacetotal surface settlement settlement evolution evolution for two for soil two columns. soil columns.

Figure 18. Local settlement evolution in each of the cells of two soil columns. (a = b = 0.7 m; c = 0.05 Figure 18. Local settlement evolution in each of the cells of two soil columns. (a = b = 0.7 m; c = 0.05 m; m; d = 4 m). d = 4 m).

Water 2018, 10, 242 16 of 20 Water 2018, 10, x FOR PEER REVIEW 16 of 20

Figure 19. Surface settlement evolution in two soil columns. (a = b = 0.7 m; c = 0.05 m; d = 4 m). Figure 19. Surface settlement evolution in two soil columns. (a = b = 0.7 m; c = 0.05 m; d = 4 m). 4.3. Third Scenario: Influence of the Smear Zone and the Discharge Capacity Limitation of the PVD 4.3. Third Scenario: Influence of the Smear Zone and the Discharge Capacity Limitation of the PVD The third application addresses the use of limited discharge capacity drains, as well as the The third application addresses the use of limited discharge capacity drains, as well as the influence of the smear zone. The geometric characteristics and ground and drain properties are influence of the smear zone. The geometric characteristics and ground and drain properties are shown in Table 5. The geometric values of the problem (a, b, c, and td) have been selected so that the shown in Table5. The geometric values of the problem (a, b, c, and t ) have been selected so that solutions of SICOMED_2018 can be compared with the results obtainedd by Hansbo et al. [16] and the solutions of SICOMED_2018 can be compared with the results obtained by Hansbo et al. [16] and Barron [34], in the same problem but using a two-dimensional (2D) radial geometry, with an Barron [34], in the same problem but using a two-dimensional (2D) radial geometry, with an equivalent equivalent diameter of the PVD and a diameter of influence of 0.062 m and 0.95 m, respectively. The diameter of the PVD and a diameter of influence of 0.062 m and 0.95 m, respectively. The physical physical soil and drain properties are identical. Regarding av and e, any pair of values that provides soil and drain properties are identical. Regarding av and e, any pair of values that provides the same the same consolidation coefficients is valid, having adopted the values of av = 1.224 × 10−5 m2/N and e consolidation coefficients is valid, having adopted the values of a = 1.224 × 10−5 m2/N and e = 1. = 1. Finally, ∆Q = 10.000 N/m2. v Finally, ∆Q = 10.000 N/m2. Table 5. Geometry and physical properties of the problem. Table 5. Geometry and physical properties of the problem. Parameter/Soil Property Value Units Parameter/SoilH (draining Property length) Value15 Unitsm 2a (separationH (draining length)between PVDs) 15 0.785 m m 2b2a (separation (separation betweenbetween PVDs) rows PVDs) 0.785 0.739 m m 2b (separation between rows PVDs) 0.739 m 2c (PVD width) 0.0924 m 2c (PVD width) 0.0924 m d td (PVDt (PVD thickness) thickness) 0.005 0.005 m m PVDPVD equivalent equivalent diameter diameter 0.0620 0.0620 m m SmearSmear length length (normal (normal to the to PVDthe PVD plain) plain) 0.0693 0.0693 m m kh (horizontal hydraulic conductivity of the undisturbed soil) 0.03 m/year kh (horizontal hydraulic conductivity of the undisturbed soil) 0.03 m/year kh,s (horizontal hydraulic conductivity of the smear zone) 0.01 m/year kh,s (horizontal hydraulic conductivity of the smear zone) 0.01 m/year kv (vertical hydraulic conductivity of the undisturbed soil) 0.009 m/year kvk v,s(vertical(vertical hydraulic hydraulic conductivity conductivity of of the the smear undisturbed zone) soil) 0.003 0.009 m/year m/year 2 kv,s (verticalcv,h (horizontal hydraulic consolidation conductivity coefficient) of the smear zone) 0.5 0.003 m/yearm /year c (vertical consolidation coefficient) 0.15 2 cv,zv,h (horizontal consolidation coefficient) 0.5 mm2/year/year × × Grid size (Nx,Ny,Nz) 17 16 15 2 cv,z (vertical consolidation coefficient)3 0.15 m /year Drain hydraulic conductivity for qw = 20 m /year 43,290 m/year Grid size (Nx, Ny, Nz) 3 17 × 16 × 15 Drain hydraulic conductivity for qw = 10 m /year 21,645 m/year 3 3 DrainDrain hydraulic hydraulic conductivity conductivity for q forw = q 5w m = /year20 m /year 10,822.543,290 m/year m/year Drain hydraulic conductivity for qw = 10 m3/year 21,645 m/year Drain hydraulic conductivity for qw = 5 m3/year 10,822.5 m/year Four consolidation scenarios have been simulated, comparing the average degree of consolidation achieved in three typical times: 0.5, one, and two years (Table6). In the first case, in which the Four consolidation scenarios have been simulated, comparing the average degree of smear effects are not considered and the drain is supposed to have an infinite discharge capacity, consolidation achieved in three typical times: 0.5, one, and two years (Table 6). In the first case, in the results reproduce those obtained analytically by Barron [34] under the free strain hypothesis, which the smear effects are not considered and the drain is supposed to have an infinite discharge capacity, the results reproduce those obtained analytically by Barron [34] under the free strain hypothesis, with relative deviations below 3% for the three times. The other three cases assume the

Water 2018, 10, 242 17 of 20 with relative deviations below 3% for the three times. The other three cases assume the same smear length but consider different PVD discharge capacities. For them, the comparison is established with the results obtained by Hansbo et al. [16], based on their empirical expressions for the average degree of consolidation. Deviations, particularly for relatively high times, can be considered acceptable, taking into account that these are not numerical or analytical results. These deviations grow slightly to 11.6% for small times and low discharge capacities, perhaps because the models degrees of freedom (3D for SICOMED_2018 and 2D radial for Hansbo) are not the same. On the other hand, the results are coherent with respect to the correlation between the average degree of consolidation and the inﬂuence of both the smear effects and the discharge capacity limitation of the drain.

Table 6. Comparison of the average degree of consolidation between SICOMED_2018 and Hansbo et al. [16] and Barron’s [34] solutions.

t = 0.5 Years t = 1 Years t = 2 Years Case SICOMED SICOMED SICOMED Barron Hansbo Barron Hansbo Barron Hansbo (% Error) (% Error) (% Error) Case 1 68.1% 89.6% 98.9% No smear, limited 66.3% - 89.2% - 98.6% - (2.71%) (0.45%) (0.30%) discharge capacity Case 2 44.5% 68.8% 90.2% Smear zone, - 46.0% - 69.8% - 90.5% 3 (−3.26%) (−1.43%) (−0.33%) qw = 20 m /year Case 3 39.1% 62.5% 85.5% Smear zone, - 41.8% - 63.3% - 86.0% 3 (−6.46%) (−1.26%) (−0.58%) qw = 10 m /year Case 4 30.5% 50.9% 74.8% Smear zone, - 34.5% - 53.9% - 77.9% 3 (−11.6%) (−5.57%) (−3.98%) qw = 5 m /year

5. Final Comments and Conclusions SICOMED_2018 has been created as free software for solving 3D consolidation problems with partial or totally penetrating vertical drains, in heterogeneous soils and considering the influence of the smear effects and the discharge capacity limitation of the drain. All the unknowns of interest for the civil engineer, such as the excess pore pressure evolution, the average degree of consolidation, local, and total settlements and even a video animation of the soil surface deformation, can be obtained with SICOMED_2018 in a relatively simple way. Therefore, it can be considered a program with professional, educational, and research applications. Following network simulation method rules, SICOMED_2018 designs network models (electrical circuits) whose governing equations (with spatial discretization and retaining time as a continuous variable) are formally equivalent to the discretized equations of the consolidation problem. These network models are solved by an electrical circuit resolution code such as Ngspice, directly providing the solutions to the consolidation problem. The program has been developed in a pleasant communication environment, both for the data entry and the graphical presentation of results, under a classic windows environment programmed with MATLAB, a code whose license is not free but allows the distribution of the programs created with it. On the other hand, for the numerical calculation the powerful computational algorithms integrated in Ngspice, an electrical circuits resolution free code that provides practically exact solutions to the models designed by SICOMED_2018, have been used. The combination of these codes has allowed us to obtain completely free software. To illustrate the operation of the program, three applications, corresponding to real cases, are presented. The first one deals with the simulation and resolution of a consolidation scenario with the PVD distribution already imposed (a classic problem), showing the evolution of the main variables of interest (excess pore pressure, average degree of consolidation, and local and total settlements). The second one concerns an optimization problem in a soil formed by three strata Water 2018, 10, 242 18 of 20 with different properties, in which it is sought to find the optimum PVD distribution in order to reduce the characteristic time of consolidation to a certain target value. Finally, the third application serves as a verification of the program against the analytical and empirical solutions provided by other authors, studying the influence that the smear zone and the discharge capacity limitation of the drain have on the consolidation problem. Deviations are reduced in all cases, showing the versatility and reliability of the program. Finally, from the work presented here, a series of challenges that could be addressed in future research are suggested. On the one hand, in the field of soil consolidation, this research could be completed (or improved) by implementing additional options in the program, such as analysis based on real tabulated data, or even the possibility of addressing inverse problems. On the other hand, the wide experience of the ‘network simulation’ research group of the UPCT opens up the possibility of using a similar methodology in the creation of free software in disciplines as diverse as seepage, atmospheric dispersion of pollutants, and hydrologic engineering, among others.

Acknowledgments: The authors acknowledge the professors Francisco Alhama and Juan Francisco Sánchez, from the Applied Physics Department of the Technical University of Cartagena, for their help both in the development of the network models and in the elaboration of the graphic interface. The authors also thank the reviewers for the useful comments to improve the manuscript. Author Contributions: Gonzalo García-Ros and Iván Alhama conceived and designed the software and wrote this paper, while Manuel Cánovas performed the numerical simulations and analyzed the results obtained. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Nomenclature a semi-separation between prefabricated vertical drains (m) 2 av coefficient of compressibility (m /N) b semi-separation between rows of prefabricated vertical drains (m) c Semi-width of prefabricated vertical drain (m) C capacity value of a capacitor (F) Ci capacitor connected between the central node of cell i and the common ground node 2 cv coefficient of consolidation (m /s) 2 cv,h horizontal coefficient of consolidation (m /s) 2 cv,x horizontal coefficient of consolidation in x spatial direction (m /s) 2 cv,y horizontal coefficient of consolidation in y spatial direction (m /s) 2 cv,z vertical coefficient of consolidation (m /s) d depth of penetration of prefabricated vertical drain (m) e void ratio (dimensionless) eo initial void ratio (dimensionless) H soil thickness or draining length of the water in vertical direction (m) IC current that flows through a capacitor (A) IR current that flows through a resistance (A) JC current flow associated with a capacitor (A) JR current flow associated with a resistance (A) kd hydraulic conductivity of the PVD (m/s) kh horizontal hydraulic conductivity of the undisturbed soil (m/s) kh,s horizontal hydraulic conductivity of the smear zone (m/s) kv vertical hydraulic conductivity of the undisturbed soil (m/s) kv,s vertical hydraulic conductivity of the smear zone (m/s) n normal direction to the boundary surface Nx number of cells in x direction Ny number of cells in y direction Nz number of cells in z direction q uniform load applied to the ground surface (N/m2) 3 qw discharge capacity of the PVD for unit gradient of pressure (m /s) R value of electrical resistance (Ω) Rj resistance connected between the central node of cell i and node j t time independent variable (s) td PVD thickness (m) to characteristic time of the consolidation process (s) u excess pore water pressure (N/m2) U local degree of settlement (dimensionless) Water 2018, 10, 242 19 of 20

U average degree of settlement (dimensionless) 2 ui excess pore pressure in node i (N/m ) 2 uo initial excess pore pressure (N/m ) VC voltage between the terminals of a capacitor (V) VR voltage between the terminals of a resistance (V) x long horizontal spatial coordinate (m) X value of the spatial coordinate x (m) y wide horizontal spatial coordinate (m) Y value of the spatial coordinate y (m) z vertical spatial coordinate (m) Z value of the spatial coordinate z (m) 3 γw speciﬁc weight of water (N/m ) 0 2 σ f ﬁnal effective pressure (N/m ) 0 2 σ o initial effective pressure (N/m )

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