Powerful Software to Simulate Soil Consolidation Problems with Prefabricated Vertical Drains

Transcription

1 water Article Powerful Stware to Simulate Soil Consoliation Problems with Prefabricate Vertical Drains Gonzalo García-Ros 1, *, Iván Alhama 1 an Manuel Cánovas 2 ID 1 Civil Engineering Department, Technical University Cartagena, Paseo Alfonso XIII 52, Cartagena, Spain; ivan.alhama@upct.es 2 Metallurgical an Mining Engineering Department, Universia Católica el Norte, Av. Angamos 0610, Antagasta (Región e Antagasta), Chile; manuel.canovas@ucn.cl * Corresponence: gonzalo.garcia@upct.es Receive: 2 February 2018; Accepte: 23 February 2018; Publishe: 27 February 2018 Abstract: The present work escribes program Simulation Consoliation with Vertical Drains (SICOMED_2018), a tool for solution consoliation processes in heterogeneous soils, with totally or partially penetrating prefabricate vertical rains (PVD) an consiering both effects smear zone, generate when introucing rain into groun, an limitation in ischarge capacity rain. In orer to provie a completely free program, coe Next-Generation Simulation Program with Integrate Circuit Emphasis (Ngspice) has been use as a numerical tool while Matrix Laboratory (MATLAB) coe was use to program an create an interface with user through interactive screens. In this way, SICOMED_2018 is presente as an easy-to-use an intuitive program, with a simple graphical interface that allows user to enter all soil properties an geometry problem without having to resort to a complex stware package that requires programming. Illustrative applications escribe both versatility program an reliability its numerical solutions. Keywors: consoliation stware; vertical rains; heterogeneous soils; smear; rain resistance 1. Introuction The use specific stware for unerstaning complex physical phenomena involve in engineering problems has become an inispensable tool in moern teaching, mainly ue to high costs involve in laboratory tests. These programs, in spite effort an knowlege especially programming an calculation that ir evelopment require, are generally esigne by university epartments mselves as an interisciplinary work among several areas knowlege. The basic objective is to create an interactive computer-to-user communication program that, with minimal help, enables learning objectives to be met in a shorter time than woul be necessary with laboratory tests. In this sense, esign se programs avoiing concept black box allows user to access choice a large number parameters associate with programming an esign moel, with purpose etermining potential influence m in numerical results. The network simulation research group UPCT (Technical University Cartagena) has significant experience in stware evelopment, in fiels mechanical engineering an couple flow an transport processes, such as: (i) Program for Designing Simple Fins (PRODASIM) an Program for Heat Conuction (PROCCA-09), for assembling transient heat transfer in fins [1]; (ii) Flow an Transport Simulator (FATSIM-A), for simulation flow an transport processes [2]; an (iii) Couple Orinary Differential Equations by Network Simulation (CODENS_13), for simulation mechanical moels governe by couple orinary ifferential equations [3]. All m are available an have been use as tools for numerous research works [4,5], but require a license to Water 2018, 10, 242; oi: /w

2 Water 2018, 10, use Personal Computer Simulation Program with Integrate Circuit Emphasis (Pspice), which is responsible for performing numerical calculations. In this work we present free program SICOMED_2018 [6], a coe for simulation transient problems soil consoliation with use prefabricate vertical rains to accelerate settlement process, easy to use but with numerous applications within fiel civil engineering. The possibility selecting not only value any physical parameters involve, with heterogeneous soils up to three layers, but also those associate with gri size an numerical computation parameters allows user to aress specific consoliation scenarios etermining separate influence each parameters involve as well as to solve optimization problems such as relationship between consoliation time an epth rain. On or han, powerful graphic output environment program allows us to visualize an interpret results in a irect an simple way. For se reasons, SICOMED_2018 is presente as a complete stware package, with novel contributions, that joins or specific programs for learning ory an practice soil consoliation such as Consoliation Testing an Multimeia (CTM) [7]. The performance SICOMED_2018, which can be acquire through UPCT (Technical University Cartagena), requires installation free coe Ngspice [8]. The graphical communication environment has been evelope with MATLAB (uner a license that allows its istribution), a very wiesprea coe at a scientific an pressional level that has highly evelope tools to create user interfaces, while powerful computational algorithms Ngspice provie nearly exact solution network moel consoliation process, whose esign is base on analogy between electric an physical quantities. The esign numerical moel follows rules network simulation metho [9], a tool wiely known in scientific worl for implementation network moels. Each term in equation is an electric current that is balance with or terms in a common noe volume element whose voltage is solution potential quantity (pressure) governing equation. A mesh size orer 50 volume elements, in one-imensional (1D) problems, reuces errors with respect to analytical solutions to below 1% [10], which is acceptable for this type engineering problem. Illustrative applications are presente to emonstrate efficiency an reliability program. 2. Funamentals Soil Consoliation with Prefabricate Vertical Drains an Network Moel Prefabricate vertical rains (PVD) are a system soil rainage commonly use in civil engineering in problems soil consoliation in which goal is to accelerate final settlement. When a fully saturate clay soil is subjecte to action loas on its surface, it begins to experience a ecrease in volume or settlement over a generally long perio, in a transitory process commonly known as consoliation. This volume reuction is ue to graual expulsion part water containe in vois between soil particles, through one or more raining bounaries (generally across groun surface), as excess pore pressure initially generate by applie loa is transmitte to soil, increasing effective pressure on it. In real scenarios, with groun consierable thickness an low permeability, main problem facing civil engineer is long consoliation times neee to reach final settlement (on orer years). For this reason, installation PVD is a goo solution to reuce time consoliation a soil, as we are significantly shortening rainage length water. Assuming hyposes postulate by Terzaghi [11] (i) both water an soil particles are assume incompressible; (ii) groun self-weight is neglecte; (iii) excess pore pressure (u) is cause by application external loa (q), constant over time; (iv) flui movement in porous meium is assume to obey law Darcy; (v) soil skeleton oes not creep uner action a constant effective pressure; (vi) change in soil thickness in relation to its total volume (hyposis 1 + e constant) is consiere negligible; (vii) re is uniform applie loa, so that soil eformation only occurs in vertical irection; (viii) geotechnical parameters soil, inclue in consoliation coefficient c v, remain constant; an (ix) soil rests on an impermeable

3 Water 2018, 10, Water 2018, 10, x FOR PEER REVIEW 3 20 stratum so that water rains only by groun surface (water flow only in vertical irection), governing equation an bounary conitions for 1D linear soil consoliation problem are given by ( u t = 2 ) c u v z 2 (1) t =c u z (1) u (z=0,t) = 0 Surface rainage conition (2) u (, ) = 0 Surface rainage conition (2) u u = 0 Impermeable bottom ege (3) z (z=h,t) = 0 Impermeable bottom ege (3) z (, ) u (z,t=0) = u o = q Initial conition. (4) u (, ) =u = q Initial conition. (4) The analytical solution for a single layer soil can be foun in several works [12,13], proviing solutions The analytical for local solution egree for settlement a single layer (U) for soil any can epth be foun an instant in several time, works as well [12,13], as proviing average solutions egree for settlement local (U) egree at any settlement time. (U) for any epth an instant time, as well as average egree The installation settlement (U ) PVD, at any mae time. geosyntic material, allowing rainage through m, involves transformation The installation PVD, 1D problem mae into geosyntic a three-imensional material, (3D) allowing problem, rainage since through irection m, or involves flow water transformation occurs in any three 1D problem spatial irections into a three-imensional (x, y, z). By aopting (3D) a problem, rectangular since layout irection (Figure 1), or which flow allows water us occurs to obtain in volume any elements three spatial rectangular irections geometry, (x, y, z). By mamatical aopting a rectangular moel for layout typical (Figure section 1), (Figure which 2) allows is governe us to obtain by volume following elements equations: rectangular geometry, mamatical moel for typical section (Figure 2) is governe by following equations: ( u t = 2 ) ( c v,z z ) ( c v,x x ) u u c v,y 2 Governing equation (5) t =c, u z +c, u x +c, u y Governing equation (5) u (x,y,z=0,t) u (,,, ) = =u u (0 x c,y=0,0 z,t) (,,, ) = = 0 0 Soil surface an rainage area PVD (6) (6) ( ) u 0 (7) n = 0 Impermeable eges (7) rest bounaries u (,,, ) (x,y,z,t=0) =u = u o Initial conition, (8) (8) where u represents partial erivative in a irection normal to bounary surface. n RECTANGULAR DISTRIBUTION LONGITUDINAL AND CROSS SECTIONS 2c 2c H z + 2b 2b 2a 2a Figure D 3D rectangular rectangular consoliation consoliation problem problem with with prefabricate prefabricate vertical vertical rains. rains. Rectangular Rectangular layout. layout. Physical Physical scheme scheme (left) an (left) section an section moel (right). moel (right). Assuming isotropy in (x, y) horizontal irections (a common feature soil behavior), Equation (5) can be simplifie to =c, +c, +. (9)

4 Water 2018, 10, Assuming isotropy in (x, y) horizontal irections (a common feature soil behavior), Equation (5) can be simplifie to ( u t = 2 ) ( c u v,z z ) u c v,h x u Water 2018, 10, x FOR PEER REVIEW y 2. 4 (9) 20 At this point, it itis isimportant to note that, as asin in linear ory Terzaghi [11], consoliation coefficients cv,z c v,z an cv,h c v,h governing Equation (9) (9) have been assume constant. As Asis iswell known, this is not entirely true, since in practice soil properties (kv, v, kh, k h, e an av) a v ) that are groupe by se coefficients (c v,z k v(1+e) γ w a v, c v,h k h(1+e), = ( ), c γ w a v ) are not but, = ( ) ) are not constant, but y vary uring consoliation process, a phenomenon that that began beganto to be be analyze analyze many many years years ago ago by Davis by Davis an an Raymon Raymon [14] an [14] many an many or or authors authors time. time. At At present present re re are arenumerous publications an anresearch that incorporate non-linearity se se coefficients in in mamatical mamatical moels, moels, although although it is also it is true also that, true for that, forusual usual practical practical values values surface surface applie applie loa (q), loa (q), changes changes in properties in properties that give that rise give riseconsoliation consoliation coefficients coefficients (cv,z an (c v,z cv,h) anare c v,h small ) are small (an (anvariations variations in se in se properties properties may even may even compensate compensate for each for each or), or), so that so that final final result resultis ispractically constant consoliation coefficients. In Ineffect, non-linearity se coefficients grows higher σ f/σ σ f/σ o quotient, o quotient, a ratio a ratio that, that, though though it oretically it can can reach reach any any value, value, in in practice practice very very rarely rarely excees excees value value 2 (or, 2 (or, at at most, most, 2.5); 2.5); a avalue valuebelow which ifference between solutions provie by by linear an non-linear moels is ispractically insignificant [15]. In Inthis way, smearing effects prouce by introuction vertical rain are reflecte in in mamatical moel in ina very simple way, since, in practice, soil property that is isaffecte by smear, for purposes consoliation coefficient, is is hyraulic conuctivity [16 18]. So, an for area soil that is isconsiere to be affecte by smear effects, hyraulic conuctivity values smear zone (k (kv,s an kkh,s) ) must simply be assigne. RECTANGULAR DISTRIBUTION SYMMETRY CONDITIONS c b 0 a Figure 2. Symmetry conitions rectangular layout. Figure 2. Symmetry conitions rectangular layout. The set Equations (6) (9) has a complex analytical solution (from point view management The set for Equations civil engineer), (6) (9) has so a complex it is presume analytical much solution more (from appropriate point to solve view m by numerical management methos. for civil engineer), so it is presume much more appropriate to solve m by numerical methos Network Moel Network Moel The network metho is technique for stuy an numerical solution any physical The network metho is a technique for stuy an numerical solution any physical processes that can be efine by mamatical moel. Starting from latter, proceure processes that can be efine by a mamatical moel. Starting from latter, proceure consists two stages: (i) elaborate network moel or electrical circuit equivalent to process, consists two stages: (i) elaborate a network moel or electrical circuit equivalent to process, an (ii) numerically simulate moel by means an aequate electrical circuits resolution coe. an (ii) numerically simulate moel by means an aequate electrical circuits resolution coe. Its applications are multiple; González-Fernánez [9] uses network metho in solving transport Its applications are multiple; González-Fernánez [9] uses network metho in solving transport problems through membranes, heat transfer an electrochemical systems. In recent years it has been successfully applie in or engineering fiels such as elastic waves [19], material resistance [20], corrosion [21], magneto-hyroynamics [22], flow an transport [4], tribology [23], mechanical problems an chaos [24]. On or han, among future research lines in which network metho can be use we fin: seepage [25], real-life case moels in hyrologic engineering [26 29],

5 Water 2018, 10, problems through membranes, heat transfer an electrochemical systems. In recent years it has been successfully applie in or engineering fiels such as elastic waves [19], material resistance [20], corrosion [21], magneto-hyroynamics [22], flow an transport [4], tribology [23], mechanical problems an chaos [24]. On or han, among future research lines in which network metho can be use we fin: seepage [25], real-life case moels in hyrologic engineering [26 29], Water 2018, 10, x FOR PEER REVIEW 5 20 an atmospheric ispersion pollutants [30,31], some m alreay in evelopment phase. establishing The formal a corresponence equivalence between network epenent moel variables an physical problem, process pressure is that an both water are flow governe in by physical same moel, equations an (iscretize electrical variables in space, but retaining circuit, time voltages, as a an continuous currents. variable), That is, both establishing moels aare corresponence governe by between same equations epenent referre variables to a volume problem, (or cell) element pressurean by water same flow iniscrete physical equations moel, for an bounary electrical conitions. variablesconsequently, circuit, voltages, errors an from currents. simulation That is, are bothonly moels attributable are governe to geometric by same mesh equations size, so that referre for an toacceptable a volume number (or cell) element cells ( an by orer same 50 in iscrete 1D scenarios, equations 4000 for for 3D bounary problems) conitions. errors are Consequently, much less than 1% errors in linear fromproblems simulation [10]. The are powerful only attributable computation to coes geometric for mesh numerical size, so that resolution for an acceptable circuits provie number cells exact ( solution orer se 50 in thanks 1D scenarios, to an optimal 4000selection for 3D problems) calculation errors are time much steps, less impose than 1% by in linear Ngspice problems coe or [10]. by The user. powerful computation coes for numerical resolution circuits provie exact solution sein thanks terms to an optimal first spatial selection erivative, calculation Equation (9) time is written steps, impose in form by Ngspice coe or by user. In terms =c first spatial erivative, Equation (9) is written in form, +c, +c, (10), {[ ] [ ] } {[ ] [ ] } { [ ] [ } where z u t an = c u v,z z u z enote z cell z + z output + c u v,h an u z x input x locations x + x in + c u v,h z-irection, u x y y an y + y same, for (10) x ]y an x in x-irection an for y an y in y-irection. where Diviing z + an z enote meium into Nx cell output Ny an Nz input volume locations elements in z-irection, size Δx an Δy Δz, same an for using x + an xnomenclature in x-irection Figure an3, for y + above an yequation in can y-irection. be written in form Diviing meium into N x N y N z volume elements size x y z, an using u 1 nomenclature Figure 3, above u u equation 1 can+ be c, written in form u u or or t =c, z u u z z x u u x x {[ 2 ] [ 2 ]} {[ 2 ] [ 2 ]} u 1 u t = c i+ z u i u v,z z z i u i z 1 u z + c i+ x u i u v,h x x i u i x (11) 1 x +c 2 {[ 2 ] [ ]} 2 2, (11) y u u u u y y 1 +c v,h y 2 2 u t = [ = ] [ u i+ z u i u i u i z ( ) ( z) 2 ( z) 2 2cv,z ( ) 2cv,z,, ] + + u i+ y u i y 2 u i u i y y 2 u i+ x u i u i u i x ( ) ( x) 2 ( ) ( x) 2 2c v,h 2c v,h,, + + u i+ y u i u i u i y ( y) 2 ( ) ( y) 2 2c v,h 2c v,h ( ),.. (12) (12), z + y + x + u i+ z u i+ y z u i- x u i u i+ x u i- y u i- z y x Figure Figure Nomenclature Nomenclature element element cell cell 3D 3D rectangular rectangular consoliation consoliation moel. moel. Each terms in above equation is a flow: j = u t, j = u u u u u u ( z), j = ( z), j = ( x), 2c, 2c, 2c, u u u u u u (13)

6 Water 2018, 10, Each terms in above equation is a flow: j C = u t, j R+ z = u i+ z u i ( z) 2 2cv,z, j R z = u i u i z ( z) 2 2cv,z j R+ y = u i+ y u i ( y) 2 2c v,h so that Equation (12) can be written as a flow balance:, j R+ x = u i+ x u i ( x) 2 2c v,h, j R x = u i u i x ( x) 2 2c v,h,, j R y = ui ui y ( y) 2 2c v,h (13) j C = j R+ z j R z + j R+ x j R x + j R+ y j R y. (14) In electric analogy, by establishing associations between variables so that excess pore pressure (u) is equivalent to electrical voltage (V) while temporal (first erivative) an spatial changes (secon erivative) u are two electric currents (J), we have: j R+ z = u i+ z u i ( z) 2, (15) 2c v,z which is equivalent to Ohm s law, which relates current (I R ) in a resistance with voltage (V R ) at its ens (terminals): I R = V R R. (16) Thus, implementation Equation (15) is carrie out by resistance between central noe an upper ege cell (Figure 3). The value this resistance is By same reasoning, term R i+ z = ( z)2 2c v,z. (17) j R z = u i u i z ( z) 2 2c v,z (18) is implemente by a new resistor same value as Equation (17) between central noe an lower ege cell. Finally, term j C = u (19) t is equivalent to equation that relates electric current at a capacitor (I C ) to voltage or potential ifference at its ens (V C ): ( ) VC I C = C. (20) t Thus, implementation this term Equation (12) is carrie out by a capacitor C value unity connecte between central noe an an outer noe (which will be common reference noe for all cells). In short, values electrical evices that implements moel volume element, whose network is epicte in Figure 4, are C = 1, R i+ z = R i z = ( z)2 2c v,z, R i+ x = R i x = ( x)2 2c v,h, R i+ y = R i y = ( y)2 2c v,h. (21) Notice that, physically, an increase in consoliation coefficient ecreases value resistances an makes consoliation process faster, as expecte. Each cell omain is connecte to ajacent cells by means ieal electrical contacts. As regars bounaries, a secon-type Neumann (homogeneous) conition is aopte at

7 Water 2018, 10, impermeable eges, which is simply implemente by a resistance infinite value, while a first-type Dirichlet conition (constant pressure) is assume at raining bounaries (soil surface an rainage area PVD), which is implemente by a constant voltage source zero value. Finally, rain ischarge capacity (q w ) limitation cause by unit graient pressure at rain is moele by fixing rain permeability (hyraulic conuctivity) to require value. Note that this solution allows us to increase flow water through rain as water moves Water towars 2018, 10, x exit FOR borer PEER REVIEW rain x u i+ z u i+ y z u i- x R i- x R i- y R i+ z R i+ y u i u i+ R i+ x x u i- y R i- z C i u i- z y Figure 4. Network moel element cell (3D rectangular consoliation). Figure 4. Network moel element cell (3D rectangular consoliation). 3. The Simulation Program 3. The Simulation Program The objective this program (SICOMED_2018) is simulation 3D consoliation The objective this program (SICOMED_2018) is simulation 3D consoliation problem problem with partially or totally penetrating PVD, in heterogeneous soils forme by one, two, or with partially or totally penetrating PVD, in heterogeneous soils forme by one, two, or three layers an three layers an taking into account effects smear zone an ischarge capacity taking into account effects smear zone an ischarge capacity rain. It must be an rain. It must be an easy-to-use stware for user but able to provie powerful numerical easy-to-use stware for user but able to provie powerful numerical calculations. This requirement calculations. This requirement entails creation a practical an simple graphical interface, where entails creation a practical an simple graphical interface, where ata entry an choice ata entry an choice ifferent options is performe in an orerly, intuitive, an guie way. ifferent options is performe in an orerly, intuitive, an guie way. Also, program shoul Also, program shoul inclue a complete guie [32] that allows user to unerstan inclue a complete guie [32] that allows user to unerstan successive steps in ata entry, successive steps in ata entry, simulation an representation results. simulation an representation results. With regar to simulation possibilities, user can use SICOMED_2018 both to esign an With regar to simulation possibilities, user can use SICOMED_2018 both to esign an to optimize installation PVD in groun, since program allows us to know, in a simple to optimize installation PVD in groun, since program allows us to know, in a simple way, great variety results that will allow engineers to make ecisions that most suit ir way, great variety results that will allow engineers to make ecisions that most suit ir nees: evolution settlements on soil surface, excess pore pressure at any point groun, nees: evolution settlements on soil surface, excess pore pressure at any point groun, an average egree settlement, among ors. an average egree settlement, among ors. Figure 5 shows a simplifie scheme SICOMED_2018 flowchart The Input Data an Network Design 3.1. The Input Data an Network Design Data entry is carrie out through following three winows that appear successively as y Data entry is carrie out through following three winows that appear successively as y are complete (Figures 6 8). The start-up screen (Figure 6), allows entry problem geometry are complete (Figures 6 8). The start-up screen (Figure 6), allows entry problem geometry (values a, b, an c in Figure 2), efinition smear zone, an esire gri size in plant. In (values a, b, an c in Figure 2), efinition smear zone, an esire gri size in plant. In secon winow (Figure 7), geometry strata (thickness an vertical gri size), parameters secon winow (Figure 7), geometry strata (thickness an vertical gri size), parameters to efine consoliation coefficients (both unisturbe an smear zone), epth to efine consoliation coefficients (both unisturbe an smear zone), epth PVD, an its ischarge capacity are introuce. Finally, in thir screen (Figure 8), parameters PVD, an its ischarge capacity are introuce. Finally, in thir screen (Figure 8), parameters relate to simulation such as value uniform applie loa, initial an final calculation relate to simulation such as value uniform applie loa, initial an final calculation times, maximum time step between each iteration, an a relative tolerance parameter use by times, maximum time step between each iteration, an a relative tolerance parameter use by Ngspice to achieve convergence are introuce. As a unit measure time, year (y) has been Ngspice to achieve convergence are introuce. As a unit measure time, year (y) has been chosen, a more appropriate magnitue than secon(s) for this type problem, while for or magnitues units international system have been maintaine.

8 Water 2018, 10, chosen, a more appropriate magnitue than secon(s) for this type problem, while for or magnitues units international system have been maintaine. Water Figure 2018, 10,5x shows FOR PEER REVIEW 8 20 a simplifie scheme SICOMED_2018 flowchart. Water 2018, 10, x FOR PEER REVIEW 8 20 Figure 5. Operating scheme SICOMED_2018. Figure 5. Operating scheme SICOMED_2018. Figure 6. SICOMED_2018 start-up screen. Figure 6. SICOMED_2018 start-up screen. Figure 6. SICOMED_2018 start-up screen.

9 Water 2018, 10, Water 2018, 10, x FOR PEER REVIEW 9 20 Water 2018, 10, x FOR PEER REVIEW 9 20 Figure 7. Secon ata entry screen. Figure 7. Secon ata entry screen. Figure 7. Secon ata entry screen. Figure 8. Thir ata entry screen. Figure 8. Thir ata entry screen. Figure 8. Thir ata entry screen.

10 Water 2018, 10, x FOR PEER REVIEW Water 2018, 10, Simulation an Output Data To increase flexibility an effectiveness program, user can avoi entering ata by 3.2. Simulation working with an anor Outputfile Data previously save. For this, SICOMED_2018 inclues options saving an loaing ata. Once all ata have been entere or loae simulation can start. To o this, To increase flexibility an effectiveness program, user can avoi entering ata by SICOMED_2018 creates network moel file in a specific source coe that is run in Ngspice. At working en with anor simulation file previously user can access save. For results this, given SICOMED_2018 in a graphic way. inclues SICOMED_2018 options fers saving an loaing up to six ata. possibilities Once all results ata have representation been entere (Figure or loae 9), conveniently simulation arrange canso start. that To o this, SICOMED_2018 geotechnical creates engineer can network have all moel necessary file ininformation a specific source in a simple, coeprecise that isform: run in Ngspice. At en simulation user can access results given in a graphic way. SICOMED_2018 fers up 1. Excess pore pressure in a given column soil to six possibilities 2. Excess pore results pressure representation in a given point (Figure soil 9), conveniently arrange so that geotechnical engineer 3. can Average haveegree all necessary settlement information in a simple, precise form: 4. Local settlements in a given column soil 1. Excess pore pressure in a given column soil 5. Total settlement in a given point surface 2. Excess 6. Surface pore settlements pressure in animation. a given point soil 3. Average egree settlement Some se representations are shown in illustrative applications following section, 4. Local which settlements have been in selecte a given with column aim soil presenting all potentialities an utilities 5. Total SICOMED_2018, settlement such in a given as: (i) point analysis classic surface consoliation problems; (ii) real scenarios PVD 6. Surface layout optimization; settlements animation. (iii) inclusion smearing an rain ischarge capacity effects; an (iv) verification results. Figure 9. Representation selection screen. Figure 9. Representation selection screen. 4. Applications Some se representations are shown in illustrative applications following section, 4.1. which First Scenario: have been Consoliation selecte a with One-Layer aim Soil with presenting PVD all potentialities an utilities SICOMED_2018, This application such as: refers (i) analysis to a fixe classic one-layere consoliation problems; scenario (ii) in real which scenarios PVD layout PVDis layout optimization; alreay impose. (iii) inclusion The geometry smearing problem an an rain ischarge soil properties capacity are shown effects; in Table an 1 (iv) (note verification that results. PVD thickness, t, is not given, as this parameter oes not have an influence when smear an rain ischarge capacity effects are not consiere, as in this case). The final time simulation 4. Applications has been set at four years, while chosen time step is 0.01 years. Finally, a gri has been chosen, enough to guarantee precision in results, with relative errors below 1% First Scenario: Consoliation a One-Layer Soil with PVD Table 1. Geometry problem an soil properties. This application refers to a fixe one-layere consoliation scenario in which PVD layout is Thickness (H) 3 m alreay impose. The geometry problem an soil properties are shown in Table 1 (note that Separation between PVDs (2a) 2 m PVD thickness, t, is not Separation given, between as this parameter rows PVDs oes (2b) not have 1.6 an m influence when smear an rain ischarge capacity effects are not PVD consiere, with (2c) as in this case). The 0.1 m final time simulation has been set at four years, whiledepth chosen penetration time step PVD is 0.01 () years. Finally, 3 m a gri has been chosen, enough to guarantee precision Applie in loa results, (q) with relative30 errors kn/mbelow 2 1%. Initial voi ratio (eo) 1.23 Table 1. Geometry problem an soil properties. Thickness (H) 3 m Separation between PVDs (2a) 2 m Separation between rows PVDs (2b) 1.6 m PVD with (2c) 0.1 m Depth penetration PVD () 3 m Applie loa (q) 30 kn/m 2 Initial voi ratio (e o ) 1.23

11 Water 2018, 10, Table 1. Cont. Water 2018, 10, x FOR PEER Compressibility REVIEW coefficient (a v ) m 2 /kn Vertical hyraulic conuctivity (k v ) m/year Water 2018, 10, x FOR Horizontal PEER REVIEW hyraulic Compressibility conuctivity coefficient (k h ) (av) m 2 /kn m/year Vertical consoliation coefficient (c v,z ) 0.94 m 2 /year Vertical hyraulic conuctivity (kv) m/year Horizontal consoliation Compressibility coefficient coefficient (c v,h ) (av) mm 2 /year Horizontal hyraulic conuctivity (kh) m/year /kn Vertical hyraulic conuctivity (kv) m/year Vertical consoliation coefficient (cv,z) 0.94 m 2 /year Horizontal hyraulic conuctivity (kh) m/year Horizontal consoliation coefficient (cv,h) 2.56 m 2 /year Vertical consoliation coefficient (cv,z) 0.94 m 2 /year Horizontal consoliation coefficient (cv,h) 2.56 m At this point it is important to note that SICOMED_2018 has /year option to irectly enter values consoliation coefficients (cv,z an cv,h), allowing only analysis problem in At this point it is important to note that SICOMED_2018 has option to irectly enter terms uration process (options 1, 2, an 3 in Figure 9), or provie values soil values consoliation coefficients (cv,z an cv,h), allowing only analysis problem in parameters (av, eo, kv an kh) from which se coefficients are euce Equation (22) allowing terms uration process (options this secon option to perform an analysis c v,z = k 1, 2, an 3 in Figure v(1 + e) problem in, c v,h = k 9), or provie values soil terms h(1 + e) settlement (options 4, 5, an 6 in parameters (av, eo, kv an kh) from which se coefficients are euce Equation (22) allowing Figure 9): this secon option to perform an analysis wa v problem in terms settlement (options 4, 5, an 6 in Figure 9): c, = ( ), c, = ( ). (22) Once simulation is run, with c a, computation = ( ), c, time = ( ) 10 min. on an Intel Core i CPU (22) 4.00 GHz computer, user can access results (Figure 9). The excess pore pressure can be represente Once simulation is run, with a computation time 10 min on an Intel Core i CPU 4.00 for all cells one or two soil columns, given to program by ir (X, Y) coorinates (Figure GHz computer, user can access results (Figure 9). The excess pore pressure can be represente 10). As can be seen, issipation excess pore pressure (Figure 11) occurs faster in those points for all cells one or two soil columns, given to program by ir (X, Y) coorinates (Figure soil closer to raining eges, soil surface, an PVD. The program also allows (option 2) 10). As can be seen, issipation excess pore pressure (Figure 11) occurs faster in those points representation evolution this unknown in one, two, or three locations, inicate to soil closer to raining eges, soil surface, an PVD. The program also allows (option 2) SICOMED_2018 by its (X, Y, Z) coorinates. representation evolution this unknown in one, two, or three locations, inicate to SICOMED_2018 by its (X, Y, Z) coorinates. At this point it is important to note that SICOMED_2018 has option to irectly enter values consoliation coefficients (c v,z an c v,h ), allowing only analysis problem in terms uration process (options 1, 2, an 3 in Figure 9), or provie values soil parameters (a v, e o, k v an k h ) from which se coefficients are euce Equation (22) allowing this secon option to perform an analysis problem in terms settlement (options 4, 5, an 6 in Figure 9): wa v. (22) Once simulation is run, with a computation time 10 min on an Intel Core i CPU 4.00 GHz computer, user can access results (Figure 9). The excess pore pressure can be represente for all cells one or two soil columns, given to program by ir (X, Y) coorinates (Figure 10). As can be seen, issipation excess pore pressure (Figure 11) occurs faster in those points soil closer to raining eges, soil surface, an PVD. The program also allows (option 2) representation evolution this unknown in one, two, or three locations, inicate to SICOMED_2018 by its (X, Y, Z) coorinates. Figure 10. Selection soil columns to be represente an time interval. Figure 10. Selection soil columns to be represente an time interval. Figure 10. Selection soil columns to be represente an 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 (N z = 20).

12 Water 2018, 10, Water 2018, 10, x FOR PEER REVIEW The evolution average egree settlement ( egree consoliation average over all Water The 2018, evolution 10, x FOR PEER REVIEW average egree settlement ( egree consoliation average over 12 all 20 soil columns) is shown in Figure 12. Thus, it can be observe that 90% consoliation is reache soil Water columns) 2018, 10, x is FOR shown PEER REVIEW in Figure 12. Thus, it can be observe that 90% consoliation is reache after 1.30 after years. The 1.30 evolution years. average egree settlement ( egree consoliation average over all soil columns) The evolution is shown in Figure average 12. egree Thus, it settlement can be observe ( egree that 90% consoliation consoliation average is reache over all after soil columns) 1.30 years. is shown in Figure 12. Thus, it can be observe that 90% consoliation is reache after 1.30 years. Figure 12. Average egree settlement evolution. Figure 12. Average egree settlement evolution. Options 4 6 Figure Figure 9 are also 12. Average available egree (Figures settlement 13 15) since evolution. parameters av, eo, kv, an kh are Options given. Note 4 6that Figure soil 9columns are Figure alsocloser 12. available Average to egree (Figures PVD will settlement 13 15) reach since evolution. final parameters settlement before a v, e(figure o, k v, an 14), k h are given. although Options Note that value 4 6 soil Figure columns this unknown 9 are also closer is available to same (Figures PVD for all will 13 15) reach columns: since parameters final settlement m. av, eo, kv, an kh are before (Figure 14), given. although Options Note that value4 6 soil columns closer to PVD will reach final settlement before (Figure 14), this Figure unknown 9 are also is available same(figures for all 13 15) columns: since parameters m. av, eo, kv, an kh are although given. Note that value soil this columns unknown closer is to same PVD for all will reach columns: final settlement m. before (Figure 14), although value this unknown is same for all columns: m. Figure 13. Evolution local settlement in each cells a soil column (Nz = 20). Figure 13. Evolution local settlement in each cells a soil column (Nz = 20). Figure 13. Evolution local settlement in each cells a soil column (N z = 20). Figure 13. Evolution local settlement in each cells a soil column (Nz = 20). Figure 14. Surface settlement evolution two soil columns. Figure 14. Surface settlement evolution two soil columns. Figure 14. Surface settlement evolution two soil columns. Figure 14. Surface settlement evolution two soil columns.

13 Water 2018, 10, Water 2018, 10, x FOR PEER REVIEW Figure 15. Screenshot settlement evolution animation on groun surface. Figure 15. Screenshot settlement evolution animation on groun surface. The last representation option (Figure 15) is a vieo animation settlement evolution whole The soil last surface. representation As representation option (Figure options, 15) is auser vieo can animation choose time settlement interval to evolution isplay an number whole soil surface. frames per As secon representation animation options, (for faster canor choose slower isplay). interval to isplay an number frames per secon animation (for faster slower isplay) Secon Scenario: Optimization PVD Layout 4.2. Secon Scenario: Optimization PVD Layout In this application we analyze an accelerate process consoliation a three-layere soil In this application we analyze an accelerate process consoliation a three-layere soil (whose properties are given in Table 2) on which an embankment is going to be built, transmitting to (whose properties are given in Table 2) on which an embankment is going to be built, transmitting to groun a uniform loa 60 kn/m 2. Using SICOMED_2018, ifferent layouts (epth groun a uniform loa 60 kn/m penetration an separation PVDs) will 2. Using SICOMED_2018, ifferent layouts (epth penetration be simulate (with ifferent final an step times, epening an separation PVDs) will be simulate (with ifferent final an step times, epening on case), on case), looking for optimal solution from point view consoliation time. A looking for optimal solution from point view consoliation time. A gri has 30 gri has been chosen (0.2 m high in each cell). been chosen (0.2 m high in each cell). Table 2. Thickness an properties three strata. Table 2. Thickness an properties three strata. S1 (Upper S2 (Intermeiate S3 (Lower Soil Property S Soil Property Stratum) 1 (Upper S 2 (Intermeiate Stratum) S 3 Stratum) (Lower Stratum) Stratum) Stratum) Thickness (m) H Thickness (m) H Initial voi voi ratio ratio e o eo Compressibility coefficient coefficient (m 2 /kn) (m 2 /kn) a v av Vertical hyraulic conuctivity (m/year) k v Vertical Horizontal hyraulic conuctivity (m/year) (m/year) k kv h Horizontal Vertical consoliation hyraulic coefficient conuctivity (m 2 /year)(m/year) c v,z kh Horizontal consoliation coefficient (m Vertical consoliation coefficient 2 /year) (m 2 c v,h /year) cv,z Horizontal consoliation coefficient (m 2 /year) cv,h To aress this problem it is necessary to set a efinition for consoliation time (or characteristic time, To t o [33]), aress as well this as problem to fix ait target is necessary value to this set time a efinition to be achieve for with consoliation use PVDs. time (or So, characteristic we efine time, characteristic to [33]), time as well as as to time fix that a target takes value soil this to reach time an to average be achieve egree with settlement use PVDs. (U) 90%. So, we In aition, efine PVDs characteristic must reuce time this as time time to a value that takes below two soil years. to reach In this an way, average we procee egree to settlement perform (U ) simulation 90%. In aition, problem PVDs inmust most reuce unfavorable this time to scenario, a value that below is, when two years. rein arethis no way, PVDswe installe procee into perform groun. For simulation this case (Figure problem 16), 90% in most average unfavorable egreescenario, settlement that is, when achieve re after are more no PVDs thaninstalle 30 years, in a value groun. too high For for this work case in(figure civil engineering. 16), 90% average egree settlement is achieve after more than 30 years, a value too high for work in civil engineering.

14 Water 2018, 10, Water 2018, 10, x FOR PEER REVIEW Water 2018, 10, x FOR PEER REVIEW Figure 16. Average egree settlement for case with no PVDs. Figure Figure Average Average egree egree settlement settlement for for case case with with no no PVDs. PVDs. Figure 17 shows local settlement evolution an entire soil column, which is ifferent for each Figure stratum The shows simulation local local also settlement provies evolution a final surface an an entire settlement entire soil soil value column, which 0.53 which m. is is ifferent for for each stratum. each stratum. The simulation The simulation also provies also provies a final a surface final surface settlement settlement value value m. m. Figure 17. Local settlements evolution for case with no PVDs (Nz = 30). Figure 17. Local settlements evolution for case with no PVDs (Nz = 30). Figure 17. Local settlements evolution for case with no PVDs (N z = 30). In orer to reuce consoliation time, PVDs separate every 2 m (both at x an y spatial irections) In orer are to installe. reuce The consoliation with time, PVD PVDs is 0.1 m, separate a typical every commercial 2 m (both value at x for an this y type spatial irections) In orer rain geosyntic are to reuce installe. origin. The consoliation with As epth time, penetration PVD is PVDs 0.1 m, separate we a begin typical every with commercial 2 m (both = 1 m (partially value at for x this an penetrating type y spatial irections) rain to geosyntic are installe. first layer). origin. The with As As reuction epth penetration PVD is 0.1 m, to is not enough, we abegin typical we with commercial increase = 1 m penetration (partially value for penetrating this type up to 4 m rain rain (partially to geosyntic penetrating first layer). origin. rain As As to reuction epth intermeiate penetration to is not stratum) enough, we begin an we 6 increase with = m (totally 1 m penetrating penetration (partially rain). up penetrating to 4 The m rain (partially results to for penetrating first consoliation layer). rain As time to reuction (to) intermeiate are shown t o is in not stratum) Table enough, 3. an we 6 m increase (totally penetrating penetration rain). up to The 4 m (partially results for penetrating consoliation rain to time intermeiate (to) are shown stratum) in Table an 3. 6 m (totally penetrating rain). The results for consoliation Table 3. time Values (t o ) are to as shown a function in Table PVD 3. penetration (a = 1 m; b = 1 m; c = 0.05 m). Table 3. Values to as a function PVD penetration (a = 1 m; b = 1 m; c = 0.05 m). Table 3. Values t o as adepth function PVD (m) penetration to (Years) (a = 1 m; b = 1 m; c = 0.05 m). Depth PVD (m) to (Years) Without rain 32 Depth PVD Without (m) 1 rain t o (Years) Without rain

15 Water 2018, 10, x FOR PEER REVIEW Water 2018, 10, Since none stuie options fits two-year objective, it is necessary to bring PVDs closer. Thus, for cases = 6 m an = 4 m, a set simulations in which both istance between PVDs an Since separation none between stuierows options fits PVDs are two-year reuce objective, will be itsimulate is necessary with to bring SICOMED_2018. PVDs closer. Thus, Characteristic for cases times for = each 6 m an layout = are 4 m, summarize a set simulations in Table 4. in which both istance between PVDs an separation between rows PVDs are reuce will be simulate with SICOMED_2018. Characteristic Table times 4. Values for each to layout as a function are summarize PVD penetration in Tablean 4. surface layout (c = 0.05 m). Depth PVD (m) Table 4. Values t o as a function PVD penetration an surface layout (c = 0.05 m). Without rain 32 Depth 1 PVD (a = (m) b = 1 m) 27 t o (Years) Without 4 (a rain = b = 1 m) (a = b = 1 m) 27 4 (a = b 6 = (a 1 = m) b = 1 m) (a = 6 b (a = 1= m) b = 0.9 m) (a = b 6 = (a 0.9 = b m) = 0.8 m) (a = b = 0.8 m) (a = b 4 = (a 0.8 = b m) = 0.8 m) (a = b 4 = (a 0.7 = b m) = 0.7 m) (a = b = 0.65 m) 4 (a = b = 0.65 m) to (Years) In view se se results, results, two two alternatives alternatives emerge emerge as a solution as a solution to problem: to (i) problem: totally penetrating (i) totally rains penetrating everyrains 1.6 m every (2 linear 1.6 metersm (2 linear PVD meters per square PVD meter per square soil meter surface), soil which surface), provies which a characteristic provies a characteristic time 1.9time years; 1.9 an years; (ii) partially an (ii) partially penetrating penetrating rains (4rains m) every (4 m) 1.4 every m ( m linear (1.75 meters linear meters PVD per PVD square per square meter meter soil surface), soil surface), which which provies provies a characteristica characteristic time time 2.0 years. 2.0 Figures years. Figures 18 an18 19an illustrate 19 illustrate last alternative last alternative showing, showing, respectively, respectively, local settlements local settlements an total an surface total surface settlement settlement evolution evolution for two for soil two columns. soil columns. Figure 18. Local settlement evolution in each cells two soil columns. (a = b = 0.7 m; c = 0.05 Figure 18. Local settlement evolution in each cells two soil columns. (a = b = 0.7 m; c = 0.05 m; m; = 4 m). = 4 m).

16 Water 2018, 10, Water 2018, 10, x FOR PEER REVIEW Figure 19. Surface settlement evolution in two soil columns. (a 0.7 m; 0.05 m; m). Figure 19. Surface settlement evolution in two soil columns. (a = b = 0.7 m; c = 0.05 m; = 4 m) Thir Scenario: Influence Smear Zone an Discharge Capacity Limitation PVD 4.3. Thir Scenario: Influence Smear Zone an Discharge Capacity Limitation PVD The thir application aresses use limite ischarge capacity rains, as well as The thir application aresses use limite ischarge capacity rains, as well as influence smear zone. The geometric characteristics an groun an rain properties are influence smear zone. The geometric characteristics an groun an rain properties are shown in Table 5. The geometric values problem (a, b, c, an t) have been selecte so that shown in Table 5. The geometric values problem (a, b, c, an t solutions SICOMED_2018 can be compare with results obtaine ) have been selecte so that by Hansbo et al. [16] an solutions SICOMED_2018 can be compare with results obtaine by Hansbo et al. [16] an Barron [34], in same problem but using a two-imensional (2D) raial geometry, with an Barron [34], in same problem but using a two-imensional (2D) raial geometry, with an equivalent equivalent iameter PVD an a iameter influence m an 0.95 m, respectively. The iameter PVD an a iameter influence m an 0.95 m, respectively. The physical physical soil an rain properties are ientical. Regaring av an e, any pair values that provies soil an rain properties are ientical. Regaring a same consoliation coefficients is vali, having v an e, any pair values that provies same aopte values av = m 2 /N an e consoliation coefficients is vali, having aopte values a = 1. Finally, Q = N/m 2. v = m 2 /N an e = 1. Finally, Q = N/m 2. Table 5. Geometry an physical properties problem. Table 5. Geometry an physical properties problem. Parameter/Soil Property Value Units Parameter/Soil H (raining Property length) Value15 Units m 2a (separation H (raining length) between PVDs) m 2b 2a (separation between PVDs) rows PVDs) m 2b (separation between rows PVDs) m 2c (PVD with) c (PVD with) m t t (PVD (PVD thickness) thickness) m PVD PVD equivalent iameter iameter m Smear length length (normal (normal to to PVD PVD plain) plain) m k kh h (horizontal hyraulic conuctivity unisturbe soil) 0.03 m/year (horizontal hyraulic conuctivity unisturbe soil) 0.03 m/year k h,s (horizontal hyraulic conuctivity smear zone) 0.01 m/year kh,s k v (vertical (horizontal hyraulic hyraulic conuctivity conuctivity unisturbe smear soil) zone) m/year m/year kv k v,s (vertical hyraulic conuctivity smear unisturbe zone) soil) m/year kv,s (vertical c v,h (horizontal hyraulic consoliation conuctivity coefficient) smear zone) m/year m 2 ccv,h v,z (vertical (horizontal consoliation consoliation coefficient) coefficient) m 2 /year Gri size (N cv,z (vertical consoliation x, N y, N z ) coefficient) 0.15 m 2 /year Drain hyraulic conuctivity q w = 20 m 3 /year 43,290 m/year Drain hyraulic conuctivity Gri size (Nx, forny, q w = Nz) 10 m 3 /year 21, m/year Drain hyraulic conuctivity for qfor qw = 20 m 3 w = 5 m 3 /year /year 10, ,290 m/year Drain hyraulic conuctivity for qw = 10 m 3 /year 21,645 m/year Drain hyraulic conuctivity for qw = 5 m 3 /year 10,822.5 m/year Four consoliation scenarios have been simulate, comparing average egree consoliation achieve in three typical times: 0.5, one, an two years (Table 6). In first case, in which Four consoliation scenarios have been simulate, comparing average egree smear effects are not consiere an rain is suppose to have an infinite ischarge capacity, consoliation achieve in three typical times: 0.5, one, an two years (Table 6). In first case, in results reprouce those obtaine analytically by Barron [34] uner free strain hyposis, which smear effects are not consiere an rain is suppose to have an infinite ischarge capacity, results reprouce those obtaine analytically by Barron [34] uner free strain hyposis, with relative eviations below 3% for three times. The or three cases assume

17 Water 2018, 10, with relative eviations below 3% for three times. The or three cases assume same smear length but consier ifferent PVD ischarge capacities. For m, comparison is establishe with results obtaine by Hansbo et al. [16], base on ir empirical expressions for average egree consoliation. Deviations, particularly for relatively high times, can be consiere acceptable, taking into account that se are not numerical or analytical results. These eviations grow slightly to 11.6% for small times an low ischarge capacities, perhaps because moels egrees freeom (3D for SICOMED_2018 an 2D raial for Hansbo) are not same. On or han, results are coherent with respect to correlation between average egree consoliation an influence both smear effects an ischarge capacity limitation rain. Table 6. Comparison average egree consoliation between SICOMED_2018 an Hansbo et al. [16] an Barron s [34] solutions. Case Barron t = 0.5 Years t = 1 Years t = 2 Years Hansbo SICOMED (% Error) Barron Hansbo SICOMED (% Error) Barron Hansbo SICOMED (% Error) Case 1 No smear, limite ischarge capacity 66.3% % (2.71%) 89.2% % (0.45%) 98.6% % (0.30%) Case 2 Smear zone, q w = 20 m 3 /year % 44.5% ( 3.26%) % 68.8% ( 1.43%) % 90.2% ( 0.33%) Case 3 Smear zone, q w = 10 m 3 /year % 39.1% ( 6.46%) % 62.5% ( 1.26%) % 85.5% ( 0.58%) Case 4 Smear zone, q w = 5 m 3 /year % 30.5% ( 11.6%) % 50.9% ( 5.57%) % 74.8% ( 3.98%) 5. Final Comments an Conclusions SICOMED_2018 has been create as free stware for solving 3D consoliation problems with partial or totally penetrating vertical rains, in heterogeneous soils an consiering influence smear effects an ischarge capacity limitation rain. All unknowns interest for civil engineer, such as excess pore pressure evolution, average egree consoliation, local, an total settlements an even a vieo animation soil surface eformation, can be obtaine with SICOMED_2018 in a relatively simple way. Therefore, it can be consiere a program with pressional, eucational, an research applications. Following network simulation metho rules, SICOMED_2018 esigns network moels (electrical circuits) whose governing equations (with spatial iscretization an retaining time as a continuous variable) are formally equivalent to iscretize equations consoliation problem. These network moels are solve by an electrical circuit resolution coe such as Ngspice, irectly proviing solutions to consoliation problem. The program has been evelope in a pleasant communication environment, both for ata entry an graphical presentation results, uner a classic winows environment programme with MATLAB, a coe whose license is not free but allows istribution programs create with it. On or han, for numerical calculation powerful computational algorithms integrate in Ngspice, an electrical circuits resolution free coe that provies practically exact solutions to moels esigne by SICOMED_2018, have been use. The combination se coes has allowe us to obtain completely free stware. To illustrate operation program, three applications, corresponing to real cases, are presente. The first one eals with simulation an resolution a consoliation scenario with PVD istribution alreay impose (a classic problem), showing evolution main variables interest (excess pore pressure, average egree consoliation, an local an total settlements). The secon one concerns an optimization problem in a soil forme by three strata

18 Water 2018, 10, with ifferent properties, in which it is sought to fin optimum PVD istribution in orer to reuce characteristic time consoliation to a certain target value. Finally, thir application serves as a verification program against analytical an empirical solutions provie by or authors, stuying influence that smear zone an ischarge capacity limitation rain have on consoliation problem. Deviations are reuce in all cases, showing versatility an reliability program. Finally, from work presente here, a series challenges that coul be aresse in future research are suggeste. On one han, in fiel soil consoliation, this research coul be complete (or improve) by implementing aitional options in program, such as analysis base on real tabulate ata, or even possibility aressing inverse problems. On or han, wie experience network simulation research group UPCT opens up possibility using a similar methoology in creation free stware in isciplines as iverse as seepage, atmospheric ispersion pollutants, an hyrologic engineering, among ors. Acknowlegments: The authors acknowlege pressors Francisco Alhama an Juan Francisco Sánchez, from Applie Physics Department Technical University Cartagena, for ir help both in evelopment network moels an in elaboration graphic interface. The authors also thank reviewers for useful comments to improve manuscript. Author Contributions: Gonzalo García-Ros an Iván Alhama conceive an esigne stware an wrote this paper, while Manuel Cánovas performe numerical simulations an analyze results obtaine. Conflicts Interest: The authors eclare no conflict interest. Nomenclature a semi-separation between prefabricate vertical rains (m) a v coefficient compressibility (m 2 /N) b semi-separation between rows prefabricate vertical rains (m) c Semi-with prefabricate vertical rain (m) C capacity value a capacitor (F) C i capacitor connecte between central noe cell i an common groun noe c v coefficient consoliation (m 2 /s) c v,h horizontal coefficient consoliation (m 2 /s) c v,x horizontal coefficient consoliation in x spatial irection (m 2 /s) c v,y horizontal coefficient consoliation in y spatial irection (m 2 /s) c v,z vertical coefficient consoliation (m 2 /s) epth penetration prefabricate vertical rain (m) e voi ratio (imensionless) e o initial voi ratio (imensionless) H soil thickness or raining length water in vertical irection (m) I C current that flows through a capacitor (A) I R current that flows through a resistance (A) J C current flow associate with a capacitor (A) J R current flow associate with a resistance (A) k hyraulic conuctivity PVD (m/s) k h horizontal hyraulic conuctivity unisturbe soil (m/s) k h,s horizontal hyraulic conuctivity smear zone (m/s) k v vertical hyraulic conuctivity unisturbe soil (m/s) k v,s vertical hyraulic conuctivity smear zone (m/s) n normal irection to bounary surface N x number cells in x irection N y number cells in y irection N z number cells in z irection q uniform loa applie to groun surface (N/m 2 ) q w ischarge capacity PVD for unit graient pressure (m 3 /s) R value electrical resistance (Ω) R j resistance connecte between central noe cell i an noe j t time inepenent variable (s) t PVD thickness (m) t o characteristic time consoliation process (s) u excess pore water pressure (N/m 2 ) U local egree settlement (imensionless)

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