MODELACIÓN NUMÉRICA EN INGENIERÍA CIVIL APLICADA A: CARGAS DINÁMICAS DE PAVIMENTOS VISCO-‐ELÁSTICOS, Y MODELACIÓN DEL PROCESO
DE BIO-‐GROUTH EN SUELOS
PRESENTADO POR:
DAVID ALEJANDRO CASTRO CRUZ
ASESORES DEL PROYECTO
BERNARDO CAICEDO HORMAZA
FERNANDO LÓPEZ CABALLERO
BOGOTÁ, COLOMBIA
2014
AGRADECIMIENTOS
Puntualmente quiero dar las gracias a mi familia que me ha apoyado siempre en las decisiones que tomo. Siempre han sido mi ejemplo para esforzarme cada día. Quiero agradecer a Dios por la oportunidad de servir que ahora tengo al iniciar mi vida profesional.
A la universidad de los Andes que es una institución con la que estoy muy agradecido y que ha impactado mi vida formándome como persona y como profesional. La universidad es un espacio de crecimiento donde encontré todos los recursos para ser una mejor persona y adquirir las herramientas para servir a los demás. Por esto agradezco a todos los que componen esta institución. A mis profesores, a mis compañeros, a las directivas, y demás personas que hacen de la universidad de los Andes un lugar tan especial.
Quiero agradecer a los aportantes y trabajadores del programa quiero estudiar, programa que me permitió acceder a un lugar de educación tan alto como la universidad durante mi pregrado. También estoy muy agradecido con la universidad por los diversos proyectos en los que trabajé. Con estos logré aplicar mis conocimientos y aprender cosas nuevas. Además, me permitieron cursar mi maestría dándome el espacio, el tiempo, y el apoyo financiero necesario. Quiero agradecer especialmente a los profesores con quienes he trabajado y considero me desarrollado profesionalmente. Al profesor Bernardo Caicedo, al profesor Mauricio Sánchez y las demás profesores por sus concejos y lo que he aprendido de ellos. Especialmente quiero agradecer al profesor Fernando López Caballero del Ecolé Centrale de París, quien me ayudó no solo con mi investigación sino con lo que necesité en Francia.
Muchas gracias.
Introducción
La modelación en ingeniería civil es una importante herramienta para el análisis de situaciones complejas. La modelación permite estimar los resultados previamente lo cual hace que esta sea usada para la toma de decisiones y manejo de eventos en la ingeniería. Los trabajos presentados abordan dos aplicaciones de la modelación, pavimentos y en la generación de Biogrouth en el suelo.
El proceso de análisis por medio de la modelación numérica permite controlar los inputs de un modelo matemático que describe a un fenómeno físico. Esto permite realizar múltiples ensayos a diferentes condiciones completamente controladas. Lo que facilita la comprensión del problema y el análisis de múltiples alternativas. El análisis numérico permite realizar múltiples ensayos a un bajo costo tanto monetario como humano en comparación a la mayoría de ensayos físicos. Esto se debe a que las múltiples pruebas físicas demandan una mayor cantidad de espacio, materiales, tiempo de trabajo en la mayoría de los casos.
La evaluación de múltiples alternativas permite realizar análisis más complejos. Por ejemplo, los análisis de sensibilidad donde se busca determinar la importancia de cada parámetro en la respuesta del modelo, son muy importantes para la toma de decisiones. Por otro lado, se pueden realizar análisis de propagación de incertidumbre, donde se realiza un análisis sobre el desconocimiento del problema y sus condiciones. Este análisis resulta muy importante en la ingeniería debido al desconocimiento y incertidumbre que se tiene de muchos aspectos en la ingeniería.
En este trabajo se presenta primero un modelo para un material continuo con características visco-‐elásticas. Este es aplicado a pavimentos y al análisis del paso de ejes tandem y tridem en pavimentos. La segunda parte, muestra un análisis sobre el proceso de Biogrouth o MICP, que es una técnica de remediación de suelos por medio de bacterias. Adicionalmente, se hace un análisis probabilista que involucra complejos análisis de sensibilidad los cuales son FAST, y RBD-‐FAST. Aborda el problema con un método de campos aleatorios y realiza un análisis de resultados sobre todo esto.
Dynamic analysis of axes single, tandem, and tridem on viscoelastic
pavements, with a model in the time domain.
David Alejandro Castro Cruz, Bernardo Caicedo Hormaza, and Julián Tristancho
Abstract
The pavements support big loads transfers by axes tandem and tridem. They use special configurations for them axes. When the consecutive loads are applied to viscoelastic material, the material change the answer in dynamic situation. Hence in this work the axes tandem and tridem are evaluated in special model. It uses differences finites and solves the motion equation to viscoelastic behavior in time domain.
Introduction
In this paper is present a viscoelastic model in time domain for pavements that is able of applying moving loads for model different configuration of trucks specially. The model is constructed with the method of finite differences; they are applied for model propagation of mechanical waves in viscoelastic materials.
The pavements needs support big loads transfer by the normal traffic. The trucks give more critical loads of all traffic. Furthermore, in Colombia an important part of trucks exceed the allowable loads
(Macea, Luis and Alvarez 2013). The trucks have special configuration in their tires how tandem or tridem, and the majority of configurations have in common the application of consecutive loads. The answer of materials viscoelasticity for consecutive loads is dependent of many factor how the velocity, the distance, and the magnitude into consecutive loads. In this paper is present a numerical model for study this cases very common in the pass of trucks in the pavements.
In many precedent models of pavements, the loads are applied on same points. The magnitudes of these loads are variable according with a fit done with Fourier's series. This application method doesn't study some produced effects by consecutive loads. When tire passes by a point, and
previous tire change the behavior of pavement due to viscoelasticity. Furthermore, in the approximation of the load the pavement supports stress of compression and tension in some case. These effects of approximation modify the material’s behavior (Dai and You 2007). By this reason was build a model of pavement able of study the move of loads is important.
The numerical models involve a numerical error. In the differences finites, this error is proportional to size of differential in length and in time. Is possible reducing the numerical error for the equations of propagation of waves if in the model is optimized. In previous studies is present how do it for elastic materials (Ohminato and Chouet 1997). Use these studies; the equations that reduce the error and optimized the calculus time for viscoelastic materials are presented in this work.
To evaluate the dynamic loads was build a special model. This model resolves the equation of movement by the difference finite method in the time domain. The material of the model can be viscoelastic and characterized by Prony's Series
(Dai and You 2007). The equations use and their solutions are show in this paper. Also, this paper shows a verification of model with analytic solutions and with the program ALIZE. The results of models for single axis, tandem axis, and tridem axis are analyzed.
Numerical Model
When some force excites the material, the material changes his configuration. In this model the mechanical changes are studies.
Figure 1. Shape of the answer in some element with an external load.
The materials in the model are assumed as materials without spaces or discontinuities. By this reason, the propagation waves are by the material exclusively. Mechanically, in continuous materials all relative displacement into differential elements involves some deformation in the material. By this reason, in the continuum there's relationship into displacements and deformation for differential volumes. The model solves this relationship with finite difference method.
Motion equations
The mechanical effect by events can be perceived to distance of the source. This success is due to energy propagation. All mechanical transmission of energy is domain by the second law of Newton. The moment conservation in a continuum derives for differential volumes in this.
𝜌𝜕!𝑢! 𝜕𝑡! =
𝜕𝜎!
𝜕ℎ! +
𝜕𝜏!"
𝜕ℎ! +
𝜕𝜏!"
𝜕ℎ! +𝑓!
In this equation "u" is the displacement, "σ" and "τ" are stresses axial and shear respectively, and "f" is a value of energy loss. The subscript "i", "j", and "k" denote the direction of properties. The model solves the motion equation with finite difference method. The equation in the algorithm results in these equations for the three directions.
𝜌
Δ𝑡!· 𝑢!!,!!!,!−2·𝑢!!,!,!+𝑢!,!,!!!!
≈ 𝜎!! !,!,!− 𝜎!! !!!,!,! Δ𝑥
+ 𝜏!" !
!,!!!,!− 𝜏!" !
!,!,! Δ𝑦
+ 𝜏!"! !,!,!!! − 𝜏!"! !,!,!
Δ𝑧 + 𝑓! !,!" 𝜌
Δ𝑡!· 𝑣!!,!,!!! −2·𝑣!,!,!! +𝑣!!!,!!,!
≈ 𝜎!
!
!,!,!− 𝜎!! !,!!!,!
Δ𝑦
+ 𝜏!"
!
!!!,!,! − 𝜏!"! !,!,!
Δ𝑥
+ 𝜏!"
!
!,!,!!!− 𝜏!"! !,!,!
Δ𝑧 + 𝑓! !,!" 𝜌
Δ𝑡!· 𝑤!,!!!!,!−2·𝑤!,!,!! +𝑤!,!,!!!! ≈ 𝜎!! !,!,!− 𝜎!! !,!,!!!
Δ𝑧
+ 𝜏!"! !!!,!,!− 𝜏!"! !,!,!
Δ𝑥
+ 𝜏!"
!
!,!!!,!− 𝜏!"! !,!,!
Δ𝑦 + 𝑓! !,!" In the above equations "u", "v", and "w" are displacements in "x", "y" and "z" respectively. The algorithm calculates the displacement in next time (t+1). This solution finds a new position in the grid points. The points have a special configuration. The point stresses need to be between the displacement points to get a better solution. This form grid reduces a computational error and it reduces the number calculations. The Figure 2 shows the configuration for a cubic.
!
Body!
External load
displacement Rotation
Figure 2. Position of calculated stresses and displacements. A. Equal points identified by convention B. distribution of points (Ohminato and Chouet 1997).
Constitutive Model
The mechanical properties in the materials determine the behavior of wave propagation. This effect is associated in the constitutive models. They relate the strains with stress. The constructed model took a viscoelastic model. These models relate the stress and strains with their derivatives in the time. By this, these models use special configuration of dashpots and elastic components. The used model of Maxwell-‐Wiechert is shows in the Figure 3. It can to recreate the behavior with good aproximation (Delépine, et al. 2009).
Figure 3. Viscoelastic model of Maxwell-‐Wiechert (Mun and Lee 2011).
The Maxwell-‐Wiecher model has a system of elements in parallel. The first element is a spring and its behavior is lineal.
𝜎
=
𝐶
·
𝜀
The equation shows the model of a spring. "C" is a matrix of constants and "ε" is vector of strains. The other elements have a Maxwell configuration. This configuration consists in a dashpot and a string in series. The behavior of Maxwell elements is descripted in the next equation (Delphin Monsia 2011).
∂ε ∂t = σi ηi + 1 Ei
·∂σi
∂t
Furthermore, each element can be characterized with Prony series (Mun and Lee 2011). When the configuration is in parallel, the strain of each element is equal to global strain.
𝜀=𝜀!
The reaction of system is equal to summation of reactions of each element.
The previous equations derived in a relationship into stress, strain, and time. These equations are solved with finite difference method. The approximate solution is showed here.
σ =εt·E∞+
εt−εt−1+σt−1E i Δt ηi + 1 Ei
∑
Program Aspects
The constructed algorithm consists in three parts. The first part advances in the time and it actualizes all variables. The second part calculates the movements in each point of the grid. Finally, the third calculates the reactions and stress by the strains in the material. The program was developed in Fortran 90.
Model tests
The consolidation of materials is a know phenomenon. By this reason, the solution model and the analytic solution are compared in the Figure 4.
Figure 4. Consolidation of material by gravity effect.
Boussinesq deduced analytic solutions for the propagation stresses (Poulos and Davis 1974). This solution is compared with model results in the Figure 5. The model achieve approximate the analytic solution. Only in the boundary of model the solution have differences. This is due to the boundary conditions.
!10000$ 10000$ 30000$ 50000$ 70000$ 90000$ 110000$
0$ 1$ 2$ 3$ 4$ 5$
Str
ess&(P
a)&
Depth&(m)&
Figure 5. Vertical stresses due to a superficial load.
Additionally, the model evaluated a static load on a pavement structure. The results are compared with the software Alize. This is showed in the Figure 6, here the points are edge layer. Alize can to calculate the values in the edge. The program calculates near of edge. This difference does the results approximately equals.
Figure 6. Comparison between Alize and model answer.
In general the verifications from model were good. Hence the model is taken to evaluate other systems with dynamic analysis. Recommendable does experimental comparisons to calibrate the model in special with the viscoelastic parameters.
Viscoelastic model
To model viscoelastic behavior is used a Maxwell-‐ Wiechert model. This model can fit the answer of material in different conditions. The Figure 7 in up part shows the behavior of a sample of test with constant strain (relaxing test). In the bottom is showed a test with constant stress (creep
Figure 7. Answer of relaxing test and creep compliance. Compression in negative part and traction in positive part.
The used viscoelastic material uses twelve Maxwell’s elements and one spring. They are identifying in the Table 1.
i Ei(Mpa) ρi
1 3341.7 3.00E-‐11 2 932.6 3.00E-‐10 3 4379.3 3.00E-‐09 4 3869.9 3.00E-‐08 5 5641.3 3.00E-‐07 6 5677.4 3.00E-‐06 7 5930.3 3.00E-‐05 8 4844.8 3.00E-‐04 9 3379.3 3.00E-‐03 10 2121.4 3.00E-‐02 11 743.2 3.00E-‐01 12 451.3 3.00E+00 13 64.7 3.00E+01 14 93.7 3.00E+02
Einf 211.3
Table 1. Viscoelastic parameters.
Usually, each dashpot, for Prony's Series, each dashpot element is related with the spring element. The equation is showed here.
𝜂 =𝐸 · 𝜌
!60000$ !50000$ !40000$ !30000$ !20000$ !10000$ 0$
0$ 1$ 2$ 3$ 4$ 5$
Str ess&(KP a)& Depht&(m)& Boussinesq$ Model$ !400$ !200$ 0$ 200$ 400$ 600$ 800$ 1000$
S$ 1!2$ 2!1$ 2!3$ 3!2$
Es fu er zo )H or iz on ta l)( kP a) ) Capa)de)Pavimento) Alice$ Modelo$ !120% !100% !80% !60% !40% !20% 0% !2E!08% !1.8E!08% !1.6E!08% !1.4E!08% !1.2E!08% !1E!08% !8E!09% !6E!09% !4E!09% !2E!09% 0%
0% 0.05% 0.1% 0.15% 0.2% 0.25% 0.3% 0.35% 0.4% 0.45% 0.5%
Esf ue rz o)(Pa)) de for m ac ión) (m /m )) Tiempo)(s)) !180% !160% !140% !120% !100% !80% !60% !40% !20% 0% !1.2E!08% !1E!08% !8E!09% !6E!09% !4E!09% !2E!09% 0%
0% 0.05% 0.1% 0.15% 0.2% 0.25% 0.3% 0.35% 0.4% 0.45% 0.5%
Esf ue rz o)(Pa)) de for m ac ión) (m /m )) Tiempo)(s))
Model of axis tandem and tridem
The axis tandem and tridem are used for trucks and heavy vehicles. They are the principal loads for pavements. With the explicated model is evaluated this load types. The evaluated model uses a simple configuration and the results are taken from center of model in some points. The points and the configuration layer are showed in the Figure 8. The points are in boundaries of each layer.
Figure 8. Evaluation points and layer configuration of structure model.
The model evaluates three cases: single, tandem and tridem axes. The next figures show the answers. The Figure 9 shows the comparison between deformations when the velocity of axis is 20Km/h. The Figure 10 shows 80Km/h case.
Figure 9. Strains with a velocity of 20Km/h in bottom part of pavement layer in C point. Blue, green, and red are horizontal direction, transversal direction, and vertical or gravity direction.
Figure 10. Strains with a velocity of 80Km/h in bottom part of pavement layer in C point. Blue, green, and red are horizontal direction, transversal direction, and vertical or gravity direction.
The previous figures show the deformation in three directions with traction in positive direction. Low velocities have an answer lower than high velocities. The obtained strains are lower to 20km/h in all directions. When the situation is evaluated with tandem axis to velocity of 40Km/h, the Figure 11 shows the answer.
Figure 11. Strains by tandem axis effect with a velocity of 40Km/h in bottom part of pavement layer in C point. Blue, green, and red are horizontal direction, transversal direction, and vertical or gravity direction.
The Figure 11 shows how the second axis with same load, it causes a mayor effect in traction (blue line).
Tridem case causes the same effect. The first axis has mayor effect than the other two axes in this direction; they have very similar effect. It is showed in the Figure 12.
Figure 12. Strains by tridem axis effect with a velocity of 40Km/h in bottom part of pavement layer in C point. Blue, green, and red are horizontal direction, transversal direction, and vertical or gravity direction.
The effect of first axis changes the conditions of pavement. The approximation effect creates compression in longitudinal direction before and after of arrival of the axis. This is showed for blue line in the Figure 9 and Figure 10. When pass more of one axis, it generates high compressions between axes. It changes the initial conditions to second axis. When other axis arrives to pavement it is in high compression; so the final traction is lesser than the first traction. Other explication is when the first axis pass, this may extend the curvature radius to consecutives axis. After to pass the first axis, the pavement does not recover its form by viscoelastic behavior. It does that the second axis passes on some previous curvature. This reduces the longitudinal strain to consecutives axis after of the first axis. It is showed in the Figure 13.
Figure 13. Shape of curvature for tandem axis effect.
The results show that the effect of first axis is mayor than other axes. This means that the multiplied configurations don't cause special damages by consecutive action.
Advantages of the model
The model develops problems in time domain. By this, the load hasn't to be approximated. The model does an analysis in time domain for any load; therefore, the load hasn't to be constant. The solution doesn’t depend of all mechanical history of material. This allows studying and applying complex conditions for any load, because the model only uses the last displacement or stresses to advance in the time.
The model studies the viscoelastic behavior with Maxwell-‐Wiechert model. Hence the study may make good approximations for laboratory dates and real conditions. Furthermore, the average can be controlled by the user with the selected grid and chased time variation.
Recommendations
The model can study complex materials with appropriate dates. Hence, future works should use dates from laboratory. This allows doing comparisons and calibrations to improve the model. The method of finite differences gives the answer in some points, so approximation functions are recommendable to find the answer in the limits of the layers.
Conclusions
The study determinates that the effects after of first axis are lower for traction in the bottom of pavement layer. By this, the configuration tandem and tridem don't cause special damages by consecutive load actions. Furthermore, the configurations with more of one axis create an effect of distribution that helps the application of big loads on pavements.
The model can evaluate different conditions in viscoelastic materials. These need to be character by Prony's series. After the model can evaluate the answer material in different conditions. The model can reproduce the hardening and the relaxing from a viscoelastic material.
The evaluation with dynamic loads in pavements allows study the approximation effects. They are important because cause an inverse answer before of the axes arrival. In viscoelastic behavior as pavements, this effect is important in the answer.
Bibliography
Macea, Luis, Guentes Luis, and Allex Alvarez. "Evaluación de los factores camión de los vehículos comerciales de carretera que circulan por la red vial principal colombiana."
Revista de la facultad de ingeniería de la universidad de
antioquía, Marzo 2013: 57-69.
Ohminato, Takao, and Bernart Chouet. "A Free-Surface Boundary Condition for Including 3D Topography in the Finite-Difference Method." Bulletin of the Seismological
Society of America 87, no. 2 (April 1997): 494-515.
Dai, Qingli, and Zhanping You. "Prediction of Creep Stiffness of Asphalt Mixture with Micromechanical Finite-Element and Discrete-Finite-Element Models." JOURNAL OF
ENGINEERING MECHANICS ASCE, February 2007:
163-173.
Poulos, H. G., and E. H. Davis. Elastics solutions for soil
Delépine, Nicolas, Luca Lenti, Guy Bonnet, and Jean François Semblat. "Nonlinear Viscoelastic Wave Propagation: An Extension of Nearly Constant Attenuation Models." JOURNAL OF ENGINEERING MECHANICS
135 (November 2009): 1305-1314.
Mun, Sungho, and Sangyum Lee. "Identification of Viscoelastic Functions for Hot-Mix Asphalt Mixtures Using a Modified Harmony Search Algorithm." American
Society of Civil Engineers, April 2011: 139-148.
Delphin Monsia, Marc. "A Simplified Nonlinear Generalized Maxwell Model for Predicting the Time Dependent Behavior of Viscoelastic Materials." World
Journal of Mechanics 1 ( 2011): 158-167.
PARTE II
Research at the improvement of soil
properties by calcium carbonate
precipitation
Research work:
David Alejandro Castro Cruz
Supervised by:
Fernando Lopez-Caballero
´
Ecole Centrale des Arts et Manufactures
Laboratoire de M´
ecanique des Sols, Structures et Mat´
eriaux
Chapter 1
Introduction
The work is part of the research process in the precipitation of calcium car-bonate on soils. The calcium carcar-bonate is a constructed solid substance by bacteria in a process call microbial induced carbonate precipitation (MICP). The bacteria do a transformation of urea, water and calcium chloride in other substances as calcium carbonate. The calcium carbonate is deposed between particles of soil. It is a solid and it builds bridges acting as cementing, also it reduces the porosity in the soil. The inputs of the reaction are fluid sub-stances or they can be dissolves in water. This let deposits the calcium carbonate in the soil without big modification as excavations or perforations, only for a process of injection and propagation. By this reason, the MICP is the appropriated procedure in many occasions when the soil allows flow easily.
MICP involves many substances and many events as hydrology nomenons, biological phenomenons, chemistry phenomenons and physics phe-nomenons. By this, the models made in the bibliography have different sup-poses and different procedures. In this work the models are compared. Fur-ther by the same reason, that the models of MICP have many phenomenons, they have several inputs. In this work is done a sensibility analysis to stab-lish the key inputs in the model. It allows improve the making-decision and focus the study.
The uncertainty parameters need to be considered in the models. By this reason, this work contributes with a probabilistic model. The probabilistic evaluation evaluation needs do a evaluation with random fields for some soil parameters as the porosity. The work evaluates probabilistic answers; it will
improve the future analysis in other research process.
The project contributes with a analysis probabilistic of a MICP process. This supports the research done in the Ecole Centrale de arts et manufactures de Paris. The final model is able of integrate different supposes with easy changes in its code, and the analysis of result is fast and automatic. It support the futures research in the Biogrout process.
Chapter 2
Model - MICP
This is a remediation technique that improve the soil by biologic and chem-istry processes. The Figure 2.1 shows the injection of the necessary initial chemical substances in the field. The injection needs a permeable soil. By this reason the method is used usually in sand or soils with big particles. (J. T. DEJONG and PAASSEN, 2013)
2.1
Chemical process of MICP
The calcium carbonate is a substance composed of calcium, carbon, and
oxy-gen (CaCO3). This is a substance present in organic process, and in the soils
weathering processes. It is presented in a rocks and it is produced by organic reactions. The presence of calcium carbonate in soil improves the properties for civil works (B.C. Martinez and Ginn, 2014). Additionally, the injection of the substance is done without big manipulation in the soils (Cheng and Cord-Ruwisch, 2012). It is a principal difference and advantage with other remedation techniques.
The calcium carbonate is a substance produce in the soils by bacteria.
It substance is a solid that improve the properties of soil ˙The bacteria
in-duce the reaction and creation of this substance. The reaction needs Urea (CO(N H2)2), calcium chloride (CaCl2), and water(H2O) initially. These substances are dissolved with water and the fluid is injected in the soil joined with bacteria. The reaction that domain the process is this.
Figure 2.1: Injection on soil of inputs for MICP process.(Paassen, 2011)
CO(N H2)2(aq)+Ca2+(aq)+2H2O(l)
bacteria
−−−−→ 2N H4+(aq)+CaCO3(s) (2.1)
In the previous reaction, the initial substances are Urea (CO(N H2)2)
cal-cium and water. Urea is a aqueous substance produced by many organisms in their metabolic processes. The Urea is used also as fertilizer in the agri-culture process. The substance has appearance aqueous and it is dissolved
in presence of water. The calcium (Ca2+) is the ion obtained from calcium
chloride. This is a substance very special by his easy dissolution at water, and it is stranger for substances derived from calcium. By this reason, this is used in many process that use calcium in the reactions.
The resulting substances are ammonium that it reacts with the chlorine
from calcium chloride and they make the ammonium chloride (N H4Cl). This
is a source of nitrogen for some fertilizers and other uses. The substance is aqueous and it dissolves in water. This has to be removed of soil. The last substance is the calcium carbonate. All the process is done in two steps. The firs is showed in the next equation.
CO(N H2)2(aq) + 2H2O(l)
bacteria
−−−−→2N H4+(aq) +CO23−(aq) (2.2)
In the previous reaction, the Urea (CO(N H2)2) and the water are
Figure 2.2: Action of bacteria on soils. (?)
only bacteria activate the process. The products react after with other sub-stances. The second reaction is this.
Ca2+(aq) +CO23−(aq)→CaCO3(s) (2.3)
The calcium (Ca2+) is obtained from calcium chloride, this reacts with
the carbonate ions. The product of this reaction is the calcium carbonate. This material is a solid that appears as crystal; these particles make bridges between soil particles. The Figure 2.2 shows a zoom of this generation.
The results react with calcium chloride (CaCl2). Additionally, also is
created a component of ammonium chloride. All process is known as micro-biology induced calcite precipitation (MICP). This is active only in presence of microbial material. Many types of bacteria can start the reaction always that they have suitable enzymes. Sporosarcina pasteurii is the most studied by this purpose. They work by some controlled time and after they die. Therefore, they are very appropriated for the MICP.
2.2
Numerical model
Some experimental carbonate precipitation is showed in the Figure 2.3. Pre-viously the implicated substances are presented. The urea, the calcium, and
Figure 2.3: Calcium carbonate production in soils (Paassen, 2011).
the ammonium are substances that are dissolves in water or with aqueous behaviour. By this reason, the transport of these substances is modulated how a contaminant. The differential equation by this system is show here.
R·θ· ∂C
∂t =∇ ·(θD· ∇C)−q· ∇C+qsCs−( ∂θ
∂t +∇ ·q)·C+θmr (2.4)
In these equation,θ is the porosity, C is the substance concentration
dis-solve in the fluid. D is the dispersion tensor,q is the Darcy velocity,qs and
Cs is the flux and concentration of some source, r is the reaction rate, and
m is a natural number of production in the reaction. In case of the inputs
the number will be negative. The retarding factor R is associated with the
concentration sorption by the soil particles.
This equation models the propagation of some dissolved substance in some fluid. In this study case, the equation is used to model the propagation and
reaction of urea, calcium, and ammonium. The first term R does reference
to retarding factor, which is associated with the concentration sorption by the soil particles. The estimation of this parameter is present as:
R= 1 + ρb
θ ∂C
∂C (2.5)
Whereρb is the bulk density, and C is the sorpted concentration. Hence,
Figure 2.4: Mechanic dispersion on soils (Sanchez, 2012).
The first term after of the equal in the equation 2.4 models the dispersion. The dispersion phenomenon is due to two principal reasons. The mechanic dispersion is when the molecules of the fluid advance to differential velocities by the interaction with the particles, and when they change of direction by the same reason. The Figure 12 has an outline of the mechanic dispersion. The other reason is by diffusion, it is due to moving of molecules in every fluid. The action of this is very lower than mechanic dispersion. Consequently these reasons, the dispersion tensor is formed with the next equations in 1D and 2D(Sanchez, 2012).
D=αL·v+D∗ (2.6)
In 2D the tensor is defined as:
Dij = (αL−αT) vivj
|v| +δijαT X
i v2
i
|v| +δijD
∗
(2.7)
WhereαL and αT are the longitudinal and transversal dispersivities, and
δij is the Kronecker delta. D∗ is the coefficient of diffusion.
In the other hand, if is supposed that the calcium carbonate is static, this appears in the soil only by chemistry reaction. In mathematical terms these is showed in the next equation.
∂CCaCO3
∂t =mCaCO3θr (2.8)
These equation presents the calcium carbonate concentration (CCaCO3)
in units of mass per volume. By this reason the variable mCaCO3 is used; it
the previously.
The model constructed here suppose that the propagation is uncoupled from other physics. Hence the variation in the porosity of medium is equal to inverse variation by the calcium carbonate generation. The next equation expresses this.
∂θ ∂t =−
1 ρCaCO3 ·
∂CCaCO3
∂t (2.9)
Other supposed is that the total volume of fluid doesn’t change, by this the fluid is assumed as incompressible and the volume changes by the reaction are despised. In this condition by mass conservation the flux is due to changes in the porosity.
∇ ·q=−∂θ
∂t (2.10)
By Darcy law the flux is obtained with the pressure as:
−∇ ·(k
µ(∇p+ρgez)) =
CaCO3
ρCaCO3
θr (2.11)
The variableez is 1 in the analysis in the gravity direction. The variable
k is the intrinsic permeability, this is model in the soil for big particles as:
k = (dm) 2
180 θ3
(1−θ)2 (2.12)
This equation is an empirical relation present in the bibliography. The density variation is estimated with empirical relation. These depend of the substance concentrations in the fluid. The next equation presents the density in kilogram per cubic meters, in relation with the substance concentration express in kilo mole per cubic meters.
ρ= 1000 + 15.4996Curea+ 86.7338CCa2+ + 15.8991CN H4+ (2.13)
The analysis of previous equations, result:
−∇ ·(k
µ(∇p+ρgez)) =
mCaCO3
ρCaCO3
Figure 2.5: Calcium carbonate production in soils.
2.3
Verification of model
In the bibliography is suppose that the rate of attached in the substances is
small and negligible, hence in all cases R = 1. By this reason the equation
2.4 for propagation of substances result in the urea case, where the reaction
number m = −1 is showed in the equation 2.15, it is negative because the
Urea is a input. In case of the calcium, the situation is too similar to Urea.
θ·∂C
urea/Ca2+
∂t =∇ ·(θD· ∇C
urea/Ca2+
)−q· ∇Curea/Ca2+
−θr (2.15)
The ammonium is modelled as:
θ·∂C N H4+
∂t =∇ ·(θD· ∇C
N H+4)−q· ∇CN H +
4 + 2θr (2.16)
The bibliography (Van Wijngaarden et al., 2011) after or validation pro-cess they do a numerical model and it is compared with the results from analytic solution. After they use the same model in different conditions, where they take a domain in 1D with one meter of length. The boundary conditions are showed in the Figure 2.5 for each model. The used parameters are showed in the 2.6.
The first condition induces the inputs with constant flux. The results of the model with constant flux are showed in the Figure 2.7 for urea concen-tration. The circles show the values obtained in the bibliography (Van Wi-jngaarden et al., 2011). The line is the urea in the model built for this work.
Figure 2.6: Parameter used in the validation models.
Figure 2.7: Urea concentration in the model.
The Figure 4.4 shows the same comparison with the calcium carbonate ob-tained.
The second case uses a differential pressure for induced the flux. In this case, the results for urea and calcium carbonate are showed in the figures 4.5 and 6.7. The Figure 2.11 shows the comparison of two dimensional model.
These figures validate the constructed model for this work. In the bibli-ography, the model was validated with the analytic solution in some simple case. By this reason, the coincidence validates the construction of the model but it doesn’t validate the supposes of the model.
Figure 2.8: Calcium carbonate concentration in the model.
Figure 2.10: Calcium carbonate concentration in the model.
Figure 2.11: Calcium carbonate concentration in the model. Left bibliogra-phy model, right model done.
Chapter 3
Initial probabilistic theory
The actual engineering does design with different mathematical process. They need many inputs and these are measures and estimates in many pro-cesses. Each process has an uncertainty associated. For example, the ac-curacy instruments, the change properties with the time, or the mistakes in the measures are some kinds of source of uncertainty in the engineer process. To analyse this important aspect many methods are used. Most popular method is Monte Carlo simulation. This evaluates the problem wit a lot different conditions and it determines a probabilistic distribution by the an-swer. However, Monte Carlo uses random inputs and it explores the domain without previous study, by this this method requires much simulation and his result analysis is limited to other methods.
Here are presented two methods alternatives. they can do sensitive anal-ysis for non-lineal systems (see Figure 3.1). Additionally, they explore the space with organizer method than Monte Carlo. It helps to reach problem convergence with less iterations. They are showed in this chapter.
3.1
The Fourier Amplitude Sensitivity Test
(FAST)
This method boards the uncertainty problem in complex models. It consists in a similar procedure to Monte Carlo method because it evaluates many times the model with different parameters. But the FAST method does not generate purely randomly combinations. By this reason, FAST obtains
prob-Figure 3.1: Sensitivity analysis in the answer of some model with many input.
abilities more accurate than with other methods. Additionally, the FAST method allows do a sensibility analysis; it determines the contribution from uncertain local of some variable in the final uncertain of model. With this method is possible quantify non-linear relationships. It is an advantage with past models as Monte Carlo or the correlation determination (Saltelli et al., 1999).
The first step of this method is to select the study variables. Each variable
of study is identified with the index m. After of choose the study variables,
the variable values in all iterations are generated. The indexiidentifies each
iteration. The third step is define the model of study, for example.
Yi =f(Xi1, X 2 i, ..., X
M
i ) (3.1)
Where M is the number of parameters in probabilistic evaluation, and f
is the model. Xim is obtained with the next equation.
This is a sinusoidal wave equation related with the frequency ωm; it is
different for each variable. si is defined according to the simulation number
N. The next equation show how it is defined.
si+1−si = 2πsi ∈[−π, π] (3.3)
The functiong transform the sinusoidal wave to probabilistic distribution
related with the parameter. In this work this process is done in two steps.
The first is obtain a uniformly distribution ui. It is done with the next
equation.
umi = 0.5 + 1
π ·arcsin(sin(ωmsi)) (3.4)
Finally, each parameterXm
i is obtained by evaluatingumi in inverse
prob-ability distribution associated with the parameter.
Xim =CDF−1(umi ) (3.5)
All frequencies must be independent for all variables; this is guaranteed when the frequency is not a linear combination ( it is not obtained with addition or subtraction of the other frequencies). Finally, the model is run with each combination, and the answers are evaluated with Fourier’s analysis. The analysis shows the energy associated with each frequency in the answer. When some variable is most important, the energy of the answer in the associated frequency is greater. Hence, the sensibility analysis accounts the energy in the variable frequency and in his depend frequencies. The set depend frequencies are defined as:
wm =z·wm ∈z = 1,2, ...,∞ (3.6)
After of obtain the energy in for the variables (Em), the energy is divide
by the total energy (Et). The percentage of importance in the model of some
variable (sm) is determinates as:
Em =
∞
X
z=1
F(z·ωm) (3.7)
Et =
∞
X
z=1
0 20 40 60 80 100 −2
0 2 4 6 8 10x 10
8 Fourier´s spectrum
frecuency
Figure 3.2: Fourier analysis for the answer in the FAST analysis.
sm = Em
Et
(3.9)
WhereF is the function of Fourier spectrum. The process is present with
a simple example. For the next model with two variablesx and ywhere xis
clearly most important.
Z =x+xy (3.10)
The frequencies for the study variablesxandyare 23 and 28 respectively.
After of do the process the Fourier’s analysis obtained is showed in the Figure 3.2. It shows that for large frequencies the peaks have least energy, by this is possible despise the energy in high frequencies.
The importance obtained for x and y are x= 0.8708 y= 0.0968. This
result can be validated with analytic solution of variance, where the impor-tances are 0.9 and 0.1 respectively.
3.2
Random balance design by the Fourier
amplitude sensitivity test (RBD-FAST)
The analysis FAST needs to be careful to choose the frequencies for each vari-able; when the study take many variable the frequencies need to be very high, hence the simulation number is also very high. By this reason, is proposed a new system of FAST. In this system all variables have the same frequency, but the values are disorder of random form for each variable. After the model is run with different combinations. Finally the answer is reorder for each variable and the Fourier’s analysis is done. RBD-FAST determines the importance of each variable, but this analysis doesn’t evaluate the combined importance (S. Tarantolaa and Mara, 2006)
Similarly to FAST analysis, the generation of each variable has some fre-quency, but in this case only exist one frequency for all variables. By this reason, is done a random permutation and at each variable is called in
dif-ferent ordersi. For example, in the last model, the first iterationx can take
i= 45 whiley can takei= 13 and so there is a different combination to each
iteration. After of run the model the answer has not some order or wave shape for do a Fourier’s analysis. By this, The sensibility analysis requires reorder the results for each variable. For example, if the random permutation
generates in the first iteration for some variable x the value i = 45, when
will do the analysis ofx, the answer of first iteration becomes to fortieth-fifth
answer in a new array. When all answers are ordered the Fourier analysis is done, and the same analysis is applied. Similarly, when some variable is important, the Fourier’s analysis show a high energy at the frequencies asso-ciated with the frequency choose for the system.
For the same model of the equation 3.10 the Fourier analysis is showed
here for variablexand y. In this case the frequency taken was 5. The Figure
3.3 showed the Fourier analysis for x and in the Figure 3.4 the analysis for
y. The importances obtained are 0.8704 and 0.0983 forxand y respectively. These importances are very similar than obtained with FAST model.
The analysis RBD-FAST has similar results to FAST analysis. But it
has a dependence of random generation. Furthermore, it requires leaser
0 5 10 15 20 25 0
1 2 3 4 5 6 7 8 9 10x 10
8 Fourier´s spectrum
frecuency
Figure 3.3: Fourier analysis for the answer reorder for xin RBD-FAST
anal-ysis.
0 5 10 15 20 25
0 1 2 3 4 5 6 7 8 9 10x 10
7 Fourier´s spectrum
frecuency
Figure 3.4: Fourier analysis for the answer reorder for yin RBD-FAST
Chapter 4
Analysis probabilistic of Model
Study
The model is done to evaluate the Biogrout process, here is evaluated the situation. The result are present in next sections. The sensitivity analysis about the model and the analysis of uncertainty is showed here.
4.1
Uncertainty propagation
In this sections is showed the answer of the model to simulation. The pa-rameters select to change are:
• Initial porosity.
• Saturation factor.
• Longitudinal dispersivity.
• Max rate of reaction.
• Longitudinal viscosity.
• Size of particles.
• Time reaction max.
Each variable has the same mean to the validation model in the Figure 4.1. The dispersion is given by variation coefficient of 10%. All variables are taken
as uniform distributions. The number of simulation used are 1000 in both cases, FAST and RBD-FAST. The first result in the Figure 4.1 shows the uniform variable for each parameter in the diagonal. The other squares show the relation between variables. They show the independence of variables with both methods, and that they explore all domain uniformly. Is necessary re-member that both methods generate uniform distribution between zero and one. After it is transformed.
The limits are showed in the Figure 4.2 for many cases. The mean is showed with black line, and the value of mean more and less one standard deviation is marked by blue error bars. These graphs show that the disper-sion is upper with the time and with the calcium carbonate increment. Both methods give like answers. The means and variances are same, Even the limits are very similar in all case.
Each point in each time has different answers. Hence, the probabilistic distribution change with the time. The distribution obtained are very similar with both methods. The Figure 4.3 shows it with a sample for some point in some time. The shape distribution change with the time, most cases have Weitbull or beta distribution for calcium carbonate distribution.
The model can give probabilistic solution in each time for each position for all variables. It will let realize futures research in other aspects as the supposes from program. With the obtained results can conclude that the analysis uncertainty is necessary to study the MICP problem. The proba-bilistic distributions change with the time and the space and the standard deviation is big in all case. By this, a deterministic solution does not do a good evaluation of system.
4.2
Sensitivity analysis
The study will helps to future researches for the Biogrout processes. The process of FAST and RBD-FAST are used for the evaluation. The system evaluated is show in the Figure 2.5. The study was done about seven pa-rameter. The importances of each parameter in different points in the space are showed in the Figure 4.4. The Figure 4.5 shows the importances of the variables in the time. From these figures set that the important parameters in the model are: the max reaction rate, the max time reaction, the viscosity,
Figure 4.2: Limits, mean and standard deviation for concentration of calcium carbonate. Left column is answer for FAST model, right column is answer for RBD-FAST model.
Figure 4.3: CDF for some point in some time. Blue CDF is obtained with RBD-FAST and red CDF with FAST method.
and initial porosity.
This work deepens the study of important variables. The porosity is a variable that can not be equal in all domain how is supposed. Hence the study with random fields is done. Others important variables are parame-ters of reaction rate model. By this, the reaction rate model is studied with more accurately in the next chapter. Furthermore, this study validates some supposes as the constant viscosity. The results show low importance of vis-cosity, and if the viscosity model is used, it increases the computational cost of model.
Figure 4.4: Importance with the distance at different times. The continue line is the Fast analysis, The discontinuous line is the RBD-FAST analysis.
Figure 4.5: Importance with the time on different points. The continue line is the Fast analysis, The discontinuous line is the RBD-FAST analysis.
Chapter 5
Analysis on the reaction rate
The dependency of the results to reaction rate parameters shows the impor-tance of the model of reaction. The model get the rate of transformation from inputs to outputs. To remember the inputs are:
• The Urea.
• The chloride of calcium.
• Water.
The transformation of these substances depend of the bacteria activity. The activity of them is modified by the pH of the medium, the presence of Urea, the number of bacteria, the affinity of the bacteria with the soil, and many factors that can be used on the model.
The most important things are the quantity of inputs, the bacteria died and the changes in the pH; this work does a study on these aspect specially. It shows three alternative models for the reaction rate, they is based in dif-ferent supposes that are showed in this section.
The model of study wants estimate the scope of calcium carbonate gener-ation. Therefore, the model takes a domain of ten meters and it is modelled for 200 hours. The other parameter are same to previous models. Addition-ally, for each reaction model is evaluated in a domain of two dimensions. They are showed in the annexes.
5.1
Reaction rate time depending.
A simple model estimates that the reaction rate decrees with the time. It sim-plifies the dead of the bacteria, and the generation of ammonium to change the pH of the medium. The reaction model is showed the next equation. In
the model tb is 38.5 hours.
r=rmax
Curea Km+Curea
exp(−t/tb) (5.1)
This model considers that the reaction changes with exponential depen-dence in the time, so it is a simple model. It doesn’t contemplate the ingress of new bacteria with the flux. The Figure 5.1 shows the results for this model in a domain of 10 meters in one dimension. It shows in the left that after of decay time, the problem is flux problem without reaction to change the substance quantities. The other graph shows the decay porosity is same after of some time.
5.2
Reaction rate model integrate with a
bac-teria model.
This model evaluates the presence of bacteria with life in the medium. The bacteria are transported in the flux as some contaminant product, hence the equation is so similar to 2.4, but it has new components as the rate of bacteria
dead. It is showed in the next equation for live bacteria (B).
∂B
∂t =−∇(B)·Vs− 1
θB∇(θVs)− 1
θ∇(θBVr−θD∇B)− B
θ ∂θ
∂t−kattB−kdB (5.2) The last equation shows the dominant function for bacteria with life in
the fluid. This work supposes that the rate of attached is zero (katt = 0).
Also here is supposed that the solid particles have not movement (Vs = 0).
Equally the supposes for volume conservation are supposed. The previous equation is transformed in:
θ· ∂B
Figure 5.1: Model of reaction rate time-depending, left result of Urea-Calcium and right porosity answer.
The parameters for dispersion tensor (D) are the same as for the urea and calcium (S. Fauriel, 2012). The bacteria with life induce the generation of calcium carbonate, so the bacteria number changes the maximal reaction rate with same conditions in others factors. The relation between bacteria and the max reaction rate is showed in the next equation.
rmax =usp·(
1−θ
θ B
0
+B) (5.4)
Where usp is the initial specific urea activity, it is determinate with
ex-perimental dates. In general, it depends of the affinity bacteria-soil. B0 is
the bacteria attached, but in for this work it is supposed as zero. The
con-structed model takes usp as 2·10−5kmol/(m3sOD) (S. Fauriel, 2012). The
boundary condition in γ1 is of 5OD forB (see Figure ...). Finally, the model
for reaction rate in each point is obtained as:
r =rmax
Curea Km+Curea
(5.5)
The answer for this model on the calcium carbonate generation is showed in the Figure 5.2. Unlike of time depending model the convergence is not reached in the simulation time, but the models shows a high reaction in the boundary of ingress. In this point the urea presence is imposed, and the number of live bacteria equally. By this reason the ingress point has high rate of reaction and a low porosity with the time. The reaction scope is lower than 1 meter. However, the scope can be modified with initial parameters as the number of ingress bacteria, or the magnitude of flux on the flux, this only is an example.
5.3
Reaction rate model that contemplate the
inhibitor material.
This model contemplates the change of pH in the medium. The changes of the pH deactivates the bacteria action and it reduces the reaction rate. The pH or acidity level is changed by the presence of ammonium (M. Fidaleo, 2003). It is generated in the MICP process and it should be extracted to
Figure 5.2: Model of reaction rate with bacteria, left result of Urea-Calcium and right porosity answer.
Figure 5.3: Model of reaction rate with inhibitor material, left result of Urea-Calcium and right porosity answer.
protect the soil. The next equation show the model used with some inhibitor material for the reaction, as the chloride of ammonium.
r =rmax C urea
Km+Curea
Kp Kp+CN H4
(5.6)
In this casekp is the dissociation constant, it is taken from the
bibliogra-phy as 13mol/m3 (M. Fidaleo, 2003). With this model the answer in porosity
is showed in the Figure 5.3. It shows that the reaction is done a big length with this model. However the porosity decay is low in comparative with other models. In the ingress point of inputs and of extraction of ammonium there is a reaction very high. It is due to the imposed condition for the ammonium
(CNH4 = 0) and the imposition of inputs.
5.4
Comparative answers and analysis of
al-ternatives.
Three alternative models for reaction model were showed in previous section. In this section the answers are evaluated between them in 2D. The models
Figure 5.4: 2D model sample. Black line with Γ2 condition and blue line
with Γ1 condition.
in 1d don’t study the extraction or gravity action. By this, a model of two dimension is done, the models in 2 dimensions are showed in 5.4 with
bound-ary conditions. The black line has the condition Γ2 shows in the Figure 2.5
with hydrostatic pressure conditions. The blue line has Γ1 condition. The
initial values are the same to last model with hydrostatic pressure.
The substances are injected with constant flux. This is very low in com-parative with a normal injection. The table of water to make the flux is lower than 40 cm all time. By this, the scope is reduced and the model is reduced. In real application the scope and the power injection can be upper. The parameters are estimated from different bibliographies, in futures works need to do empirical determinations for the parameters and can validate the model.
The Figure 5.5 shows the reaction rate after of 200 hours. Each model founds the reaction rate with a different way. The first figure with bacteria model found an high rate after in this time. The reaction converge with the distance when the substance injection is constant. The second figure shows the reaction with inhibitor material. It is high in the injection sector because in the injection additional the ammonium is extracted in this model. Hence the concentration of ammonium is low in this sector so the reaction is high. With this model the reaction decay with the time except in the injection sector, the far points have high reaction until high quantities of ammonium
are generated. The time depending model in the last figure has a reaction of zero at 200 hours because it depends of the absolute time, the new substances with new bacteria aren’t contemplated.
The Figure 5.6 shows the final porosity with 200 hours of reaction and simulation. The answers are consequent with the reaction figures. The most reduction is with bacteria because this model has high reactions with the time. The inhibitor material model gives low porosity near to injection point by the same previous reason. The time depending model has high porosities because after of some time the reaction is annulled in all model.
Is necessary do experimental work to select the best reaction model. Also the parameters of the models need be estimated of real situation to validate the model.
Figure 5.5: Reaction rate in kmol
m3s in 2D model with 200 hours for reaction
rate. From up to down, Bacteria model, inhibitor material model, and time depending model.
Chapter 6
Study using random fields.
The problem has many parameters with characteristics to be modulated with random fields. This chapter explicates and apply the random fields to the study problem.
6.1
Theory of random fields.
The random fields are spaces where the distribution of some variable is re-lated with the position in the space. In geotechnical for example, the cohesion is relates with the position, and is stranger that near soil elements have very different values of cohesion. To generate random numbers in these cases are used random fields.
Exist many types of random fields, in this document only is showed the stationary fields. These are fields where the parameters do not change for all domain, so all nodes have the same mean and same variance. Additionally, the showed random fields are related by distance and never with time that is other alternative. In general the studies of random fields can be not sta-tionary; but this work only uses stationary random fields.
When two variables are related, they have a correlation factor. With this value is possible generate random numbers for two correlated variables. In random fields case any variable is related with the value of the others variables in the space. The relations with the neighborhood depend of the
Figure 6.1: Sample of theoretical variogram (Allen L. Jones, 2002).
distance in this study. To show this the variograms (γh) are used; they show
the variance between the variation dates between two points separate by any
distance. This distance or lag is showed in axis x of variogram. When the
distance is zero should not exist variation, by this the variogram start in zero with distance zero. When the distance is very long, in the sill the variagram shows the variance of the variable, because far points are independent. The variogram shape is different in each study case. In the Figure 6.1 shows a theoretic variogram.
The variograms done with empiric dates can be rather different from Fig-ure 6.1. For example, the FigFig-ure 6.2 shows an empirical variogram. This was done with few dates represents by the squares in the graph. In some cases the empirical variograms are upper than the variance; it is due to mistakes of measures or few dates.(Allen L. Jones, 2002)
Is necessary fitting the variograms with some model to do studies. In the Figure 6.3 are showed some common models. Each model generates some type of distribution associated in each point. In general, all models depend
of the sill (σ2), the nugget (a2), and the scale of fluctuation or correlation
(hr) here is called correlation length. The sill is the variance that needs to
generate in the random field. The nugget is a measure of the minimum vari-ability in close points; this value is near to zero and is due to mistakes in measures or the discontinuity in close point variability. Finally, the scale of