The aim of the thesis is to study the dynamics of water in soils under freezing conditions. The seasonal creation of ice in underlying soils can cause the upward movement of the ground surface. This process is called "Frost Heave" and generates lot of engineering problems creating damage to roads, pipelines and infrastructures for billion dollars. The study of literature highlight the complexity of the process which is a hydro-thermalmechanical problem. The system of main governing equations is very complex both for the number of equations and for the non-linearity of the equations. Many authors propose different approaches in describing from a mathematical point of view the frost heave process. The exhaustive study of the State of Art suggested to classify the different models in two main group. The first one, introduces models that studies saturated condition. These models are very specific studying the dynamics of frost heave process, the creation of the lenses and the cryostatic suction. The soil is subdivided in various layers (below, and upper the ice lens) with different equations and hypothesis governing each layer. Various mechanical models are presented but the governing equations are written in a not rigorous way. Moreover,the pressure exerted on the soil skeleton, at pore-size dimension, is often calculated with empirical formulation based on strong approximations. Indeed, it is not taken into consideration the pressure exerted due the thermal expansion of freezing water. The second group of models, studies unsaturated soils in freezing conditions and considers the frost heave process completely negligible. Indeed, there is a lack of a mechanical model: the pressure and stress field in freezing conditions are not analyzed and not considered. Another common approximation regards the ice gauge pressure which is considered equal to zero. In other word, the pressure exerted by the ice in freezing conditions is considered negligible. However, these models are based on rigorous formulation: the mass conservation equations and energy equations are written in a rigorous and complete mathematical form. On the basis of the State of Art analysis, the first stage of this work concerns a mathematical study of a system of equations which can be applied both for saturated and unsaturated soils and which take into consideration the change of phase phenomena. The aim of this system of equations is to include the frost heave process and the ice pressure exerted on the solid skeleton in the equations; utilizing a rigorous mathematical formulation. The main governing equations are studied and written such as to consider the ice pressure contribute and the cryostatic suction. The complexity of the system implies the necessity of more elaborate studies and advanced numerical and computational instrument to solve entirely it. So, the experimental and numerical part of the study is focused on the resolution of an easier case. In particular, considering saturated conditions, the energy equation is solved both for soils and water. Moreover, considering the pore-size dimension a coupled thermo-mechanical model is presented to study the pressure exerted on the soil skeleton by the freezing water. Focusing on a water volume, the temperature trend and pressure exerted in freezing conditions are studied experimentally and with coupled numerical simulations. As a proof a concept, it has been founded a correlation between the pressure exerted at pore-size dimension and the pressure exerted by the soil registered with the experimental test. A innovative laboratory experiment has been designed and built in all its component to register temperature and pressure data in freezing sample of soil and water. Considering a water sample of 5 cm of diameter and 2,5 cm of height, a pressure of 10151.91 kPa (20000 N) has been registered. This is not the maximum force that can be exerted by the water because the test was stopped since the chosen load sensor can support a maximum force of 20000 N. From the soils sample test the maximum pressure registered is of 130.43 kPa (corresponding to a force of 255 N). The experimental results follow the theory: in soils the freezing temperature is below 0 C, reaching a value of -4 C in some experiments. The temperature trend in a soil probe has been simulated solving the energy equations for soils, using the software COMSOL. The model manage to simulate with a certain precision the experimental results. The comparison has been done looking at the temperature trend registered and simulated in a point. A coupled thermo-mechanical model has been utilized to simulate the pressure exerted by water in freezing condition. The comparison with the experimental results shows that , under specific boundary conditions, the model is able to reproduce the pressure trend. The last comparison with the experimental results has been done considering the pressure exerted by freezing soils. It has been considered the force exerted by a volume of water of the same dimension of the water volume presents on the soil sample (so considering the sample porosity). Comparing this estimated values of soil force with the experimental values, it can be seen that they are comparable and of the same order of magnitude. So, with further studies, this can be a way of proceeding to evaluated the pressure exerted on solid skeleton, taking into account of ice pressure. Looking at the future applications of the entire project, the laboratory experiment set-up can be easily modified to do a huge range of studies such as considering different type of soils in different conditions. For instance, it can be added a liquid water reserve to take into account of the cryostatic suction. The numerical modeling it is at a first stage for soils, but can be implemented to solve also the hydrological part of the phenomena. The energy equations gives as an output the temperature and change of mass parameter for each time. These two unknowns of the general system can be inserted into the mass conservation equation and Clapeyron equation. Moreover, further researches on the proposed mechanical model can lead to the evaluation of another unknown of the system: the pressure exerted on the solid skeleton. The study in every aspects (mathematical, experimental and numerical) has a huge potentiality to reach a better understand of freezing soils phenomena and it set the basis for an exhaustive and complete model which can represents a possible way of resolution of this complex problem.
Lo scopo della tesi é lo studio della dinamica dell’acqua nei suoli in condizioni di congelamento.La formazione di ghiaccio nel sottosuolo puó causare il movimento verso l’alto della superficie del suolo. Questo processo é chiamato "Frost Heave", o criosollevamento, ed é causa di notevoli problematiche ingegneristiche, ad esempio danni alle strade, allecondotte e alle infrastrutture. L’ analisi della letteratura evidenzia la complessitá di questo processo che consiste in un problema termo-idro-meccanico. Il sistema di equazioni che governano questo fenomeno é molto complesso sia per il numero di equazioni che per la non linearitá delle equazioni stesse. Molti autori, durante gli anni, hanno proposto diversi approcci per descrivere matematicamente il problema. Lo studio approfondito della letteratura suggerisce una classificazione dei diversi modelli in due gruppi. Da una parte, abbiamo i modelli che studiano il suolo in condizioni sature. Questi modelli risultano molto specifici, analizzando nel dettaglio il fenomeno del criosollevamento, creazione delle lenti e "cryostatic suction". In questi modelli il terreno é suddiviso in diversi strati (al di sotto, e sopra la lente di ghiaccio) e ognuno di questi livelli presenta diverse equazioni e ipotesi che lo governano. I modelli comprendono uno studio meccanico ma le equazioni non sono spesso scritte seguendo una trattazione matematica rigorosa. Un’altra criticitá é collegata alla pressione esercitata sullo scheletro solido, a livello di dimensione del pori, che é spesso calcolata con una formulazione empirica. Inoltre,non tiene conto della forza risultante dall’espansione termica durante il cambio di fase. D’altra parte, i modelli che studiano terreni insaturi, considerano totalmente trascurabile il processo del criosollevamento. Infatti, vi é la mancanza di un modello meccanico e le forze generate in condizioni di congelamento non sono analizzate. Un’altra approssimazione comune riguarda la pressione relativa del ghiaccio, che é considerata uguale a zero. In altre parole, la pressione esercitata dal ghiaccio in condizioni di congelamento viene considerata trascurabile. Sulla base dello studio dello Stato dell’Arte, la prima fase di questo lavoro di tesi riguarda uno studio matematico di un sistema di equazioni che puó essere applicato sia per i suoli saturi che insaturi e che tiene in considerazione il fenomeno del cambiamento di fase. Sono state studiate e scritte le principali equazioni che governano il processo, in modo tale da considerare la pressione esercitata dal ghiaccio e la "criostatic sunction" . La complessitá del sistema matematico implica la necessitá di studi piú elaborati e avanzati strumenti numerico e computazionali per la sua completa risoluzione. Per questa ragione, la parte sperimentale e la parte numerica di questo lavoro si concentrano sulla risoluzione di un caso semplificato. In particolare,considerando condizioni sature, l’equazione di energia é risolta sia per i suoli che per l’acqua. Inoltre, considerando lo studio a livello della dimensione dei pori, viene presentato un modello termo-meccanico per studiare la pressione esercitata sullo scheletro del suolo dall’acqua in fase di congelamento. Considerando un volume d’acqua, sono state studiate sperimentalmente e con un modello termo-meccanico, sia l’andamento della temperatura che la pressione esercitata in condizioni di congelamento. L’obiettivo di questo modello é la simulazione della pressione esercitata dal ghiaccio in fase di congelamento a dimensione di poro. Inoltre, come primo passo per uno studio piú approfondito, é stata trovata una correlazione tra i valori di pressione a scala di poro e i valori di pressione esercitati dal terreno ottenuti sperimentalmente. Per effettuare le prove sperimentali, e’stato progettato e costruito un esperimento di laboratorio innovativo, per registrare i dati di temperatura e pressione in campione di acqua e suolo in condizioni di congelamento. Considerando un campione d’acqua di 5 cm di diametro e 2.5 cm di altezza, é stata registrata una pressione di 10151,91 kPa (20000 N). Questa non é la forza massima che puó essere esercitata dal provino poiché il test é stato interrotto, in quanto la cella di carico scelta poteva supportare una forza massima di 20000 N. Dall’esame del campione di suolo la massima pressione registrata é di 130,43 kPa (corrispondente a una forza di 255 N). I risultati sperimentali seguono la teoria: nel suolo la temperatura di congelamento é inferiore a 0C, in particolare, in alcuni esperimenti, raggiunge un valore di -4 C. Un modello termo-meccanico é stato utilizzato per simulare le pressioni esercitate dall’acqua in condizioni di congelamento. Dal confronto con i risultati sperimentali si evince che il modello é in grado di simulare l’andamento delle pressioni dell’acqua, considerando oppurtune condizioni al contorno. Inoltre, é stata valutata la forza esercitata da un volume d’acqua di grandezza pari al volume d’acqua presente nei pori del terreno (quindi considerando la porositá). Confrontando questi valori di forza con le forze esercitate dal terreno misurate sperimentalmente, si puó vedere che esiste una correlazione. I valori trovati sono confrontabili e dello stesso ordine di grandezza. Alla luce di questi risultati, piú approfonditi studi su questa relazione, possono portare alla valutazione delle pressioni esercitate sul terreno. L’equazione di conservazione dell’energia é stata risolta anche per il suolo, per fare un confronto tra l’andamento della temperatura simulato con i risultati sperimentali. Confrontando le due curve, si puó vedere che la simulazione ha la stessa tendenza dei dati sperimentali. Analizzando le applicazioni future dell ’intero progetto, il set up dell’esperimento di laboratorio puó essere adottato per fare una vasta gamma di studi, ad esempio: la considerazione di diversi tipi di terreno in condizioni diverse. Inoltre il sistema può essere facilmente modificato per tener conto della "criostatic sunction", aggiungendo una riserva idrica liquida. La modellazione numerica é ad uno stadio iniziale per quanto riguarda i suoli, ma puó essere implementata per risolvere anche la parte idrologica del fenomeno.I risultati della simulazione bidimensionale dell’equazione dell’energia nel terreno, danno come risultato il valore di temperatura in ogni punto del suolo e in ogni momento. Inoltre fornisce come risultati la variazione del contenuto di acqua dovuta al congelamento, in tal modo si conosce la massa d’acqua e la massa di ghiaccio presenti in ogni istante. L’equazione di energia fornisce come uscita la temperatura e il cambio del parametro di massa per ogni istante. Queste due incognite del sistema generale possono essere inserite nell’ equazione di conservazione della massa e nell’equazione di Clapeyron. Inoltre, con successivi studi sul modello meccanico proposto, si puó arrivare alla valutazione di un’altra incognita del problema: la pressione esercitata sullo scheletro solido. Lo studio svolto in questa tesi, in tutti gli aspetti (matematico, sperimentale e numerico), presenta una grande potenzialitá per raggiungere una migliore comprensione del fenomeno di congelamento del suolo e fonda la base per un modello esaustivo e completo che puó essere una chiave di risoluzione di questo complesso problema.
Soil water dynamics with phase changes : mathematical modeling, numerical simulations and laboratory experiments
DEIDDA, CRISTINA
2016/2017
Abstract
The aim of the thesis is to study the dynamics of water in soils under freezing conditions. The seasonal creation of ice in underlying soils can cause the upward movement of the ground surface. This process is called "Frost Heave" and generates lot of engineering problems creating damage to roads, pipelines and infrastructures for billion dollars. The study of literature highlight the complexity of the process which is a hydro-thermalmechanical problem. The system of main governing equations is very complex both for the number of equations and for the non-linearity of the equations. Many authors propose different approaches in describing from a mathematical point of view the frost heave process. The exhaustive study of the State of Art suggested to classify the different models in two main group. The first one, introduces models that studies saturated condition. These models are very specific studying the dynamics of frost heave process, the creation of the lenses and the cryostatic suction. The soil is subdivided in various layers (below, and upper the ice lens) with different equations and hypothesis governing each layer. Various mechanical models are presented but the governing equations are written in a not rigorous way. Moreover,the pressure exerted on the soil skeleton, at pore-size dimension, is often calculated with empirical formulation based on strong approximations. Indeed, it is not taken into consideration the pressure exerted due the thermal expansion of freezing water. The second group of models, studies unsaturated soils in freezing conditions and considers the frost heave process completely negligible. Indeed, there is a lack of a mechanical model: the pressure and stress field in freezing conditions are not analyzed and not considered. Another common approximation regards the ice gauge pressure which is considered equal to zero. In other word, the pressure exerted by the ice in freezing conditions is considered negligible. However, these models are based on rigorous formulation: the mass conservation equations and energy equations are written in a rigorous and complete mathematical form. On the basis of the State of Art analysis, the first stage of this work concerns a mathematical study of a system of equations which can be applied both for saturated and unsaturated soils and which take into consideration the change of phase phenomena. The aim of this system of equations is to include the frost heave process and the ice pressure exerted on the solid skeleton in the equations; utilizing a rigorous mathematical formulation. The main governing equations are studied and written such as to consider the ice pressure contribute and the cryostatic suction. The complexity of the system implies the necessity of more elaborate studies and advanced numerical and computational instrument to solve entirely it. So, the experimental and numerical part of the study is focused on the resolution of an easier case. In particular, considering saturated conditions, the energy equation is solved both for soils and water. Moreover, considering the pore-size dimension a coupled thermo-mechanical model is presented to study the pressure exerted on the soil skeleton by the freezing water. Focusing on a water volume, the temperature trend and pressure exerted in freezing conditions are studied experimentally and with coupled numerical simulations. As a proof a concept, it has been founded a correlation between the pressure exerted at pore-size dimension and the pressure exerted by the soil registered with the experimental test. A innovative laboratory experiment has been designed and built in all its component to register temperature and pressure data in freezing sample of soil and water. Considering a water sample of 5 cm of diameter and 2,5 cm of height, a pressure of 10151.91 kPa (20000 N) has been registered. This is not the maximum force that can be exerted by the water because the test was stopped since the chosen load sensor can support a maximum force of 20000 N. From the soils sample test the maximum pressure registered is of 130.43 kPa (corresponding to a force of 255 N). The experimental results follow the theory: in soils the freezing temperature is below 0 C, reaching a value of -4 C in some experiments. The temperature trend in a soil probe has been simulated solving the energy equations for soils, using the software COMSOL. The model manage to simulate with a certain precision the experimental results. The comparison has been done looking at the temperature trend registered and simulated in a point. A coupled thermo-mechanical model has been utilized to simulate the pressure exerted by water in freezing condition. The comparison with the experimental results shows that , under specific boundary conditions, the model is able to reproduce the pressure trend. The last comparison with the experimental results has been done considering the pressure exerted by freezing soils. It has been considered the force exerted by a volume of water of the same dimension of the water volume presents on the soil sample (so considering the sample porosity). Comparing this estimated values of soil force with the experimental values, it can be seen that they are comparable and of the same order of magnitude. So, with further studies, this can be a way of proceeding to evaluated the pressure exerted on solid skeleton, taking into account of ice pressure. Looking at the future applications of the entire project, the laboratory experiment set-up can be easily modified to do a huge range of studies such as considering different type of soils in different conditions. For instance, it can be added a liquid water reserve to take into account of the cryostatic suction. The numerical modeling it is at a first stage for soils, but can be implemented to solve also the hydrological part of the phenomena. The energy equations gives as an output the temperature and change of mass parameter for each time. These two unknowns of the general system can be inserted into the mass conservation equation and Clapeyron equation. Moreover, further researches on the proposed mechanical model can lead to the evaluation of another unknown of the system: the pressure exerted on the solid skeleton. The study in every aspects (mathematical, experimental and numerical) has a huge potentiality to reach a better understand of freezing soils phenomena and it set the basis for an exhaustive and complete model which can represents a possible way of resolution of this complex problem.File | Dimensione | Formato | |
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https://hdl.handle.net/10589/136279