Prediction of petro-physical properties for carbonates

This thesis is concerned with the inversion of lattice pore-network model parameters of carbonate rocks using only the capillary pressure, and then the use of the inverted parameters to predict the water-flooding relative permeabilities of the carbonate rocks. Background: There has been a tendency to claim that pore-network modelling using three-dimensional micro-computed tomography or 3D mathematically created images can predict imbibition relative permeabilities for wettabilities other than strongly water/oil-wetting. This is based on the flexibility for matching data, which is a weakness of pore-network modelling. The method proposed in this thesis is important because a large percentage of the porosity in carbonates is microporosity. Conclusions: We applied stochastic inversion of lattice pore-network model parameters using Hamiltonian Dynamics (Hamiltonian Monte Carlo) to three carbonate rock samples and we predicted water-flooding relative permeabilities with good accuracy by using as constraint only routinely obtained data, such as mercury intrusion capillary pressure (MICP) and oil/water capillary pressure. We found that there is a strong correlation between the amount of microporosity and the volume exponent parameter. This suggests that when microporosity is ignored, the volume exponent will tend to be systematically strongly underestimated. HMC found large variability in wettability that causes mid-sized pores to be invaded at the same level of pressure as larger pores. The coexistence of these events reduces the tendency for preferential flow through large pores, resulting in more uniform flow at the pore scale compared with the case in which flow is restricted only to large pores. Mid-sized pores have an important effect on the connectivity because they could have higher contact angles than larger pores. Therefore, they do not spontaneously imbibe and shield larger pores, improving water-flooding displacement. The wettability of micropores could better explain the gentle curvature of the imbibition water relative permeability compared with the generally assumed mixed-wet large wettability model. The importance of the maximum and minimum observed capillary pressure is directly connected to accounting for the full pore-size distribution. Thus, the common assumption that microporosity can be ignored is unsatisfactory. The ranges of advancing contact angles obtained from the HMC inversion were wider than the ranges of apparent advancing contact angles obtained with analytical determinations in previous studies, and in one case our results were contradictory to the analytical determination. It follows that variability in advancing and receding contact angles is not reflected in the apparent contact angle data outside porous media. Apparent contact angle data outside porous media cannot completely characterise the wettability in porenetwork models because the data does not capture the contact angle variability in porous media. The existence of wetting films depends on the maximum capillary pressure during drainage, and thus wettability alteration during ageing. Our results suggest that matching both connate water at the maximum drainage capillary pressure before ageing and matching residual oil at the minimum imbibition capillary pressure leads to better estimation of the advancing and receding variability in the contact angles

Partial differential equations for self-organization in cellular and developmental biology

Understanding the mechanisms governing and regulating the emergence of structure and heterogeneity within cellular systems, such as the developing embryo, represents a multiscale challenge typifying current integrative biology research, namely, explaining the macroscale behaviour of a system from microscale dynamics. This review will focus upon modelling how cell-based dynamics orchestrate the emergence of higher level structure. After surveying representative biological examples and the models used to describe them, we will assess how developments at the scale of molecular biology have impacted on current theoretical frameworks, and the new modelling opportunities that are emerging as a result. We shall restrict our survey of mathematical approaches to partial differential equations and the tools required for their analysis. We will discuss the gap between the modelling abstraction and biological reality, the challenges this presents and highlight some open problems in the field

Competitive Nucleation in Nanoclusters

Nucleation is the primary mechanism by which systems change phase and it plays a major role in the formation of new materials in nature and industrially. In particular, experiments and molecular dynamic simulations have shown that nanoclusters, at the same initial conditions, freeze to different structures through a competitive process. Understanding the mechanism of nucleation requires the knowledge of the reaction coordinate, which consists of a set of variables that accurately describe the formation of the critical nucleus. In classical nucleation theory (CNT), the embryo size is solely used as the reaction coordinate, but this does not capture the formation of different structures in a competitive nucleation event. Competitive nucleation is modeled using a two dimensional Potts model undergoing heterogeneous nucleation on to a nanoscale impurity. The rates of formation of the different stable phases are calculated using transition state theory and compared with the rates obtained from the mean first passage time and survival probability methods. Transition state theory is shown to predict the rates to the different structures under various conditions when the nucleation barrier is correctly normalized relative to the metastable state. A multiple path maximum likelihood analysis, (MPMLA), is developed to extract accurate reaction coordinates to the different phases. The results show that the linear combination of size and surface area of a given component is the accurate variable that describes the transition to the phase. Molecular dynamics simulations are used to study the competitive freezing of gold nanoclusters for a range of cluster sizes and temperatures. Measuring the probability of observing each cluster type in an ensemble of freezing events, along with the overall rate at which liquid drops freeze to any structure, allows the rate of formation for each structure to be calculated. The rate of formation of icosahedral structures is about an order of magnitude higher than the rates for other structures. Also, as the size of the cluster increased, the rate of formation of icosahedral structure decreased while that of decahedral and FCC structures increases. The MPMLA is applied to the transition path ensembles to obtain the best reaction coordinate for the different transitions. Order parameters such as size, the Steinhardt bond orientational parameters, local order parameters such as Qe, ratio of local atom type in the largest embryo, and structural order parameters are tested as reaction coordinates. A linear combination of size, the fcc-fcc correlation parameter, and the Qe provided the maximum estimate for the liquid-icosahedral transition, making it the best reaction coordinate. The critical embryo for this transition consists of bulk fcc-type atoms arranged in a small group, and capped by surface 111 atoms to form a tetrahedron. There is at least one 5-fold symmetric cap for this critical embryo. For the liquid-decahedral transition, the linear combination of size, Qe and the Ihedge- correlation parameter is the best reaction coordinate. Analysis of the critical embryo shows the formation of blocks of bulk fcc atoms. The number of fcc-type atoms in these blocks is greater than those observed in the case of the icosahedral transition, hence, the Qe parameter has a stronger effect. There is also the presence of and the Ihedge atoms positioned to form the 5-fold cap. The formation of the FCC structures follows the growth of the bulk fcc atoms with a corresponding elimination of the 5-fold facets. Hence, the linear combination of Qe, the Ihbulk-Hcpbulk and the Ihedge- correlation parameters, is the best reaction coordinate that describes the formation of FCC clusters

Modeling of metal nanocluster growth on patterned substrates and surface pattern formation under ion bombardment

This thesis addresses the metal nanocluster growth process on prepatterned substrates, the development of atomistic simulation method with respect to an acceleration of the atomistic transition states, and the continuum model of the ion-beam inducing semiconductor surface pattern formation mechanism. Experimentally, highly ordered Ag nanocluster structures have been grown on pre-patterned amorphous SiO^2 surfaces by oblique angle physical vapor deposition at room temperature. Despite the small undulation of the rippled surface, the stripe-like Ag nanoclusters are very pronounced, reproducible and well-separated. The first topic is the investigation of this growth process with a continuum theoretical approach to the surface gas condensation as well as an atomistic cluster growth model. The atomistic simulation model is a lattice-based kinetic Monte-Carlo (KMC) method using a combination of a simplified inter-atomic potential and experimental transition barriers taken from the literature. An effective transition event classification method is introduced which allows a boost factor of several thousand compared to a traditional KMC approach, thus allowing experimental time scales to be modeled. The simulation predicts a low sticking probability for the arriving atoms, millisecond order lifetimes for single Ag monomers and ≈1 nm square surface migration ranges of Ag monomers. The simulations give excellent reproduction of the experimentally observed nanocluster growth patterns. The second topic specifies the acceleration scheme utilized in the metallic cluster growth model. Concerning the atomistic movements, a classical harmonic transition state theory is considered and applied in discrete lattice cells with hierarchical transition levels. The model results in an effective reduction of KMC simulation steps by utilizing a classification scheme of transition levels for thermally activated atomistic diffusion processes. Thermally activated atomistic movements are considered as local transition events constrained in potential energy wells over certain local time periods. These processes are represented by Markov chains of multi-dimensional Boolean valued functions in three dimensional lattice space. The events inhibited by the barriers under a certain level are regarded as thermal fluctuations of the canonical ensemble and accepted freely. Consequently, the fluctuating system evolution process is implemented as a Markov chain of equivalence class objects. It is shown that the process can be characterized by the acceptance of metastable local transitions. The method is applied to a problem of Au and Ag cluster growth on a rippled surface. The simulation predicts the existence of a morphology dependent transition time limit from a local metastable to stable state for subsequent cluster growth by accretion. The third topic is the formation of ripple structures on ion bombarded semiconductor surfaces treated in the first topic as the prepatterned substrate of the metallic deposition. This intriguing phenomenon has been known since the 1960\'s and various theoretical approaches have been explored. These previous models are discussed and a new non-linear model is formulated, based on the local atomic flow and associated density change in the near surface region. Within this framework ripple structures are shown to form without the necessity to invoke surface diffusion or large sputtering as important mechanisms. The model can also be extended to the case where sputtering is important and it is shown that in this case, certain \\lq magic\' angles can occur at which the ripple patterns are most clearly defined. The results including some analytic solutions of the nonlinear equation of motions are in very good agreement with experimental observation

Domain coarsening and interface kinetics in the Ising model

In this thesis, I investigate in detail two basic problems in nonequilibrium statistical mechanics. First, if a spin system such as a kinetic Ising model or a kinetic Potts model is quenched from supercritical temperature to subcritical temperature, how does the system coarsen, and what complexities arise as the system descends in energy toward one of its equilibrium states? Second, if a kinetic Ising model is evolved from a deterministic initial condition at zero temperature, how do the domain interfaces evolve in time? I first study the nonconserved coarsening of the kinetic spin systems mentioned above. The coarsening of a 2d ferromagnet can be described exactly by an intriguing connection with continuum critical percolation. Furthermore, careful simulations of phase ordering in the 3d Ising model at zero temperature reveal strange nonstatic final states and anomalously slow relaxation modes, which we explain in detail. I find similarly rich phenomena in the zero-temperature evolution of a kinetic Potts model in 2d, where glassy behavior is again manifest. We also find large-scale avalanches in which clusters merge and dramatically expand beyond their original convex hulls at late times in the dynamics. Next, I study the geometrically simpler problem of the evolution of a single corner interface in the Ising model. We extend prior work by investigating the Ising Hamiltonian with longer interaction range. We solve exactly the limiting shapes of the corner interface in 2d for several interaction ranges. In 3d, where analytical treatments are notoriously difficult, we develop novel methods for studying corner interface growth. I conjecture a growth equation for the interface that agrees quite well with simulation data, and I discuss the interface's surprising geometrical features. In the summary, I discuss the broader implications of our findings and offer some thoughts on possible directions for future work