Numerical simulations of viscoplastic flow for wellbore cementing applicationsGebäude 33, Raum 0401 (Werner-Heisenberg-Weg 39, 85577 Neubiberg)

When drilling an oil well, cement provides a barrier to unwanted hydrocarbon flow, isolates permeable zones in the foundation, and supports and protects the borehole casing. Many challenges exist when displacing cement into position, and numerical simulation is often used to predict flow patterns and fluid interfaces. Due to challenging well conditions and non-Newtonian (viscoplastic) rheological models used to describe the nature of the fluids, highly accurate numerical solutions are difficult to obtain in an efficient manner. This constitutes a significant challenge for ensuring safe and efficient analysis of a vital drilling operation.

We aim to address the aforementioned issues by building a new cementing simulation tool. In addition to revisiting the underlying mathematical formulation of the physical problem, the key challenges associated with the project are modelling of viscoplastic rheology; performing simulations in a low-speed flow regime; treatment of nontrivial, time-dependent geometries; capturing interfaces between displacing and displaced fluids accurately; and the achievement of fast run times through massive parallelisation and adaptive mesh refinement. The latter is necessary due to the disparate length and time scales of the problem at hand.

In this talk, I will introduce the problem at hand, before going into some more detail on the choice of computational methods employed. Specifically, I will elucidate how viscoplasticity can be treated using regularisation techniques; why the level-set algorithm is suitable for the fluid interfaces; and how we gained access to adaptive mesh refinement and state-of-the-art parallelisation on large clusters through the BoxLib/AMReX code libraries.

Sampling, Embedding and Inference for Lévy-driven CARMA ProcessesBC1 2.01.10 (Parkring 11, 85748 Garching)

If Y is a CARMA (continuous time ARMA) process, determination of the Δ-sampled sequence is of particular importance because the observed values of the process Y are often available only at regularly spaced time intervals. The exact determination of the sampled ARMA sequence for low-order CARMA processes is discussed, together with the converse problem of finding (when possible) a CARMA process from which a given ARMA process can be regarded as the Δ-sampled sequence. These questions and their implications for inference from regularly spaced observations of a CARMA process are discussed, together with inference based on unequally spaced data.

Path regularity of the solution to the stochastic heat equation with Lévy noiseBC1 2.01.10 (Parkring 11, 85748 Garching)

We investigate the path properties of the solution to the stochastic heat equation with Lévy noise. Viewed as a function-valued stochastic process in time, we establish the existence of a càdlàg modification of the solution in certain fractional Sobolev spaces. Furthermore, also the regularity of sections, for fixed time or fixed space, is analyzed. In both cases, we find critical values such that noises with a smaller Blumenthal-Getoor index typically lead to continuous sections, while noises with a larger Blumenthal-Getoor index typically lead to sections that are unbounded on any non-empty open subset. (This is joint work with Robert Dalang and Thomas Humeau, both at EPFL)

Site percolation in high dimensionsB 252 (Theresienstr. 39, 80333 München)

Percolation usually studies the random sub-lattice consisting of occupied bonds. In contrast to the majority of the literature, we study site percolation and restrict attention to the hypercubic lattice Zd, where each site is occupied with probability p 2 [0; 1]. A key result in high-dimensional percolation is the so-called infrared bound that has been proven by Hara and Slade in 1990 for bond percolation. The aim of this presentation is to explain how the infrared bound can be proven for site percolation. The proof makes use of a combinatorial expansion technique, the so-called percolation lace-expansion. Our lace-expansion analysis covers the nearest-neighbor model.

The critical 1-arm exponent for the Ising model on Cayley treesB 252 (Theresienstr. 39, 80333 München)

The ferromagnetic Ising model is one of the most extensively studied models from statistical mechanics. Here, we consider a regular tree as the underlying graph. We consider subtrees of depth n with fixed plus-valued boundary spins and investigate the expected spin value of the root of the tree with respect to the Gibbs measure on the subtree. It is well known that in the absence of an external field, the model undergoes a phase transition. At critical temperature, the influence of a plus boundary condition fades away when we take the limit in the distance between the root and the boundary of the subtree, i.e. the expected root spin is converging towards zero for n to infinity. Our main goal is to quantify the rate of this convergence. A short introduction to the Ising model will be given at the beginning of the talk.