## Computational and Applied Mathematics Seminar

## Location and Time

University of Wyoming Ross Hall 247, Fridays from 3:10-4:00 (unless otherwise stated).

## Support

The CAM seminar series is currently supported through volunteers and the financial contributions by the UW Mathematics Department, MGNet.org, and and an energy grant from ExxonMobil. The speaker's entire expenses are paid by the sponsor(s) if noted.

## Schedule

For Fall 2016, the speakers are as follows:

Date Speaker From/Note September 9 Xiukun Hu University of Wyoming September 20** Mookwon Seo University of Wyoming October 6*** ^{++}Hong Wang University of South Carolina October 14 ^{+}Dongwoo Sheen Seoul National University October 21 Zachary J. Lebo University of Wyoming October 28 ^{+}Xiaoming He Missouri University of Science and Technology November 4 Man-Chung Yeung University of Wyoming November 18 Zhangxin Chen University of Calgary November 21* Jerry Schuster King Abdullah University of Science & Technology and University of Wyoming * Monday (not Friday) and joint with the Analysis Seminar, **Tuesday in RH 241, ***Thursday in AG 1032

Funded by

^{+}MGNet.org,^{++}an Energy grant from ExxonMobilI am looking for speakers for Spring 2017!

The schedule, titles, and abstracts from Spring 2016 are here.

## Titles and Abstracts

## September 9

## Inversion of Seismic Data

Xiukun Hu, Department of Mathematics, University of WyomingIn this talk, I will give an introduction to seismic inversion. I will mainly talk about migration, which is a process to relocate seismic events in data domain to the underground location. Three different migration methods will be discussed including Diffraction Migration, Reverse Time Migration, and Least Square Migration. Full Waveform Inversion (FWI) then will be derived. We will discuss briefly the problems FWI may bring and how people in the field deal with it.

## September 20

## Alternative Models for Water Infiltration and Oil Reservoirs in the Ground

Mookwon Seo, Department of Mathematics, University of WyomingThis presentation has two main topics. One is for the water flow equation in unsaturated porous medium. Another is for water flooding models in subsurface porous medium filled with oil.

For the first topic, Richards equation is generally used to estimate isothermal flows through an incompressible, variably saturated porous medium. However, Richards equation has a term that causes difficulties in two important cases: a water-saturated soil and for sharp infiltration fronts. We include physical reasons of those problems and derive from basic principles an alternative equation for Richards equation that does not include the problematic term. We also show how the alternative equation differs from Richards equation and demonstrate that the alternative model can be solved numerically in a far superior and computationally efficient manner that is one or two orders of magnitude faster than the common Richards equation solver Hydrus-1D on sample problems. A new fast algorithm is described in detail. In order to illustrate the new model's realism, we compare its predictions in simulations with a fixed water table 1 m below the land surface in Panama and Singapore from real rainfall data. The history matching clearly demonstrates the superiority of our new method.

In the second topic, we introduce an alternative water flooding model for oil reservoirs, which is similar to the Buckley-Leverett model. The Buckley-Leverett model is a two-phase flow equation with zero capillary pressure. A new numerical model of the alternative water flooding model is described using an one-dimensional memory allocation technique similar to the one in the first topic. To verity the alternative model, we compare the analytic solution of the Buckley-Leverett equation, a numerical solution of the Buckley-Leverett equation using the implicit pressure, explicit saturation formulation (IMPES) with a finite difference discretization, and the implementation of the alternative water flooding model. Once again, we can demonstrate clear superiority over standard methods.

## October 6

## Fractional Partial Differential Equations: Modeling, Numerical Methods and Mathematical Analysis

Prof. Hong Wang, Department of Mathematics, University of South CarolinaTraditional second-order diffusion PDEs model Fickian diffusion processes, in which the particles follow Brownian motion. However, many diffusion processes were found to exhibit anomalous diffusion behavior, in which the probability density functions of the underlying particle motions are characterized by an algebraically decaying tail and so cannot be modeled properly by second-order diffusion PDEs.

Fractional PDEs provide a powerful tool for modeling these problems, as the probability density functions of anomalous diffusion processes satisfy these equations. Fractional PDEs present new difficulties that were not encountered in the context of integer-order PDEs. Computationally, the numerical methods for space-fractional PDEs generate dense matrices. Direct solvers were traditionally used, which require O(

N^{3}) computations per time step and O(N^{2}) memory, whereNis the number of unknowns.Mathematical difficulties include the loss of coercivity of the Galerkin formulation for variable-coefficient problems, non-existence of the weak solution to inhomogeneous Dirichlet boundary-boundary value problems, and low regularity (the solution to homogeneous Dirichlet boundary-value problem of a one-dimensional fractional PDE with constant coefficient and source term is not in the Sobolev space

H^{1}.We present the recent developments of accurate and efficient numerical methods for space-fractional PDEs, which has optimal storage and almost linear computational complexity. We will also address mathematical issues such as well-posedness and regularity of the problems and their impact on the convergence behavior of numerical methods.

## October 14

## A Hybrid Two-Step Finite Element Method for Flux Approximation

Prof. Dongwoo Sheen, Department of Mathematics, Seoul National UniversityWe present a new two–step method based on the hybridization of mesh sizes in the traditional mixed finite element method. On a coarse mesh, the primary variable is approximated by a standard Galerkin method, whose computational cost is very low. Then, on a fine mesh, an

H(div) projection of the dual variable is sought as an accurate approximation for the flux variable.Our method does not rely on the framework of traditional mixed formulations, the choice of pair of finite element spaces is, therefore, free from the requirement of the inf-sup stability condition. More precisely, our method is formulated in a fully decoupled manner, still achieving an optimal error convergence order. This leads to a computational strategy much easier and wider to implement than the mixed finite element method. Additionally, the independently posed solution strategy allows us to use different meshes as well as different discretization schemes in the calculation of the primary and flux variables.

We show that the finer mesh size

hcan be taken as the square of the coarse mesh sizeH, or a higher order power with a proper choice of parameter δ. This means that the computational cost for the coarse-grid solution is negligible compared to that for the fine-grid solution. In fact, numerical experiments show an advantage of using our strategy compared to the mixed finite element method. Some guidelines to choose an optimal parameter δ are also given. In addition, our approach is shown to provide an asymptotically exact a posteriori error estimator for the primary variablepinH^{1}norm.## October 21

## Aerosols and Deep Convective Clouds: Have We "Resolved" the Problem?

Prof. Zachary J. Lebo, Atmospheric Science Department, University of WyomingThe potential effects of changes in aerosol loading on deep convective cloud systems have received considerable attention in the recent literature, focusing on the response in precipitation, storm strength, lightning frequency, etc. In this talk, a review of the responses will be presented, focusing on two potential microphysical pathways by which an aerosol perturbation may modify the cloud and dynamical characteristics (i.e., the conventional “invigoration” mechanism and the modification of cold pool strength and/or structure). A detailed comparison between the simulated effects due to changes in aerosol loading and changes in environmental characteristics will be discussed; the effects due to changes in aerosol loading will be shown to be masked by even small changes in environmental characteristics. This conclusion suggests that observing such effects will be extremely difficult. Furthermore, the sensitivity of the model results to grid spacing will be explored and analyzed using a novel approach; this is an important consideration because many of the effects of changes in the aerosol loading are likely reliant on the entrainment/detrainment characteristics of deep convective cloud systems. Lastly, recent numerical solutions will be presented to confirm the cloud-resolving model simulations and pinpoint the processes in which deep convective cloud characteristics are likely to be most susceptible to changes in aerosol loading. These numerical solutions combined with the 3D cloud-resolving simulations will demonstrate that the “invigoration” mechanism is insignificant for strong deep convective cloud systems.

## October 28

## Dual-Porosity-Stokes Model and Finite Element Method for Coupling Dual-Porosity Flow and Free Flow

Prof. Xiaoming He, Department of Mathematics and Statistics, Missouri University of Science and TechnologyWe propose and numerically solve a new model considering confined flow in dual-porosity media coupled with free flow in embedded macro-fractures and conduits. Such situation arises, for example, for fluid flows in hydraulic fractured tight/shale oil/gas reservoirs. The flow in dual-porosity media, which consists of both matrix and micro-fractures, is described by a dual-porosity model. And the flow in the macro-fractures and conduits is governed by the Stokes equation. Then the two models are coupled through four physically valid interface conditions on the interface between dual-porosity media and macro-fractures/conduits, which play a key role in a physically faithful simulation with high accuracy. All the four interface conditions are constructed based on fundamental properties of the traditional dual-porosity model and the well-known Stokes-Darcy model. The weak formulation is derived for the proposed model and the well-posedness of the model is analyzed. A finite element semi-discretization in space is presented based on the weak formulation and four different schemes are then utilized for the full discretization. The convergence of the full discretization with backward Euler scheme is analyzed. Four numerical experiments are presented to validate the proposed model and demonstrate the features of both the model and numerical method, such as the optimal convergence rate of the numerical solution, the detail flow characteristics around macro-fractures and conduits, and the applicability to the real world problems.

## November 4

## A Spectral Projection Preconditioner for Solving Ill Conditioned Linear Systems

Prof. Man-Chung Yeung, Department of Mathematics, University of WyomingWe present a preconditioner based on spectral projection that is combined with a deflated Krylov subspace method for solving ill conditioned linear systems of equations. Our results show that the proposed algorithm requires many fewer iterations to achieve the convergence criterion for solving an ill conditioned problem than a Krylov subspace solver. In our numerical experiments, the solution obtained by the proposed algorithm is more accurate in terms of the norm of the distance to the exact solution of the linear system of equations.

## November 18

## Unconventional Oil and Gas Reservoir Modeling and Simulation

Prof. Zhangxin Chen, Mathematics Department, University of CalgaryMathematical models have widely been used to predict, understand, and optimize complex physical processes in modeling and simulation of multiphase fluid flow in petroleum reservoirs. These models are important for understanding the fate and transport of chemical species and heat. With this understanding the models are then applied to the needs of the petroleum industry to design enhanced oil and gas recovery strategies.

While mathematical modeling and computer simulation have been successful in their applications to the recovery of conventional oil and gas, there exist a lot of challenges in their applications to unconventional oil and gas modeling. As conventional oil and gas reserves dwindle, the recovery of unconventional oil and gas resources (such as heavy oil, oil sands, tight oil and gas, and shale oil and gas) is now the center stage. For example, enhanced heavy oil/oil sands recovery technologies are an intensive research area in the petroleum industry, and have recently generated a battery of recovery methods, such as cyclic steam stimulation (CSS), steam assisted gravity drainage (SAGD), vapor extraction (VAPEX), in situ combustion (ISC), hybrid steam-solvent processes, and other emerging recovery processes; horizontal well and hydraulic fracturing technologies have been very successful in the production of tight and shale oil and gas reservoirs. This presentation will give an overview on challenges encountered in modeling and simulation of unconventional oil and gas reservoirs. It will also present some case studies for applications of recovery processes to heavy oil and shale gas reservoirs.

## November 21

## The Boom and Bust Cycles of Full Waveform Inversion: Is FWI a Bust, a Boom, or Becoming a Commodity?

Prof. Gerard Schuster, King Abdullah University of Science & TechnologyThe history of seismic inversion is similar to that of the US economy: periodic boom cycles followed by downturns. Seismic waveform inversion started with a bang in the early 1980s with the theoretical foundations established by Albert Tarantola and his colleagues in France. Tarantola’s theory claimed that all of the waveforms in recorded seismic traces can be inverted to give unprecedented images of the Earth’s geological properties. This claim seemed to be numerically verified by Peter Mora at Stanford University who showed dramatic velocity and density tomograms after applying full waveform inversion (FWI) to synthetic data. This accelerated the boom cycle to a peak in 1990 with the promises of complete inversion of the Earth’s geological properties by the FWI black box. At an SEG talk, Mora spectacularly predicted that computers in the near future would one day simulate wave propagation in 3D earth models faster than the waves propagated in the earth, implying fast and reliable inversion of the earth’s parameters from seismic data. This bubble of excitement collapsed in dramatic fashion by 1992, mainly because FWI did not reliably work on field data. In addition, the 1990 computers were not yet up to the expensive task of applying FWI to large data sets. From 1992 to about 2006, FWI underwent a dramatic recession with an FWI unemployment rate of more than 90%. Around 2009, FWI started a dramatic rebirth with work by both BP and Exxon that showed FWI can yield a profitable return in producing reliable velocity tomograms. Moreover, Exxon showed that this could be done in a cost-effective way using multisource FWI where a large blended suite of N>>3 shot gathers could be imaged with just three forward modeling operations. Thus began the latest boom cycle of FWI with almost full employment that continues today. Is there more hype than substance? How long will the FWI boom last? Is it too late to invest your research efforts and make a profit? Or is FWI becoming a commodity? Did Mora’s prediction come true that computers can simulate 3D wave propagation faster than they actually propagate in the Earth? Come to this talk to hear the latest forecasts.