## Miscible displacement

**COMP-M** has developed efficient algorithms for modeling one-component single-phase flow in porous media. The motion of a miscible fluid through the rock is also called **miscible displacement**. Modeling this complex fluid flow is crucial to applications related to both Energy and Environment. For instance, production of hydrocarbons is increased by the use of enhanced oil recovery techniques, which is based on the injection of miscible fluids in the reservoirs.

The incompressible miscible displacement problem is governed by a system of nonlinear coupled partial differential equations. The fluid mixture moves as a single phase modeled by Darcy’s law. The first equation in the system is the pressure equation. The second equation in the system is a transport equation with a diffusion-dispersion matrix coefficient that depends nonlinearly on the velocity. The unknowns are the fluid mixture pressure and velocity as well as the concentration of the solvent fluid. The miscible displacement problem has several numerical challenges. The equations are coupled via the fluid viscosity for the pressure equation and via the velocity and diffusion-dispersion coefficients for the concentration equation. One important challenge is that the diffusion-dispersion matrix is not bounded, and thus the convergence analysis of the numerical method for the concentration equation is difficult. Most of the published literature on the analysis of algorithms simply assumes that the diffusion-dispersion coefficient is uniformly bounded above, independently of the velocity. In our most recent work, we do not make this boundedness assumption and we refer to this problem as **the miscible displacement problem with low regularity**. We proved convergence of a high order scheme (both in time and in space) of the numerical solution to the weak solution as the mesh size and time step tend to zero. The underlying discretizations are interior penalty discontinuous Galerkin (IPDG) methods. We also analyzed the scheme combining IPDG and mixed finite element methods.

Viscous fingering in porous media characterizes the flow instability phenomenon that occurs when a fluid with low viscosity is used to displace a fluid with high viscosity. As the result of the viscosity difference, a tiny perturbation can be amplified exponentially, which triggers a finger-like pattern in the fluid concentration profile during the fluid displacement. **COMP-M** has simulated the viscous fingering fluid instability during the miscible displacement process in porous media, on structured and unstructured grids. The numerical model incorporates decoupling in time, discontinuous Galerkin method of high order, flux reconstruction, and parallel implicit solvers to produce an accurate and efficient predictive tool for finger growth. The numerical model does not suffer from grid orientation, accurately measures finger growth rate.

Our publications include discussions on the effect of grid orientation and anisotropic permeability using high order discontinuous Galerkin method in contrast with cell-centered finite volume method. The study of the parallel implementation shows the scalability and efficiency of the method on parallel architecture. We also verify the simulation result on highly heterogeneous permeability field from the SPE10 model.

## Coupled free-flows with porous media flows

Multiphysics problems are characterized by complex physical processes occurring in different spatial regions. In order to model different physics, the computational domain is subdivided into several subdomains, in which different types of flow and transport phenomena are investigated. The basic mathematical equations are derived from balance equations of continuum mechanics that express conservation laws for mass, momentum, and energy of an arbitrary volume moving with the fluid. Interface conditions are important in capturing the right physics. We assume continuity of normal component of velocity, balance of forces across the interface and the Beavers-Joseph-Saffman condition.

**COMP-M** has developed convergent numerical models that couple free flows with porous media flows. The free flow region is characterized by the steady-state incompressible Navier-Stokes equations whereas the porous media region is modeled by single phase flow. This project helps track pollutants that leak in rivers and lakes and eventually reach groundwater. We proved convergence of various schemes by deriving explicit a priori error estimates; these schemes are the standard IPDG schemes, the classical finite element method and the more recent strongly conservative numerical methods that use divergence-conforming spaces for the fluid velocities. Two-grid solutions are compared with monolithic solutions and we showed that the computational time for the two-grid solution is significantly smaller than the time for the fully coupled solution.

Our group has also obtained existence of weak solutions to these coupled flow problems, for both steady-state and time-dependent cases.

Our most recent contribution to the topic is the formulation and analysis of a numerical scheme for the coupled time-dependent Navier-Stokes equations with single phase flow. The time dependence makes the theoretical error analysis more challenging.

## Pore-scale modeling

Phase-field models are becoming increasingly popular in hydrodynamics for modeling multi-phase fluid flow below Darcy scale. Instead of sharp interfaces, phase transitions appear as diffuse finite-thickness transition regions. The Cahnâ€“Hilliard equation describes phase separation (the alignment of a system into spatial domains predominated by one of the two components) of an immiscible binary mixture at constant temperature in the presence of a mass constraint and dissipation of free energy. It is a stiff, fourth-order, nonlinear parabolic partial differential equation, which may serve as a prototype phase-field problem as intermediate step towards models that take other or additional phenomena into account, e.g., miscibility or multiple components. The wettability characters of the solid surface is reflected by a contact angle between the diffuse interface and the surface.

The Cahn-Hilliard equations are coupled with incompressible time-dependent Navier-Stokes equations to simulate immiscible two-phase flow in the rock at the pore scale.

## Modeling hemodynamics and congenital heart defect

**COMP-M** has developed and analyzed mathematical models of blood flow in whole-body circulatory system that includes major organs and blood vessels, as well as the heart and its valves. The mathematical models have been applied to modeling of single ventricle heart defects, such as **hypoplastic left heart syndrome** (HLHS). Left untreated, HLHS is fatal. Even with surgical repair, the mortality rate is 15%. Children with HLHS are hemodynamically unstable as a result of mixing between oxygenated and de-oxygenated blood, which often results in cardiorespiratory arrest.

## Incompressible Navier-Stokes

**COMP-M** has formulated and analyzed interior penalty discontinuous Galerkin methods for solving the incompressible Navier-Stokes equations, both steady-state and time-dependent. We have obtained inf-sup condition for the discontinuous polynomial spaces of degree k for velocity and degree k-1 for pressure. We have developed splitting schemes as well as sub grid Eddy viscosity schemes. By enriching the discrete space with bubbles, we have obtained a convergent DG scheme that is penalty-free.

Our most recent contribution to the development of stable methods is the continuous P1 element with piecewise constants for velocity, combined with piecewise constants for pressure. The resulting method is referred to as the enriched Galerkin method. We have formulated a stable and convergent scheme.

## Intestinal edema

Intestinal edema refers to the excess accumulation of fluid in the interstitial spaces of the intestinal wall tissue. This condition can arise in patients with gastroschisis, inflammatory bowel disease and cirrhosis, as well as in patients receiving resuscitative fluid treatments after traumatic injuries. The main problem for a patient with intestinal edema is that the condition causes ileus, a decrease in intestinal transit due to decreased intestinal smooth muscle contractility. Decreased intestinal transit often leads to longer hospital stays and recovery times for patients and in extreme cases can be fatal. The link between edema and ileus is unknown, and is thus the motivation for developing mathematical models to explore this phenomenon.

Our numerical model uses the poroelasticity equations and allows for heterogeneities in the intestine,

which corresponds to the different layers of the intestine (mucosa, submucosa, muscularis externa, serosa). The rate at which fluid enters the interstitium via the vascular system and the rate at which fluid is removed from the interstitium by the lymphatic system

are given by the Starling-Landis equation and the Drake-Laine equation.

**COMP-M** has developed robust numerical models of the intestine, that have been validated by experimental data.

New results include an improved view into the effects of resuscitation on the hydrostatic pressure profile of edematous rats, effects on lumenal volume attenuation, relative fluid gain, and an estimation of the impacts of both acute edema, and resuscitation, on intestinal motility.

## Multiphase Flow in Porous Media

Multiphase flows in porous media occur in the production of hydrocarbons from reservoirs, or in the clean-up of contaminated groundwater sites. The mathematical models are coupled nonlinear systems of partial differential equations that are challenging to solve. Some numerical challenges are: heterogeneity of the porous media, degeneracy of the relative mobilities and mass transfer between phases. For instance, absolute permeability can vary over several orders of magnitude in a geological formation.

**COMP-M** has developed high order and locally mass conservative methods for incompressible two-phase, three-phase and black-oil models. Our publications show the advantages of using discontinuous polynomial approximations of high degree. The spatial discretization is based on interior penalty discontinuous Galerkin methods, or on hybridizable discontinuous Galerkin methods. For time stepping, we have compared sequential and fully implicit formulations. For two-phase flow, we have analyzed and implemented several schemes using any combination of wetting and non-wetting phase pressure and phase saturation. Our method for solving black-oil uses for primary unknowns, the liquid phase pressure, the aqueous phase saturation and the total mass fraction of gas component.

## Coupled flow and geomechanics

Biot’s linear poroelasticity model is used in the oil and gas industry to simulate the interactions between the fluid flow and the deformation of the rock. We have proposed decoupled and sequential numerical methods based on either finite element methods or discontinuous Galerkin methods. Unlike other splitting approaches, our method is not iterative, which results in a speed-up of the computational time. We have obtained a convergence analysis and we have applied the method to large reservoirs.