# Master projects at SINTEF ICT

## General background

SINTEF ICT in Oslo has a research group that focuses on development of computational methods for simulation of flow and transport in subsurface porous rock formations. The primary applications are within CO2 storage and production of hydrocarbon. We work with industry-standard simulation technology as well as a large variety of specialized numerical numerical methods (multiscale methods, unstructured grids, grid coarsening, consistent discretizations, fast nonlinear solvers, open-source software, CO2 storage, streamline methods).

To enable other researchers to benefit from our efforts, we have compiled an open-source MATLAB toolbox (http://www.sintef.no/MRST) , which is released under the GNU General Public License (GPL). The toolbox has become quite popular and is used by hundreds of students and researchers all over the world. The group is also one of the key contributors to the OPM initiative (http://www.opm-project.org), whose main goal is to provide a suite of open-source simulation tools for porous media flow developed in C++. (Statoil is the primary industry partner).

In the following we will outline various project proposals that are all related to ongoing research activities in our group. In general, we look for students who are independent and hard-working and who want to be part of a thriving research group consisting of approximately ten researchers. Almost all the proposed projects will enable the student to work at the international research front.

Contact: Knut [dash] Andreas [dot] Lie [at] sintef [dot] no, Xavier [dot] Raynaud [at] sintef [dot] no

## Simulation and Machine Learning:

### Quantification of geological uncertainties

Subsurface geological data suffer from high uncertainty. In this master project, we propose to develop a methodology to directly assess the effect of the uncertainty in the geological data on the cumulative production. Fast in-house simulation tools based on reduced models will be made available to the student. They will be used to process a a large set of reservoir realizations and compute the corresponding outputs in term of production. The student will also be encouraged to develop his/her own methods based on machine learning to increase the efficiency of the simulations.

### Machine learning and parameter upscaling

The high computation time of fine resolution simulations severely limits the use of optimization algorithm for cumulative output such as the net present value of a oil reservoir. Proxi models, parameter upscaling, model reduction methods have been used to reduce the computation time while preserving acceptable accuracy. In this project, we invite the student to assess the use of machine learning in the design of proxi and parameter upscaling. The idea is to use a set of high-fidelity simulations as a training set in a machine-learning algorithm. Once the training phase is over, the machine should be able to compute very rapidly reliable approximations of the response of the system. This new simulation capability could then be used to significantly speed up optimization algorithms.

[1] S. Krogstad, X. Raynaud and H.M. Nilsen, *Reservoir management
optimization using well-specific upscaling and control switching* Comput Geosci
(2016)

[2] S. Trehan, K. Carlberg and L.J. Durlofsky, *Error estimation for surrogate
models of dynamical systems using machine learning*, Int. J. Numer. Meth. Engng
(2016)

### Mesh-less simulations of two phase flow.

More description coming …

## High-order conservative methods for unstructured grids

Heterogeneity of the rock formation naturally leads to geological models with irregular cells. In this project, we invite the student to study and implement high-order schemes for the pressure equation for irregular meshes. We will focus on schemes that are locally conservative, which means that the numerical fluxes satisfies a discrete conservation law. In this way, the high-order fluxes can directly be used in multiphase reservoir simulations.

[1] Ø. Klemetsdal. *The virtual element method as a common
framework for finite element and finite difference methods - Numerical and
theoretical analysis*. Master thesis NTNU, 2016.

[2] A. Ern, M. Vohralík *Four closely related equilibrated flux reconstructions
for nonconforming finite elements*, Comptes Rendus Mathematique, 2013

## Multiscale solvers:

Flow and transport in porous rock formations exhibit true multiscale behavior in the sense that the rock’s ability to transmit fluids can vary several orders of magnitude across short lengths. At the same time, the equation defining the pressure is elliptic/parabolic (depending on the model) and has global features. As a consequence, solving the pressure equation is often the limiting factor when solving large simulation problems. Multiscale methods have been proposed as a way to overcome this issue. The key idea is to construct a set of prolongation operators that map between unknowns associated with a fine underlying grid that describes the rock structure and unknowns on a coarser grid used for dynamic simulation. The prolongation operators (think of basis functions in finite-element methods) are computed numerically by solving localized flow problems. This way, local fine-scale variations can be systematically and correctly accounted for when constructing a reduced coarse-scale problem to study the macro-scale displacement driven by global forces.

Our group has been working with a variety of different multiscale methods with an emphasis on formulating methods suitable for general grids with high contrast media. Our main activity is currently funded by Schlumberger and focuses on formulating multiscale methods to be used in next-generation reservoir simulators.

Contact knut [dash] Andreas [dot] Lie [at] sintef [dot] no and Olav [dot] Moyner [at] sintef [dot] no if you are interested in writing a thesis on multiscale solver.

## Advanced solvers for specific applications

The equations governing flow in porous rocks consist of a coupled set of nonlinear PDEs that typically are discretized using a fully-implicit, finite-volume method. The most common way of solving the resulting nonlinear system of discrete equations is to use some variant of Newton-Raphson’s method, which is robust and has good convergence properties. To implement Newton’s method, one typically needs to compute Jacobians of residual equations (conservation of mass between computational cells, conservation of energy over interfaces etc). For systems of equations this can quickly become cumbersome even though the derivations are relatively simple. Automatic differentiation is a technology that, at runtime, automatically computes all relevant derivatives according to a simple set of rules. The result is solvers that are easy to modify and maintain. MRST and OPM both have libraries for automatic differentiation that makes it easy to switch between different solvers for both the linear and nonlinear problems.

### Mathematical modeling and numerical simulation of external and in-bed filtering for polymer injection

Polymer injection is one of the most effective strategies to improve recovery in highly viscous oil fields. By increasing the water viscosity, polymers prevent the formation of water channels in the reservoir and therefore increase the sweep efficiency. A major challenge in polymer injection is the permeability reduction that follows in the near-well region because of filtration processes. Polymers accumulate on the walls of the well, forming so-called filter cakes, which reduce significantly the well injectivity. Inside the formation, in-bed filtering, which occurs as large polymer molecules get trapped in the pores of the rock, causes similar effects. In this project, the student is invited to set up mathematical models for external and in-bed filtering, coupled with the polymer flow equations. Preferably, visco-elastic effects and polymer degradation which are known to have a significant effect on the injectivity, will be included. The implementation of the numerical methods to solve the modeling equations and test their validity will be an essential part in the project.

### Numerical simulation of fracture formations on irregular grids, using phase field and virtual element methods

A fracture is formed along a surface, when the stress that is exerted on the rock exceeds the energy required to break the cohesion of the rock along the surface. The fractures are two-dimensional objects and the difficulty is to identify the surfaces that eventually become the support of fractures. Phase-field methods bring a solution to this problem by introducing a variable field that measures the damage of the material and the location of fractures are then identified as the regions where the phase field variable takes its maximum possible value. Elliptic partial differential equations of second or fourth order are used to model the evolution of the phase field variable. In this project, the student is invited to study and implement numerical schemes to solve these partial differential equations and couple them with the linear elasticity equations that model the behavior of the unfractured rock. In order to obtain methods that are compatible with the grids that are used in reservoir simulation, the implementation will be done using virtual elements methods as they are able to handle irregular grids.

### Advanced nonlinear solvers

Our current Newton-Raphson solver can in theory be replaced by more specialized nonlinear solvers that use problem-specific strategies to accelerate convergence and stability. The purpose of the project is to study and adapt state-of-the-art nonlinear solvers to the reservoir simulation setting. The student should familiarize him/herself with the AD framework and the alternate solvers before doing an implementation and comparison. Examples of solvers include continuation Newton methods, methods that exploit causality structures in the equations, etc.

### Efficient linear solvers

The nonlinear solver must typically compute one or more linear systems at each iteration. Solving these problems usually dominate the runtime of the simulation and thus reducing the time taken is essential. Since we know the mathematical origins of the linear systems we need to solve, it is possible to use specialized linear solvers / preconditioners tailored to each problem. Some keywords here are domain decomposition, smoothers, elliptic solvers (such as multigrid) and strategies for combining these when the linear system exhibits multiple characteristics simultaneously.

### MRST laboratory simulator

Core-flooding laboratory experiments are essential to understand and evaluate the effectiveness of displacement mechanisms for improved and enhanced oil recovery. MRST and OPM have mainly been developed to simulate fluid displacement on the reservoir scale. The model equations are (almost) the same on the reservoir and the laboratory scale, but the latter scale poses some challenges (e.g., formulation of correct boundary conditions, radial grids, etc) that have so far not been thoroughly investigated in our software. In the project, the student will develop mathematical models for selected core experiments and use the rapid prototyping capabilities in MRST to develop a dedicated laboratory simulator that includes a graphical user interface and routines for upscaling effective parameters.

### MEOR

Microbial Enhanced Oil Recovery (MEOR) is a strategy for increased oil recovery which uses live microorganisms to alter the fluid properties in the reservoir. For example, one model considers that the bacteria can emulsify near the injection site, reducing the permeability of the surrounding rock. Implementing a MEOR model can include models for bacterial growth, transport of nutrition in the reservoir and a multitude of other effects. In the project, the student will study existing literature and implement some subset of the known models and study the effects on various flow scenarios.

## CO2 Sequestration

The net European CO2 emissions from energy industries, manufacturing, and production totaled 1.95 Gt in 2011. Ongoing Carbon Capture and Storage (CCS) operations typically inject 1-2 Mt per year. To make a significant impact on climate, one needs to significantly scale up the injection operations and store hundreds of megatonnes each year. Sedimentary basins offshore Norway contain a number of saline aquifers with large volumes of pore space potentially usable for CO2 storage of this magnitude. The Norwegian Petroleum Directorate (NPD) has released two CO2 Storage Atlases that explore large-scale CO2 storage for a number of aquifers. In total, twenty-seven geological formations have been grouped into aquifers whose qualities are assessed with regard to CO2 storage potential.

At SINTEF, we have developed MRST-co2lab (http://www.sintef.no/co2lab) which offers a set of novel simulation tools that can be used to simulate likely outcomes of large-scale, long-term migration processes and estimate capacity for large-scale structural, residual, and solubility trapping. The tools have, in particular, been used to develop plausible plans for injecting hundreds of megatonnes of CO2 into the Utsira and Skade formations. However, the studies performed so far have been cursory and only intended to demonstrate the capabilities of the numerical methods. In the project, the student(s) will perform a more in-depth study that includes critical evaluation of various model assumptions (boundary conditions, etc) and focus on providing capacity estimates with a high degree of realism. More details of possible project topics are summarized below.

### Optimal well placement and long-term migration forecast:

As CO2 is less dense than resident brine, long-term migration within a geological storage formation is primarily driven by gravity, and thus largely determined by the geometrical shape of the upper confining layer of the formation. Based on this principle, simple geometry-based methods have been implemented in MRST-co2lab to forecast CO2 migration within an aquifer. Moreover, analytical solutions can be used to estimate the radial distance of the CO2 plume from its injection point, and may be used to support appropriate well placement algorithms that exploit the trapping capacity of the aquifer. The combination of these analytical solutions with migration forecasting can be used to generate back-of-the-envelope CO2 storage potentials without performing any numerical simulation. In this project, the student(s) will help to implement this approach and then assess the quality of these high-level estimates through comparison to simulation results.

### Optimal injection strategies with uncertainty quantification:

Based on rapid simulation models (VE-models), forecast models for long-term migration, nonlinear numerical optimization routines and real models of potential storage formations in the North Sea, it is possible to assess realisable storage capacities for given initial assumptions and constraints. Optimal usage scenarios are defined by placement of multiple injection wells, extraction wells (to relieve pressure), and injection/extraction patterns including optimal rates. Using this approach, practical storage capacities for formations found along the Norwegian Continental Shelf have been presented in recent publications using MRST-co2lab (10.1016/j.egypro.2016.01.033). These estimates were obtained using datasets provided by the NPD, which typically include homogeneous rock properties, etc. In this project, the student(s) will further this work, including the impact of uncertainty on such storage estimates by considering a range of rock/fluid properties, cap-rock shape, grid resolution, etc.

## Domain-specific languages for scientific computing

Equelle is a domain-specific language for the specification of simulators for systems of PDEs through a high-level syntax. The language allows the user to focus on equations and numerics while hiding the low-level details of software and hardware implementations. The language has been developed at SINTEF since 2013. See http://equelle.org for more information about the language itself.

### Optimization of arithmetic expression sequences

A performance-limiting aspect of the current implementation of Equelle, is that sequences of arithmetic expressions run up against memory-access limitations. For each read/write operations ideally many arithmetic operations should be executed. Currently Equelle generates code that only does a few or one arithmetic operation per memory access. Addressing this issue can be done using techniques such as expression templates, by changing the expression tree generated in the compiler, or a mixture of such methods.