arXiv subjects

Per Pettersson

Publications and source records attributed to Per Pettersson.

11 recordsLinked to original sources

arXiv2026

Risk sharing in cooperative game models for CO$_2$ storage with uncertain geology and pressure competition

With an increasing number of prospective geological CO$_2$ storage projects and potential pressure communication between different projects, risk sharing under uncertain geological conditions are relevant to many project operators. In this work, the project operators are modeled as agents in a stochastic cooperative game. The agents can have different risk attitudes, here defined as being willing to accept more or less uncertainty in the predicted storage of CO$_2$. This uncertainty stems from lack of knowledge of geological parameters as well as unknown future actions of competing agents, and the corresponding probability distributions need to be estimated by numerical simulation. The agents can choose to share commercial risk if collaboration is preferable to a baseline scenario of individual injection. If their operations affect each other by means of, e.g., pressure communication, there may be no unique natural definition of a baseline scenario. As a remedy, we suggest belief distributions that combine uncertainty in physical data with maximum entropy prior distributions over the sets of viable injection actions. For a realistic storage site, exemplified by the Utsira Formation in the North Sea, we present numerical results for both cases of pressure competition, and no hydraulic connections between different project operations. It is shown that risk averse agents benefit from collaboration when there is no pressure communication or other interference between agents. It is also demonstrated that pressure communication leads to large variability in the feasible injection rates, but the resulting belief distributions are nevertheless informative and useful for decision making about collaboration.

physics.soc-ph↗

arXiv2026

A high-order, structure preserving scheme for the stochastic Galerkin shallow water equations -- unification and two-dimensional extension

Recently, two independent research efforts have been made to study the stochastic Galerkin formulation of the shallow water equations. Bender and \"Offner developed entropy-conservative discontinuous Galerkin (DG) methods to solve the stochastic shallow water equations in a stochastic Galerkin framework using Roe variable transformation, while Dai, Epshteyn and collaborators proposed second-order, energy-stable and well-balanced schemes for the same class of problems with a specific projection step used inside the Galerkin projection together with high-order quadrature rules and a time-step restriction. In this paper, we provide a comprehensive comparison of the two methodologies mentioned, focusing on their theoretical properties and practical implementation aspects. We highlight shared foundational concepts and key differences of both approaches, with a particular focus on the selection of basis functions in the stochastic domain. As a highlight, we show that under specific conditions, the two formulations align, offering a unified framework that connects these distinct approaches. From our theoretical findings, we extend the development of high-order entropy conservative DG methods for the one-dimensional stochastic Galerkin shallow equations to two space dimensions; constructing entropy conservative two-point fluxes via primitive variables instead of entropy variables and applying it in our high-order DG setting. In numerical simulations, we verify and support our theoretical findings of a well-balanced and entropy-stable DG scheme which can be used to solve geophyiscal fluid flows with uncertainty.

math.NA↗

arXiv2025

Cooperative games defined by multi-objective optimization in competition for subsurface resources

We propose a novel decision making framework for forming potential collaboration among otherwise competing agents in subsurface systems. The agents can be, e.g., groundwater, CO$_2$, or hydrogen injectors and extractors with conflicting goals on a geophysically connected system. The operations of a given agent affect the other agents by induced pressure buildup that may jeopardize system integrity. In this work, such a situation is modeled as a cooperative game where the set of agents is partitioned into disjoint coalitions that define the collaborations. The games are in partition function form with externalities, i.e., the value of a coalition depends on both the coalition itself and on the actions of external agents. We investigate the class of cooperative games where the coalition values are the total injection volumes as given by Pareto optimal solutions to multi-objective optimization problems subject to arbitrary physical constraints. For this class of games, we prove that the Pareto set of any coalition structure is a subset of any other coalition structure obtained by splitting coalitions of the first coalition structure. Furthermore, the hierarchical structure of the Pareto sets is used to reduce the computational cost in an algorithm to hierarchically compute the entire Pareto fronts of all possible coalition structures. We demonstrate the framework on a pumping wells groundwater example, and nonlinear and realistic CO$_2$ injection cases, displaying a wide range of possible outcomes. Numerical cost reduction is demonstrated for the proposed algorithm with hierarchically computed Pareto fronts compared to independently solving the multi-objective optimization problems.

math.OC↗

arXiv2024

Multi-objective optimization for multi-agent injection strategies in subsurface CO$_2$ storage

We propose a novel framework for optimizing injection strategies in large-scale CO$_2$ storage combining multi-agent models with multi-objective optimization, and reservoir simulation. We investigate whether agents should form coalitions for collaboration to maximize the outcome of their storage activities. In multi-agent systems, it is typically assumed that the optimal strategy for any given coalition structure is already known, and it remains to identify which coalition structure is optimal according to some predefined criterion. For any coalition structure in this work, the optimal CO$_2$ injection strategy is not a priori known, and needs to be found by a combination of reservoir simulation and a multi-objective optimization problem. The multi-objective optimization problems all come with the numerical challenges of repeated evaluations of complex-physics models. We use versatile evolutionary algorithms to solve the multi-objective optimization problems, where the solution is a set of values, e.g., a Pareto front. The Pareto fronts are first computed using the so-called weighted sum method that transforms the multi-objective optimization problem into a set of single-objective optimization problems. Results based on two different Pareto front selection criteria are presented. Then a truly multi-objective optimization method is used to obtain the Pareto fronts, and compared to the previous weighted sum method. We demonstrate the proposed framework on the Bjarmeland formation, a pressure-limited prospective storage site in the Barents Sea. The problem is constrained by the maximum sustainable pressure buildup and a supply of CO$_2$ that can vary over time. In addition to identifying the optimal coalitions, the methodology shows how distinct suboptimal coalitions perform in comparison to the optimum.

math.NA↗

arXiv2023

Copula modeling and uncertainty propagation in field-scale simulation of CO$_2$ fault leakage

Subsurface storage of CO$_2$ is an important means to mitigate climate change, and to investigate the fate of CO$_2$ over several decades in vast reservoirs, numerical simulation based on realistic models is essential. Faults and other complex geological structures introduce modeling challenges as their effects on storage operations are uncertain due to limited data. In this work, we present a computational framework for forward propagation of uncertainty, including stochastic upscaling and copula representation of flow functions for a CO$_2$ storage site using the Vette fault zone in the Smeaheia formation in the North Sea as a test case. The upscaling method leads to a reduction of the number of stochastic dimensions and the cost of evaluating the reservoir model. A viable model that represents the upscaled data needs to capture dependencies between variables, and allow sampling. Copulas provide representation of dependent multidimensional random variables and a good fit to data, allow fast sampling, and coupling to the forward propagation method via independent uniform random variables. The non-stationary correlation within some of the upscaled flow function are accurately captured by a data-driven transformation model. The uncertainty in upscaled flow functions and other parameters are propagated to uncertain leakage estimates using numerical reservoir simulation of a two-phase system. The expectations of leakage are estimated by an adaptive stratified sampling technique, where samples are sequentially concentrated to regions of the parameter space to greedily maximize variance reduction. We demonstrate cost reduction compared to standard Monte Carlo of one or two orders of magnitude for simpler test cases with only fault and reservoir layer permeabilities assumed uncertain, and factors 2--8 cost reduction for stochastic multi-phase flow properties and more complex stochastic models.

math.NA↗

arXiv2023

Sequential Estimation using Hierarchically Stratified Domains with Latin Hypercube Sampling

Quantifying the effect of uncertainties in systems where only point evaluations in the stochastic domain but no regularity conditions are available is limited to sampling-based techniques. This work presents an adaptive sequential stratification estimation method that uses Latin Hypercube Sampling within each stratum. The adaptation is achieved through a sequential hierarchical refinement of the stratification, guided by previous estimators using local (i.e., stratum-dependent) variability indicators based on generalized polynomial chaos expansions and Sobol decompositions. For a given total number of samples $N$, the corresponding hierarchically constructed sequence of Stratified Sampling estimators combined with Latin Hypercube sampling is adequately averaged to provide a final estimator with reduced variance. Numerical experiments illustrate the procedure's efficiency, indicating that it can offer a variance decay proportional to $N^{-2}$ in some cases.

stat.ME↗

arXiv2021

Adaptive stratified sampling for non-smooth problems

Science and engineering problems subject to uncertainty are frequently both computationally expensive and feature nonsmooth parameter dependence, making standard Monte Carlo too slow, and excluding efficient use of accelerated uncertainty quantification methods relying on strict smoothness assumptions. To remedy these challenges, we propose an adaptive stratification method suitable for nonsmooth problems and with significantly reduced variance compared to Monte Carlo sampling. The stratification is iteratively refined and samples are added sequentially to satisfy an allocation criterion combining the benefits of proportional and optimal sampling. Theoretical estimates are provided for the expected performance and probability of failure to correctly estimate essential statistics. We devise a practical adaptive stratification method with strata of the same kind of geometrical shapes, cost-effective refinement satisfying a greedy variance reduction criterion. Numerical experiments corroborate the theoretical findings and exhibit speedups of up to three orders of magnitude compared to standard Monte Carlo sampling.

math.NA↗

arXiv2019

Mathematical Modeling, Laboratory Experiments, and Sensitivity Analysis of Bioplug Technology at Darcy Scale

In this paper we study a Darcy-scale mathematical model for biofilm formation in porous media. The pores in the core are divided into three phases: water, oil, and biofilm. The water and oil flow are modeled by an extended version of Darcy's law and the substrate is transported by diffusion and convection in the water phase. Initially there is biofilm on the pore walls. The biofilm consumes substrate for production of biomass and modifies the pore space which changes the rock permeability. The model includes detachment of biomass due to water flux and death of bacteria, and is implemented in MRST. We discuss the capability of the numerical simulator to capture results from laboratory experiments. We perform a novel sensitivity analysis based on sparse-grid interpolation and multi-wavelet expansion to identify the critical model parameters. Numerical experiments using diverse injection strategies are performed to study the impact of different porosity-permeability relations in a core saturated with water and oil.

physics.app-ph↗

arXiv2018

Level Set Methods for Stochastic Discontinuity Detection in Nonlinear Problems

Stochastic physical problems governed by nonlinear conservation laws are challenging due to solution discontinuities in stochastic and physical space. In this paper, we present a level set method to track discontinuities in stochastic space by solving a Hamilton-Jacobi equation. By introducing a speed function that vanishes at discontinuities, the iso-zero of the level set problem coincide with the discontinuities of the conservation law. The level set problem is solved on a sequence of successively finer grids in stochastic space. The method is adaptive in the sense that costly evaluations of the conservation law of interest are only performed in the vicinity of the discontinuities during the refinement stage. In regions of stochastic space where the solution is smooth, a surrogate method replaces expensive evaluations of the conservation law. The proposed method is tested in conjunction with different sets of localized orthogonal basis functions on simplex elements, as well as frames based on piecewise polynomials conforming to the level set function. The performance of the proposed method is compared to existing adaptive multi-element generalized polynomial chaos methods.

math.NA↗

arXiv2018

A pore-scale model for permeable biofilm: numerical simulations and laboratory experiments

In this paper we derive a pore-scale model for permeable biofilm formation in a two-dimensional pore. The pore is divided in two phases: water and biofilm. The biofilm is assumed to consist of four components: water, extracellular polymeric substances (EPS), active bacteria, and dead bacteria. The flow of water is modeled by the Stokes equation whereas a diffusion-convection equation is involved for the transport of nutrients. At the water/biofilm interface, nutrient transport and shear forces due to the water flux are considered. In the biofilm, the Brinkman equation for the water flow, transport of nutrients due to diffusion and convection, displacement of the biofilm components due to reproduction/dead of bacteria, and production of EPS are considered. A segregated finite element algorithm is used to solve the mathematical equations. Numerical simulations are performed based on experimentally determined parameters. The stress coefficient is fitted to the experimental data. To identify the critical model parameters, a sensitivity analysis is performed. The Sobol sensitivity indices of the input parameters are computed based on uniform perturbation by $\pm 10 \%$ of the nominal parameter values. The sensitivity analysis confirms that the variability or uncertainty in none of the parameters should be neglected.

physics.flu-dyn↗

arXiv2016

Stochastic Galerkin Framework with Locally Reduced Bases for Nonlinear Two-Phase Transport in Heterogeneous Formations

The generalized polynomial chaos method is applied to the Buckley-Leverett equation. We consider a spatially homogeneous domain modeled as a random field. The problem is projected onto stochastic basis functions which yields an extended system of partial differential equations. Analysis and numerical methods leading to reduced computational cost are presented for the extended system of equations. The accurate representation of the evolution of a discontinuous stochastic solution over time requires a large number of stochastic basis functions. Adaptivity of the stochastic basis to reduce computational cost is challenging in the stochastic Galerkin setting since the change of basis affects the system matrix itself. To achieve adaptivity without adding overhead by rewriting the entire system of equations for every grid cell, we devise a basis reduction method that distinguishes between locally significant and insignificant modes without changing the actual system matrices. Results are presented for problems in one and two spatial dimensions, with varying number of stochastic dimensions. We show how to obtain stochastic velocity fields from realistic permeability fields and demonstrate the performance of the stochastic Galerkin method with local basis reduction. The system of conservation laws is discretized with a finite volume method and we demonstrate numerical convergence to the reference solution obtained through Monte Carlo sampling.

math.NA↗