Efficient Uncertainty Quantification for Biotransport in Tumors With Uncertain Material Properties

2018 ◽  
Author(s):  
Alen Alexanderian ◽  
William Reese ◽  
Ralph C. Smith ◽  
Meilin Yu
Keyword(s):  
Porous Media ◽  
Random Field ◽  
Material Properties ◽  
Pressure Field ◽  
Input Parameter ◽  
Gaussian Random Field ◽  
Porous Media Flow ◽  
Expansion Technique ◽  
Low Dimensional ◽  
Permeability Field

We consider modeling of single phase fluid flow in heterogeneous porous media governed by elliptic partial differential equations (PDEs) with random field coefficients. Our target application is biotransport in tumors with uncertain heterogeneous material properties. We numerically explore dimension reduction of the input parameter and model output. In the present work, the permeability field is modeled as a log-Gaussian random field, and its covariance function is specified. Uncertainties in permeability are then propagated into the pressure field through the elliptic PDE governing porous media flow. The covariance matrix of pressure is constructed via Monte Carlo sampling. The truncated Karhunen–Loève (KL) expansion technique is used to decompose the log-permeability field, as well as the random pressure field resulting from random permeability. We find that although very high-dimensional representation is needed to recover the permeability field when the correlation length is small, the pressure field is not sensitive to high-oder KL terms of input parameter, and itself can be modeled using a low-dimensional model. Thus a low-rank representation of the pressure field in a low-dimensional parameter space is constructed using the truncated KL expansion technique.

scholarly journals Model Input and Output Dimension Reduction Using Karhunen–Loève Expansions With Application to Biotransport

2019 ◽  
Vol 5 (4) ◽  
Author(s):  
Alen Alexanderian ◽  
William Reese ◽  
Ralph C. Smith ◽  
Meilin Yu
Keyword(s):  
Dimension Reduction ◽  
Pressure Field ◽  
Input Parameter ◽  
Gaussian Random Field ◽  
Elliptic Partial Differential Equation ◽  
Heterogeneous Material ◽  
Correlation Lengths ◽  
Reduced Order ◽  
Output Dimension ◽  
Permeability Field

Abstract We consider biotransport in tumors with uncertain heterogeneous material properties. Specifically, we focus on the elliptic partial differential equation (PDE) modeling the pressure field inside the tumor. The permeability field is modeled as a log-Gaussian random field with a prespecified covariance function. We numerically explore dimension reduction of the input parameter and model output. Specifically, truncated Karhunen–Loève (KL) expansions are used to decompose the log-permeability field, as well as the resulting random pressure field. We find that although very high-dimensional representations are needed to accurately represent the permeability field, especially in presence of small correlation lengths, the pressure field is not sensitive to high-order KL terms of the input parameter. Moreover, we find that the pressure field itself can be represented accurately using a KL expansion with a small number of terms. These observations are used to guide a reduced-order modeling approach to accelerate computational studies of biotransport in tumors.

Geostatistical representation of multiscale heterogeneity of porous media through a Generalized Sub-Gaussian model

2020 ◽  
Author(s):  
Alberto Guadagnini ◽  
Monica Riva ◽  
Shlomo P. Neuman ◽  
Martina Siena
Keyword(s):  
Porous Media ◽  
Random Field ◽  
Gaussian Random Field ◽  
Gaussian Model ◽  
Mineral Dissolution ◽  
Heterogeneous Structure ◽  
Natural Systems ◽  
Multi Scale ◽  
Geochemical Parameters ◽  
Heavy Tailed

<p>Characterization of spatial heterogeneity of attributes of porous media is critical in several environmental and industrial settings. Quantities such as, e.g., permeability, porosity, or geochemical parameters of natural systems are typically characterized by remarkable spatial variability, their degree of heterogeneity being typically linked to the size of observation/measurement/support scale as well as to length scales associated with the domain of investigation. Here, we address the way stochastic representations of multiscale heterogeneity can be employed to assess documented manifestations of scaling of statistics of hydrological and soil science variables. As such, we focus on perspectives associated with interpretive approaches to scaling of the main statistical descriptors of heterogeneity observed at diverse scales. We start from the geostatistical framework proposed by Riva et al. (2015), who rely on the representation of the heterogeneous structure of hydrological variables by way of a Generalized Sub-Gaussian (GSG) model. The latter describes the random field of interest as the product of a zero-mean, generally (but not necessarily) multi-scale Gaussian random field (<em>G</em>) and a subordinator (<em>U</em>), which is independent of <em>G</em> and consists of statistically independent identically distributed non-negative random variables. The underlying Gaussian random field generally displays a multi-scale (statistical) nature which can be captured, for example, through a geostatistical description based on a Truncated Power Variogram (TPV) model. In this study we (<em>i</em>) generalize the original GSG model formulation to include alternative distributional forms of the subordinator and (<em>ii</em>) apply such a theoretical framework to analyze datasets associated with differing processes and observation scales. These include (<em>i</em>) measurements of surface topography of a (millimeter-scale) calcite sample resulting from induced mineral dissolution and (<em>ii</em>) neutron porosity data sampled from a (kilometer-scale) borehole. We finally merge all of the above mentioned elements within a geostatistical interpretation of the system based on the GSG approach where a Truncated Power Variogram (TPV) model is employed to represent the underlying correlation structure. By doing so, we propose to rely on these models to condition the spatial statistics of such fields on multiscale measurements via a co-kriging approach.</p><p><strong>References</strong></p><p>Riva, M., S.P. Neuman, and A. Guadagnini (2015), New scaling model for variables and increments with heavy-tailed distributions, Water Resour. Res., 51, 4623-4634, doi:10.1002/2015WR016998.</p>

scholarly journals Estimation of a Permeability Field within the Two-Phase Porous Media Flow Using Nonlinear Multigrid Method

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Tao Liu ◽  
Jie Song
Keyword(s):  
Porous Media ◽  
Multigrid Method ◽  
Computational Cost ◽  
Estimation Method ◽  
Porous Media Flow ◽  
Computational Accuracy ◽  
Two Phase ◽  
Nonlinear Multigrid ◽  
Nonlinear Multigrid Method ◽  
Permeability Field

Estimation of spatially varying permeability within the two-phase porous media flow plays an important role in reservoir simulation. Usually, one needs to estimate a large number of permeability values from a limited number of observations, so the computational cost is very high even for a single field-model. This paper applies a nonlinear multigrid method to estimate the permeability field within the two-phase porous media flow. Numerical examples are provided to illustrate the feasibility and effectiveness of the proposed estimation method. In comparison with other existing methods, the most outstanding advantage of this method is the computational efficiency, computational accuracy, and antinoise ability. The proposed method has a potential applicability to a variety of parameter estimation problems.

scholarly journals Multilevel and quasi-Monte Carlo methods for uncertainty quantification in particle travel times through random heterogeneous porous media

2017 ◽  
Vol 4 (8) ◽  
pp. 170203 ◽  
Author(s):  
D. Crevillén-García ◽  
H. Power
Keyword(s):  
Porous Media ◽  
Monte Carlo ◽  
Uncertainty Quantification ◽  
Input Parameter ◽  
Computational Cost ◽  
Gaussian Random Field ◽  
Heterogeneous Porous Media ◽  
Quasi Monte Carlo ◽  
Average Travel Time ◽  
The Mean

In this study, we apply four Monte Carlo simulation methods, namely, Monte Carlo, quasi-Monte Carlo, multilevel Monte Carlo and multilevel quasi-Monte Carlo to the problem of uncertainty quantification in the estimation of the average travel time during the transport of particles through random heterogeneous porous media. We apply the four methodologies to a model problem where the only input parameter, the hydraulic conductivity, is modelled as a log-Gaussian random field by using direct Karhunen–Loéve decompositions. The random terms in such expansions represent the coefficients in the equations. Numerical calculations demonstrating the effectiveness of each of the methods are presented. A comparison of the computational cost incurred by each of the methods for three different tolerances is provided. The accuracy of the approaches is quantified via the mean square error.

Convective mixing in homogeneous porous media flow

2017 ◽  
Vol 2 (1) ◽  
Author(s):  
Jia-Hau Ching ◽  
Peilong Chen ◽  
Peichun Amy Tsai
Keyword(s):  
Porous Media ◽  
Porous Media Flow ◽  
Convective Mixing ◽  
Homogeneous Porous Media

Unsteady and porous media flow of reactive non-Newtonian fluids subjected to buoyancy and suction/injection

2015 ◽  
Vol 25 (7) ◽  
pp. 1682-1704 ◽  
Author(s):  
Tirivanhu Chinyoka ◽  
Daniel Oluwole Makinde
Keyword(s):  
Heat Transfer ◽  
Porous Media ◽  
Boundary Conditions ◽  
Flow Velocity ◽  
Flow System ◽  
Finite Difference Methods ◽  
Porous Media Flow ◽  
Convective Boundary Conditions ◽  
Convective Boundary ◽  
Content Type

Purpose – The purpose of this paper is to examine the unsteady pressure-driven flow of a reactive third-grade non-Newtonian fluid in a channel filled with a porous medium. The flow is subjected to buoyancy, suction/injection asymmetrical and convective boundary conditions. Design/methodology/approach – The authors assume that exothermic chemical reactions take place within the flow system and that the asymmetric convective heat exchange with the ambient at the surfaces follow Newton’s law of cooling. The authors also assume unidirectional suction injection flow of uniform strength across the channel. The flow system is modeled via coupled non-linear partial differential equations derived from conservation laws of physics. The flow velocity and temperature are obtained by solving the governing equations numerically using semi-implicit finite difference methods. Findings – The authors present the results graphically and draw qualitative and quantitative observations and conclusions with respect to various parameters embedded in the problem. In particular the authors make observations regarding the effects of bouyancy, convective boundary conditions, suction/injection, non-Newtonian character and reaction strength on the flow velocity, temperature, wall shear stress and wall heat transfer. Originality/value – The combined fluid dynamical, porous media and heat transfer effects investigated in this paper have to the authors’ knowledge not been studied. Such fluid dynamical problems find important application in petroleum recovery.

Porous media flow of poly(ethylene oxide)/sodium dodecyl sulfate mixtures

1999 ◽  
Vol 42 (1) ◽  
pp. 109-116 ◽  
Author(s):  
C. M. DaRocha ◽  
L. G. Patruyo ◽  
N. E. Ramírez ◽  
A. J. Müller ◽  
A. E. Sáez
Keyword(s):  
Porous Media ◽  
Sodium Dodecyl Sulfate ◽  
Ethylene Oxide ◽  
Sodium Dodecyl ◽  
Porous Media Flow ◽  
Dodecyl Sulfate ◽  
Poly Ethylene Oxide ◽  
Poly Ethylene
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