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Title: Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion

Abstract

There are two types of analytical solutions of temperature/concentration in and heat/mass transfer through boundaries of regularly shaped 1-D, 2-D, and 3-D blocks. These infinite-series solutions with either error functions or exponentials exhibit highly irregular but complementary convergence at different dimensionless times, td. In this paper, approximate solutions were developed by combining the error-function-series solutions for early times and the exponential-series solutions for late times and by using time partitioning at the switchover time, td0. The combined solutions contain either the leading term of both series for normal-accuracy approximations (with less than 0.003 relative error) or the first two terms for high-accuracy approximations (with less than 10-7 relative error) for 1-D isotropic (spheres, cylinders, slabs) and 2-D/3-D rectangular blocks (squares, cubes, rectangles, and rectangular parallelepipeds). This rapid and uniform convergence for rectangular blocks was achieved by employing the same time partitioning with individual dimensionless times for different directions and the product of their combined 1-D slab solutions. The switchover dimensionless time was determined to minimize the maximum approximation errors. Furthermore, the analytical solutions of first-order heat/mass flux for 2-D/3-D rectangular blocks were derived for normal-accuracy approximations. These flux equations contain the early-time solution with a three-term polynomial in √td andmore » the late-time solution with the limited-term exponentials for rectangular blocks. The heat/mass flux equations and the combined temperature/concentration solutions form the ultimate kernel for fast simulations of multirate and multidimensional heat/mass transfer in porous/fractured media with millions of low-permeability blocks of varying shapes and sizes.« less

Authors:
ORCiD logo [1];  [1];  [1];  [1]
  1. Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Publication Date:
Research Org.:
Montana State Univ., Bozeman, MT (United States); Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
USDOE Office of Fossil Energy (FE)
OSTI Identifier:
1430907
Alternate Identifier(s):
OSTI ID: 1409860; OSTI ID: 1476561
Grant/Contract Number:  
FC26-05NT42587; DE‐AC02‐05CH11231; AC02-05CH11231
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Water Resources Research
Additional Journal Information:
Journal Volume: 53; Journal Issue: 11; Journal ID: ISSN 0043-1397
Publisher:
American Geophysical Union (AGU)
Country of Publication:
United States
Language:
English
Subject:
42 ENGINEERING

Citation Formats

Zhou, Quanlin, Oldenburg, Curtis M., Rutqvist, Jonny, and Birkholzer, Jens T. Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion. United States: N. p., 2017. Web. doi:10.1002/2017WR021040.
Zhou, Quanlin, Oldenburg, Curtis M., Rutqvist, Jonny, & Birkholzer, Jens T. Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion. United States. https://doi.org/10.1002/2017WR021040
Zhou, Quanlin, Oldenburg, Curtis M., Rutqvist, Jonny, and Birkholzer, Jens T. 2017. "Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion". United States. https://doi.org/10.1002/2017WR021040. https://www.osti.gov/servlets/purl/1430907.
@article{osti_1430907,
title = {Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion},
author = {Zhou, Quanlin and Oldenburg, Curtis M. and Rutqvist, Jonny and Birkholzer, Jens T.},
abstractNote = {There are two types of analytical solutions of temperature/concentration in and heat/mass transfer through boundaries of regularly shaped 1-D, 2-D, and 3-D blocks. These infinite-series solutions with either error functions or exponentials exhibit highly irregular but complementary convergence at different dimensionless times, td. In this paper, approximate solutions were developed by combining the error-function-series solutions for early times and the exponential-series solutions for late times and by using time partitioning at the switchover time, td0. The combined solutions contain either the leading term of both series for normal-accuracy approximations (with less than 0.003 relative error) or the first two terms for high-accuracy approximations (with less than 10-7 relative error) for 1-D isotropic (spheres, cylinders, slabs) and 2-D/3-D rectangular blocks (squares, cubes, rectangles, and rectangular parallelepipeds). This rapid and uniform convergence for rectangular blocks was achieved by employing the same time partitioning with individual dimensionless times for different directions and the product of their combined 1-D slab solutions. The switchover dimensionless time was determined to minimize the maximum approximation errors. Furthermore, the analytical solutions of first-order heat/mass flux for 2-D/3-D rectangular blocks were derived for normal-accuracy approximations. These flux equations contain the early-time solution with a three-term polynomial in √td and the late-time solution with the limited-term exponentials for rectangular blocks. The heat/mass flux equations and the combined temperature/concentration solutions form the ultimate kernel for fast simulations of multirate and multidimensional heat/mass transfer in porous/fractured media with millions of low-permeability blocks of varying shapes and sizes.},
doi = {10.1002/2017WR021040},
url = {https://www.osti.gov/biblio/1430907}, journal = {Water Resources Research},
issn = {0043-1397},
number = 11,
volume = 53,
place = {United States},
year = {Tue Oct 24 00:00:00 EDT 2017},
month = {Tue Oct 24 00:00:00 EDT 2017}
}

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Works referenced in this record:

Fast-converging steady-state heat conduction in a rectangular parallelepiped
journal, August 2002


Mathematical modeling of tracer behavior in short-term experiments in fissured rocks
journal, July 1990


Green's functions, temperature and heat flux in the rectangle
journal, October 2001


First-passage-time transfer functions for groundwater tracer tests conducted in radially convergent flow
journal, January 2000


Field-scale effective matrix diffusion coefficient for fractured rock: Results from literature survey
journal, August 2007


Behavior of the mass transfer coefficient during the MADE-2 experiment: New insights: MASS TRANSFER DURING THE MADE-2 EXPERIMENT
journal, February 2008


Tracer experiments in fractured rocks: Matrix diffusion and the validity of models
journal, August 1993


The transport of heat in an oil layer caused by the injection of hot fluid
journal, March 1955


Double-Porosity Models for a Fissured Groundwater Reservoir With Fracture Skin
journal, July 1984


On the theory of tracer experiments in fissured rocks with a porous matrix
journal, July 1985


Models for simulating salt movement in aggregated field soils
journal, September 1986


Accurate and Efficient Simulation of Fracture–Matrix Interaction in Shale Gas Reservoirs
journal, December 2014


Multiple-Rate Mass Transfer for Modeling Diffusion and Surface Reactions in Media with Pore-Scale Heterogeneity
journal, June 1995


Solution of an initial-boundary value problem for heat conduction in a parallelepiped by time partitioning
journal, October 2002


Contaminant transport in fractured porous media: Analytical solution for a single fracture
journal, June 1981


Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]
journal, January 1960


A Practical Method for Modeling Fluid and Heat Flow in Fractured Porous Media
journal, February 1985


Evidence of Multi-Process Matrix Diffusion in a Single Fracture from a Field Tracer Test
journal, June 2006


On the late-time behavior of tracer test breakthrough curves
journal, December 2000


Tracer tests in a fractured dolomite: 2. Analysis of mass transfer in single-well injection-withdrawal tests
journal, May 2001


Verification solution for partial heating of rectangular solids
journal, September 2004


Improving convergence of summations in heat conduction
journal, January 2007


The Behavior of Naturally Fractured Reservoirs
journal, September 1963


Conduction in rectangular plates with boundary temperatures specified
journal, September 2008


Dead-End Pore Volume and Dispersion in Porous Media
journal, March 1964


Contaminant transport in fractured porous media: Analytical solutions for a system of parallel fractures
journal, December 1982


Intrinsic verification methods in linear heat conduction
journal, August 2006


Matrix-fracture transfer shape factors for dual-porosity simulators
journal, November 1995


Accurate early-time and late-time modeling of countercurrent spontaneous imbibition: ACCURATE MODELING OF SPONTANEOUS IMBIBITION
journal, August 2016


A general real-time formulation for multi-rate mass transfer problems
journal, January 2009


Heating and Cooling Rectangular and Cylindrical Solids
journal, May 1936


Convergent radial dispersion: A Laplace transform solution for aquifer tracer testing
journal, March 1989


Mass Transfer Studies in Sorbing Porous Media I. Analytical Solutions1
journal, January 1976


Modeling the transport of solutes influenced by multiprocess nonequilibrium
journal, September 1989


Methodology to Generate Accurate Solutions for Verification in Transient Three-Dimensional heat Conduction
journal, June 2002


On matrix diffusion: formulations, solution methods and qualitative effects
journal, June 1998


Convergent Radial Dispersion: A Note on Evaluation of the Laplace Transform Solution
journal, December 1991


A time-convolution approach for modeling heat exchange between a wellbore and surrounding formation
journal, December 2011


A new coupling term for dual-porosity models
journal, February 1990