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 »
- Authors:
-
- 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}
}
Web of Science
Works referenced in this record:
Fast-converging steady-state heat conduction in a rectangular parallelepiped
journal, August 2002
- Crittenden, Paul E.; Cole, Kevin D.
- International Journal of Heat and Mass Transfer, Vol. 45, Issue 17
Mathematical modeling of tracer behavior in short-term experiments in fissured rocks
journal, July 1990
- Maloszewski, Piotr; Zuber, Andrzej
- Water Resources Research, Vol. 26, Issue 7
Green's functions, temperature and heat flux in the rectangle
journal, October 2001
- Cole, Kevin D.; Yen, David H. Y.
- International Journal of Heat and Mass Transfer, Vol. 44, Issue 20
First-passage-time transfer functions for groundwater tracer tests conducted in radially convergent flow
journal, January 2000
- Becker, Matthew W.; Charbeneau, Randall J.
- Journal of Contaminant Hydrology, Vol. 40, Issue 4
Field-scale effective matrix diffusion coefficient for fractured rock: Results from literature survey
journal, August 2007
- Zhou, Quanlin; Liu, Hui-Hai; Molz, Fred J.
- Journal of Contaminant Hydrology, Vol. 93, Issue 1-4
Behavior of the mass transfer coefficient during the MADE-2 experiment: New insights: MASS TRANSFER DURING THE MADE-2 EXPERIMENT
journal, February 2008
- Guan, Jianyong; Molz, Fred J.; Zhou, Quanlin
- Water Resources Research, Vol. 44, Issue 2
Tracer experiments in fractured rocks: Matrix diffusion and the validity of models
journal, August 1993
- Maloszewski, Piotr; Zuber, Andrzej
- Water Resources Research, Vol. 29, Issue 8
The transport of heat in an oil layer caused by the injection of hot fluid
journal, March 1955
- Lauwerier, H. A.
- Applied Scientific Research, Vol. 5, Issue 2-3
Double-Porosity Models for a Fissured Groundwater Reservoir With Fracture Skin
journal, July 1984
- Moench, Allen F.
- Water Resources Research, Vol. 20, Issue 7
Accurate Simulation of Non Darcy Flow in Stimulated Fractured Shale Reservoirs
conference, April 2013
- Rubin, Barry
- SPE Western Regional Meeting
On the theory of tracer experiments in fissured rocks with a porous matrix
journal, July 1985
- Małoszewski, P.; Zuber, A.
- Journal of Hydrology, Vol. 79, Issue 3-4
Models for simulating salt movement in aggregated field soils
journal, September 1986
- van Genuchten, M. Th.; Dalton, F. N.
- Geoderma, Vol. 38, Issue 1-4
Accurate and Efficient Simulation of Fracture–Matrix Interaction in Shale Gas Reservoirs
journal, December 2014
- Cai, Long; Ding, Didier-Yu; Wang, Cong
- Transport in Porous Media, Vol. 107, Issue 2
Multiple-Rate Mass Transfer for Modeling Diffusion and Surface Reactions in Media with Pore-Scale Heterogeneity
journal, June 1995
- Haggerty, Roy; Gorelick, Steven M.
- Water Resources Research, Vol. 31, Issue 10
Solution of an initial-boundary value problem for heat conduction in a parallelepiped by time partitioning
journal, October 2002
- Yen, David H. Y.; Beck, James V.; McMasters, Robert L.
- International Journal of Heat and Mass Transfer, Vol. 45, Issue 21
Contaminant transport in fractured porous media: Analytical solution for a single fracture
journal, June 1981
- Tang, D. H.; Frind, E. O.; Sudicky, E. A.
- Water Resources Research, Vol. 17, Issue 3
Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]
journal, January 1960
- Barenblatt, G. I.; Zheltov, Iu. P.; Kochina, I. N.
- Journal of Applied Mathematics and Mechanics, Vol. 24, Issue 5
A Practical Method for Modeling Fluid and Heat Flow in Fractured Porous Media
journal, February 1985
- Pruess, Karsten
- Society of Petroleum Engineers Journal, Vol. 25, Issue 01
Evidence of Multi-Process Matrix Diffusion in a Single Fracture from a Field Tracer Test
journal, June 2006
- Zhou, Quanlin; Liu, Hui-Hai; Bodvarsson, Gudmundur S.
- Transport in Porous Media, Vol. 63, Issue 3
On the late-time behavior of tracer test breakthrough curves
journal, December 2000
- Haggerty, Roy; McKenna, Sean A.; Meigs, Lucy C.
- Water Resources Research, Vol. 36, Issue 12
Convergent Radial Dispersion in a Double-Porosity Aquifer with Fracture Skin: Analytical Solution and Application to a Field Experiment in Fractured Chalk
journal, August 1995
- Moench, Allen F.
- Water Resources Research, Vol. 31, Issue 8
Tracer tests in a fractured dolomite: 2. Analysis of mass transfer in single-well injection-withdrawal tests
journal, May 2001
- Haggerty, Roy; Fleming, Sean W.; Meigs, Lucy C.
- Water Resources Research, Vol. 37, Issue 5
Approximate solutions for diffusive fracture-matrix transfer: Application to storage of dissolved CO 2 in fractured rocks : APPROXIMATE MASS TRANSFER SOLUTION
journal, February 2017
- Zhou, Quanlin; Oldenburg, Curtis M.; Spangler, Lee H.
- Water Resources Research, Vol. 53, Issue 2
Verification solution for partial heating of rectangular solids
journal, September 2004
- Beck, James V.; Haji-Sheikh, A.; Amos, Donald E.
- International Journal of Heat and Mass Transfer, Vol. 47, Issue 19-20
Improving convergence of summations in heat conduction
journal, January 2007
- Beck, James V.; Cole, Kevin D.
- International Journal of Heat and Mass Transfer, Vol. 50, Issue 1-2
The Behavior of Naturally Fractured Reservoirs
journal, September 1963
- Warren, J. E.; Root, P. J.
- Society of Petroleum Engineers Journal, Vol. 3, Issue 03
Conduction in rectangular plates with boundary temperatures specified
journal, September 2008
- Beck, James V.; Wright, Neil T.; Haji-Sheikh, A.
- International Journal of Heat and Mass Transfer, Vol. 51, Issue 19-20
Dead-End Pore Volume and Dispersion in Porous Media
journal, March 1964
- Coats, K. H.; Smith, B. D.
- Society of Petroleum Engineers Journal, Vol. 4, Issue 01
Contaminant transport in fractured porous media: Analytical solutions for a system of parallel fractures
journal, December 1982
- Sudicky, E. A.; Frind, E. O.
- Water Resources Research, Vol. 18, Issue 6
Analytical solutions for pressure perturbation and fluid leakage through aquitards and wells in multilayered-aquifer systems: ANALYTICAL SOLUTIONS IN MULTILAYERED-AQUIFER SYSTEMS
journal, October 2011
- Cihan, Abdullah; Zhou, Quanlin; Birkholzer, Jens T.
- Water Resources Research, Vol. 47, Issue 10
Intrinsic verification methods in linear heat conduction
journal, August 2006
- Beck, James V.; McMasters, Robert; Dowding, Kevin J.
- International Journal of Heat and Mass Transfer, Vol. 49, Issue 17-18
Matrix-fracture transfer shape factors for dual-porosity simulators
journal, November 1995
- Lim, K. T.; Aziz, K.
- Journal of Petroleum Science and Engineering, Vol. 13, Issue 3-4
Testing and parameterizing a conceptual model for solute transport in a fractured granite using multiple tracers in a forced-gradient test: MULTIPLE TRACER TESTING IN FRACTURED ROCK
journal, December 2003
- Reimus, Paul; Pohll, Greg; Mihevc, Todd
- Water Resources Research, Vol. 39, Issue 12
Accurate early-time and late-time modeling of countercurrent spontaneous imbibition: ACCURATE MODELING OF SPONTANEOUS IMBIBITION
journal, August 2016
- March, Rafael; Doster, Florian; Geiger, Sebastian
- Water Resources Research, Vol. 52, Issue 8
A general real-time formulation for multi-rate mass transfer problems
journal, January 2009
- Silva, O.; Carrera, J.; Dentz, M.
- Hydrology and Earth System Sciences, Vol. 13, Issue 8
A Semi-Analytical Solution for Large-Scale Injection-Induced Pressure Perturbation and Leakage in a Laterally Bounded Aquifer–Aquitard System
journal, October 2008
- Zhou, Quanlin; Birkholzer, Jens T.; Tsang, Chin-Fu
- Transport in Porous Media, Vol. 78, Issue 1
Heating and Cooling Rectangular and Cylindrical Solids
journal, May 1936
- Newman, Albert B.
- Industrial & Engineering Chemistry, Vol. 28, Issue 5
A closed-form analytical solution for thermal single-well injection-withdrawal tests: ANALYTICAL SOLUTION FOR THERMAL SWIW TESTS
journal, March 2012
- Jung, Yoojin; Pruess, Karsten
- Water Resources Research, Vol. 48, Issue 3
Convergent radial dispersion: A Laplace transform solution for aquifer tracer testing
journal, March 1989
- Moench, Allen F.
- Water Resources Research, Vol. 25, Issue 3
Mass Transfer Studies in Sorbing Porous Media I. Analytical Solutions1
journal, January 1976
- van Genuchten, M. Th.; Wierenga, P. J.
- Soil Science Society of America Journal, Vol. 40, Issue 4
Transport upscaling in heterogeneous aquifers: What physical parameters control memory functions?: HETEROGENEITY EFFECTS ON MEMORY FUNCTION
journal, December 2008
- Willmann, M.; Carrera, J.; Sánchez-Vila, X.
- Water Resources Research, Vol. 44, Issue 12
Modeling the transport of solutes influenced by multiprocess nonequilibrium
journal, September 1989
- Brusseau, M. L.; Jessup, R. E.; Rao, P. S. C.
- Water Resources Research, Vol. 25, Issue 9
Methodology to Generate Accurate Solutions for Verification in Transient Three-Dimensional heat Conduction
journal, June 2002
- McMasters, Robert L.; Dowding, Kevin J.; Beck, James V.
- Numerical Heat Transfer, Part B: Fundamentals, Vol. 41, Issue 6
On matrix diffusion: formulations, solution methods and qualitative effects
journal, June 1998
- Carrera, Jesús; Sánchez-Vila, Xavier; Benet, Inmaculada
- Hydrogeology Journal, Vol. 6, Issue 1
What controls the apparent timescale of solute mass transfer in aquifers and soils? A comparison of experimental results: CONTROLLING THE TIMESCALE OF SOLUTE MASS TRANSFER
journal, January 2004
- Haggerty, Roy; Harvey, Charles F.; Freiherr von Schwerin, Claudius
- Water Resources Research, Vol. 40, Issue 1
Convergent Radial Dispersion: A Note on Evaluation of the Laplace Transform Solution
journal, December 1991
- Moench, Allen F.
- Water Resources Research, Vol. 27, Issue 12
A time-convolution approach for modeling heat exchange between a wellbore and surrounding formation
journal, December 2011
- Zhang, Yingqi; Pan, Lehua; Pruess, Karsten
- Geothermics, Vol. 40, Issue 4
A new coupling term for dual-porosity models
journal, February 1990
- Dykhuizen, R. C.
- Water Resources Research, Vol. 26, Issue 2