@article{a8438f637e9348b59fc4860c64e6d73c,

title = "Solvable Model for Dynamic Mass Transport in Disordered Geophysical Media",

abstract = "We present an analytically solvable model for transport in geophysical materials on large length and time scales. It describes the flow of gas to a complicated absorbing boundary over long periods of time. We find a solution to this model using Green's function techniques, and apply the solution to three absorbing networks of increasing complexity.",

author = "M. Marder and Behzad Eftekhari and Patzek, {Tadeusz W.}",

note = "Funding Information: The essential behavior of the examples presented so far could be examined without much difficulty from a continuum perspective [14] or using finite element programs such as comsol . For a final example, depicted in Fig. 4(a) , we present a geometrically complex structure produced by 550 intersecting horizontal and vertical cracks with a power-law length distribution motivated by geophysical data [15] . The probability that a fracture has length l is proportional to l - 2.2 . The interior portion, shown with gray scale in the figure, has 3080 sites in light gray that are rock, interspersed by interior absorbers in darker gray. Absorbers forming the exterior of the network are colored in black. The 3080 eigenvalues and eigenvectors of the interior problem needed for Eq. (4) can be computed in seconds on a single processor. The sum rule Q ˙ interior ( 0 ) = 3264 provides a check on the accuracy of these computations. The integrand of Eq. (7) has hundreds of narrow spikes that make its computation more difficult although still tractable. Counting up the number of exterior faces on the absorbers gives the sum rule Q ˙ exterior ( 0 ) = 1556 , but the integral of the continuous spectrum gives only 1546.53. This is because the external problem has localized eigenfunctions precisely at λ = - 4 . Their weight can be determined by taking η = 10 - 8 in Eq. (7) and performing the integration in a small neighborhood of λ = - 4 . The contribution from the exterior localized eigenfunctions is 9.47, and adding this to the integral of the continuous spectrum finally exhausts the sum rule. 4 10.1103/PhysRevLett.120.138302.f4 FIG. 4. Gas absorption into network of criss-crossing fractures. (a) Network of absorbers created by 550 horizontal and vertical cracks. (b) Continuous spectrum for network in part (a). The sharp peaks come from small features on the exterior of the network shielded by extended arms. (c) Total gas production from network in part (a). For large f = ln ( - λ ) , the continuous spectrum assumes the asymptotic form of Eq. (10) with τ 2 = . 025 33 , τ 1 = - 0.3947 , τ 0 = 1.7877 . This form is all one needs for the very long time behavior. The continuous spectrum appears in Fig. 4(b) and the production rate as a function of time, summing contributions from the discrete and continuous spectra, appears in Fig. 4(c) . This solution illustrates but certainly does not exhaust possible applications of methods from disordered electronic materials to geophysical transport. Possible extensions to complement large-scale simulations are as follows: deduction of fracture geometry from production history, extraction of long-time behavior without reference to fine details, extension to three dimensions, removal of the assumption of complete homogeneity outside fractures, inclusion of thermodynamic properties of real gas, and transport of multiple fluid phases. Improved concepts for geophysical transport are needed if we are to make the most of existing hydrocarbon resources during the inevitable transition to other forms of energy [16] . B. E. acknowledges funding from the King Abdullah University of Science and Technology. We thank David DiCarlo, Carlos Torres-Verdin, and Larry Lake for useful comments as the work was progressing. Qian Niu helpfully pointed out the possibility of localized modes in the continuum. [1] 1 P. W. Anderson , Phys. Rev. 109 , 1492 ( 1958 ). PHRVAO 0031-899X 10.1103/PhysRev.109.1492 [2] 2 S. R. Broadbent and J. M. Hammersley , Math. Proc. Cambridge Philos. Soc. 53 , 629 ( 1957 ). MPCPCO 0305-0041 10.1017/S0305004100032680 [3] 3 A. Hunt , R. Ewing , and B. Ghanbarian , Percolation Theory for Flow in Porous Media ( Springer , New York, 2014 ). [4] 4 T. Patzek , F. Male , and M. Marder , Proc. Natl. Acad. Sci. U.S.A. 110 , 19731 ( 2013 ). PNASA6 0027-8424 10.1073/pnas.1313380110 [5] 5 D. L. Turcotte , E. 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( Oxford , New York, 1959 ). [15] 15 B. Eftekhari , Ph.D. thesis, The University of Texas at Austin , 2016 . [16] 16 M. Marder , T. Patzek , and S. W. Tinker , Phys. Today 69 , 46 ( 2016 ). PHTOAD 0031-9228 10.1063/PT.3.3236 Publisher Copyright: {\textcopyright} 2018 American Physical Society.",

year = "2018",

month = mar,

day = "29",

doi = "10.1103/PhysRevLett.120.138302",

language = "English (US)",

volume = "120",

journal = "Physical Review Letters",

issn = "0031-9007",

publisher = "American Physical Society",

number = "13",

}