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1. The Transition from Darcy to Nonlinear Flow in Heterogeneous Porous Media: I—Single-Phase Flow

   https://doi.org/10.1007/s11242-024-02070-3  

   Arbabi, Sepehr ; Sahimi, Muhammad ( March 2024 , Transport in Porous Media)

   Using extensive numerical simulation of the Navier–Stokes equations, we study the transition from the Darcy’s law for slow flow of fluids through a disordered porous medium to the nonlinear flow regime in which the effect of inertia cannot be neglected. The porous medium is represented by two-dimensional slices of a three-dimensional image of a sandstone. We study the problem over wide ranges of porosity and the Reynolds number, as well as two types of boundary conditions, and compute essential features of fluid flow, namely, the strength of the vorticity, the effective permeability of the pore space, the frictional drag, and the relationship between the macroscopic pressure gradient$${\varvec{\nabla }}P$$$\nabla P$and the fluid velocityv. The results indicate that when the Reynolds number Re is low enough that the Darcy’s law holds, the magnitude$$\omega _z$$${\omega }_z$of the vorticity is nearly zero. As Re increases, however, so also does$$\omega _z$$${\omega }_z$, and its rise from nearly zero begins at the same Re at which the Darcy’s law breaks down. We also show that a nonlinear relation between the macroscopic pressure gradient and the fluid velocityv, given by,$$-{\varvec{\nabla }}P=(\mu /K_e)\textbf{v}+\beta _n\rho |\textbf{v}|^2\textbf{v}$$$-\nabla P=(\mu /K_e)v+{\beta }_n\rho {\left|v\right|}^{2}v$, provides accurate representation of the numerical data, where$$\mu$$$\mu$and$$\rho$$$\rho$are the fluid’s viscosity and density,$$K_e$$$K_e$is the effective Darcy permeability in the linear regime, and$$\beta _n$$${\beta }_n$is a generalized nonlinear resistance. Theoretical justification for the relation is presented, and its predictions are also compared with those of the Forchheimer’s equation.

    

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2. Eulerian Walks in Temporal Graphs

   https://doi.org/10.1007/s00453-022-01021-y  

   Marino, Andrea ; Silva, Ana ( August 2022 , Algorithmica)

   An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graphGat least (resp. exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. But what if Euler had to take a bus? In a temporal graph$$\varvec{(G,\lambda )}$$$(G,\lambda )$, with$$\varvec{\lambda : E(G)}\varvec{\rightarrow } \varvec{2}^{\varvec{[\tau ]}}$$$\lambda :E\left(G\right)\to 2^{\left[\tau \right]}$, an edge$$\varvec{e}\varvec{\in } \varvec{E(G)}$$$e\in E\left(G\right)$is available only at the times specified by$$\varvec{\lambda (e)}\varvec{\subseteq } \varvec{[\tau ]}$$$\lambda \left(e\right)\subseteq \left[\tau \right]$, in the same way the connections of the public transportation network of a city or of sightseeing tours are available only at scheduled times. In this paper, we deal with temporal walks, local trails, and trails, respectively referring to edge traversal with no constraints, constrained to not repeating the same edge in a single timestamp, and constrained to never repeating the same edge throughout the entire traversal. We show that, if the edges are always available, then deciding whether$$\varvec{(G,\lambda )}$$$(G,\lambda )$has a temporal walk or trail is polynomial, while deciding whether it has a local trail is$$\varvec{\texttt {NP}}$$$\mathrm{NP}$-complete even if$$\varvec{\tau = 2}$$$\tau =2$. In contrast, in the general case, solving any of these problems is$$\varvec{\texttt {NP}}$$$\mathrm{NP}$-complete, even under very strict hypotheses. We finally give$$\varvec{\texttt {XP}}$$$\mathrm{XP}$algorithms parametrized by$$\varvec{\tau }$$$\tau$for walks, and by$$\varvec{\tau +tw(G)}$$$\tau +tw\left(G\right)$for trails and local trails, where$$\varvec{tw(G)}$$$tw\left(G\right)$refers to the treewidth of$$\varvec{G}$$$G$.

    

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3. Hindcasting injection-induced aseismic slip and microseismicity at the Cooper Basin Enhanced Geothermal Systems Project

   https://doi.org/10.1038/s41598-022-23812-7  

   Wang, Taiyi A. ; Dunham, Eric M. ( November 2022 , Scientific Reports)

   There is a growing recognition that subsurface fluid injection can produce not only earthquakes, but also aseismic slip on faults. A major challenge in understanding interactions between injection-related aseismic and seismic slip on faults is identifying aseismic slip on the field scale, given that most monitored fields are only equipped with seismic arrays. We present a modeling workflow for evaluating the possibility of aseismic slip, given observational constraints on the spatial-temporal distribution of microseismicity, injection rate, and wellhead pressure. Our numerical model simultaneously simulates discrete off-fault microseismic events and aseismic slip on a main fault during fluid injection. We apply the workflow to the 2012 Enhanced Geothermal System injection episode at Cooper Basin, Australia, which aimed to stimulate a water-saturated granitic reservoir containing a highly permeable ($$k = 10^{-13} - 10^{-12}$$$k={10}^{-13}-{10}^{-12}$$$\hbox {m}{^2}$$$\text{m}{}^2$) fault zone. We find that aseismic slip likely contributed to half of the total moment release. In addition, fault weakening from pore pressure changes, not elastic stress transfer from aseismic slip, induces the majority of observed microseismic events, given the inferred stress state. We derive a theoretical model to better estimate the time-dependent spatial extent of seismicity triggered by increases in pore pressure. To our knowledge, this is the first time injection-induced aseismic slip in a granitic reservoir has been inferred, suggesting that aseismic slip could be widespread across a range of lithologies.

    

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4. Polynomial-time algorithms for multimarginal optimal transport problems with structure

   https://doi.org/10.1007/s10107-022-01868-7  

   Altschuler, Jason M. ; Boix-Adserà, Enric ( August 2022 , Mathematical Programming)

   Multimarginal Optimal Transport (MOT) has attracted significant interest due to applications in machine learning, statistics, and the sciences. However, in most applications, the success of MOT is severely limited by a lack of efficient algorithms. Indeed, MOT in general requires exponential time in the number of marginalskand their support sizesn. This paper develops a general theory about what “structure” makes MOT solvable in$$\mathrm {poly}(n,k)$$$\mathrm{poly}(n,k)$time. We develop a unified algorithmic framework for solving MOT in$$\mathrm {poly}(n,k)$$$\mathrm{poly}(n,k)$time by characterizing the structure that different algorithms require in terms of simple variants of the dual feasibility oracle. This framework has several benefits. First, it enables us to show that the Sinkhorn algorithm, which is currently the most popular MOT algorithm, requires strictly more structure than other algorithms do to solve MOT in$$\mathrm {poly}(n,k)$$$\mathrm{poly}(n,k)$time. Second, our framework makes it much simpler to develop$$\mathrm {poly}(n,k)$$$\mathrm{poly}(n,k)$time algorithms for a given MOT problem. In particular, it is necessary and sufficient to (approximately) solve the dual feasibility oracle—which is much more amenable to standard algorithmic techniques. We illustrate this ease-of-use by developing$$\mathrm {poly}(n,k)$$$\mathrm{poly}(n,k)$-time algorithms for three general classes of MOT cost structures: (1) graphical structure; (2) set-optimization structure; and (3) low-rank plus sparse structure. For structure (1), we recover the known result that Sinkhorn has$$\mathrm {poly}(n,k)$$$\mathrm{poly}(n,k)$runtime; moreover, we provide the first$$\mathrm {poly}(n,k)$$$\mathrm{poly}(n,k)$time algorithms for computing solutions that are exact and sparse. For structures (2)-(3), we give the first$$\mathrm {poly}(n,k)$$$\mathrm{poly}(n,k)$time algorithms, even for approximate computation. Together, these three structures encompass many—if not most—current applications of MOT.

    

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5. On the stationarity for nonlinear optimization problems with polyhedral constraints

   https://doi.org/10.1007/s10107-023-01979-9  

   di Serafino, Daniela ; Hager, William W. ; Toraldo, Gerardo ; Viola, Marco ( May 2023 , Mathematical Programming)

   For polyhedral constrained optimization problems and a feasible point$$\textbf{x}$$$x$, it is shown that the projection of the negative gradient on the tangent cone, denoted$$\nabla _\varOmega f(\textbf{x})$$${\nabla }_{\Omega }f\left(x\right)$, has an orthogonal decomposition of the form$$\varvec{\beta }(\textbf{x}) + \varvec{\varphi }(\textbf{x})$$$\beta \left(x\right)+\phi \left(x\right)$. At a stationary point,$$\nabla _\varOmega f(\textbf{x}) = \textbf{0}$$${\nabla }_{\Omega }f\left(x\right)=0$so$$\Vert \nabla _\varOmega f(\textbf{x})\Vert $$$\Vert {\nabla }_{\Omega }f\left(x\right)\Vert$reflects the distance to a stationary point. Away from a stationary point,$$\Vert \varvec{\beta }(\textbf{x})\Vert $$$\Vert \beta \left(x\right)\Vert$and$$\Vert \varvec{\varphi }(\textbf{x})\Vert $$$\Vert \phi \left(x\right)\Vert$measure different aspects of optimality since$$\varvec{\beta }(\textbf{x})$$$\beta \left(x\right)$only vanishes when the KKT multipliers at$$\textbf{x}$$$x$have the correct sign, while$$\varvec{\varphi }(\textbf{x})$$$\phi \left(x\right)$only vanishes when$$\textbf{x}$$$x$is a stationary point in the active manifold. As an application of the theory, an active set algorithm is developed for convex quadratic programs which adapts the flow of the algorithm based on a comparison between$$\Vert \varvec{\beta }(\textbf{x})\Vert $$$\Vert \beta \left(x\right)\Vert$and$$\Vert \varvec{\varphi }(\textbf{x})\Vert $$$\Vert \phi \left(x\right)\Vert$.

    

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