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REVIEW 3 major objections 7 minor 61 references

Probability density function (PDF) models for particle transport in porous media

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Pith's one-line read This paper derives, for the first time, a rigorous evolution equation for the Lagrangian velocity PDF in porous media, using stochastic calculus rather than empirical velocity models.

desk verdict A worthwhile theoretical derivation of a velocity-PDF evolution equation for porous media, but the equilibrium closure is tested only by self-consistency and the joint FP has a likely typo; deserves revision rather than rejection. read the letter →

arxiv 1908.01770 v2 pith:VOZNWEUX submitted 2019-08-04 physics.flu-dyn cond-mat.stat-mech

Matteo Icardi , Marco Dentz This is my paper · ORCID
classification physics.flu-dyncond-mat.stat-mech
keywords porousmediaLagrangianvelocityPDFFokker-PlanckequationstochasticdifferentialequationshydrodynamicdispersioncontinuoustimerandomwalkconditionalexpectationsPoiseuilleflow
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The authors aim to replace empirical models of how particle velocities change as solutes drift through heterogeneous porous media with a derivation from stochastic calculus. They extend the usual position-only description to a joint position–velocity probability density, write the corresponding Itô/Fokker-Planck equation, and then average over space to obtain an evolution equation for the marginal Lagrangian velocity PDF. In the equilibrium closure—valid when the solute concentration has spread uniformly—the equation closes and, for Poiseuille flow, its stationary solution is $F(v)\propto(1-u/u_0)^{-1/2}$, exactly the Eulerian velocity PDF. A sympathetic reader would care because this supplies a principled basis for continuous-time-random-walk velocity models and for closures in anomalous dispersion.

What carries the argument

The carrying object is the joint position–velocity PDF $f(x,v;t)$, which restores trajectory information lost in the concentration-only advection-diffusion description. Its evolution is generated by the Itô expansion $dU=\nabla u\,dX+\tfrac12 dX^T\nabla^2 u\,dX$ of the Lagrangian velocity, producing the coupled SDE system (5) and the Fokker-Planck equation (7) with velocity-space drift $(Gv + D\Delta u)$ and diffusion $D GG^T$. Marginalising over space moves the closure problem into the velocity-space terms, which are expressed through conditional expectations $G^{x|v}$, $\Delta u^{x|v}$, $GG^T{}^{x|v}$; Bayes's rule turns these into concentration-weighted integrals over iso-velocity surfaces. The equilibrium closure replaces the concentration by its uniform long-time value, making the coefficients functions of $v$ alone; in shear flows the marginal equation reduces to an explicit diffusion equation in velocity space whose stationary solution is the Eulerian velocity PDF.

What would settle it

Run pore-scale particle-tracking simulations in a heterogeneous 2D or 3D porous medium starting from a spatially localized injection, compute the marginal Lagrangian velocity PDF over time, and compare it with the solution of the equilibrium-closed equation (24 or its generalisation); if the predicted relaxation to the Eulerian velocity PDF is systematically slower or faster than the simulated one before the concentration becomes uniform, the closure is falsified.

Watch

Extended reading notes

Core claim

The central claim is that the marginal Lagrangian velocity PDF $F(v,t)$ satisfies a Fokker-Planck-type equation (Eq. 10) whose drift and diffusion coefficients are conditional expectations of the velocity gradient and Laplacian given velocity. Because the position PDF is just the concentration field and the velocity is a deterministic function of position, these conditional expectations can be written as integrals over the iso-velocity surfaces weighted by concentration (Bayes formula, Eq. 11). Under the equilibrium closure, the coefficients become the time-independent Eulerian conditional averages of Eq. (14), yielding the closed equation (24) in two-dimensional Hagen-Poiseuille flow. The paper shows the stationary solution of this closed equation is $F\propto(1-u/u_0)^{-1/2}$, which matches the Eulerian velocity PDF obtained by uniform sampling of the parabolic profile. This establishes, on the paper's own terms, a first-principles route from the advection-diffusion equation to Lagrangian velocity statistics, without assuming a velocity Markov model.

Load-bearing premise

The equilibrium closure assumes the solute concentration has already spread uniformly across the medium, so that time-dependent conditional averages can be replaced by their long-time, concentration-independent values; for a localized plume in a heterogeneous medium this is not yet justified, and the paper offers no direct simulation of that regime.

Editorial extensions

If this is right

  • The evolution of Lagrangian velocity statistics in steady heterogeneous flows can in principle be computed from the flow field and its gradients, not fitted to data.
  • The derived equation gives CTRW dispersion models a rigorous velocity-evolution ingredient instead of an ad hoc one.
  • In Poiseuille-type flows the equilibrium closure is exact in the long-time limit, so the closed equation reproduces the known Eulerian velocity distribution $F(v)\propto(1-u/u_0)^{-1/2}$.
  • Because the spatial advection term needs no closure, the approach shifts the modelling difficulty into velocity-space conditional averages, where explicit closures such as the IEM approximation can be applied.
  • The formulation extends naturally to inertial particles through a drag-force SDE with Stokes number, opening a route to second-order velocity models.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • I would expect the unclosed time-dependent conditional expectations to matter most in the early life of a localized plume; a direct particle-tracking test in a three-dimensional pore geometry, comparing measured $F(v,t)$ with the equilibrium-closed solution, would show how fast the closure becomes valid.
  • The same marginalisation could be applied to filtered (locally averaged) equations, not just global averages; that would produce a coupled system where concentration heterogeneities feed back into velocity statistics—an untested extension the paper leaves open.
  • Because the stationary solution is the Eulerian velocity PDF, the paper implies that any discrepancy between Lagrangian and Eulerian velocity statistics must be a transient, diffusion-driven relaxation phenomenon; this could be checked experimentally with particle tracking in bead packs.
  • The analogy with turbulent mixing suggests standard mixing closures (interaction-by-exchange-with-the-mean type) could be ported to porous-media velocity PDFs, yielding relaxation-rate parameters that are measurable from acceleration statistics.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 7 minor

Summary. The paper develops a probability-density-function (PDF) framework for particle transport in steady heterogeneous velocity fields. Starting from the Itō SDE dX = u(X) dt + sqrt(2D) dW, the authors apply Itō's lemma to U = u(X), obtain a joint position-velocity Fokker-Planck equation, and spatially marginalize it into an equation for the Lagrangian velocity PDF F(v,t) with conditional-expectation coefficients. They discuss closures: an equilibrium closure replacing conditional expectations by uniform-concentration values, a perturbation expansion around equilibrium, and an IEM/Gaussian closure. For two-dimensional shear flows they derive reduced equations, and for Hagen-Poiseuille flow they obtain the closed equation (24) whose stationary solution is the Eulerian velocity PDF, with a numerical illustration of convergence.

Significance. The paper offers a useful and well-motivated route from a particle-resolved SDE to velocity-space transport equations for Lagrangian velocity statistics in porous media, a problem of current interest for CTRW and anomalous-dispersion modeling. The Poiseuille reduction (Eqs. 22-25) is a clean exact benchmark, and the conditional-expectation and IEM closure framework is a sensible adaptation of standard PDF methods. The derivation is, however, an application of textbook Itō calculus rather than a fundamentally new technique, and the paper currently contains internal algebraic inconsistencies in the central equations (see major comments). No code or machine-checked proofs are provided; the numerical test is illustrative. If the sign and cross-derivative issues are repaired and the equilibrium closure is tested in a genuinely heterogeneous flow, the framework would be a solid contribution.

major comments (3)
  1. [Section 3, Eqs. (9)-(10), (15)-(17)] The sign of the DΔu term is incorrect. Marginalizing the conservative form (7), whose velocity-space divergence is ∇v·[(-Gv - DΔu + DGG^T∇v)f], gives ∂F/∂t = -∇v·[(G^{x|v} v + DΔu^{x|v})F] plus the diffusion term, not -∇v·[(G^{x|v} v - DΔu^{x|v})F]. The printed sign is consequential: inserting the Poiseuille data into Eq. (10) with Gv=0 and Δu = -2u0/L² yields a drift -2Du0/L² ∂F/∂u, opposite to the +2Du0/L² ∂F/∂u in Eq. (24), and the stationary solution is then not Eq. (25). Equation (22), which the authors actually solve, uses the correct sign, so the paper is internally inconsistent with its own central equation.
  2. [Section 2, Eqs. (6)-(8)] The three displayed forms of the joint Fokker-Planck equation are not equivalent as written. With G = ∇u as defined, the Itō SDE (5) has a noise matrix whose off-diagonal block is G^T, so the mixed derivative in the standard Fokker-Planck equation is 2D ∇x·∇v:(G^T f), not 2D ∇x∇v:[G f] as in Eq. (6). In addition, Eq. (8), claimed to be an equivalent non-conservative form, has no mixed derivative at all; expansion of the correct Eq. (6) retains a mixed second derivative and an additional drift containing ∇x·G. The mixed term vanishes under the spatial marginalization used in Section 3 for periodic or infinite domains, so the marginal equation can be repaired, but the joint-PDF equation and the equivalence statement need correction.
  3. [Section 3.2 and Section 5, Fig. 1] The numerical test does not validate the equilibrium closure that is the paper's main closure tool. For the Hagen-Poiseuille profile, the map y ↦ u(y) is invertible, so the conditional density f(y|u;t) is δ(y-y(u)) for all times; consequently M_{σ'}(u;t)=σ' and M_{σ²}(u;t)=σ²(y(u)) are time-independent even without the equilibrium closure, and Eq. (22) is exact. Fig. 1 therefore only demonstrates convergence of the exact reduced equation to its stationary solution. It gives no evidence about the accuracy of closure (14) in heterogeneous, non-invertible flow fields, where the conditional expectations in Eq. (10) depend on the evolving concentration. The authors should either add a direct comparison against particle tracking (or the exact unclosed dynamics) in a non-invertible flow, or explicitly limit the claims of predictive closure to the shear-flow cases.
minor comments (7)
  1. [Abstract] The phrase 'in in deterministic' should read 'in deterministic'.
  2. [Eq. (6)] For consistency with SDE (5), the x-space advection term should be u(x)f, not v f; the equality v f = u(x)f holds only on the support of the joint delta distribution.
  3. [Section 4, Eqs. (20)-(21)] Clarify that the transverse velocity component v in the shear-flow dynamics is taken to be zero; otherwise the statement that 'the term σv ∂f/∂u can be disregarded' is not justified.
  4. [Section 3.3] The sentence 'Since f*(x|v)=0' is confusing because f* is not a probability density; rephrase as an explicit assumption that the conditional-density fluctuation vanishes.
  5. [Fig. 1 and Eq. (24)] State the no-flux boundary conditions in velocity space and give the numerical time-stepping details for the Chebfun solution; the paper cites Chebfun but provides no discretization or convergence information.
  6. [Section 5] The claim 'For the first time, in this work, we propose a rigorous approach...' should be softened, since velocity-PDF/Fokker-Planck derivations based on Itō calculus and conditional expectations have a long history in turbulence and stochastic coarse-graining.
  7. [Throughout] Typographical errors: 'Ito' for 'Itō' throughout, 'Smoluchowki' in Section 5, 'would noto be fully closed' in Section 3.4, and inconsistent notation for dimensionless times in Fig. 1.

Circularity Check

2 steps flagged · score 4.0 of 10

Central evolution-equation derivation is self-contained, but the equilibrium closure makes the stationary Eulerian velocity PDF an input; the Poiseuille result is a consistency condition rather than an independent prediction.

  1. self definitional [Section 3.2, Eqs. (14)-(15)]
    "g(x)^{e|v} = ∫_{Ω_v} g(x)δ(v-u(x)) ce / Fe(v) dx ... The stationary version of Eq. 10 ... is therefore the equation satisfied by the equilibrium velocity PDF Fe(v) which, in this case, it is equivalent to the Eulerian velocity PDF, i.e. the velocity PDF obtained by sampling uniformly the whole domain."

    The equilibrium conditional expectations in Eq. (14) are normalized by F_e(v), which is exactly the quantity that Eq. (15) is supposed to determine. Substituting g^{e|v} = A_g(v)/F_e(v) into Eq. (15) cancels F_e and reduces the stationary equation to an identity inherited from the uniform stationary concentration c_e. Thus the equivalence between F_e and the Eulerian velocity PDF is an input of the closure, not a derived prediction.

  2. self definitional [Section 4.2, Eqs. (24)-(25)]
    "The stationary solution of this equation is given by: F∝(1-u/u0)^{-1/2} (25) This solution is, in fact, consistent with the fact that, when particles are uniformly spread throughout the channel, their velocity is simply given by the inverse function of the velocity profile."

    The closed Poiseuille equation (24) is constructed from the equilibrium closure, whose coefficients are σ'(y(u)) and σ²(y(u)) evaluated on the inverse velocity profile. Its stationary solution (25) is precisely c_e/|u'(y(u))|, the Eulerian velocity PDF implied by uniform concentration. The numerical test in Fig. 1 only checks convergence to this same input; it cannot validate the closure for finite-time evolution. Therefore Eq. (25) is a consistency condition by construction, not an independent prediction.

full rationale

The core derivation of the marginal velocity-PDF equation (10) is self-contained: starting from the Itô SDE (1), Eqs. (4)-(5) apply Itô's lemma, and Eq. (6) is the corresponding Fokker-Planck equation; marginalizing over space gives the unclosed Eq. (10) with conditional expectations. No fitted parameter or self-citation is load-bearing in that part. The circularity is confined to the equilibrium closure and its use in the Poiseuille example. Eq. (14) defines the equilibrium conditional expectations using F_e(v) in the denominator, and Eq. (15) then states that F_e satisfies the stationary equation; substituting (14) into (15) cancels F_e and reduces to an identity inherited from the uniform stationary concentration. Likewise, Eq. (24) is built from the inverse velocity profile and its stationary solution (25) is exactly c_e/|u'(y(u))|, the Eulerian velocity PDF implied by that same uniform concentration. Fig. 1 therefore checks self-consistency, not an independent prediction. This warrants a moderate score: the central evolution-equation derivation is independent, but one advertised result—the equilibrium Eulerian velocity PDF—is an input of the closure by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The derivation rests on standard stochastic calculus and on modeling assumptions about steady non-inertial incompressible flow. The equilibrium closure is an ad-hoc simplification that makes the velocity-PDF equation closed but is only expected to hold near equilibrium. No new physical entities are introduced.

assumptions (5)
  • standard math Ito's lemma and the standard Fokker-Planck correspondence apply to the derived joint SDE.
    Used in Section 2 to derive the joint position-velocity SDE (Eq. 5) and the Fokker-Planck equation (Eq. 6); standard result in stochastic calculus.
  • domain assumption The flow field is steady, incompressible, and non-inertial, so particle velocity equals the local fluid velocity U = u(X) at all times.
    Stated in the note after Eq. (8) and used throughout; if particles have inertia or the flow is unsteady, the derived equations change.
  • domain assumption The velocity field is smooth enough for the second-order Ito expansion (bounded velocity gradients and second derivatives).
    Required for the Taylor expansion in Eq. (4); the paper assumes Gaussian-correlated smooth fields in Section 4, which may fail at pore-scale no-slip boundaries.
  • ad hoc to paper Equilibrium closure: time-dependent conditional expectations G^{x|v}, ∆u^{x|v}, and GG^T^{x|v} are replaced by their uniform-concentration equilibrium values.
    Introduced in Section 3.2 and used in Section 4.2 (Eq. 24); this is an uncontrolled approximation expected to hold only near equilibrium.
  • ad hoc to paper IEM/Gaussian closure: velocity and velocity gradient are jointly Gaussian, leading to linear conditional expectations in Eqs. (18)-(19).
    Proposed in Section 3.4 as an option; not tested numerically and not used in the main example.

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Cite this review

Pith. "Pith review of Probability density function (PDF) models for particle transport in porous media." pith.science (2026). https://pith.science/paper/VOZNWEUX

@misc{pith2026190801770,
  author       = {Pith},
  title        = {Pith review of: Probability density function (PDF) models for particle transport in porous media},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VOZNWEUX}},
  note         = {Machine review of arXiv:1908.01770}
}
read the original abstract

Mathematical models based on probability density functions (PDF) have been extensively used in hydrology and subsurface flow problems, to describe the uncertainty in porous media properties (e.g., permeability modelled as random field). Recently, closer to the spirit of PDF models for turbulent flows, some approaches have used this statistical viewpoint also in pore-scale transport processes (fully resolved porous media models). When a concentration field is transported, by advection and diffusion, in a heterogeneous media, in fact, spatial PDFs can be defined to characterise local fluctuations and improve or better understand the closures performed by classical upscaling methods. In the study of hydrodynamical dispersion, for example, PDE-based PDF approach can replace expensive and noisy Lagrangian simulations (e.g. trajectories of drift-diffusion stochastic processes). In this work we derive a joint position-velocity Fokker-Planck equation to model the motion of particles undergoing advection and diffusion in in deterministic or stochastic heterogeneous velocity fields. After appropriate closure assumptions, this description can help deriving rigorously stochastic models for the statistics of Lagrangian velocities. This is very important to be able to characterise the dispersion properties and can, for example, inform velocity evolution processes in Continuous Time Random Walk (CTRW) dispersion models. The closure problem that arises when averaging the Fokker Planck equation shows also interesting similarities with the mixing problem and can be used to propose alternative closures for anomalous dispersion.

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