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Geometric evolution as a source of discontinuous behavior in soft condensed matter
James E. McClure
Virginia Polytechnic Institute & State University, Blacksburg
Steffen Berg
Shell Global Solutions International B.V. Grasweg 31, 1031HW Amsterdam, The Netherlands
Ryan T. Armstrong
University of New South Wales, Sydney
(Dated: June 11, 2019)
Geometric evolution represents a fundamental aspect of many physical phenomena. In this paper
we consider the geometric evolution of structures that undergo topological changes. Topological
changes occur when the shape of an object evolves such that it either breaks apart or converges
back into itself to form a loop. Changes to the topology of an object are fundamentally discrete
events. We consider how discontinuities arise during geometric evolution processes by characteriz-
ing the possible topological events and analyzing the associated source terms based on evolution
equations for geometric invariants. We show that the discrete nature of a topological change leads
to discontinuous source terms that propagate to physical variables.
PACS numbers: 92.40Cy, 92.40.Kf, 91.60.Tn, 91.60.Fe
Geometric structure frequently influences the behav-
ior of soft matter systems. Micro-emulsions, liquid crys-
tals, biological tissues, and flow through porous media
are each governed by closely coupled geometric, ther-
modynamic and mechanical processes. The properties
and behavior of soft condensed matter are very much de-
pendent on structure due to the presence of interfaces,
membranes, and other energy barriers. In this work we
focus specifically on interactions that result from topo-
logical changes, which are alterations to how a structure
is connected. While the importance of structural effects
is frequently obvious, the mathematical consequences of
topological change have been overlooked in the develop-
ment of physical theory. Core theoretical results used to
formulate arguments about the behavior of physical sys-
tems routinely rely on explicit assumptions of continuity
and differentiability [1–4]. However, topological changes
are not smooth. Understanding how to model the asso-
ciated structural dynamics is critical to develop effective
models, both from the geometric perspective and with re-
spect to the impact on thermodynamic and mechanical
variables.
Discontinuous effects due to topological changes have
been noted previously for fluid systems. Droplet coa-
lescence involves what is essentially the simplest possi-
ble change in connectivity: the merging of two fluid re-
gions. Droplet coalescence is associated with the forma-
tion of a singularity, which has been considered in detail
from an experimental perspective [5–8]. Coalescence can
dominate the physical behavior from the largest length
scales to the smallest, with applications ranging from
meteorology to nano-technology, linking with fundamen-
tal wetting mechanisms and spreading [9–13]. The ba-
sic collision-coalescence mechanism is illustrated in Fig.
1. Within emulsions such droplet interactions occur fre-
quently. In emulsions droplet interactions with films can
inhibit coalescence, implying that the structural proper-
ties of the films separating droplets are also critical [14–
16]. Such mechanisms are also operative in the behavior
of lipid membranes in both biological and engineered sys-
tems [17]. The effect of fluid structure is also evident for
flows in porous media, where rapid topological changes
that occur on a fast timescale lead to persistent geometric
effects that can dominate macroscopic behavior [18, 19].
FIG. 1. Collision-coalescence mechanism within a micro-
emulsion. After two droplets collide, a bridge forms at the
coalescence point. The larger droplet formed by the event
evolves to attain a new equilibrium configuration based on
the minimum potential energy.
This paper considers geometric evolution in the gen-
eral sense, noting that all connectivity transitions occur
as discrete events. At the simplest level these may be
reduced to local coalescence and snap-off phenomena, al-
beit with distinctly more variability. We show that sin-
gularities resulting from geometric transitions occur even
in well-connected structures, leading to an accumulation
of discontinuous transitions that influence overall macro-
scopic behavior. An example is considered based on the
flow of immiscible fluids in porous media, demonstrating
that snap-off and coalescence events cause jump condi-
tions in the time derivative of the fluid pressures. Specific
arXiv:1906.04073v1 [cond-mat.soft] 10 Jun 2019
2
contributions of this work are as follows: (1) character-
ize geometric evolution processes, showing that in gen-
eral both continuous and discontinuous transformations
must be considered; (2) classify the possible topological
changes in a three-dimensional system; (3) describe how
topological change contributes discrete source terms in
the fundamental equations that describe geometric evolu-
tion; and (4) demonstrate that geometric discontinuities
propagate to physical variables.
THEORY
Geometric structure emerges as a consequence of forces
acting between the molecules within a system. Particu-
lar molecular configurations lead to local minimum for
the potential energy, and the system will spontaneously
self-segregate toward such energetically favorable config-
urations. From the microscopic perspective, the system
dynamics are entirely determined from the phase coordi-
nates related to the molecular degrees of freedom. Let us
consider a closed system with two distinguishable types
of particle, i∈ {a, b}. The classical state of the system is
represented based on the position q(i)
kand velocity ˙q(i)
k
for each particle, r=q(i)
k,˙q(i)
k, with k∈ {1, . . . , N (i)}.
Geometric structure arises when the molecular system is
treated in an averaged way based on classical thermody-
namics and continuum mechanics.
Within the interface region, the composition of the
molecular system differs from the bulk phases, as shown
in Fig. 2. The Gibbs dividing surface is defined to con-
struct particular regions within the system, allowing sep-
arate treatment for three-dimensional phase regions and
two-dimensional interface regions [20]. Since the dividing
surface is determined based on the component densities,
the phase regions depend only the position of the par-
ticles. The two sets formed based on the choice of the
Gibbs dividing surface may be formally represented as
Ωi(t) = Ωiq(i)
kfor i∈ {a, b}.(1)
It follows that invariant geometric measures of Ωidepend
only on the particle coordinates. Since the Gibbs dividing
surface separates the system into a countable number of
entities, an inherently discrete element is introduced into
the classical description of the system.
Sub-division of the system into phase and interface re-
gions is relied upon in the classical thermodynamic de-
scription of heterogeneous systems [21]. For the case at
hand, the internal energy of the system can be written
as
U=X
i∈{a,b}TiSi−piVi+µikNik +γab Aab ,(2)
where the relevant quantities for each phase i∈ {a, b}
are the temperature Ti, the entropy Si, the pressure pi,
FIG. 2. The Gibbs dividing surface separates the interface
region to obtain the sharp representation used in classical
thermodynamics.
the volume Vi, the chemical potential µik and the num-
ber of particles of type k,Nik . The energy due to the
interfacial arrangement is given based on the interfacial
tension γab and the associated surface area separating
the two regions. Geometry is therefore introduced into
description of the internal energy of the system. The av-
erage pressure within the associated phase regions can be
determined as
pi=RΩipdV
RΩidV .(3)
where the pressure pat any point in the system is de-
fined based on arguments of statistical mechanics. The
spatial average given in Eq. 3 ensures that the internal
energy of the system is properly captured. Analogous ar-
guments can be made to define spatial averages for other
thermodynamic quantities [22]. Due to the structural di-
vision of the system based on the Gibb’s dividing surface,
each quantity in Eq. 2 is either explicitly or implicitly
dependent on the geometric structure of the problem.
Geometric characterization
Assumptions regarding the geometric structure of a
system provide the basis to establish key concepts re-
lated to differentiability, symmetry, and conservation.
Geometric results determined for three-dimensional ob-
jects represent the possible shapes that structures can
attain, and derived results also govern how structures
can evolve. Hadwiger’s characterization theorem estab-
lishes that only four invariant measures are needed to
describe the structure of a three-dimensional object [23].
Our work proceeds based on these invariant measures.
A kinematic evolution equation for the volume of a set
3
Ωiis provided by the Minkowski-Steiner formula. If we
consider rolling a sufficiently small ball with diameter δ
around Ωi, the change in volume is predicted as a linear
function of the scalar invariants of the boundary Γi
V(Ωi⊕δζ)−V(Ωi) = α1Aiδ+α2Hiδ2+α3χiδ3,(4)
where ζis the unit ball and the coefficients αkfor k∈
1,2,3 are determined based on the structure of Ωi[24].
The scalar invariants are the surface area Ai, the mean
width Hi, and Euler characteristic χi. The mean width
is defined based on the integral of mean curvature
Hi=ZΓi
κ1+κ2
2dS (5)
where κ1and κ2are the principal curvatures along the
surface. Euler characteristic is directly proportional to
the total curvature, which includes contributions from
the Gaussian curvature of the object boundary Γias
well as the geodesic curvature associated with any non-
smooth portions of the boundary ∂Γi,
4πχi=ZΓi
κ1κ2dS +Z∂Γi
κgdC (6)
where κgis the geodesic curvature. The Euler character-
istic provides the link to the topology of an object, which
is the basis for using Euler characteristic to measure con-
nectivity [25, 26]. Euler characteristic is related to the
alternating sum of the Betti numbers,
χi=B0−B1+B2.(7)
where B0is the number of connected components, B1
is the number of loops and B2is the number of cavities
enclosed within the object.
The Minkowski-Steiner formula is only valid for suffi-
ciently small values of δ. The coefficients α1, α2and α3
are particular functions of Ωiand different coefficients
may be obtained for different structures. However, re-
cent work suggests that structures with identical geo-
metric measures will be associated with identical coef-
ficients when considered across the state space of many
possible geometric structures [27]. For additional math-
ematical background pertaining to the scalar invariants,
the reader is referred to the works of Federer [24] and
Klain [28]. Detailed reviews pertaining to the applica-
tion to characterise the behavior of structures are also
available [29–33]. Here we seek to understand how par-
ticular structures and their associated invariant measures
change with time.
Geometric evolution: a simple example
Consider first the change in volume that results when
growing a sphere with radius rto the larger sphere r+δr
V(Ωi⊕δrζ )−V(Ωi) = 4π
3(r+δr)3−4π
3r3
=Aiδr +Hi(δr)2+4π
3χi(δr)3
(8)
where expressions for the geometric invariants associated
with a sphere have been inserted: Ai= 4πr2,Hi= 4πr
and χi= 1. Comparing with Eq.4 we quickly see that the
coefficients are α1= 1, α2= 1 and α3= 4π/3. Next we
consider a torus with major radius Rand minor radius r
V(Ωi⊕δrζ )−V(Ωi)=2π2(r+δr)2R−2π2r2R
=Aiδr +Hi(δr)2(9)
using the fact that for the torus Ai= 4π2rR,Hi= 2π2R
and χi= 0. Again referring to Eq.4, we see that α1= 1
and α2= 1. Even though χi= 0 for a torus, identical
coefficients are obtained for the two structures.
An important challenge associated with modeling geo-
metric evolution is that changes in the connectivity of an
object occur as discrete events; χi(t) is not a continuous
function. This means that while the volume, surface area
and mean width will necessarily be continuous functions,
the Euler characteristic will evolve based on a series of
discrete jumps. To illustrate this behavior we consider
the geometric evolution of a torus into a sphere as shown
in Fig. 3. This relatively simple problem is significant
because it defines a change in connectivity. The initial
torus has χ= 0; for the final sphere χ= 1. Based on
Eq. 7 we can see that for the torus B0= 1, B1= 1 and
B2= 0. The single handle disappears at time t= 1 and
B0= 1, B1= 0 and B2= 0. The change in connectivity
occurs at the instant the hole in the center of the torus
closes, representing a discontinuity in the geometric evo-
lution process. We now examine the ramifications for this
change. Specifically, we will show that the discontinuity
in the time evolution for the Euler characteristic leads to
non-smooth behavior for the other geometric invariants.
The torus is defined based on a surface of revolution
for a circle with minor radius rthat is displaced from the
origin based on the major radius R. We choose initial
minor radius r0= 1/4 with ran increasing function of
time,
r(t) = r0+t
4.(10)
We enforce r+R= 1 such that the object has constant
unit width. For time t < 1 the circle has radius r < R
and the surface of revolution is a torus. At time t= 1 the
circle has radius r=R= 1/2 and the surface becomes a
spindle torus for 1 < r < 3. At t= 3 the object is the
unit sphere. The associated three-dimensional structures
are shown in Fig. 3.
4
To compute the geometric invariants, we rely on ex-
pressions derived for surfaces of revolution. The circle is
parameterized by
x(s) = R+rcos(s) (11)
y(s) = rsin(s).(12)
Since the surface of revolution is self-intersecting for r >
R, the limits of integration are defined based on the angle
α=arccos(R/r) if r > R
0 otherwise. (13)
The principle curvatures are
κ1=1
( ˙x2+ ˙y2)3/2−¨x˙y+ ˙x¨y,(14)
κ2=1
( ˙x2+ ˙y2)1/2
˙y
x.(15)
The surface area is
A(t)=2πZπ−α
−π+α
x(s)p˙x2+ ˙y2ds
= 2π[Rr(2π−2α)+2r2sin(α)] (16)
For time t < 1, α= 0 and the expression reduces to
A= 4πRr. For time t= 3, the expression for a sphere
is obtained, A= 4πr2, since R= 0. Based on a similar
calculation we can calculate the mean width
H(t) = πZπ−α
−π+α
(κ1+κ2)xp˙x2+ ˙y2ds
= 2πR(π−α)+4πr sin(α).(17)
Eqs. 16 and 17 are plotted in Fig. 3, clearly showing
that while A(t) and H(t) are both continuous functions,
the time derivatives are discontinuous at the instant that
the hole in the middle of the torus closes (t= 1), which is
also associated with a jump condition in the Euler char-
acteristic from χ= 0 to χ= 1. Predicting when such
topological changes occur requires information regarding
the overall geometric structure of an object.
Due to the topological change, a one-to-one mapping
cannot be defined for the geometric evolution at t= 1.
Prior to the topological change we can identify the ring
of points at the interior boundary of the torus x, y :x2+
y2−(R−r)2= 0; z= 0. At time t= 1 the ring closes
such that the entire ring of points are mapped to the
origin, which means that the associated mapping is not
one-to-one. The geometric evolution can be considered
based on a continuous deformation of the structure for
t < 1 and then again for t > 1, but not for t= 1. This
underscores the fact that geometric evolution should be
generally considered as a sequence of processes divided
into two distinct categories:
1. Continuous local geometric flow at constant topol-
ogy as given by the time dependent mapping Gt
k:
Ω→Ω where Gt
kis one-to-one for time tk−1<t<
tk.
FIG. 3. The evolution of a torus into a sphere. The center hole
of the torus closes at t= 1 causing a jump condition in the
Euler characteristic. Plots showing the (c) surface area and
(d) mean width as a function of time based on the evolution
of a torus into a sphere. The hole in the center of the torus
closes at time t= 1, resulting in time dependence that is non-
smooth. Changes in connectivity are linked to a breakdown
in local smoothness, presenting a fundamental challenge to
modeling the geometric evolution and associated physics.
2. Discrete geometric transformations defined by G∗
k:
Ω→Ω occurring at time tkand corresponding
to a topological change that cannot be represented
based on a bijection.
By chaining together a sequence of such mappings k∈
{1,2, . . . , K}, it is possible to model geometric evolution
processes of arbitrary complexity.
Classification of Topological Changes
Only a limited number of topologically distinct struc-
tures are possible within a particular dimensional space
[48]. In a three-dimensional system, all possible topolog-
ical states can be reduced to the analysis of three funda-
mental shapes: the sphere, the torus, and the spherical
shell. These shapes are linked to the Betti numbers: the
sphere is a single connected component and links to B0;
the torus has a single loop and links to B1; the spherical
shell contains a single cavity and links to B2. To form
more complex structures, these shapes can be placed in
a system, first gluing objects together as needed, then
stretching and deforming the resulting object until a de-
sired structure is obtained. Based on this it is possible
to characterize the possible topological changes that may
occur during geometric evolution, i.e. the discontinuous
maps G∗
k. These eight possibilities are depicted in Fig.
4.
Objects on left side of Fig. 4 can be produced by con-
tinuously stretching and deforming a a sphere (i.e. home-
omorphisms). The depicted topological changes increase
or decrease the Euler characteristic by exactly one. Mov-
5
FIG. 4. Discontinuous geometric maps in three-dimensions
fall into eight categories based on homeomorphism. The as-
sociated topological changes increase or decrease Euler char-
acteristic by exactly one.
ing left to right for case A, the snap-off mechanism gener-
ates two regions homeomorphic to a sphere from a single
such region. The consequence is to increase Euler char-
acteristic by one due to the corresponding increase in B0.
The coalescence mechanism corresponds to the opposite
situation, where two regions merge together to destroy
one connected component. The pair of transformations
labeled as B rely on the same underlying mechanism, in
this case either forming or destroying a loop. Moving left
to right, the Euler characteristic decreases by one due to
a corresponding increase to B1. Snap-off has the opposite
effect.
For four transformations given by A and B the singu-
larities are isolated to particular points, which represent
the sub-regions of the set for which the mapping is not
bijective. For the four transformations given by C and
D, the singularity involves a ring structure. For the pair
of transformations labeled as C, the puncture mechanism
forms a ring of points at the location of the singularity.
This transformation increases B1and decreases χ. The
closure mechanism involves the collapse of a ring of points
to the singularity, destroying the loop. This is the evo-
lution considered in Fig. 3. The pair of transformations
shown in Fig. 4D shows that the closure mechanism can
also form a cavity starting from a bowl-shaped region.
This increases both B2and χby one. The puncture
mechanism can re-open the hole to destroy the cavity.
For each of the eight cases, the associated mappings are
not bijective. Noting that a single object can undergo
multiple such transformations, we can always identify a
neighborhood of points that surround the singular point,
effectively isolating each discontinuity in space and time.
Geometric evolution: a hierarchical perspective
One can further gain insight into the nature of geo-
metric discontinuities by considering the time evolution
based on a hierarchical view of Eq. 4. Given an arbitrary
closed, three-dimensional object we consider continuous
changes in volume such that the first derivative can be
defined with respect to time. From Eq. 4 we can easily
see that if δis sufficiently small
V(Ωi⊕δζ)−V(Ωi) = α1Ai(t)δ+O(δ2).(18)
That is, the infinitesimal change in volume is entirely
given by the movement of the boundary, and the surface
area is the only boundary invariant on which the volume
change depends. The same insight can be obtained by
applying the Reynolds transport theorem to the region
Ωito predict the time rate of change
dVi
dt =ZΓi
(wi·ni)dS , (19)
where niis the outward normal to the boundary and
wiis the boundary velocity. The time derivative for the
surface area and mean width can also be identified from
boundary boundary integrals. Since the change in surface
area due to the deformation of a local surface element is
determined by the curvature of the surface element [49],
dAi
dt =ZΓi
κ1+κ2
2(wi·ni)dS , (20)
and similarly for the mean curvature,
dHi
dt = 2 ZΓi
κ1κ2(wi·ni)dS . (21)
Eqs. 19 – 21 describe how the geometric invariants evolve
during the continuous portions of the geometric evolution
of an object. The integrals capture both the growth and
the deformation of the object boundary. A topological
source term can be identified in Eq. 21 due to the de-
pendence on the Gaussian curvature.
Noting that a differential equation cannot be derived
for Euler characteristic, it is useful to predict the topo-
logical state based on the geometric state function
χi=χ(Vi, Ai, Hi).(22)
Previous work indicates that the geometric state can be
predicted uniquely for complex structures based on a re-
lationship between the four geometric invariants [47]. In
6
the context of geometric evolution, expressing the rela-
tionship as a predictor of the topological state is par-
ticularly useful; together with Eqs. 19–21, an equation
is associated with each of the four geometric invariants.
While the time evolution is given generally from integral
equations, approximate forms have also been explored as
a way to apply the result to particular physical systems
[50].
An intuitive basis for how topological source terms
arise within geometric evolution can be established based
on further consideration of the Minkowski-Steiner for-
mula. Based on Eq. 19 it is natural to define the average
boundary displacement as
ξ≡RΓi(wi·ni)dS
Ai
.(23)
For the special case where the boundary velocity in the
normal direction is constant on Γi,ξ=wi·niand we
can make a direct link with Eq. 18. For a volume change
over some time ∆tit is evident that δ=ξ∆tand αi= 1,
meaning that Eqs. 19 – 21 may be expressed as
dVi
dt =Ai(t)ξ, (24)
dAi
dt =α2Hi(t)ξ, (25)
dHi
dt =α3
α2
χi(t)ξ. (26)
That is, to predict the time evolution we need to know the
coefficients α2and α3as well as the Euler characteristic
χi. Based on these arguments it is intuitive to express
the Minkowski-Steiner formula in a hierarchical manner,
V(Ωi⊕δζ)−V(Ωi) = hAi+α2Hi+α3
α2
χiδδiδ . (27)
Eqs. 24– 26 offer insight into how the jump condi-
tion due to change in Euler characteristic propagates to
other variables. From Eq. 26 we can deduce that Hi(t)
is not differentiable whenever the topology of the sys-
tem changes, which is implied by the dependence on χi.
Ai(t) inherits a discontinuity in the second-order time
derivative from Hi. Both results are confirmed based on
inspection of Fig. 3. It can also be seen that since the
spindle torus does not satisfy the requirement of posi-
tive reach, α2and α3are time-dependent. However, this
apparently does not alter the associated differentiability
class for Hior Ai. Of course, more favorable smoothness
properties should be expected in the regions of the time
where no topological changes occur.
Discontinuous aspects of geometric evolution represent
a critical challenge to modeling physical behavior. Sys-
tems that undergo topological change will involve non-
smooth aspects with respect to the time evolution for
the geometric invariants. An essential question is the
extent to which geometric effects impact continuity and
differentiability for thermodynamic and mechanical vari-
ables. Noether’s theorem provides a formal basis for the
continuous description of physical systems based on dif-
ferential equations [1, 2]. Noether’s theorem focuses on
invariant properties for systems that undergo continuous
group transformations (i.e. Lie groups), relying explicitly
on invertibility. Typical proofs of the Poincar´e recurrence
relation also rely on the invertability of group transforma-
tions [3]. The continuous transformations Gt
kfall within
this category. These correspond to continuous deforma-
tion of an object at constant topology, which proceed
during a finite interval of time. Topological changes are
associated with a discontinuous map G∗
k, which fall into
a class of transformations that are explicitly excluded
from Noether’s arguments. The implication is that even
though Noether’s arguments apply to the molecular sys-
tem, the choice to represent the behavior of a heteroge-
neous system based on the classical thermodynamic de-
scription will effectively destroy the underlying symme-
try within the defined regions. Particular consequences
become evident when non-stationary processes are con-
sidered. Differentiability of the terms that appear in
Eq. 2 is required to develop essential results of non-
equilibrium thermodynamics, which in turn play a major
role in the description of soft matter systems [35]. Topo-
logical effects must be treated carefully, since the associ-
ated discontinuities undermine the symmetry arguments
that form the basis for continuum-scale descriptions.
RESULTS
The flow of immiscible fluids in porous media are
strongly influenced by geometric and topological effects
[39–47]. Capillary forces typically dominate such flows,
and these forces are directly impacted by changes to
fluid connectivity. Coalescence events always involve
the local destruction of interfacial curvature, which in-
stantaneously alters accompanying capillary forces. The
Laplace equation relates the pressure difference between
of the adjoining fluids and the capillary pressure given
by the product of the interfacial tension γand mean cur-
vature:
pn−pw=γ
2κ1+κ2.(28)
The Laplace equation holds only at mechanical equilib-
rium, and a rapid change to the interface mean curvature
will cause an immediate force imbalance. The associated
discontinuity is directly evident from Eq. 5 and the as-
sociated dynamics. The coalescence event illustrated in
Fig. 1 provides an intuitive example. Immediately prior
to a coalescence event, the interface curvature is positive
based on the outward pointing normal vector relative to
the object boundary. Coalescence leads to the formation
of a bridge, which causes the sign of the curvature to
7
change from positive to negative. The local disruption
to the geometric structure creates a force imbalance that
is analogous to puncturing a balloon with a pin; coales-
cence effectively ruptures the interface. The local fluid
pressures must respond rapidly to correct the force imbal-
ance. However, the timescale for this to occur is coupled
to geometric evolution. Mechanical equilibrium cannot
be re-established until the interfaces attain an appropri-
ate shape. Detailed studies of the coalescence mechanism
show that the flow behavior and geometric evolution are
coupled on a very short timescale [7].
We consider fluid displacement within a porous
medium as a means to explore how geometric changes
influence physical behavior. A periodic sphere packing
was used as the flow domain. Using a collective rear-
rangement algorithm, eight equally-sized spheres with di-
ameter D= 0.52 mm were packed into a fully-periodic
domain 1 ×1×1 mm in size. The system was discretized
to obtain a regular lattice with 2563voxels. Simulations
of fluid displacement were performed by injecting fluid
into the system so that the effect of topological changes
could be considered in detail. Initially the system was
saturated with water, and fluid displacement was then
simulated based on a multi-relaxation time (MRT) im-
plementation of the color lattice Boltzmann model [54].
To simulate primary drainage, non-wetting fluid was
injected into the system at a rate of Q= 50 voxels per
timestep, corresponding to a capillary number of 4.7×
10−4. A total of 500,000 timesteps were performed to
allow fluid to invade the pores within the system, forming
the sequence of structures shown in Fig. 5 a–c. The first
pores are invaded at t= 0.19, which is just after the
system overcomes the entry pressure. The fluid does not
yet form any loops, since a pore must be invaded from two
different directions before a loop will be created. Haines
jumps and loop creation can be considered as distinct
pore-scale events that may coincide in certain situations.
At time t= 0.34 the first loop is formed. As primary
drainage progresses additional loops form such that the
total curvature tends to decrease as non-wetting fluid is
injected.
At the end of primary drainage the flow direction was
reversed to induce imbibition, with water injected at a
rate of Q= 20 voxels per timestep. Sequences from the
displacement are shown in Fig. 5 d–f. As water imbibes
into the smaller pores in the system, there is a well-known
tendency to snap off non-wetting fluid between connec-
tions to larger adjacent pores where imbibition has not
yet occurred. This is known as the Roof snap-off mecha-
nism [55]. Snap-off breaks loops within the system, caus-
ing the Euler characteristic to increase. The order that
loops snap off during imibition is not the reverse of the
order that the loops form during drainage. This is be-
cause the drainage process is dominated by the size of the
throats whereas imibition is governed by the size of the
pores. Eventually, the sequence of snap-off events can
FIG. 5. Coalescence and snap-off events alter the topology
of fluids within porous media: (a)-(c) non-wetting fluid loops
formed during drainage lead to a decrease in Euler charac-
teristic; (d)-(f) imibition destroys fluid loops due to snap-off,
eventually trapping non-wetting fluid within the porespace;
(h)-(i) trapped fluids reconnect during secondary drainage.
cause non-wetting fluid to become disconnected from the
main region, leading to trapping as seen at time t= 1.86.
The presence of trapped fluid distinguishes secondary
drainage from primary drainage. The sequence of fluid
configurations shown in Fig. 5 depicts the sequence of
coalescence events that reconnect trapped fluid. Due to
the presence of trapped fluid, the order that loops are
formed within the porespace does not match the order
observed during primary drainage.
Structural changes that occur during displacement are
quantitatively captured based on the evolution of the Eu-
ler characteristic. As shown in Fig. 6, Euler characteris-
tic always takes on an integer value due to the relation-
ship to number of loops and disconnected regions – it is
not possible to create a partial loop. The Euler character-
istic consequently evolves based on a sequence of discrete
jump conditions that align precisely with the connectiv-
ity events shown in in Fig. 5 a-i. As loops form during
drainage, Euler characteristic decreases. As loops are de-
stroyed during imbibition, Euler characteristic increases.
We see clearly from Fig. 6 that the evolution of the Euler
characteristic can be considered as piece-wise continuous,
with the continuous portions being constant. Predicting
the evolution for the Euler characteristic is therefore en-
tirely a question of being able to predict when the jump
conditions will occur.
Of particular interest is the manner in which the dis-
crete events captured by the Euler characteristic influ-
8
FIG. 6. Euler characteristic evolves based on a sequence of
jump conditions that correspond to connectivity events within
the porespace. Euler characteristic decreases due to coales-
cence events and increases due to snap-off.
ence other physical variables. Fig. 6 shows how the fluid
pressures behave during displacement. Primary and sec-
ondary drainage each show a rapid spike in the pressure
difference due to the entry pressure. The entry sequence
along primary drainage is labeled as A–C and matches
the configurations from Fig. 7a. The high pore entry
pressure is consistent with expected behavior that the
largest energy barriers in the capillary dominated system
are due to to pore entry [53]. The pore entry sequence
along secondary drainage is labeled H – J. On secondary
drainage, the entry pressure is slightly reduced due to
the presence of trapped fluid. This shows that the first
coalescence event depicted in Fig. 5a–c corresponds ex-
actly to the maximum fluid pressure difference observed
during pore entry. This is distinct from the initial pore
entry observed during primary drainage, because it is the
fluid coalescence event that causes the time derivative of
the pressures to change. In the absence of coalescence
a Haines jump can be viewed as locally smooth process
that happens on a very fast timescale. On the other hand
coalescence events are not locally smooth. Indeed, non-
smooth disruptions to the capillary pressure landscape
are observed each time a coalescence event occurs.
Like coalescence, snap-off events are associated with
clear jump conditions in the bulk fluid pressures. The
snap-off sequence shown in Fig. 5 is labeled as D–F in
Fig. 7. As loops of fluid are destroyed, corresponding
fluctuations in the fluid pressure difference are observed.
The largest disruption is associated with the snap-off
event. The capillary pressure for the connected and dis-
connected regions of non-wetting fluid is shown in Fig.
7b. Consistent with the observation of other authors,
fluid is trapped at a higher capillary pressure than what
is measured in the connected fluid phases at the instant of
snap-off [55]. As soon as snap-off occurs, the time deriva-
tive of the pressure difference between the two connected
fluid phases immediately jumps from strongly negative to
strongly positive. This occurs as each fluid region relaxes
toward its preferred equilibrium state, which is possible
FIG. 7. Response of the capillary pressure to connectivity
events: (a) the difference between the pressure of the con-
nected water and non-wetting fluid show that non-smooth
disruptions to the capillary pressure landscape align with con-
nectivity events identified by changes in Euler characteris-
tic (circles); (b) snap-off causes large disruptions in capillary
pressure, with non-wetting fluid trapped at higher capillary
pressure than the bulk fluid.
because the two fluid regions are no longer mixing af-
ter the snap-off event. The final pressure for the trapped
region is determined based on geometry, with the fluid as-
suming a minimum energy configuration within the pores
where it is trapped.
For both drainage and imbibition processes, changes
in connectivity frequently correspond to a rapid jump in
the time derivative of the pressure. The magnitude of the
jump varies significantly, with events that most signifi-
cantly alter the fluids ability to mix causing the largest
disruptions. This is consistent with a local breakdown of
the ergodic hypothesis. Physically, the equilibrium state
that the system relaxes to is immediately altered when a
connectivity change occurs. This is most evident in the
snap-off process, because the trapped fluid relaxes to-
ward a separate equilibrium capillary pressure, which is
only possible due to the snap-off event. While the relax-
ation process itself is not instantaneous, the associated
change in the time derivative is an instantaneous jump
condition.
The behavior of the surface area and mean width yields
further insight into the effects of the connectivity dynam-
ics. The time evolution for these variables is shown in
Fig. 8. Since coalescence and snap-off events impact
only a small fraction of the surface area, local disconti-
9
FIG. 8. Time evolution for the surface area and mean width.
Connectivity events cause jump conditions in the mean width,
which drive discontinuous behavior in the bulk fluid pressures.
nuities have a negligible impact on the total surface area.
On the other hand, the effect of the local discontinuity on
the mean curvature is significant. Since coalescence and
snap-off events occur at the fluid meniscus, a strong im-
pact on the capillary pressure is unavoidable – the total
curvature and mean curvature are coupled. The impli-
cation is that the discontinuity in the capillary force due
to connectivity events is directly responsible for the cor-
responding behavior in the bulk fluid pressures. Based
on Fig. 8 one can clearly identify both the discrete and
continuous portions of the geometric evolution process.
CONCLUSIONS
Inherently discontinuous aspects of geometric evolu-
tion processes arise due to the use of the Gibbs di-
viding surface in classical thermodynamic descriptions.
Key questions arise regarding the application of continu-
ous theory to systems that undergo topological changes.
We classify the geometric discontinuities into eight cate-
gories, noting that any geometric transformations that do
not involve a topological change are continuous. The con-
tinuous and discontinuous sub-regions for geometric evo-
lution can be chained together to describe how complex
structures evolve. It is demonstrated that discontinuities
resulting from topological change imply that the system
dynamics will be piece-wise continuous. Expressions are
provided to describe geometric evolution, with the topo-
logical state of the system being inferred from a state
relationship relating the four geometric invariants. Not-
ing that Noether’s theorem is derived based on explicit
assumptions of differentiability, geometric discontinuities
result in symmetry-breaking when the singular points
are considered. Results of classical thermodynamics link
the geometric invariants with standard thermodynamic
quantities. Discontinuous geometric effects can thereby
propagate to the associated physical variables. The con-
sequences are demonstrated for flow of immiscible fluids
in porous media, which confirm that the time derivative
for the fluid pressures is discontinuous in the vicinity of
a topological change. We show that these discontinuities
are mechanical in nature, resulting from a discontinuity
in the mean curvature at the fluid boundary. Further
work is needed to explore the consequences of discon-
tinuous phenomena for statistical thermodynamics and
continuum-mechanical descriptions of the system behav-
ior within this broad class of problems. Efforts to under-
stand how topological changes impact energy dissipation
are of particular importance.
Acknowledgements
‘An award of computer time was provided by the De-
partment of Energy Summit Early Science program. This
research also used resources of the Oak Ridge Leader-
ship Computing Facility, which is a DOE Office of Sci-
ence User Facility supported under Contract DE-AC05-
00OR22725.
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