[681.1.16.5] This appendix considers some computational aspects of our theory to aid readers simulating experiments with hysteresis. [681.1.16.6] It clarifies fundamental differences (locality vs. nonlocality) between the theory presented here and a traditional hysteresis model. [681.1.16.7] The natural assumption in eq. (2) , that breakup or coalescence of ganglia is proportional to the rate and direction of saturation changes, is neither equivalent nor related to the traditional hysteretic extensions of capillary pressure or relative peremabilities.
[681.1.17.1] The origin of hysteresis in the present theory (see [9, 10]) differs fundamentally from traditional hysteresis models such as the model utilized in [15]. [681.1.17.2] Traditional hysteresis models require to store for each location inside the sample the pressure and saturation history (i.e. the reversal points, where the process switches between drainage and imbibition). [681.1.17.3] In our theory such pressure and saturation histories are not needed. [681.1.17.4] Instead, contrary to traditional hysteresis models, our theory allows to compute the future state of the porous medium, given only the knowledge of its present state. [681.1.17.5] In other words: While traditional hysteresis models are nonlocal in time (and thus require to memorize the systems history), our theory is local in time.
[681.1.18.1] In practical computations the locality of our theory translates into reduced storage requirements and a more straightforward implementation. [681.1.18.2] Table 2 below lists the fields (i.e. the position and time dependent quantities) necessary to compute the future time evolution of the system.
[681.1.19.1] Both approaches need a pressure and saturation field
at the present time instant .
[681.1.19.2] Our theory needs in addition the unknowns
to completely specify the present state of the system.
[681.1.19.3] This amounts to two additional state variables at
each collocation point.
[681.1.19.4] Traditional hysteresis models need in addition
historic values, one for pressure
and one for saturation, at each collocation point
.
[681.1.19.5] The number
is the number of
time instants
at which reversals
occur at position
.
[681.1.19.6] A reversal is a switching between drainage and imbibition
at the collocation point
.
[681.1.19.7] The number
depends on the nesting or not of scanning curves.
[681.1.19.8] The number
and the time instants
are not known in advance.
[681.1.19.9] In [15]
it is assumed ad hoc that nested loops do not
occur and that the last two reversals are sufficient
to avoid pumping effects.
[681.1.19.10] This uncontrolled approximation might fail
for experiments with cyclic pressure changes
where nested scanning loops are expected to
occur locally.
[681.1.19.11] In the general case [15] expect that
the last four or five reversals are sufficient.
[681.1.20.1] Finally, it may be of interest for
practical computations that the model of
[15] postulates explicitly
and implicitly numerous
functional relations between the variables and unknowns
characterizing the state of the system.
[681.1.20.2] Examples are not only the capillary pressure-saturation
relationship or the relative permeability-saturation
relation, but also the functional relations between
the various effective, apparent, entrapped and
historic saturations and pressures
,
,
,
,
and
appearing in [15].
[681.1.20.3] The functional forms for these relationships are
postulated purely theoretically and have, apparently,
been tested by inverse fitting but not yet by a direct
experimental test.
[681.1.20.4] The large number of such functional relations and the
freedom to parametrize them results in so many
possible fit parameters for the model of [15]
that a meaningful comparison to other approaches based
on the number of free fit parameters becomes difficult.
this theory | [15] |
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