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[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 t. [681.1.19.2] Our theory needs in addition the unknowns S_{2},S_{4} 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 2l(x_{i}) historic values, one for pressure and one for saturation, at each collocation point x_{i}. [681.1.19.5] The number l(x_{i}) is the number of time instants t_{j}(x_{i}),j=1,...,l(x_{i}) at which reversals occur at position x_{i},i=1,...,N. [681.1.19.6] A reversal is a switching between drainage and imbibition at the collocation point x_{i}. [681.1.19.7] The number l(x_{i}) depends on the nesting or not of scanning curves. [681.1.19.8] The number l(x_{i}) and the time instants t_{j}(x_{i}) 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 \overline{S}_{w}, \overline{\overline{S}}_{w},h_{{nw}}, \overline{S}_{{nt}}, \overline{S}_{{nr}}, \sideset{{}^{\Delta}}{{}_{{nwj}}}{\mathop{h}} and \sideset{{}^{\Delta}}{{}_{j}}{\mathop{\overline{S}_{w}}} 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.

Table 2: List of unknowns needed at a given time instant t to compute the future time evolution for the mathematical model in this paper as compared to the mathematical model of [15]. Quantities corresponding to each other appear in the same row. The arguments x_{i} denote N discretized positions, i.e. collocation points i=1,...N of the numerical simulation. The notation of in the right column follows [15], the notation in the left column is that of this paper. The time instants t_{j}(x_{i}),j=1,...,l(x_{i}) are the time instant of the j-th reversal at position x_{i}. The number l(x_{i}) of reversals (nested scanning curves) depends on position x_{i}.
this theory [15]
{S_{{\mathbb{W}}}}(x_{i},t) S_{w}(x_{i},t)
P_{3}(x_{i},t) h_{{nw}}(x_{i},t)
\sideset{{}^{\Delta}}{{}_{{\; l(x_{i})-1}}}{\mathop{\overline{S}_{w}}}(x_{i},t_{{l(x_{i})-1}}(x_{i}))
\sideset{{}^{\Delta}}{{}_{{nw\; l(x_{i})-1}}}{\mathop{h}}(x_{i},t_{{l(x_{i})-1}}(x_{i}))
\sideset{{}^{\Delta}}{{}_{{\; l(x_{i})}}}{\mathop{\overline{S}_{w}}}(x_{i},t_{{l(x_{i})}}(x_{i}))
\sideset{{}^{\Delta}}{{}_{{nw\; l(x_{i})}}}{\mathop{h}}(x_{i},t_{{l(x_{i})}}(x_{i}))