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VIII Discussion

A Theoretical predictions

[] This article has introduced theoretical predictions from eq. (27) shown in Figure 3 for continuous mode capillary desaturation that until now seem to have remained unnoticed within the established traditional theory of twophase flow. [] The predictions illustrated in Figure 3 provide a quantitative basis to discuss deviations observed in capillary desaturation experiments. [] They help to establish limits of validity for the traditional theory as well as for continuous mode desaturation. [] Predictions require precise knowledge of {P_{\mathrm{c}}}(S), {k^{r}_{{\mathbb{W}}}}(S) and {k^{r}_{{\mathbb{O}}}}(S) emphasizing the importance and need for reliable special core analysis of high quality.

B Protocol dependence

[] The efficiency of residual oil recovery during waterflooding depends not only on the balance of forces, but also on other factors, such as the distribution of fluids inside the medium and/or the desaturation protocol. [] The difference between continuous mode and discontinuous mode desaturation is known in the literature and it may change the critical capillary number (breakpoint) by several decades (see e.g. [40]). [] The present paper suggests for the first time equally strong differences between the DO/IWI/G-protocoland the DO/WI-protocol. [] For the latter protocol the breakpoint is sometimes found decades above unity. [] Further studies of protocol dependence are encouraged to corroborate and clarify such differences and their origin.

C Plateau saturation

[] If the saturation or desaturation process is experimentally reproducible one expects for the CO/WI- and CO/OI-protocols that [page 10, §0]

\displaystyle\mathbb{W}(t_{i}) \displaystyle\approx\mathbb{W}(t_{j}) (59a)
\displaystyle|\mathbb{W}(t_{i})| \displaystyle\approx|\mathbb{W}(t_{j})| (59b)
\displaystyle S(t_{i}) \displaystyle\approx S(t_{j})\approx S^{{\mathrm{p}}} (59c)

holds in the limit where Q_{i}\to 0 and Q_{j}\to 0 are both very small. [] The saturation S^{{\mathrm{p}}} denotes the plateau saturation. [] It is seen from Figures 3 as well as 4 that the plateau saturation will in general differ from S_{{\mathbb{O}\,\mathrm{r}}}. [] It fulfills either S^{{\mathrm{p}}}\geq 1-S_{{\mathbb{W}\,\mathrm{i}}} or S^{{\mathrm{p}}}\leq S_{\mathrm{z}}, where S_{\mathrm{z}} is defined as the zero

\displaystyle{P_{\mathrm{c}}}^{{\mathrm{imb}}}(S_{\mathrm{z}})=0 (60)

on the capillary pressure curve for secondary imbibition. [] The actual value is expected to depend on the protocol.

Figure 4: Theoretically predicted capillary saturation curves for oil injection into continuous oil according to the CO/OI-protocol (56) at F=1. The curve with crosses corresponds to drainage, the solid line without symbols to imbibition.

D Computation of P_rb from image analysis

[] The stationary pore scale pressure fields P_{i}(\widetilde{\mathbf{x}}) (i=\mathbb{W},\mathbb{O}) are generally assumed to represent equilibrium pressures for local thermodynamic equilibrium although the stationary phase velocities v_{i} may be nonzero. [] Even at v_{i}=0 the pressures cannot be constant throughout the pore space \mathbb{P}, because curved interfaces exist in local thermodynamic equilibrium within \mathbb{P}.

[] The macroscopic capillary pressure {P_{\mathrm{c}}}(S,v) from eq. (24) measured in experiments for a sample with saturation S at constant flow velocity v is typically measured using pressure sensors located in the oil and water reservoirs outside the sample. [] These pressure sensors average the local equilibrium pressure field P_{i}(\widetilde{\mathbf{x}}) over the surface \partial\mathbb{D} of the sensor. [] Assuming that the local equilibrium pressures P_{i}(\widetilde{\mathbf{x}};S) depend parametrically only on S, but are independent of v, one has

{P_{\mathrm{c}}}(S)=\frac{1}{|\partial\mathbb{D}_{\mathbb{O}}|}\int\limits _{{\partial\mathbb{D}_{\mathbb{O}}}}P_{\mathbb{O}}(\widetilde{\mathbf{x}};S)\mathrm{d}^{2}\widetilde{\mathbf{x}}-\frac{1}{|\partial\mathbb{D}_{\mathbb{W}}|}\int\limits _{{\partial\mathbb{D}_{\mathbb{W}}}}P_{\mathbb{W}}(\widetilde{\mathbf{x}};S)\mathrm{d}^{2}\widetilde{\mathbf{x}} (61)

where \partial\mathbb{D}_{i} denote the sensor surface located in the reservoir of phase i. [] In [5] a pore scale capillary pressure {P_{\mathrm{c}}}^{{**}} is computed from image analysis of oil clusters as a surface area weighted average of mean curvatures over the fluid-fluid-interface during the imaging intervals without flow (v=0). [] In other words a formula such as

{P_{\mathrm{c}}}^{{**}}(S,0,\mathbf{\Pi};w)=\frac{\sigma _{{\mathbb{W}\mathbb{O}}}}{|\partial(\mathbb{W}\cap\mathbb{O})|}\int\limits _{{\partial(\mathbb{W}\cap\mathbb{O})}}w(\widetilde{\mathbf{x}})\kappa(\widetilde{\mathbf{x}})\mathrm{d}^{2}\widetilde{\mathbf{x}} (62)

with a weighting function w(\widetilde{\mathbf{x}}) was used to estimate P_{\mathrm{b}}. [] The weighting function is based on estimating the fluid/fluidinterfacial area which in turn depends on the method and parameters of discretizing the interface. [] Here \kappa(\widetilde{\mathbf{x}}) is the estimate of the local curvature of the interface. [] The computed capillary pressures {P_{\mathrm{c}}}^{{**}}(S,0,\mathbf{\Pi};w) reported in Figure 2(a) of [5] range between 250\;\mathrm{Pa} and 380\;\mathrm{Pa}. [] This would suggest a value of

\displaystyle P_{\mathrm{b}}(\mathbf{\Pi};w)=315\;\mathrm{Pa} (63)

for the characteristic pressure P_{\mathrm{b}}. [] We emphasize that clusters at much higher local {P_{\mathrm{c}}} have been observed in the experiments. [] The weighting function w emphasizes the largest cluster and this cluster had very low mean curvature. [] A precise mathematical relation between {P_{\mathrm{c}}} and {P_{\mathrm{c}}}^{{**}} cannot be given without a rigorous connection between the microscopic Newton and Laplace law and the macroscopic generalized Darcy law (cf. Section VI). [] Equations (61) and (62) above do not establish such a connection because the domains of integration are disjoint, i.e. \partial\mathbb{D}_{i}\cap\partial(\mathbb{W}\cap\mathbb{O})=\emptyset. [] The derivation of macroscopic capillary properties of porous media expressed through {P_{\mathrm{c}}}(S) from microscopic knowledge of the curvature field \kappa(\widetilde{\mathbf{x}}) remains a challenge.

E Cooperative dynamics and inertial effects

[] Inertial effects and cooperative dynamics of mesoscaleclusters have been visualized using recent advances in X-ray microcomputed tomography synchrotron beamlines [46]. [] The cooperative dynamics is believed to be related to leading and trailing menisci connected via viscous pressure gradients. [] These inertial effects are not visible on the scale of a single pore. [] In a porous medium burst-type events are observed as reported by [47] or [48]. [] In [46] it is shown that large events seem to be more frequent than suggested by simple percolation models{}^{5}\
5: Simple percolation models [49], while containing a diverging length scale at the pecolation transition, are difficult to apply in the present context, because of very strong geometric correlations and because invasion percolation models are limited to the slow drainage limit.
[] While a single pore-scale event occurs over the millisecond time scale [50], i.e. displacement in a single pore, the occurrence of multiple spatially correlated events have been observed to decay over the second time scale [46]. [] These observations might possibly indicate the emergence of a mesoscopic time or length scale intermediate between pore size \ell and system size L. [] Recent experimental evidence from [11] suggests that cluster lengths are flow rate dependent and widely distributed, ranging from many hundreds of pores down to a single pore. [] If such a cluster length or other mesoscopic length and/or time scale exists, its upper and lower limits are unknown at present. [] This challenges also the interpretation of laboratory-based results.