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I Introduction

[page 1, §1]   
[] A capillary desaturation experiment measures the volume of fluid flowing out from a porous medium as a result of injecting a second fluid (immiscible with the first) at varying injection rates. [] Displacement of a nonwetting fluid by a wetting fluid is of great importance for hydrocarbon production processes[1, 2, 3].

[] Most capillary desaturation experiments are carried out in “discontinuous mode”. [] A discontinuous mode desaturation typically starts from a sample filled with nonwetting fluid and increases the injection rate of the wetting fluid in steps, i.e. with a step function protocol in time. [] It is called “discontinuous mode” because the configuration of the residual fluid is discontinuous in the sense of not connected from inlet to outlet. [] On the other hand in a “continuous mode” desaturation one starts always from a continuous initial configuration in which the resident fluid is hydraulically connected to the inlet and outlet. [] Rather than increasing the injection rate sequentially the sample is emptied, refilled and then injected at with a new (increased) rate. [] Experiments in continuous mode are more laborious and expensive than discontinuous mode experiments. [] Many experiments report a constant plateau saturation of the remaining resident fluid at small injection rates followed by a gradual decrease at higher injection rates, as seen for example in Figures 3-17, 3-18, 3-19, pp 70–73 of [3].

[] Dimensionless (or scaling) groups are algebraic combinations of the constitutive parameters from the mathematical model. [] Examples, such as the Reynolds or Peclet number, arise in many fields of physical science. [] In capillary desaturation experiments the results are commonly given in terms of the capillary number, which quantifies the ratio of viscous to capillary forces.

[] Given the correlation between displacement efficiencyand capillary number during water flooding, it is believed that mobilization of residual nonwetting fluid depends predominantly on the competition between viscous and capillary forces [4]. [] Less importance is usually attributed to other factors such as initial fluid configuration, desaturation protocol, mesoscopic cluster interactions and/or the nature of the displacement processes (drainage vs. imbibition) during the experiment. [] One of our motivations for this work is to advance the study of these additional factors influencing capillary desaturation experiments. [] Recent experiments [5, 6, 7] have called for studies of the process dependence. [] In this work we take up such studies by analyzing these desaturation experiments. [] As a result we propose to perform not only desaturation expriments, but also saturation experiments in which the nonwetting saturation increases rather than decreases during stationary wetting fluid flow. [] Mobilization and trapping, coalescence and breakup of nonwetting fluid then determines the increase of nonwetting saturation in the same way as in desaturation experiments from the competition between viscous and capillary forces.

[] Recovery of residual oil during waterflooding depends upon how small the capillary forces become relative to viscous forces. [] Accordingly, the results of capillary desaturation experiments are generally presented as functions of various capillary numbers instead of injection rates, because capillary numbers are supposed to reflect the force balance. [] A capillary number is a dimensionless group

\displaystyle\mathrm{Ca}=\frac{\text{(viscous forces)}}{\text{(capillary forces)}} (1)

that quantifies the ratio of viscous to capillary forces. [] Ideally, the condition \mathrm{Ca}\approx 1 marks the transition from capillary dominated to viscous dominated flow. [] Numerous definitions of capillary numbers, however, do not fulfill this expectation as shown in [8, 9, 10].

[page 2, §1]    [] Herein, we present a new formulation for the force balance between viscous and capillary forces of macroscale two-phase flow that has previously been overlooked. [] This new finding is based on introducing the force balance F quantitatively into eq. (1) by writing

\displaystyle\frac{\text{(viscous pressure drop)}}{\text{(capillary pressure)}}=F(S,v,L) (2)

and investigating its dependence on saturation S, velocity v and system size L. [] This simple exercise has, to the best of our knowledge, never been done and turns up a surprising result that is applicable to capillary desaturation. [] The main new result of our investigation is eq. (27) below, which expresses the force balance F(S,v,L)=f(S,\mathrm{Ca}) quantitatively as a function of saturation and the macroscopic capillary number \mathrm{Ca} introduced in [9]. [] Setting F=F_{0} to some fixed value F_{0} generates capillary number correlations S=S_{{F_{0}}}(\mathrm{Ca}) that seem to be new. [] Capillary number correlations appear in capillary desaturation experiments, and were heretofore not predictable within the traditional theoretical framework. [] Our new findings provide a means to develop theoretical capillary desaturation curves that compare well to experimental results.

[] The article is organized as follows: [] Section II formulates the problem as a problem of length scales. [] Mobilization and recovery of residual oil produces disconnected nonwetting clusters whose average extension is determined by a balance of viscous and capillary forces. [] Section II also formulates six specific objectives that have been addressed and achieved. [] Section III introduces notation and, more importantly, formulates scale separation between microscopic pore scale and macroscopic laboratory scale. [] The mathematically precise formulation of scale separation was crucial for all of our subsequent analysis.

[] Section IV gives a brief analysis of force balance and capillary number on the pore scale. [] Section V discusses the force balance and capillary number on the sample scale uncovering eq. (27) as the main new result. [] Section VI relates the main new result to a variety of alternative definitions for capillary number and visco-capillary force balance. [] This discussion is crucial for the subsequent application of eq. (27) to experiment in Section VII. [] Equally important is the experimental fact that capillary desaturation experiments depend critically on the desaturation protocol. [] Accordingly, Section VII defines and distinguishes various experimental protocols in these experiments. [] It then applies eq. (27) in two ways: Firstly, to discuss and analyse the recent experiments of [5]. [] The result is presented in Figure 3. [] Secondly, to propose a new experiment (with the protocol specified in eq. (56)) and predict its outcome in Figure 4. [] Figures 3 and 4 are the main results demonstrating the fruitful application of eq. (27) to existing experiments as well as its predictive power for new experiments.

[] Section VIII places the main results of eq. (27), Fig. 3 and Fig. 4 in a broader context. [] It discusses protocol dependence, plateau saturation, cooperative dynamics and inertial effects. [] This discussion achieves the objectives formulated in Section II. [] The article concludes with a summary of results. [] The motivating question of length scales can, at present, not be answered with yes or no. [] However, investigating this question has uncovered new facts related to the importance of nonpercolating fluid parts for residuals, capillary desaturation and macroscopic two phase immiscible displacement.