Abstract
The capillary pressure–saturation (P c–S w) relationship is one of the central constitutive relationships used in two-phase flow simulations. There are two major concerns regarding this relation. These concerns are partially studied in a hypothetical porous medium using a dynamic pore-network model called DYPOSIT, which has been employed and extended for this study: (a) P c–S w relationship is measured empirically under equilibrium conditions. It is then used in Darcy-based simulations for all dynamic conditions. This is only valid if there is a guarantee that this relationship is unique for a given flow process (drainage or imbibition) independent of dynamic conditions; (b) It is also known that P c–S w relationship is flow process dependent. Depending on drainage and imbibition, different curves can be achieved, which are referred to as “hysteresis”. A thermodynamically derived theory (Hassanizadeh and Gray, Water Resour Res 29: 3389–3904, 1993a) suggests that, by introducing a new state variable, called the specific interfacial area (a nw, defined as the ratio of fluid–fluid interfacial area to the total volume of the domain), it is possible to define a unique relation between capillary pressure, saturation, and interfacial area. This study investigates these two aspects of capillary pressure–saturation relationship using a dynamic pore-network model. The simulation results imply that P c–S w relation not only depends on flow process (drainage and imbibition) but also on dynamic conditions for a given flow process. Moreover, this study attempts to obtain the first preliminary insights into the global functionality of capillary pressure–saturation–interfacial area relationship under equilibrium and non-equilibrium conditions and the uniqueness of P c–S w–a nw relationship.
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Abbreviations
- a nw :
-
Specific interfacial area (m−1)
- \({A_{\rm dr}^{\rm nw}}\) :
-
Interfacial area in a pore body at \({s^{\rm w}=s_i^{{\rm dr}}}\) (m2)
- \({A_i^{\rm nw}}\) :
-
Interfacial area in a pore body (m2)
- C a :
-
Capillary number
- \({K_{i j}^{\alpha}}\) :
-
Conductance of phase α in pore throat i j (m4/Pa s)
- \({K_{i,i j}^{\rm w}}\) :
-
Wetting phase conductivity in pore body i next to pore throat i j (m4/Pa s)
- l i j :
-
Length of pore throat i j (m)
- l b :
-
Length of a blob (m)
- L i :
-
Total length of the edges in the pore body i (m)
- n pb :
-
Number of pore bodies
- \({p_i^{\alpha}}\) :
-
Pressure of phase α in the pore body i (Pa)
- \({p_{i}^{\rm c}}\) :
-
Capillary pressure in the pore body i (Pa)
- \({p_{{\rm e},i j}^{\rm c}}\) :
-
Entry capillary pressure of the pore throat i j (Pa)
- \({p_{i j}^{\rm c}}\) :
-
Capillary pressure of the pore throat i j (Pa)
- \({p_{i_{\rm tip}}^{\rm c}}\) :
-
Capillary pressure of the tip of the blob in pore i (Pa)
- \({\Delta p_{i-i j}^{\rm w}}\) :
-
Wetting phase pressure drop along the blob (Pa)
- \({\bar{p}_i}\) :
-
Total pressure in the pore i (Pa)
- P c :
-
Average capillary pressure (Pa)
- \({P_{\rm global}^{\rm c}}\) :
-
Global capillary pressure (Pa)
- \({P_{{\rm e}_{\rm block}}^{\rm c}}\) :
-
Maximum entry capillary pressure of a pore throat in contact with the non-wetting phase (Pa)
- r i j :
-
Radius of the pore throat i j (m)
- \({r_{i j}^{\rm c}}\) :
-
Radius of capillary pressure in the pore throat i j (m)
- \({r_{i j}^{\rm eff}}\) :
-
Effective hydraulic radius in the pore throat i j (m)
- R i :
-
Inscribed radius in the pore body i (m)
- \({s_i^{\alpha}}\) :
-
Volume fraction of phase α in the pore body i
- \({s_{i j}^{\rm w}}\) :
-
Volume fraction of phase α in the pore throat i j
- \({s_{i,{\rm min}}^{\rm w}}\) :
-
Minimum attainable volume fraction of wetting phase in the pore body i
- \({s_i^{\rm dr}}\) :
-
Wetting phase volume fraction for an inscribed sphere
- \({s_i^{\rm imb}}\) :
-
Wetting phase volume fraction at which only one pore throat will be filled with both phases
- \({s_i^*}\) :
-
Wetting phase volume fraction in the pore body i at which, all its pore throats can be filled by both phases
- S w :
-
Average saturation of the wetting phase
- Δt i :
-
Minimum residence time in the pore body i (s)
- Δt global :
-
Global residence time over the network (s)
- V i :
-
Volume of the pore body i (m3)
- V b :
-
Volume of a blob (m3)
- q n i :
-
Flux of non-wetting phase in the pore body i (m3/s)
- \({Q_{i j}^{\alpha}}\) :
-
Flux of phase α in the pore throat i j (m3/s)
- \({{\mathbb{N}}_i}\) :
-
Total number of pore throats connected to the pore i
- β :
-
Half-corner angle in a pore throat (rad)
- Γ :
-
A geometrical coefficient for calculating the corner conductance
- κ i :
-
Curvature of the interface in the pore body i (m−1)
- σ nw :
-
Interfacial tension (N/m)
- π :
-
3.1415
- θ :
-
Contact angle (rad)
- \({\epsilon}\) :
-
Resistance factor
- μ α :
-
Viscosity of phase α (Pa s)
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Acknowledgments
The authors are members of the International Research Training Group NUPUS, financed by the German Research Foundation (DFG) and The Netherlands Organization for Scientific Research (NWO). This research was also partly funded by a King Abdullah University of Science and Technology (KAUST) Center-in-Development Award to Utrecht University (Grant No. KUK-C1-017-12).
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Joekar-Niasar, V., Hassanizadeh, S.M. Uniqueness of Specific Interfacial Area–Capillary Pressure–Saturation Relationship Under Non-Equilibrium Conditions in Two-Phase Porous Media Flow. Transp Porous Med 94, 465–486 (2012). https://doi.org/10.1007/s11242-012-9958-3
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DOI: https://doi.org/10.1007/s11242-012-9958-3