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4 Identification of capillary pressure

4.1 Hydrostatic equilibrium

[] The constitutive theory proposed above, contrary to the traditional theory, does not postulate a unique capillary pressure as a constitutive parameter function. [] On the other hand experimental evidence suggests that capillary pressure is a useful concept to correlate observations. [] To make contact with the established traditional theory it is therefore important to check whether the traditional PcSW relation can be viewed as a derived concept within the new theory.

[] Consider first the case of hydrostatic equilibrium where vi=0 for all i=1,2,3,4. [] In hydrostatic equilibrium all fluids are at rest. [] In this case the traditional theory implies SW/t=0 and SO/t=0, by mass balance eq. (11). [] The traditional momentum balance eqs. (12) can be integrated to give

PWx=PWx0+ϱWgx-x0 (31a)
POx=POx0+ϱOgx-x0 (31b)

where x0 is a point in the boundary. [] Combined with the assumption (13) one finds

PcSWx=POx-PWx (32)

implying the existence of a unique hydrostatic saturation profile SWx. [] Here Pc0=Pcx0 is the capillary pressure at x=x0. [] Experiments show, however, that hydrostatic saturation profiles are not unique. [] As a consequence the traditional theory employs multiple PcSW relations for drainage and imbibition, and this leads to difficult problems when imbibition and drainage occur simultaneously.

[] The nonlinear theory proposed here can be solved in the special case of hydrostatic equilibrium. [] Mass balance (1) now implies Si/t=0 for all i=1,2,3,4. [] Integrating eqs. (2) yields

P1x=P1x0+ϱWgx-x0 (33a)
P3x=P3x0+ΠaS1x-α-S1x0-α (33b)
P3x=P3x0+ϱOgx-x0 (33c)
P1x=P1x0+ΠbS3x-β-S3x0-β (33d)

If one identifies P1 with PW and P3 with PO then eqs. (33a) and (33b) suggest to identify Pc as P3-P1. [] Then eqs. (33c) and (33d) combined with S1=SW-S2 and S3=1-SW-S4 imply Pc=PcSW,S2,S4. [] The capillary pressure Pc depends not only on SW but also on S2 and S4 in hydrostatic equilibrium. [] In the theory proposed here it is not possible to identify a unique PcSW relation when all fluids are at rest. This agrees with experiment.

4.2 Residual decoupling approximation

[] While it is not possible to identify a unique PcSW relation in hydrostatic equilibrium such a functional relation emerges nevertheless from the present theory when the system approaches hydrostatic equilibrium in the residual decoupling approximation. [] The approach to hydrostatic equilibrium in the residual decoupling approximation (RDA) can be formulated mathematically as v4=0,v2=0 and R23=0,R41=0. [] In addition it is assumed that the velocities v1,v30 are small but nonzero. [] In the RDA mass balance becomes

S1t+S1v1=η2S2-S2*SW*-SWSWt (34a)
S2t=-η2S2-S2*SW*-SWSWt (34b)
S3t+S3v3=η4S4-S4*SO*-SOSOt (34c)
S4t=-η4S4-S4*SO*-SOSOt (34d)

Momentum balance becomes in the RDA

ϕ1P1-ϱWg=R13v3-R1+M1v1 (35a)
0=ϕ2P3+ΠcW-γP2*S2γ-1-ϱWg (35b)
ϕ3P3-ϱOg=R31v1-R3+M3v3 (35c)
0=ϕ4P1+ΠcO-δP4*S4δ-1-ϱOg (35d)

where the abbreviations

R1=R13+R14+R15 (36a)
R3=R31+R32+R35 (36b)

were used. [] Equations (34) and (35) together with eq.(16b) provide 17 equations for 12 variables (P1,P3,v1,v3 and Si,i=1,2,3,4).

[] Equations (34) and (35) can now be compared to the traditional equations (11)–(13) with the aim of identifying capillary pressure and relative permeability. [] Consider first the momentum balance eqs. (35). [] As in the traditional theory [24] viscous decoupling is assumed to hold, i.e. R31=0 and R13=0. [] Next, assuming that R1M1, R3M3, and Si0 one finds [page 6, §0]

ϕ1P1-ϱWg=-R1v1=-R1ϕWϕ1vW (37a)
P3=-ΠcW+γP2*S2γ-1+ϱWg (37b)
ϕ3P3-ϱOg=-R3v3=-R3ϕOϕ3vO (37c)
P1=-ΠcO+δP4*S4δ-1+ϱOg (37d)

where barycentric velocities vW,vO defined through

SWvW=S1v1+S2v2 (38a)
SOvO=S3v3+S4v4 (38b)

have been introduced. [] Subtracting eq. (37a) from eq. (37c), as well as eq. (37d) from eq. (37b), and equating the result gives

ΠaS1-α-ΠbS3-β+γP2*S2γ-1-δP4*S4δ-1 (39)

where eq. (22) has also been employed. [] This result can be compared to the traditional theory where one finds from eqs. (12) and (13)

ϱO-ϱWg+μWkkWrϕWvW-μOkkOrϕOvO=Pc (40)

Again this seems to imply Pc=PcSW,S2,S4 as already found above for hydrostatic equilibrium. [] However, within the RDA additional constraints follow from mass balance (34).

[] First, observe that adding (34a) to (34b) resp. (34c) to (34d) with the help of eq. (38a) yields the traditional mass balance eqs. (11). [] Next, verify by insertion that eqs. (34b) and (34d) admit the solutions

S2x,t=S2*x+S20x-S2*x (41a)
S4x,t=S4*x+S40x-S4*x (41b)

where the displacement process is assumed to start from the initial conditions

SWx,t0=SW0x (42a)
S2x,t0=S20x (42b)
S4x,t0=S40x (42c)

at some initial instant t0. [] The limiting saturations SW*, SO*, S2*,S4* are given by eqs. (29). [] They depend only on the sign of SW/t if τSW/t can be assumed to hold. [] One finds in this case

SW*=1-SOim (43a)
SO*=SOim (43b)
S2*=0 (43c)
S4*=SOim (43d)

for imbibition processes (i.e. SW/t>0), resp.

SW*=SWdr (44a)
SO*=1-SWdr (44b)
S2*=SWdr (44c)
S4*=0 (44d)

for drainage processes (i.e. SW/t<0).

[] With these solutions in hand the capillary pressure can be identified up to a constant as

Pc(SW)=12[Πa(SW-S2)-α (45)

where S2=S2SW and S4=S4SW are given by eqs. (41). [] This result holds in the RDA combined with the assumptions above. [] Furthermore, equations (37a) and (37c) are recognized as generalized Darcy laws with relative permeabilities identified as

kWrSW=2R1-1μWkϕ2SW-S22 (46a)
kOrSW=2R3-1μOkϕ21-SW-S42 (46b)

where S2=S2SW and S4=S4SW are again given by eqs. (41).

Figure 1: Hysteresis loop and drainage scanning curves for capillary pressure Pc as function of water saturation SW fitted to experimental data obtained for a water wet medium grained sand of porosity ϕ=0.34. The primary drainage curve is the dash-dotted line. The main hysteresis loop is the solid line. The dashed lines are drainage scanning curves. All eight curves have the same parameters: SWdr=0.15, SOim=0.19, α=0.52, β=0.90, γ=1.5, δ=3.5 η2=4, η4=3, Πa=1620 Pa, Πb=25 Pa, and P2*=2500 Pa P4*=400 Pa. The five scanning curves start from the boundary imbibition curve at SW=0.3,0.4,0.5,0.6,0.7 .
Figure 2: Hysteresis loop for capillary pressure Pc as function of water saturation SW as in Fig. 1. The imbibition scanning curves start from the secondary drainage curve. Parameters and line styles are identical to those in Figure 1.

4.3 Reproduction of experimental observations

[] Figure 1 visualizes the results obtained by fitting eq. (45) to experiment. [] The experimental results are depicted as triangles (primary drainage) and squares (imbibition). [] The experiments were performed in a medium grained unconsolidated water wet sand of porosity ϕ=0.34. [] Water was used as wetting fluid while air resp. TCE were used as the nonwetting fluid. [] The experiments were carried out over a period of several weeks at the Versuchseinrichtung zur Grundwasser- und Altlastensanierung (VEGAS) [page 7, §0]   at the Universität Stuttgart. [] They are described in more detail in Ref. [25]. The parameters for all the curves shown in all four figures are SWdr=0.15, SOim=0.19, α=0.52, β=0.90, γ=1.5, δ=3.5 η2=4, η4=3, Πa=1620 Pa, Πb=25 Pa, and P2*=2500 Pa P4*=400 Pa.

[] If it is further assumed that the medium is isotropic and that the matrices R1,R3 have the form

R1=R1*ϕ1-κW1 (47a)
R3=R3*ϕ3-κO1 (47b)

then the relative permeability functions are obtained from eqs. (46). [] The result for the special case κW=κO=0 is shown in Figures 3 and 4. [] The parameters R1*,R3* are chosen such that R1*=2ϕ2μW/k and R3*=2ϕ2μO/k, where μW,μO are the fluid viscosities and k is the absolute permeability of the medium. [] All other parameters for the relative permeability functions shown in Figures 3 and 4 are identical to those of the capillary pressure curves in Figures 1 and 2.

Figure 3: Hysteresis loop and drainage scanning curves for relative permeabilities kWr,kOr as function of water saturation SW. Parameters and line styles are identical to those for capillary pressure in Figures 1 and 2. Here κW=κO=0 and the parameters R1*,R3* are chosen such that R1*=2ϕ2μW/k and R3*=2ϕ2μO/k.
Figure 4: Same as Figure 3 with imbibition scanning curves starting from the secondary drainage curve. Parameters and line styles are identical to those in Figure 3.

[] Note that Figures 1 through 4 show a total of 30 different scanning curves, 5 drainage and 5 imbibition scanning curves each for Pc,kWr and kOr. [] In addition a total of 9 different bounding curves are displayed, namely the primary drainage, secondary drainage and secondary imbibition curve for Pc,kWr and kOr. [] Three more bounding curves namely primary imbibition for Pc,kWr and kOr starting from SW=0 are not shown because they are difficult to obtain experimentally for a water-wet sample. [] Of course the number of scanning curves can be increased indefinitely. [] All of these curves have the same values of the constitutive parameters. [] There is less than one parameter per curve. [] The curves shown in the figures exhibit the full range of hysteretic phenomena known from experiment. [] Nevertheless it should be kept in mind that these curves are obtained only under special approximations, and when these are not valid such curves do not exist.