Ó BirkhaÈuser Verlag, Basel, 2001

Pure appl. geophys. 158 (2001) 627±646 0033 ± 4553/01/040627 ± 20 $ 1.50 + 0.20/0

Pure and Applied Geophysics

Capillary Crack Imbibition: A Theoretical and Experimental Study Using a Hele-Shaw Cell HARTMUT SCHUÈTT1;3 and HARTMUT SPETZLER1;2

Abstract Ð We study the ®lling of horizontal cracks with constant aperture driven by capillary forces. The physical model of the crack consists of a narrow gap between two ¯at glass plates (Hele-Shaw cell). The liquid enters the gap through a hole in the bottom plate. The ¯ow is driven purely by the force acting on the contact lines between solid, liquid, and gas. We developed a theoretical model for this type of ¯ow on the basis of Darcy's law; it allows for the consideration of di€erent surface conditions. We run the experiment for two surface conditions: Surfaces boiled in hydrogen peroxide to remove initial contamination, and surfaces contaminated with 2-propanol after boiling in hydrogen peroxide. The ¯ow rate depends on the gap aperture and on the interaction of the liquid with the air and the solid surfaces: The smaller the aperture, the lower the ¯ow rate due to viscous resistance of the liquid. The ¯ow rate is also reduced when the glass surfaces are contaminated with 2-propanol. The contact line force per unit length is approximately 60% higher on clean glass surfaces than it is on glass surfaces with the 2propanol contamination. These experimental results are in agreement with our theoretical model and are con®rmed by independent measurements of the liquid-solid interaction in capillary rise experiments under static conditions with the same Hele-Shaw cell. Another aspect of this study is the distribution of the liquid for the di€erent surface conditions. The overall shape is a circular disk, as assumed in the theoretical model. However, a pronounced contact line roughness develops in case of the surfaces contaminated with 2-propanol, and air bubbles are trapped behind the contact line. A further analysis of the ¯ow regime using the capillary number and the ratio of the viscosities of the involved ¯uids (water and air) reveals that the experiments take place in the transition zone between stable displacement and capillary ®ngering, i.e., neither viscous nor capillary ®ngers develop under the conditions of the experiment. The contact line roughness and the trapped air bubbles in the contaminated cell re¯ect local inhomogeneities of the surface wettability. Key words: Capillary pressure, crack, ¯uid ¯ow, groundwater contamination, Hele-Shaw cell.

Introduction We seek to understand the e€ect of organic contaminants on the ¯ow of liquids ± usually water ± through porous media. The long-term goal is the development of a 1 Cooperative Institute for Research in Environmental Sciences (CIRES), University of Colorado, Campus Box 216, Boulder, CO 80309-0216, USA. 2 Department of Geological Sciences, University of Colorado, Campus Box 399, Boulder, CO 803090399, USA. 3 Present Address: Department of Petroleum Engineering, Heriot-Watt University, Edinburg EH14 4AS, United Kingdom. E-mail: [email protected]

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Pure appl. geophys.,

seismic groundwater monitoring system near toxic waste sites. Such a monitoring system could provide information about the in®ltration of certain contaminants into the subsurface and thus complement information inferred from established, mostly geoelectrical and electromagnetic, methods (e.g., Induced Polarization; GUEÂGUEN and PALCIAUSKAS, 1994; OLHOEFT, 1986). Other potential ®elds of application are the monitoring of oil and gas recovery and the prediction of the transport of contaminants in the vadose (partially saturated) zone in the shallow subsurface. To date we have established that traces of organic contaminants (e.g., propanol) on the interior surface of partially saturated porous materials can change the wettability of the dry regions as the pore liquid is forced to ¯ow across these initially unwetted solid surfaces (MOÈRIG et al., 1997). Fluid ¯ow can be induced by compressing partially ®lled gaps within the sample, as by a passing seismic wave (SCHUÈTT et al., 2000; WAITE et al., 1997). The energy required in the wetting of dry surfaces causes seismic energy dissipation. Increased pressure within the ¯uids in the compressed gaps sti€ens the sample and leads to higher seismic wave velocities. Both e€ects can be described in terms of the complex frequency-dependent modulus of the partially saturated system. The results of laboratory measurements on natural and arti®cial rock samples as well as on arti®cial rectangular crack models led to the development of the Restricted Meniscus Motion Model (SCHUÈTT et al., 2000; WAITE et al., 1997). This is an extension of existing models that explain the characteristics of seismic wave propagation (e.g., attenuation and dispersion) with an induced ¯uid ¯ow within the pores (BIOT, 1956a,b; MURPHY et al., 1986). These previous models do not incorporate static as well as dynamic forces on the contact lines in partially saturated media1 . The Restricted Meniscus Motion Model has recently been modi®ed from a rectangular geometry with one-dimensional ¯ow to a cylindrical geometry with radial ¯ow. The modi®ed model has been applied successfully to the results of laboratory measurements on arti®cial cylindrical (dime-shaped) cracks. It is now possible to predict the complex modulus of such cracks within the limits of such experiments (SCHUÈTT et al., 2000). A further aspect is the imbibition of liquids into gaps on di€erent length scales. Here we report on a study of this phenomenon on a cm scale in the laboratory using a Hele-Shaw cell. The liquid (water) is allowed to enter the gap between two horizontal glass plates through a hole in the center of the bottom plate. The ¯ow is driven by the capillary forces acting on the contact line between the three phases in the system: solid (glass), liquid (water), and gas (air). We run the experiment for two gap apertures (1:8  10 4 m and 1:1  10 4 m) and for two di€erent surface conditions: Glass surfaces boiled in hydrogen peroxide (H2 O2 ) to remove initial contamination (state 1), and glass surfaces contaminated with 2-propanol after boiling in H2 O2 (state 2). We use 2-propanol to correlate our results with the results 1 The Biot model was developed for fully saturated media. But the model has been extended to partial saturation by a number of authors, e.g., DOMENICO (1977), DUTTA and ODE (1979a,b), and WHITE (1975).

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of a previous study in which the e€ects of surface contamination on one-dimensional ¯uid ¯ow through glass tube and cracks were investigated (MOÈRIG et al., 1997).

Experimental Method The imbibition experiment is done using a Hele-Shaw cell (see Fig. 1). It consists of two glass plates (length 0.33 m, width 0.195 m, thickness 0.02 m) separated by two plastic shims (length 0.33 m, width 0.01 m). We use two sets of shims (1:8  10 4 m and 1:1  10 4 m thickness) that are placed along the long sides of the plates. The plates are incorporated into an aluminum frame and clamped on the long sides with three screws per side. We apply a torque of 0.6 Nm at each screw (1/4 inch diameter, 20 threads/inch). This insures a reproducible gap aperture. The gap aperture is not exactly constant, as revealed by the shape of Newton rings. When we ®ll the central part with water, however, the resulting liquid disk, up to a radius of approximately 0.08 m, is almost perfectly circular and keeps this shape over a long period (at least 15 minutes, decidedly longer than the duration of the imbibition experiment, approximately 1 minute). This indicates that the capillary force acting on the contact lines are the same in all directions, i.e., that the gap aperture can be regarded as constant for our purposes. The liquid ¯ows into the cell through a circular hole (R0 ˆ 3:2  10 4 m) in the center of the bottom plate (see Fig. 2). To ®ll the cell, the hole is connected with a round glass container 0.12 m in diameter that serves as a liquid reservoir. To start the imbibition of the gap, the cell is assembled and hydraulically connected with the reservoir via a plastic tubing with an inner diameter of 1:6  10 3 m. Initially, the

Figure 1 Sketch of the experiment. The right part of the frame is omitted. A syringe pump raises the ¯uid in the reservoir and in the inlet of the Hele-Shaw cell until imbibition occurs.

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Hartmut SchuÈtt and Hartmut Spetzler

Pure appl. geophys.,

Figure 2 Con®guration of the model gap. R0 is the radius of the centered hole. V0 is the volume of the cylinder above the hole. We assume in the theoretical model that this cylinder is ®lled initially with water, i.e., before the imbibition of the gap begins. r is the radial coordinate measured from the center; r1 and r2 ˆ R(t) are the two radii of curvature of the liquid disk, where R(t) is the radius of the liquid disk at the time t; b is the gap aperture, and h is the contact angle.

liquid level in the hole of the bottom glass plate is the same as the liquid level in the reservoir, slightly below the level of the gap. We use a syringe pump to raise the liquid level in the reservoir with a constant ¯ow rate of 2  10 6 m3 per minute. The liquid level in the hole rises correspondingly until imbibition occurs. We monitor the mass of the reservoir with a digital balance with a resolution of 0.01 g. To avoid transients, the pump is not switched o€ during the imbibition. The surface area of the reservoir is 1:13  10 2 m2 . An extraction of 10 6 m3 lowers the liquid surface approximately 10 4 m, corresponding to a pressure decrease of approximately 1 Pa. We measure the ¯ow only as long as the imbibed liquid volume is nearly a circular disk. This limits the disk radius to approximately 0.08 m. The imbibed liquid volume is approximately 4  10 6 m3 for the larger aperture and approximately 2:5  10 6 m3 for the smaller aperture. The imbibition time is approximately 10 to 20 seconds for the larger aperture and approximately 20 to 30 seconds for the smaller aperture (the time intervals re¯ect di€erent surface conditions), see Figure 4. The resulting change of the hydrostatic pressure due to the extraction of these volumes from the reservoir is then approximately 4 Pa and 2.5 Pa, respectively. The capillary pressure in the gap, Equation (17), in the case of large contact angles that are characteristic of contaminated surfaces (approximately 70 ; MOÈRIG et al., 1997) is approximately 270 Pa for the wider gap and approximately 450 Pa for the narrower gap. For clean surfaces (contact angle  30 ; MOÈRIG et al., 1997), the capillary pressure is expected to be approximately 2.5 times higher. These pressures are large compared to the change of the hydrostatic pressure during the imbibition. We therefore neglect the change in the hydrostatic pressure for both clean and contaminated surfaces and treat the reservoir as an ``in®nite'' reservoir.

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We run the experiment for two di€erent surface conditions. We use plates that were boiled in hydrogen peroxide2 to remove initial contamination, resulting in a surface state referred to as state 1. In another set of experiments, we use plates that are contaminated with 2- propanol3 after they had been cleaned (state 2). When silica surfaces (e.g., glass) are exposed to molecules that contain OH groups as well as carbon based groups (e.g., propanol), the OH groups of these organic contaminants can bond to adsorbed water molecules on the silica surface. The non-polar carbon groups have a very small anity to water molecules, producing a hydrophobic e€ect (ILER, 1979). As a result, the mobility of the contact line between the contaminated solid, water, and gas (air) is greatly reduced. This e€ect has been observed in previous studies on ¯uid ¯ow and seismic wave attenuation in partially saturated media (BOYD, 1977; BRUNNER, 1999; MOÈRIG et al., 1997; SCHUÈTT et al., 2000; WAITE et al., 1997). To clean the glass surfaces, we boil them for 1 hour in an aqueous solution of hydrogen peroxide (15% in water). This method yielded good results in a study on the meniscus behavior on glass surfaces for di€erent contamination conditions (BRUNNER, 1999). The plates are immersed fully in a Pyrex dish ®lled with the hydrogen peroxide and heated on a hot plate; the dish is covered with aluminium foil to reduce heat losses. After the boiling, the plates are allowed to cool down in the liquid before they are removed and rinsed with double distilled water. Then they are dried as described above. To contaminate the glass surfaces, we ®rst boil the plates in hydrogen peroxide, as described above. Next the cell is assembled and 2-propanol is pumped through the gap for 12 hours. We use a syringe pump with a ¯ow rate of 10 7 m3 =min. The cell is taken apart and the glass plates are allowed to dry on air until most of the propanol is evaporated. The residual liquid is removed with a Kimwipeâ .

Pressure Distribution in the Liquid The basis for the theoretical interpretation of the capillary imbibition experiments is Darcy's law for cylindrical symmetry: Q(t) ˆ

k op…r, t† A(r) ; g or

…1†

where Q(t) is the ¯ow rate, i.e., the volume that ¯ows per unit time through a cylindrical area around the center of the injection; k ˆ b2 =12 is the permeability of the gap with gap height b (ROUSE, 1946; p. 156); A(r) ˆ 2prb is the area of the mantle of the cylindrical liquid disk of radius r, if the meniscus is assumed to be

2

H2 O2 ; Fisher Chemical, Fair Lawn, NJ, USA.

3

CH3 ACHOHACH3 ; Fisher Chemical, Fair Lawn, NJ, USA.

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Hartmut SchuÈtt and Hartmut Spetzler

Pure appl. geophys.,

perpendicular to the glass plates (Fig. 2); p(r, t) is the pressure at a distance r from the center of the injection and at the time t after the start of the injection. Since the liquid is nearly incompressible, and the gap contains no sources and sinks outside the injection hole, we can make the assumption div v…r† ˆ 0 ;

…2†

where v(r) is the local radial velocity vector of the liquid a distance r away from the center of the injection4 . This is equivalent to the statement that for any time t after the start of the injection, the ¯ow rate Q is independent of the radial distance from the center of the injection, even though the ¯ow rate can change with time. Substituting the permeability, k, and the cross-sectional area, A, into Equation (1) yields: Q(t) ˆ

p b3 op…r, t† : r 6g or

…3†

Thus the product r
op…r, t†=or must be independent of the radius r in order for the ¯ow rate Q to be independent of r, i.e., r

op…r, t† ˆ f(t) : or

…4†

The integration5 Z

Zr op…~r; t† ˆ R0

f(t) o~r ~r

yields: Zr p(r, t) ˆ f(t) R0

1 d~r ˆ f(t) ln…~r†jrR0 ; ~r

…5†

where R0 is the radius of the centered hole where the liquid enters (Fig. 2). The pressure depends logarithmically on the distance from the injection point for the cylindrical geometry of the experiment. This di€ers from a linear geometry (capillary tube with constant diameter), where the pressure depends linearly on the distance from the inlet (GUEÂGUEN and PALCIAUSKAS, 1994). Equation (5) yields:   r p(r, t) ˆ f(t)‰ln…r† ln…R0 †Š ˆ f(t) ln : …6† R0 4

Equation (2) follows from the equation of continuity (e.g., FURBISH, 1997).

5

The tilde () denotes the integration variable, while r is the upper limit of the integration.

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The function f(t) can be evaluated if we consider the meniscus at the position r = R(t) (for an arbitrary time t > 0). The pressure drops from the ambient atmospheric pressure p0 outside the liquid to the capillary pressure pc just inside the liquid and rises steadily to the atmospheric pressure at the injection point (GUEÂGUEN and PALCIAUSKAS, 1994; Fig. II.9). Thus we derive the following boundary condition for the pressure inside the liquid at the meniscus:   R(t) …7† p(R(t)) ˆ f(t) ln ˆ pc ; R0 p c  : ) f(t) ˆ …8† ln R…t† R0 Finally, we obtain the complete expression for the variation of the pressure with radial distance from the injection point, r, and with time, t, that satis®es the boundary condition:   ln Rr0 p(r, t) ˆ pc   : …9† ln R…t† R0 The principal behavior of the pressure distribution inside the liquid is shown in Figure 3 for two di€erent times.

Dynamic Capillary Imbibition In order to calculate the time dependence of the ¯ow rate, we use Darcy's law, Equation (1), with the pressure distribution that was derived for the case of a radial ¯ow of an incompressible liquid, Equation (9). From this equation, we obtain for the pressure gradient: op(r, t) ˆ or

p 1 c  : ln R…t† r

…10†

R0

2

Darcy's law, with A ˆ 2prb, and k ˆ b =12, gives the following expression for the ¯ow rate Q(t): Q(t) ˆ

p b3 1 c1 p  ˆ   ; 6 g c ln R…t† ln R…t† R0

3

…11†

R0

where the constant c1 ˆ p6 bg pc depends only on the gap aperture and on the properties of the liquid, i.e., on the viscosity and ± via the capillary pressure ± on the surface tension and contact angle (see Eq. (17)). The ¯ow rate depends on the 3rd power of the gap aperture b, which is equivalent to the result for 1-D ¯ow through a

634

Hartmut SchuÈtt and Hartmut Spetzler

Pure appl. geophys.,

Figure 3 Sketch of the radial pressure distribution in the gap, p(r, t), for two di€erent times t ˆ t1 and t2 ; R0 is the radius of the centered hole. The pressure in the hole is the atmospheric pressure, pa . Across the meniscus, i.e., for r ˆ R…t1;2 †, the pressure is unsteady. The pressure di€erence is the capillary pressure, pc .

crack with constant aperture (``cubic law''; FURBISH, 1997). Furthermore, Equation (11) is in accordance with the result for cylindrical steady state ¯ow through a porous medium given by COSSE (1993, p. 41), if the pressure di€erence that drives the ¯ow is taken to be the capillary pressure, and if the permeability of the porous medium is taken to be that of a gap with parallel walls, i.e., k ˆ b2 =12. The ¯ow rate Q(t) equals the change of the liquid volume in the gap, dV(t)/dt. The liquid volume in the gap can be calculated easily for a disk with radius R(t) and thickness b:   V(t) 1=2 V(t) ˆ pR2 …t†b ) R(t) ˆ : …12† pb With equations (11) and (12), we generate a di€erential equation for the liquid volume in the gap: dV(t) c1 2c1 ˆ  1=2  ˆ ; dt ln …V(t)† 2 ln …c2 † ln V …t† c2

…13†

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1=2

with c2 ˆ …pb†1=2 R0 ˆ V0
V0 is the volume of a cylinder with the radius of the injection hole, R0 , and with the height of the gap aperture, b (Fig. 2). To solve this di€erential equation, we assume that the imbibition of the gap starts for the time t0 ˆ 0 at the radius r ˆ R0 , i.e., the gap is initially ®lled with a liquid cylinder directly above the injection hole, Thus, the lower boundary for the time integration is t0 ˆ 0, and the lower boundary for the volume integration is V0 ˆ pR20 b. We then have ZV…t†

  ~ ‡ c3 dV ~ ˆ 2c1 ln…V†

V0

where c3 ˆ

2 ln…c2 † ˆ

Zt

d~t ;

…14†

0

ln…V0 †; with the solution 

~ ln…V† ~ V

~ ~ ‡ c3 V V

V…t† V0

 t ˆ 2c1 ~t 0 :

…15†

It is easier to solve Equation (15) for t than for V(t): tˆ

1 V0 V(t)‰ln…V(t)† ‡ c4 Š ‡ ; 2c1 2c1

…16†

with c4 ˆ c3 1 ˆ ‰ln…V0 † ‡ 1Š: The imbibed volume as a function of time is represented by Equation (16). It is solved for t(V) instead of V(t) for mathematical ease. The last term in Equation (16) corresponds to the imbibition of V0 and equals for clean conditions approx. 4  10 3 sec for b ˆ 1:8  10 4 m and approximately 2:5  10 3 sec for b ˆ 1:1  10 4 m. The capillary pressure pc in the liquid depends on the two radii of curvature, …b=2†=cos…h† and R(t), where h is the contact angle (see Fig. 2). For a large contact angle of 70 , which is characteristic for contaminated glass surfaces (MOÈRIG et al., 1997), the meniscus contribution, …b=2†=cos…h†, is approximately 2:5  10 4 m, while the radius of the liquid disk, R(t), is typically of the order of 10 2 m to 10 1 m. Thus the capillary pressure is dominated by the curvature of the meniscus, described by …b=2†=cos…h†, and not by the curvature of the contact line, described by R(t) (see Fig. 2). The capillary pressure then is (GUEÂGUEN and PALCIAUSKAS, 1994; p. 34):   1 1 pc ˆ r ‡ r1 r2   1 1 2r cos…h† ; …17† ˆr  …b=2†=cos…h† R…t† b where r1 and r2 the principal radii of curvature, and r is the surface tension of the liquid. Since the two principal curvatures of the meniscus have opposite orientations (one center of curvature lies within the liquid, one lies outside), they carry opposite signs (GRIMSEHL, 1957; p. 235).

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Hartmut SchuÈtt and Hartmut Spetzler

Pure appl. geophys.,

The quantity c1 , cf. Equation (11), becomes p b2 r cos…h† : 3g

…18†

1 V(t)‰ln…V(t)† ‡ c4 Š : 2c1

…19†

c1 ˆ

The evaluation of V0 and c1 shows that the last term in Equation (16), 2Vc01 , is in the order of a few milliseconds for water (g ˆ 10 3 Pa s, r ˆ 72  10 3 N/m), even for the ``worst case'' of large contact angles (70 for contaminated surfaces; MOÈRIG et al., 1997). Since the time scale involved in our imbibition experiment is in the order of one second to one minute, we can neglect this last term. The equation that describes the capillary imbibition of the gap then becomes: tˆ

2

The ®rst step of the modeling is the calculation of the constants c1 ˆ p3 bg r cos…h† and c4 ˆ ‰ln…V0 † ‡ 1Š for given gap and liquid properties. We can then calculate the time t that is required to ®ll the gap with the volume V using Equation (19).

Capillary Rise Experiment The imbibition of the gap is driven by the force per unit length at the upper and lower contact line, f ˆ r cos…h†, which is contained in the factor c1 in Equation (19). In order to measure f independently, we performed capillary rise experiments with the Hele-Shaw cell6 . This experiment is equivalent to capillary rise experiments with a circular tube (see e.g., GUEÂGUEN and PALCIAUSKAS, 1994). We clean the plates and a large glass dish by boiling in hydrogen peroxide (concentration 15% in double distilled water) for 1 hour; we refer to this surface state as state 1. Then we assemble the cell and put it vertically in the glass dish which is ®lled with double distilled water. The water rises in the gap until an equilibrium is reached between the capillary force on the contact line (upward) and the gravitational force of the water column (downward). We measure the height of the contact line above the water level in the reservoir, Dh, for both sets of shims …b ˆ 1:8  10 4 m and 1:1  10 4 m). We then contaminate the plates with 2-propanol as described earlier (state 2) and repeat the experiment. In the case of the contaminated plates, the contact line is not exactly straight, but displays irregularities similar to the contact line roughness seen in the ¯ow experiment (Fig. 5b). On the scale of the width of the cell, however, we can visually determine a mean level of the contact line above the reservoir, which is taken as the equilibrium position of the contact line. The expression for the equilibrium in the case of a rectangular gap of aperture b is equivalent to the expression derived for the circular tube (see GUEÂGUEN and 6 We can neither use the plates directly in a tensiometer, due to their mass and size, nor have we the experimental capabilities to measure the contact angle optically.

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PALCIAUSKAS, 1994, Eq. II.7), if the length of the contact line is expressed in terms of the width of the gap instead of the radius of the capability tube. We get for the force per unit length, f: f ˆ r cos…h† ˆ

qgb Dh 2

…20†

where q is the density of water, g is the gravity, b is the gap aperture, and Dh is the height of the contact line in the cell above the water level in the reservoir7 . Inserting the proper values, we ®nally get r cos…h† ˆ 0:90 N/m2  Dh for b ˆ 1:8  10 4 m and r cos…h† ˆ 0:55 N/m2  Dh for b ˆ 1:1  10 4 m. Results 1. Capillary Rise Experiments With Equation (20) we gain the following results for the contact line force from the capillary rise experiments: A) Surface state 1 b ˆ 1:8  10

4

m

Dh ˆ 2:6  10 b ˆ 1:1  10

4

2

m

)

r cos…h† ˆ 23:4  10

3

N/m :

2

m

)

r cos…h† ˆ 26:4  10

3

N/m :

2

m

)

r cos…h† ˆ 14:4  10

3

N/m :

2

m

)

r cos…h† ˆ 16:5  10

3

N/m :

m

Dh ˆ 4:8  10 B) Surface state 2 b ˆ 1:8  10

4

m

Dh ˆ 1:6  10 b ˆ 1:1  10

4

m

Dh ˆ 3:0  10

As expected, the results depend on the contamination state of the surfaces. The results are qualitatively in accordance with the results of MOÈRIG et al. (1997). They showed that a surface contamination with 2-propanol signi®cantly hinders the ¯ow of water through rectangular and circular glass ducts in the presence of a meniscus8 . The surface contamination changes the solid-liquid-gas interactions at the contact line. This leads to a larger contact angle and thus to a reduced capillary pressure. 7 8

Note that the result is independent of the width of the gap.

Their experiments di€er from our ¯ow experiment in that they used strictly 1-D ¯ow, while we use a circular (i.e., 2-D) ¯ow pattern.

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Pure appl. geophys.,

They report contact angles inferred from capillary rise experiments in the range of 20 ±40 for water on uncontaminated glass and 40 ±70 for water on surfaces contaminated with 2-propanol. Our results for the contact line force in the case of the plates boiled in hydrogen peroxide (state 1) are smaller than the expected value for clean glass surfaces (72 mN/m to 55 mN/m; MOÈRIG et al., 1997). This suggests that we cannot remove the initial contamination from the surfaces. The experiments demonstrate, however, that an additional surface contamination with 2-propanol has a measurable and reproducible e€ect on the ¯uid ¯ow in the crack. From the values given above, we get for the ratio of the contact line force per unit length between state 1 and state 2: A) b ˆ 1:8  10

4

m r cos…h†jstate1 ˆ 1:63 : r cos…h†jstate2

B) b ˆ 1:1  10

4

m

r cos…h†jstate1 ˆ 1:60 : r cos…h†jstate2

The force per unit length at the contact line is approximately 60% higher for state 1 than it is for state 2. This result is practically independent of the aperture. The magnitude of the reduction of the contact line force caused by the surface contamination is in agreement with the ®ndings of MOÈRIG et al. (1997). It can be explained with an increase of the contact angle, e.g., from 30 for state 1 to 60 for state 2. Both angles lie will within the range reported by MOÈRIG et al. (1997). We conclude that we cannot remove a ``background'' contamination from the glass surfaces with the modes we use. However the results show that an additional surface contamination with 2-propanol reduces the force acting on the contact line. The magnitude of the reduction is reproducible and similar for the two apertures. 2. Dynamic Capillary Imbibition Experiments With the experimental setup shown in Figure 1, we measure the mass of liquid that ¯ows into the gap of the Hele-Shaw cell as a function of time for two surface conditions of the glass plates. Again, state 1 refers to surfaces boiled in hydrogen peroxide9 , state 2 refers to surfaces that were ®rst boiled in hydrogen peroxide and

9 In the search for a reproducible cleaning method we also tested RBS pF, a phosphate-free replacement for dichromate-sulfuric acid mixture (manufactured by Pierce, Rockford, Illinois, USA). We found that rinsing the surfaces with RBS pF for 12 hours leads to equivalent results in terms of the surface wettability as boiling in H2 O2 . In Figure 4, state 1 for the smaller aperture was achieved by rinsing with RBS pF.

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then contaminated with 2-propanol. The mass is converted to the corresponding volume, assuming the density of water to be 1000 kg/m3 , and plotted as symbols in Figure 4. The data are shown only in the range where our model applies, i.e., for a circular geometry of the liquid volume in the gap. This is the case for volumes of approximately 4  10 6 m3 and 2:5  10 6 m3 for the larger and smaller apertures, respectively. We look at the imbibition under two aspects: Firstly, we match the measured time-dependent liquid volume in the gap with our theoretical model (Eq. (19)). We use the driving force per unit length at the contact line, r cos…h†, as determined from the capillary rise experiment. Secondly, we look at the overall shape of the imbibed liquid and the structure of the contact lines for both contamination conditions. In Figure 4, the results for the experiments with the larger aperture …1:8
10 4 m† are shown with ®lled symbols, the corresponding model curves with full lines. It takes almost 8 seconds longer to ®ll the gap to 4  10 6 m3 in case of the propanol-

Figure 4 The liquid volume V in the gap of the Hele-Shaw cell as function of the time t for di€erent surface conditions and for two apertures b. The symbols are measured data, the lines are the results of the numerical model (Eq. (19)). We use the contact line force per unit length independently measured with the capillary rise experiment. Curve 1: surfaces boiled in hydrogen peroxide; curve 3: surfaces rinsed with RBS pF; curves 2 and 4: surfaces contaminated with 2-propanol after boiling in hydrogen peroxide.

640

Hartmut SchuÈtt and Hartmut Spetzler

Pure appl. geophys.,

contaminated surfaces. The experimental data for the smaller aperture …1:110 4 m† are shown with open symbols, while the corresponding model curves are plotted with broken lines. Due to the higher viscous forces in the narrower gap, it takes longer time to ®ll the gap to the maximum volume of approximately 2:5  10 6 m3 (i.e., to the same liquid disk radius, and thus perimeter, as in the former case). We see that also here the ¯ow in the gap is considerably hindered on contaminated plates. In summary, the results of the dynamic imbibition experiment establish that an additional surface contamination with 2-propanol reduces the ¯ow rate considerably. The results of the dynamic imbibition experiments can be explained quantitatively with the force per unit length at the contact lines that was measured independently with capillary rise experiments. 3. Flow Regime In Figures 5a and b, we display photos of the liquid distribution in the gap after the imbibition of approximately 4  10 6 m3 . This requires a time of approximately 15 seconds for surface state 1, and approximately 22 seconds for surface state 2. The overall shape is nearly circular in both cases. However, the contact lines are not smooth on a smaller scale. There are deviations on length scales that seem to be characteristic for both surface conditions. The contact lines form several arcs with estimated radii of curvature ranging from one to a few centimeters in case of surface state 1; the radii of curvature are roughly one order of magnitude smaller in case of surface state 2. The overall appearance of this contact line roughness is highly reproducible. We interpret the di€erent radii of curvature as manifestation of the di€erent contamination conditions. The large radii of curvature re¯ect more

Figure 5 Photos of the imbibed liquid for surfaces boiled in hydrogen peroxide (5a) and for surfaces contaminated with 2-propanol (5b). The gap aperture is 1:8  10 4 m and the liquid volume is approximately 4  10 6 m3 in both cases.

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homogeneous surface conditions (Fig. 5a), while the small radii of curvature re¯ect extensive variations in the surface conditions (Fig. 5b). To characterize the nature of the ¯uid ¯ow in a porous medium, LENORMAND et al. (1988) use the capillary number, C, which is the ratio of the viscous force to the capillary force in the displacing ¯uid, and the ratio of the viscosities, M: Qg2 ; Ar cos…h† g Mˆ 2 ; g1

Cˆ

…21† …22†

where Q is the volumetric ¯ow rate, A is the mantle area of the liquid disk in the gap (cf., Fig. 2), g1 and g2 are the viscosity of the nonwetting ¯uid (air) and of the wetting ¯uid (water), respectively (LENORMAND et al., 1988). In the case of the radial ¯ow, both Q and A ˆ 2pR…t†b depend on the time t[R(t) is the radius of the liquid disk]. With Equation (11) for Q(t) and Equation (17) for the capillary pressure pc , we get a capillary number as a function of the radius of the liquid disk in the gap, and thus as a function of the time: C(t) ˆ

b 1   ; 6 R(t)ln R…t†

…23†

R0

where b is the aperture of the gap and R0 ˆ 3:2  10 3 m is the radius of the inlet in the lower plate. We use Equation (19) to calculate the time that is required to ®ll the gap with the volume V. Since the volume depends on the radius of the liquid disk in the gap (assuming a strictly cylindrical geometry), we derive R(t) directly from V(t), and ®nally the capillary number C as a function of the time from Equation (23). Figure 6 shows the logarithm of the capillary number as functions of time for the two apertures. In the very ®rst phase of the imbibition …R  R0 †, log C takes on very large values. However, already 10 3 sec after the onset of the imbibition, log C has dropped to approximately 1. Due to the limited size of the Hele-Shaw cell, there is a lower limit for log C between 4 and 4:5. For the ratio of the viscosities, M, we get with g1 ˆ 18  10 6 Pa s for air (LENORMAND et al., 1988) and g2 ˆ 10 3 Pa s for water: M = 55.55, i.e., log M = 1.74. Figure 7 delineates the di€erent ¯ow regimes in the M-C plane that were investigated by (LENORMAND et al., 1988). Our imbibition experiments follow the path represented by the arrow. They all cover essentially the same range. The domain in the upper right corner is characterized by stable displacement of the less viscous ¯uid (air), where the ¯ow is stabilized by viscous forces. The contact line is smooth and the more viscous ¯uid (water) occupies most of porespace, i.e., only very few small gas ``islands'' are trapped in the porespace behind the contact line. At low capillary numbers (log C  6) the ¯ow pattern is characterized by capillary ®ngering. In contrast to viscous ®ngering, capillary ®ngers have the tendency to grow back towards the source.

642

Hartmut SchuÈtt and Hartmut Spetzler

Pure appl. geophys.,

Figure 6 The logarithm of the capillary number C as function of time for the imbibition curves shown in Figure 4. The limits of the domains are drawn following LENORMAND et al. (1988). The capillary numbers of the di€erent experiments lie in the middle of the transition zone between stable displacement and capillary ®ngering, except for the very ®rst phase of the imbibition, t  1 sec. The labels are the same as in Figure 4.

Figure 7 shows that our experiments are located in the middle of the transition zone between the stable displacement domain and the viscous ®ngering (except for the very ®rst phase of the imbibition that is negligible due its short duration). Viscous ®ngers do not develop under the conditions of the experiments. For the larger aperture and plates boiled in H2 O2 , it would take approximately 24 hours to reach the domain of capillary ®ngering (log C  6), corresponding to a liquid disc radius extending 4 m. Alternatively, the aperture must be as small as 2  10 6 m if the liquid disk radius is in the range of our experiment (0.1 m), in order to decrease the capillary number to the range where a transition to capillary ®ngering can be expected. The irregularities of the contact line that can be seen in Figures 5a and 5b might re¯ect local inhomogeneities of the surface wettability (PATERSON et al., 1995). The results of PATERSON et al. (1995) indicates that the contact line can become pinned at local wettability defects of the surface. The contact line cannot move over defects that are too large and an air bubble remains trapped on that defect. Smaller defects can hold the contact line for a while, however they are eventually wetted by the advancing liquid. The pattern of air bubbles left behind the contact line depends on

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643

Figure 7 The path of the imbibition experiments (arrow) in the M-C-plane, modi®ed after LENORMAND et al. (1988). The di€erent experiments (cf. Figs. 4 and 6), follow the same path and cover essentially the same range. They start at approximately log C = 1 and decrease to approximately log C = 4 during the experiment.

the size, density and strength of the defects as well as on the speed of the advancing contact line. The results indicate that boiling in H2 O2 leads to more homogeneous surfaces than does contamination with 2-propanol. This is re¯ected in the shape of the contact line and in the density of the entrapped air bubbles behind the contact line. The signi®cance of the pinning has not been studied further. Visually the pinning of contact lines is similar to the pinning of migrating dislocations by impurities in crystals (cf., Fig. 5 in KARATO and SPETZLER, 1990).

Summary and Conclusions We performed dynamic imbibition experiments with a horizontal Hele-Shaw cell for two-gap apertures and for di€erent surface conditions of the glass plates: We boiled the surfaces in hydrogen peroxide to remove initial contaminants. In another set of experiments, the surfaces were contaminated with 2-propanol. The ¯ow is driven by the capillary forces at the contact lines, i.e., no external pressure is applied.

644

Hartmut SchuÈtt and Hartmut Spetzler

Pure appl. geophys.,

We developed a theoretical model that predicts that liquid volume in the gap as a function of time for a strictly cylindrical symmetry. The imbibition of a horizontal crack with constant aperture can be described on the basis of Darcy's law. In the case of strictly cylindrical symmetry of the liquid in the gap, the resulting expression of the ¯ow rate, Equation (11), is equivalent to the cubic law for one-dimensional ¯ow (FURBISH, 1997) and to the result for cylindrical steady-state ¯ow through a porous medium (COSSEÂ, 1993). We performed additional capillary rise experiments with a vertical Hele-Shaw cell at the same surface conditions to measure the contact line force per unit length independently under static conditions. With these values we can explain the results of the dynamic imbibition experiments very well. The experiments show that boiling the plates in hydrogen peroxide leads to reproducible and homogeneous surface conditions, judged by the wettability of the surfaces. This procedure does not remove the surface contamination completely. Capillary rise experiments with the Hele-Shaw cell show that the force per unit length at the contact line is smaller than expected for clean glass surfaces. However, an additional contamination with 2-propanol reduces the contact line force considerably and hinders the wetting of the surfaces. The contact line force is approximately 60% higher without the additional 2-propanol contamination. These results are reproducible and in agreement with 1-D ¯ow experiments reported by MOÈRIG et al. (1997). A further analysis of the ¯ow regime, using the capillary number and the ratio of the viscosities, shows that the ¯ow takes place in the transition zone between the domain of stable displacement and the domain of capillary ®ngering (LENORMAND et al., 1988). Capillary ®ngering cannot be expected under the experimental conditions (aperture, cell size, surface conditions). The shape of the contact line and the number of air bubbles trapped behind the contact line indicate an inhomogeneous surface wettability when the glass is contaminated with 2propanol. We conclude that the contamination of the surfaces with 2-propanol has two e€ects on the capillary imbibition: Firstly, the contamination reduces the wettability of the surfaces, which is re¯ected in reduced ¯ow rates; secondly, the contamination leads to local wettability inhomogeneities causing contact line roughness and trapped air bubbles. Neither viscous nor capillary ®ngers develop under the conditions of the experiment.

Acknowledgment This work was supported by the US Department of Energy through grant DEFG03-94ER14419. The authors are very grateful for the comments and suggestions of an anonymous reviewer, which helped considerably to revise and improve the paper.

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The authors would like to thank Robert J. Glass, Flow Visualization and Processes Laboratory, Sandia National Laboratories, Albuquerque, New Mexico, USA, for giving H. SchuÈtt the possibility to spend time in this facility. Robert Glass and the sta€, in particular Don Fox, Leslie ORear, William Peplinski, and Scott Pringle, provided valuable advice concerning the use of Hele-Shaw cells and the physics of ¯uid ¯ow in cracks. REFERENCES BIOT, M. A. (1956a), Theory of Propagation of Elastic Waves in a Fluid-saturated Solid ± Part I: Lowfrequency Range, J. Acoust. Soc. Am. 28, 168±178. BIOT, M. A. (1956b), Theory of Propagation of Elastic Waves in a Fluid-saturated Solid ± Part II: Higher Frequency Range, J. Acoust. Soc. Am. 28, 179±191. BOYD, O. S. (1997), E€ects on Seismic Absorption due to Changed Pore Surface Properties Resulting from Exposure to Propanol: A Study Utilizing Arti®cial Glass Cracks, Master's Thesis, University of Colorado at Boulder, Department of Geological Sciences. BRUNNER, W. M. (1999), A Physicochemical Model of Contact Angle Hysteresis: Implications for Seismic Wave Attenuation in the Vadose Zone, Master's Thesis, University of Colorado at Boulder, Department of Geological Sciences. COSSEÂ, R., Basics of Reservoir Engineering (Gulf Publishing Company 1993). DOMENICO, S. N. (1977), Elastic Properties of Unconsolidated Porous Sand Reservoirs, Geophysics 42, 1339±1368. DUTTA, N. C., and ODEÂ, H. (1979a), Attenuation and Dispersion of Compressional Waves in Fluid-®lled Porous Rocks with Partial Gas Saturation (White Model) Ð Part I: Biot Theory, Geophysics 44, 1777± 1788. DUTTA, N. C., and ODEÂ, H. (1979b), Attenuation and Dispersion of Compressional Waves in Fluid®lled Porous Rocks with Partial Gas Saturation (White Model) Ð Part II: Results, Geophysics 44, 1789± 1805. FURBISH, D. J., Fluid Physics in Geology (Oxford University Press 1997). GRIMSEHL, E., Lehrbuch der Physik, Band 1, 17th Edition (Teubner-Verlag 1957). GUEÂGUEN, Y., and PALCIAUSKAS, V., Introduction to the Physics of Rocks (Princeton University Press 1994). ILER, R. K., The Chemistry of Silica: Solubility, Polymerization, Colloid and Surface Properties, and Biochemistry (John Wiley and Sons 1979). KARATO, S., and SPETZLER, H. (1990), Defect Microdynamics in Minerals and Solid-State Mechanisms of Seismic Wave Attenuation and Velocity Dispersion in the Mantle, Rev. Geophysics 28 (4), 399±421. LENORMAND, R., TOUBOUL, E., and ZARCONE, C. (1988), Numerical Models and Experiments on Immiscible Displacements in Porous Media, J. Fluid. Mech. 180, 165±187. MOÈRIG, R., WAITE, W., and SPETZLER, H. (1997), E€ects of Surface Contamination on Fluid Flow, Geophys. Res. Lett. 24, 755±758. MURPHY, W. F., WINKLER, K. W., and KLEINBERG, R. L. (1986), Acoustic Relaxation in Sedimentary Rocks: Dependence on Grain Contacts and Fluid Saturation, Geophysics 51, 757±766. OLHOEFT, G. R. (1986), Direct Detection of Hydrocarbon and Organic Chemicals with Ground Penetrating Radar and Complex Resistivity, Proc. of the NWWA/API Conference on Petroleum Hydrocarbons and Organic Chemicals in Ground Water ± Prevention, Detection and Restoration, Nov. 12±14, 1986, 284± 305. PATERSON, A., FERMIGIER, M., JENFFER, P., and LIMAT, M. (1995), Wetting on Heterogeneous Surfaces: Experiments in an Imperfect Hele-Shaw Cell, Phys. Rev. E 51 (2), 1291±1298. ROUSE, H., Elementary Mechanics of Fluids (Dover Publications 1946). SCHUÈTT, H., KOÈHLER, J., BOYD, O., and SPETZLER, H. (2000), Seismic Attenuation in Partially Saturated Dime-shaped Cracks, Pure Appl. Geophys. 157 (3), 435±448.

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Hartmut SchuÈtt and Hartmut Spetzler

Pure appl. geophys.,

WAITE, W., MOÈRIG, R., and SPETZLER, H. (1997), Seismic Attenuation in a Partially Saturated Arti®cial Crack due to Restricted Contact Line Motion, Geophys. Res. Lett. 24, 3309±3312. WHITE, J. E. (1975), Computed Seismic Speeds and Attenuation in Rocks with Partial Gas Saturation, Geophysics 40, 224±232. (Received January 3, 2000, accepted June 5, 2000)

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