Journal of Colloid and Interface Science 449 (2015) 392–398
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Measurement of off-diagonal transport coefficients in two-phase flow in porous media T.S. Ramakrishnan ⇑, P.A. Goode 1 Schlumberger-Doll Research, 1 Hampshire St., Cambridge, MA 02139, United States
g r a p h i c a l a b s t r a c t
Water pressure increase ( pw, kPa)
0.12
0.10
0.08
0.06 line with 0.153 slope
0.04
0.02
0.00 0.0
0.2
0.4
0.6
0.8
Gas phase pressure drop (- pg, kPa)
a r t i c l e
i n f o
Article history: Received 5 November 2014 Accepted 13 January 2015 Available online 14 February 2015 Keywords: Extended Darcy’s law Reciprocity Off-diagonal coefficients
a b s t r a c t The prevalent description of low capillary number two-phase flow in porous media relies on the independence of phase transport. An extended Darcy’s law with a saturation dependent effective permeability is used for each phase. The driving force for each phase is given by its pressure gradient and the body force. This diagonally dominant form neglects momentum transfer from one phase to the other. Numerical and analytical modeling in regular geometries have however shown that while this approximation is simple and acceptable in some cases, many practical problems require inclusion of momentum transfer across the interface. Its inclusion leads to a generalized form of extended Darcy’s law in which both the diagonal relative permeabilities and the off-diagonal terms depend not only on saturation but also on the viscosity ratio. Analogous to application of thermodynamics to dynamical systems, any of the extended forms of Darcy’s law assumes quasi-static interfaces of fluids for describing displacement problems. Despite the importance of the permeability coefficients in oil recovery, soil moisture transport, contaminant removal, etc., direct measurements to infer the magnitude of the off-diagonal coefficients have been lacking. The published data based on cocurrent and countercurrent displacement experiments are necessarily indirect. In this paper, we propose a null experiment to measure the off-diagonal term directly. For a given non-wetting phase pressure-gradient, the null method is based on measuring a counter pressure drop in the wetting phase required to maintain a zero flux. The ratio of the off-diagonal coefficient to the wetting phase diagonal coefficient (relative permeability) may then be determined. The apparatus is described in detail, along with the results obtained. We demonstrate the validity of the experimental results and conclude the paper by comparing experimental data to numerical simulation. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction
⇑ Corresponding author. 1
Present address: Arle Capital Partners.
http://dx.doi.org/10.1016/j.jcis.2015.01.029 0021-9797/Ó 2015 Elsevier Inc. All rights reserved.
In the absence of body forces, based on the early work of Wyckoff and Botset [1], Wyckoff et al. [2], and Leverett [3], the extension of the isotropic form of Darcy’s law
T.S. Ramakrishnan, P.A. Goode / Journal of Colloid and Interface Science 449 (2015) 392–398
k
v ¼ rp;
ð1Þ
l
to multiphase flow is
vi ¼
kei
li
rpi ¼
kkri
li
rpi ;
ð2Þ
where v is the superficial velocity vector, v i is the superficial velocity for phase i; k is the single phase permeability, kei and kri are the saturation dependent effective and relative permeabilities to phase i; l is the dynamic coefficient of viscosity, and p is the pressure, again, with the subscript i indicating the flowing phase of interest. With i ¼ n; w for non-wetting and wetting phases respectively, the phase pressure difference pn pw is the capillary pressure pc , whose value determines the fluid saturations Si in quasi-static phase replacement processes. Excluding hysteresis, it is common to assume that kei is determined by the fluid saturation alone, and its functional dependence on saturation may be obtained by conducting flow experiments that minimally disturb interface equilibrium at the capillary pressure of interest. Application of Eq. (2) to commonly occurring displacement processes in hydrology, and oil and gas industry, requires that over the length scale R needed for defining pc and kei , RrSi 1. Furthermore, consistent with the conjecture that each fluid flows in accordance with its own pressure gradient, the effective permeabilities are independent of the viscosity ratio. In a capillary network, with pore occupancy determined by capillarity, the notion that fluid–fluid interfaces induce only local circulatory motion and do not transfer appreciable macroscopic momentum is the basis of Eq. (2). In the presence of corners and crevices that allow for macroscopically relevant wetting-phase motion, diagonally dominant force-flux relationship is inadequate [4]. To include shear induced motion on the second fluid, Rose [5] hypothesized that an Onsager-type relationship should apply i.e.,
v n ¼ knn rpn knw rpw ; v w ¼ kww rpw kwn rpn ;
ð3Þ
and Lenormand [10] measured the off-diagonal coefficient by imposing a zero-pressure gradient on water, while pumping mercury through a sand-pack, and found coefficients smaller than 0.01. Similar experiments by Dullien and Dong [11] however resulted in much larger magnitudes than [10]. Dullien and Dong obtained non-symmetric coefficients as well, and were speculating about variability in interface configurations from one experiment to the other (wetting phase gradient as opposed to non-wetting phase gradient). It is clear from their data that there is also a saturation gradient within the sample in these experiments, and the saturation configurations were not the same in the two sets of experiments that they conducted. Several authors have speculated on the form of the diagonal coefficients, some suggesting that the off-diagonal coefficients are unequal (see e.g. [12]). The symmetry of the coefficients has been demonstrated by Auriault [13], who correctly argues that the off-diagonal coefficients are negligible if the interface is nearly rigid with zero velocities. Others such as Lasseux et al. [14] derive a reciprocal form based on volume averaging and closure conditions through essentially Auriault’s [13] method, but were unable to specify a self-consistent order of magnitude for the off-diagonal coefficients. For a viscosity ratio of unity, Rakotomalala et al. [15], computed off-diagonal coefficients that were about 1% compared to the diagonal ones. The network modeling of Goode and Ramakrishnan [4] relied first on a finite element analysis within a single pore to compute a duct conductance matrix, based on which it was shown that the off-diagonal coefficients were equal.2 However both duct conductance matrix coefficients and the effective network permeability matrix were seen to depend on both the saturation and the viscosity ratio of the flowing fluids. The coefficients were computed with stationary interfaces, and therefore a pressure perturbation was imposed on one of the fluids to compute the matrix of mobility coefficients. The general conclusion may be written compactly for quasi-static two-phase flow as
kkrc ðSw ; MÞ rpn ; lw þ ln kkrc ðSw ; MÞ kk ðS ; MÞ ¼ rpw rn w rpn ; lw þ ln ln
vw ¼
ð4Þ
where it was understood that the mobilities kij were the ratio of the effective permeabilities to viscosities; no discussion of how the viscosities appeared in the off-diagonal terms was given by Rose. Invoking the Onsager reciprocity principle, he asserted that knw ¼ kwn . A number of papers have speculated on the notion of a generalized mobility matrix for two-phase flow. For example, deGennes [6] wrote a form identical to Eqs. (3) and (4), with saturation-dependent mobilities. No specific viscosity factor appeared in his formulation. Bourbiaux and Kalaydjian [7] attempted to measure the offdiagonal coefficients by conducting a cocurrent displacement, followed by a counter-current imbibition. Since the gradients in the two phase-pressures are of opposite sign in counter-current flow, and are more or less matched in co-current flow, the measured profiles or water-cut in the two experiments would be indicative of the magnitude of the coefficients knw . Symmetry of the coefficients was assumed for processing the data, and a characteristic peak in knw was obtained. Given the viscosity ratio of about ten and the implementation of the boundary conditions, it is difficult to determine whether such an indirect measurement is an artifact or not. For example, no special effort was taken to eliminate capillary end-effects [8,9]. The consequences of end-effects are quite different for counter-current and co-current flow. Furthermore, experiments were performed in natural media (clayey sandstone), and although these were stated to be fairly homogeneous, displacement profiles indicated moderate heterogeneities, the effect of which on the interpretation of data is unclear. Zarcone
393
vn
kkrw ðSw ; MÞ
lw
rpw
ð5Þ ð6Þ
where krc is the off-diagonal or cross-coefficient, krw and krn are the wetting and the non wetting phase relative permeabilities, and M is the viscosity ratio equal to ln =lw . Note that in our formulation, the relative permeabilities and the cross-coefficients are functions of saturation and viscosity ratio, and is different from those proposed by Rose [5] and deGennes [6], where only explicit dependence on saturation alone is assumed. The dependence of the wetting fluid relative permeability (krw ) on M is however rather weak and may be ignored for practical purposes. The key conclusion of Goode and Ramakrishnan [4] was that a nonzero krc implies a dependence on M for all of the three coefficients, whereas the converse is not necessarily true. For example, when the wetting phase is disconnected, krn may depend on M, but krc may be negligible since the motion near the interfaces do not appreciably contribute to the superficial velocity. Also, the form chosen in [4] for the viscosity dependence of the off-diagonal mobility is also symmetric. This form also gives comparable magnitudes for krc with respect to the viscosity ratio. Clearly, the experimental results have varied, and while some inferences have been indirect, others have used methods where the velocities are measured, while maintaining a zero pressure gradient in one of the phases. The latter have been performed in sand-packs and consistent results with regard to the 2 Contrary to the statements given by Li et al. [16] that [4] implied invalidity of reciprocity.
394
T.S. Ramakrishnan, P.A. Goode / Journal of Colloid and Interface Science 449 (2015) 392–398
magnitude and importance of krc have not emerged. It should however be noted that experiments conducted within the domain of validity of the extended forms of Darcy’s law should satisfy symmetry.
2. Experimental concept Departing from previously published literature, the experimental apparatus discussed in this paper is designed to infer the off-diagonal coefficients in relation to the relative permeabilities directly. Inference is not based by comparing the results to a numerical model and regression. Rather than measure small velocities induced by drag, we measure a pressure drop to negate the small velocity induced by drag. Thus the objective of the experimental technique is to provide an unambiguous indication of the existence and the magnitude of krc . The exercise involves conducting a ‘‘null’’ experiment, i.e., measuring a wetting-phase pressure drop, when jDpn j=pc 1, for v w ¼ 0. At steady-state, any non-zero pressure drop, in a direction opposite to that of the non-wetting phase, implies a positive krc . The essential steps in the experiment are as follows. Bring the system to capillary pressure equilibrium by imposing a fixed pressure in the non-wetting phase over that of the wetting phase. Perturb the equilibrated system by a small Dpn , so that the saturation is virtually unaffected. Obviously Dpn has to be kept small compared to pc in order to infer krc at a given saturation. Force the wetting phase velocity to be zero. Measure Dpw =Dpn at steady-state. With regard to the choice of fluids, for many clean natural media and porous ceramics, water or a low-salinity aqueous solution is an acceptable wetting fluid. For the non-wetting fluid, we prefer to choose one that will give us a resolvable response without disturbing capillary pressure equilibrium. Besides requiring that the pressure in the non-wetting phase be small in relation to the capillary pressure, we need the wetting phase pressure increase
due to a nonzero krc to be resolved easily. The pressure drop in the wetting phase for the null experiment is
Dpw ¼
krc ðSw ; MÞ 1 ðDpn Þ: krw ðSw ; MÞ 1 þ M
ð7Þ
Now, theoretical network calculations [4] show that krc monotonically increases with M. However krc =ð1 þ MÞ decreases with viscosity ratio. Furthermore, since krw shows little variation with M, it is clear that for any saturation, the largest counter pressure gradient in the wetting phase is obtained when M is the smallest practical value, suggesting that a suitable non-wetting fluid would be nitrogen (used here) or any other inert gas. For the wetting-fluid, we used low-salinity water, measuring about 1 Xm resistivity. Therefore, water is used synonymously with the wetting fluid in the rest of the paper. 3. The apparatus The experimental set-up is illustrated in Fig. 1. It is designed to be vacuum-filled by a wetting fluid to 100% saturation, and then brought to capillary-pressure equilibrium. An arbitrary pressure drop in the gas-phase may also be imposed, and by choice the wetting phase may be allowed to flow out of the core or kept stagnant. The gas-phase pressures as well as differential gas and wetting-phase pressures are measured along with the wettingphase saturation. The operational details are given below. A porous sample of about 38 mm (1.5 in.) in diameter with a length up to 200 mm may be placed in the core-holder. The particular sample we used was 65 mm in length. The core was from Cleveland Quarry, commonly known as Berea 400 sandstone, and has a nominal permeability of 0:4 lm2 ( 400 mD). The sample was chosen for being a relatively clean naturally occurring sandstone. The core is sealed on its lateral surface with a rubber sleeve on the external side of which a compressed nitrogen pressure of about 650 kPa is applied. The transducers labeled Dpg is for measuring the drop in the gas phase (non-wetting), and has a range of ±34:5 kPa (±5 psi). The transducer is Sensotec, Model Z-882-12. Since M 0:01 and is small compared to unity, the
20cc pipette for saturation measurement
Water Line Gas Line
20cc pipette for fillup
pw
V6
V5
R2
V3 R1
V9 N 2 gas in V8
V7
pg
pg
V4
V2
Core V1
Fig. 1. The experimental apparatus for the null-flow measurement.
To vacuum
T.S. Ramakrishnan, P.A. Goode / Journal of Colloid and Interface Science 449 (2015) 392–398
395
Fig. 2. The expanded assembly that illustrates wetting and non-wetting phase inlet and outlet through the hydrophilic membrane.
magnitude of krc =krw approximately determines the measured jDpw =Dpg j. In drainage, for the four-cusp model of [4], at Sw 0:6, we expect jDpw j to be about 20 times smaller than jDpg j. A differential transducer with a range of ±3:45 kPa (±0:5 psi, Sensotec, Model Z-882-28) was therefore chosen. Each of these gauges has an accuracy specification ±0:25% of full scale, repeatability within 0.05% and hysteresis error within 0.10%. The nonlinearity component is 0.15%, and is easily taken into account. Since our transducers are used only unidirectionally for pressure difference measurement, and are subject to only a fraction of the entire range, the hysteresis errors are expected to be substantially smaller than the specification. For completeness, we have assumed two different error criteria of 0.25% and 0.10% of full-scale for indicating error bars. The ±34:5 kPa transducer is calibrated against a secondary standard traceable to NIST. For the ±3:45 kPa transducer, we used an inclined tube reference manometer with a fluid whose specific gravity was known to three digits (Dwyer Red Gage Fluid). The transducers are full-bridge strain gauges excited at 225 Hz carrier frequency, and a 10 Hz Bessel filter is used to condition the output. The amplifier and signal conditioning itself was precision calibrated to 1 part in 10,000. For the purpose of measuring and controlling phase pressures, the porous-sample or the rock core is sandwiched by two strongly hydrophilic porous plates, 6.35 mm thick. The plates have a sufficiently small pore size to prevent gas-phase intrusion of N2 for all capillary pressures of interest for the Berea sample. The ‘‘semi-permeable’’ plate’s permeability range is 0:0001 lm2 to 0:0003 lm2 and its pore size for a percolating path is sufficiently small for gas to break through water only at about 175 kPa (25 psi).3 The plate is made by CoorsTek and has a designation of P-12AC. Four evenly spaced holes are drilled through the porous plate and are aligned with holes in the nylon end-plug through which gas directly flows into or out of the core. The wetting phase can access 3 The stated bubble pressure is about 80 psi. Our experiments in water-dodecane indicated the conservative number to be closer to 30 psi. Adjusted for surface tension in fresh samples of dodecane this amounts to about 50 psi for air–water systems. A safety factor of two is used in the quoted number.
the core only through the porous plate. The central port in the nylon end-plug feeds the wetting fluid to the semi-permeable plate. A rubber gasket pressed between the nylon end-plug and the porous semi-permeable plate isolates the wetting and non-wetting fluids. This is illustrated in the expanded assembly of Fig. 2. 3.1. Operating procedure Prior to the experiment, lines to atmosphere from the coreholder past V5 and V6 are filled with water and a pipette connected for saturation measurement. The position of the interface in the pipette is noted prior to any water displacement from the core. Before commencing the experiment, the entire assembly is first helium leak-tested. With the exception of the two tanks upstream of V8, the system is then evacuated. Core seal pressure is initiated, and valves V1 and V7 are closed. V3 is then opened to fill the core and water lines with water. We estimate the core filling time by assuming Darcy’s law within the wetting phase, the velocity of which determines the frontal movement. The fill-up time T f is given by
Tf ¼
L2 lw 1 ; 2/ k p0
ð8Þ
where L is the length of the sample, / is the porosity, and p0 is the atmospheric pressure driving the wetting fluid. Gas pressure sufficiently higher than p0 is set upstream of V9 pressure and then V9 is opened to allow nitrogen gas to bubble into the two 1L chambers. The chambers, which are connected in series and partially filled with water, are used to saturate the intruding gas. This minimizes water evaporation into the gas phase within the rock. Once the core is filled i.e., after a few T f , V8 and V7 opened, and the regulator R1 is set to be slightly higher than p0 so as to displace water from the gas lines between V7 and the core, and V4 and the core. The pressure in the gas-phase past the regulator should be low to prevent gas-phase intrusion into the core. After a few minutes, valve V3 may then be shut and a step in capillary pressure may be made. The capillary pressure is controlled by the regulator R1 and measured by the pressure gauge
pg . When a change in pc is imposed, valves V5 and V6 are opened and water is permitted to leave the core. Core saturation is determined by measuring the volume of water which flows into the measuring pipette. No discernible movement in the interface position is indicative of reaching equilibrium. It is only necessary to asymptotically reach equilibrium, since the interpretation of the results is unaffected by small deviations away from it. For the gas to leak at a small rate, a gas gradient is now created by turning the metering valve V2. Since this valve may not have enough resolution to ensure a very small rate so as to have a pressure drop much smaller than the capillary pressure, we also simultaneously adjust the regulator R2 for flow bypass. Adjusting R2 gives us control over the magnitude of the gas phase pressure drop across the core. Since the lines containing R2 and the core are in parallel, the gas phase pressure difference across the core is that across R2. This pressure difference is measured by the transducer labeled Dpg . Since the change in saturation is negligible with the small imposed gradient in the gas-phase, no noticeable movement should occur in the pipette for measuring saturation. Valves V5 and V6 may be closed; there is no other water entry/exit line. At steady-state, for the wetting phase no-flow boundary conditions, the velocity v w ¼ 0. We allow for sufficient time to elapse for stabilization of ðDpg =Dpw ). Since the transducers have a dead volume of a few mL, a deflection of the transducer diaphragm is needed to measure the signal, and therefore a tiny volume of water needs to flow to the higher pressure side for an equilibrated pressure difference to be established. Following the equilibrated measurement of Dpg to Dpw at one saturation, a new saturation is achieved by closing V2, and opening V5 and V6, and adjusting R1 to a higher pc and therefore a lower Sw . The procedure is repeated to infer krc =krw ratio. According to the conventional extended form of Darcy’s law, one would expect the pressure drop in the wetting phase to be zero, when the phase velocity is zero. A non-zero Dpw indicates inadequacy of this form, and furthermore, a positive pressure drop in the wetting phase would constitute a failure of the extended Darcy’s law and the one with the off-diagonal coefficients. As explained previously, the magnitude of Dpw is reflective of the ratio krc =krw , and since M 1, we may write
Off-diagonal to diagonal (wetting phase) mobility
T.S. Ramakrishnan, P.A. Goode / Journal of Colloid and Interface Science 449 (2015) 392–398
Viscosity Ratio, M 0.01 0.1 1 10 100 1000 10000
1.5
1.0
0.5
0.0 0.2
0.4
0.6
0.8
1.0
Wetting phase saturation Fig. 3. Theoretical calculations for the ratio of the off-diagonal to diagonal wetting phase mobility for various M values. Note that the characteristic peak at very low saturations.
0.12
Water pressure increase ( pw, kPa)
396
0.10
Sw
0.54
0.08
0.06 line with 0.153 slope
0.04
0.02
0.00 0.0
0.2
0.4
0.6
0.8
Gas phase pressure drop (- pg, kPa)
ð9Þ
Fig. 4. Proportional wetting-phase pressure increase with respect to gas phase pressure drop at Sw 0:54. The best slope gives Dpw =Dpg ¼ 0:153 with an intercept of zero. There are error bars for both x and y axis based on the 0.10% and 0.25% of full-scale of the respective transducers. The best-fit suggests that the error bars exaggerate the inaccuracy of the measurement.
The ratio of off-diagonal mobility to the direct coefficient krc =½krw ð1 þ MÞ as computed by Goode and Ramakrishnan [4] is shown in Fig. 3. The curve is plotted for a viscosity ratio ranging from M ¼ 0:01 to M = 10,000. Since the viscosity of N2 at experimental conditions is approximately 0:0175 mPa s, M ¼ 0:01 is representative of the gas–water system. As M ! 0, there is little sensitivity to M, and therefore the precise value of it is inconsequential. For Sw > 0:6, the ratio decreases below 1%, and the response is not expected to be resolvable. Therefore only one measurement, at about Sw ¼ 0:75, was performed. As expected, for Sw ¼ 0:75, the response was below the resolution of the transducer and the instrument amplifier, indicating that krc ð0:75; 0Þ krw ð0:75; 0Þ for the Berea sample. Following the measurement at Sw ¼ 0:75; pc was increased, equilibrating at a saturation of Sw ¼ 0:54. At this saturation, since a nonzero krc was clearly evident, further tests were conducted to test the veracity of the data. As per Eq. (9), any increase in Dpg should lead to a proportional increase in Dpw , as long as Dpg pc . Measurements at four different Dpg were conducted,
and for each, the equilibrated Dpw is shown in Fig. 4. Numerical values of these are also shown in Table 1. As seen in Fig. 4, at an Sw of 0.54, the approximate ratio of krc to krw is 0.153. Given the minute pressure differences imposed and measured, it is remarkable that the data are consistent with very little scatter. For a saturation of Sw ¼ 0:54; pg pw was about 10.34 kPa. The elevated gas pressure leads to a gradual dissolution of gas into water. Thus, water within the porous medium has a higher N2 chemical potential compared to the water saturating the porous semi-permeable plate or the lines connected to the differential transducer. Within the porous plate, gas remains in the liquid phase, because the entry capillary pressure within the plate is much higher than that of the Berea sandstone. However, through diffusion, upon reaching a sufficient concentration of nitrogen in the tubing water where its equilibrium pressure in the gas phase would be greater than the water pressure, nitrogen bubbles would spontaneously appear. We call this as gas break-through. At the point of gas-breakthrough, the experiment is terminated, the core is cleaned and reassembled to restart the process.
krc ðSw ; 0Þ Dp w: krw ðSw ; 0Þ Dpg 4. Results
397
Table 1 Saturation, non-wetting and wetting phase pressure drops and mobility ratio. Pressures and error bars are in kPa; Eðpg Þ and eðpg Þ are 0.25% and 0.10% transducer errors. The error bars for the mobility ratio are shown in Fig. 6 for both 0.25% and 0.10% errors in differential pressure. The error shown for the mobility ratio in the table is based on 0.10% error in pressure, with the upper and lower values in parenthesis. Sw
Dpg
Eðpg Þ
eðpg Þ
Dpw
Eðpw Þ
eðpw Þ
krc =½krw ð1 þ MÞ
0.75
0.490
0.086
0.034
0.000
0.0086
0.0034
0.54
0.317
0.086
0.034
0.050
0.0086
0.0034
0.668
0.086
0.034
0.104
0.0086
0.0034
0.520
0.086
0.034
0.078
0.0086
0.0034
0.415
0.086
0.034
0.062
0.0086
0.0034
0.000 (0.000, 0.008) 0.157 (0.132, 0.189) 0.156 (0.143, 0.170) 0.150 (0.134, 0.168) 0.149 (0.130, 0.172) 0.160 (0.141, 0.180) 0.387 (0.354, 0.445) 0.374 (0.346, 0.409) 0.738 (0.653, 0.817)
0.53
0.459
0.086
0.034
0.073
0.0086
0.0034
0.44
0.379
0.086
0.034
0.150
0.0086
0.0034
0.525
0.086
0.034
0.197
0.0086
0.0034
0.352
0.086
0.034
0.256
0.0086
0.0034
0.40
Off-diagonal mobility/Wetting phase mobility
T.S. Ramakrishnan, P.A. Goode / Journal of Colloid and Interface Science 449 (2015) 392–398
1.5
1.0
0.5
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Water saturation Fig. 6. The ratio of the off-diagonal mobility to the wetting phase mobility at various saturations is plotted alongside the results of the numerical network model for a qualitative comparison. The thicker error bars with wide caps are with 0.10% error in the transducer. The thinner error bars with narrow caps are with 0.25% error in differential pressures. The data are also presented in numerical form in Table 1 along with the errors based on transducer specfications.
0.5
0.16
M = 0.01, Network calculation Experiment, gas-water system
0.5
0.4 Sw = 0.44
0.14 0.4
0.2
0.06
R a tio
0.08
0.3 0.3
Ratio - pg pw
0.2
0.2
0.1
P h a se p re ssu re d ro p , k P a
0.3 Ratio - pg pw
Phase pressure drop, kPa
0.10
Ratio
0.4
Sw = 0.53
0.12
0.1
0.04 0.1 0.02
0.0 0.00
0
0.0 2
4
6
8
10
Time, hr Fig. 5. Non-wetting and wetting phase pressure drops for v w ¼ 0 at a saturation of Sw ¼ 0:53. The ratio is Dpw =Dpg .
20
25
Fig. 7. Non-wetting and wetting phase pressure drops for stationary wetting phase at a saturation of Sw ¼ 0:44. The ratio is Dpw =Dpg .
0.5 0.7 Sw = 0.40
0.6
0.4
0.5 0.3 0.4 0.2
0.3
Ratio - pg pw
0.2 0.1
0.1
P h a se p re ssu re d ro p s, k P a
Following reassembly, to check for repeatability, we attempted to go back to a saturation of Sw ¼ 0:54 and obtain a repeat point. For the same gas pressure over the wetting phase pressure, a saturation of Sw ¼ 0:53 was actually achieved. The steady-state ratio of Dpw =Dpg was 0.16, in excellent agreement with the previously acquired data at Sw ¼ 0:54. We also show the time dependence of the pressure drops in water and gas in Fig. 5. Stabilization in the ratio of the pressures occurs in a few hours at Sw ¼ 0:53. Subsequent to the repeat point, we increased the gas pressure and reached a saturation of 0.44. Again two different Dpg were imposed in sequence, each resulting in nearly the same ratio of the phase pressure drops. The lowest saturation that we could reach was 0.4. This corresponded to a capillary pressure of about 34.5 kPa (5 psi). Above this pressure, gas break-through occurred before Dpw stabilization, and therefore it was impossible to measure krc . All of the stabilized pressure drop data are summarized in Table 1. The computed off-diagonal to diagonal mobility ratio for
15
Time, hr
10
R a tio
0
5
0.0 0
5
10
15
20
25
30
35
Time, hr Fig. 8. Non-wetting and wetting phase pressure drops for v w ¼ 0 at a saturation of Sw ¼ 0:40. The ratio is Dpw =Dpg .
398
T.S. Ramakrishnan, P.A. Goode / Journal of Colloid and Interface Science 449 (2015) 392–398
the wetting phase is presented in Fig. 6. The continuous curve is obtained from network modeling of 4-cusp pores [4] and has been included purely for comparative purposes; it is certainly not suggested that the experimental data should quantitatively agree with the results of the network modeling. The approach to the stabilized pressure drop for Sw ¼ 0:44 and Sw ¼ 0:40 are given in Figs. 7 and 8. The data indicate that the off-diagonal coefficient in two-phase flow is consistent with momentum transfer at the fluid interface. Momentum transfer due to the presence of the wetting fluid in corners in the pores intruded by the non-wetting phase is the primary reason for nonzero elements in the transport matrix. The measured magnitudes are in qualitative agreement with the theoretical calculations. A point worth discussing is the contribution of krc to the flow of reservoir fluids. If one assumes that in large scale reservoir flow rpw rpn , then below about Sw of 0.5, we see that a significant fraction of the water flow is due to the off-diagonal coefficient. But given that krw itself is small compared to krn for small Sw , the off-diagonal contribution, while being significant compared to the diagonal one, is nevertheless small in relation to the nonwetting phase flow magnitude. The non-wetting gas phase flow’s diagonal term is also significantly larger than the off-diagonal one, both because krc is small compared to krn at low Sw , and lw is much larger than gas-phase viscosity. Thus, unless one is interested in remedying small water flow in gas production, the off-diagonal terms do not impact overall flow rate in co-current flow. 5. Summary We have presented a null method to measure off-diagonal transport coefficients for two-phase flow in porous media. At capillary equilibrium, a perturbation in one of the phase pressures may be imposed to enable a small pressure gradient, while keeping the second phase stationary. The resulting positive gradient in the direction of flow of the first phase is a measure of the off-diagonal
coefficient. The technique is powerful because a positive pressure gradient in the stationary phase is a direct indication of the inadequacy of the presently practiced two-phased extended Darcy formulation. For conditions of gas–water flow, our experimental data are consistent with the expected momentum transfer between the wetting phase in films and the non-wetting phase. The data indicate that in gas–water flow, for intermediate saturations, one may not neglect the drag induced by the gas phase on the more viscous wetting fluid. Eqs. (3) and (4) with nonzero krc is necessary. Acknowledgment We thank Schlumberger for allowing us to perform this work and permitting us to publish the material. References [1] R.D. Wyckoff, H.G. Botset, Physics 7 (1936) 325–345. [2] M. Muskat, R.D. Wyckoff, H.G. Botset, M.W. Meres, Trans. AIME 123 (1937) 69– 96. [3] M.C. Leverett, Trans. AIME 132 (1939) 149–171. [4] P. Goode, T.S. Ramakrishnan, AIChE J. 39 (7) (1993) 1124–1134. [5] W. Rose, in: Fundamentals of Transport Phenomena in Porous Media, 1972. [6] P.G. deGennes, Phys. Chem. Hydrodyn. 4 (1983) 175–185. [7] B.J. Bourbiaux, F.J. Kalaydjian, in: 63rd Annual Technical Conference and Exhibition of SPE, 1988. [8] R. Collins, Flow of fluids: through porous materials, Reinhold chemical engineering series, Reinhold Pub. Copr., 1961. [9] C. Marle, Multiphase Flow in Porous Media, Éditions technip, 1981. [10] C. Zarcone, R. Lenormand, C. R. l’Academie. Sci., Ser. II 318 (11) (1994) 1429– 1435. [11] F.A.L. Dullien, M. Dong, Trans. Porous Media 25 (1) (1996) 97–120. [12] D. Avraam, A. Payatakes, in: Multiphase Flow in Porous Media, Springer, 1995, pp. 135–168. [13] J.-L. Auriault, Trans. Porous Media 2 (1) (1987) 45–64. [14] D. Lasseux, M. Quintard, S. Whitaker, Trans. Porous Media 24 (2) (1996) 107– 137. [15] N. Rakotomalala, D. Salin, Y. Yortsos, Appl. Sci. Res. 55 (2) (1995) 155–169. [16] H. Li, C. Pan, C.T. Miller, Phys. Rev. E 72 (2) (2005) 026705.
Contents lists available at ScienceDirect
Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Measurement of off-diagonal transport coefficients in two-phase flow in porous media T.S. Ramakrishnan ⇑, P.A. Goode 1 Schlumberger-Doll Research, 1 Hampshire St., Cambridge, MA 02139, United States
g r a p h i c a l a b s t r a c t
Water pressure increase ( pw, kPa)
0.12
0.10
0.08
0.06 line with 0.153 slope
0.04
0.02
0.00 0.0
0.2
0.4
0.6
0.8
Gas phase pressure drop (- pg, kPa)
a r t i c l e
i n f o
Article history: Received 5 November 2014 Accepted 13 January 2015 Available online 14 February 2015 Keywords: Extended Darcy’s law Reciprocity Off-diagonal coefficients
a b s t r a c t The prevalent description of low capillary number two-phase flow in porous media relies on the independence of phase transport. An extended Darcy’s law with a saturation dependent effective permeability is used for each phase. The driving force for each phase is given by its pressure gradient and the body force. This diagonally dominant form neglects momentum transfer from one phase to the other. Numerical and analytical modeling in regular geometries have however shown that while this approximation is simple and acceptable in some cases, many practical problems require inclusion of momentum transfer across the interface. Its inclusion leads to a generalized form of extended Darcy’s law in which both the diagonal relative permeabilities and the off-diagonal terms depend not only on saturation but also on the viscosity ratio. Analogous to application of thermodynamics to dynamical systems, any of the extended forms of Darcy’s law assumes quasi-static interfaces of fluids for describing displacement problems. Despite the importance of the permeability coefficients in oil recovery, soil moisture transport, contaminant removal, etc., direct measurements to infer the magnitude of the off-diagonal coefficients have been lacking. The published data based on cocurrent and countercurrent displacement experiments are necessarily indirect. In this paper, we propose a null experiment to measure the off-diagonal term directly. For a given non-wetting phase pressure-gradient, the null method is based on measuring a counter pressure drop in the wetting phase required to maintain a zero flux. The ratio of the off-diagonal coefficient to the wetting phase diagonal coefficient (relative permeability) may then be determined. The apparatus is described in detail, along with the results obtained. We demonstrate the validity of the experimental results and conclude the paper by comparing experimental data to numerical simulation. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction
⇑ Corresponding author. 1
Present address: Arle Capital Partners.
http://dx.doi.org/10.1016/j.jcis.2015.01.029 0021-9797/Ó 2015 Elsevier Inc. All rights reserved.
In the absence of body forces, based on the early work of Wyckoff and Botset [1], Wyckoff et al. [2], and Leverett [3], the extension of the isotropic form of Darcy’s law
T.S. Ramakrishnan, P.A. Goode / Journal of Colloid and Interface Science 449 (2015) 392–398
k
v ¼ rp;
ð1Þ
l
to multiphase flow is
vi ¼
kei
li
rpi ¼
kkri
li
rpi ;
ð2Þ
where v is the superficial velocity vector, v i is the superficial velocity for phase i; k is the single phase permeability, kei and kri are the saturation dependent effective and relative permeabilities to phase i; l is the dynamic coefficient of viscosity, and p is the pressure, again, with the subscript i indicating the flowing phase of interest. With i ¼ n; w for non-wetting and wetting phases respectively, the phase pressure difference pn pw is the capillary pressure pc , whose value determines the fluid saturations Si in quasi-static phase replacement processes. Excluding hysteresis, it is common to assume that kei is determined by the fluid saturation alone, and its functional dependence on saturation may be obtained by conducting flow experiments that minimally disturb interface equilibrium at the capillary pressure of interest. Application of Eq. (2) to commonly occurring displacement processes in hydrology, and oil and gas industry, requires that over the length scale R needed for defining pc and kei , RrSi 1. Furthermore, consistent with the conjecture that each fluid flows in accordance with its own pressure gradient, the effective permeabilities are independent of the viscosity ratio. In a capillary network, with pore occupancy determined by capillarity, the notion that fluid–fluid interfaces induce only local circulatory motion and do not transfer appreciable macroscopic momentum is the basis of Eq. (2). In the presence of corners and crevices that allow for macroscopically relevant wetting-phase motion, diagonally dominant force-flux relationship is inadequate [4]. To include shear induced motion on the second fluid, Rose [5] hypothesized that an Onsager-type relationship should apply i.e.,
v n ¼ knn rpn knw rpw ; v w ¼ kww rpw kwn rpn ;
ð3Þ
and Lenormand [10] measured the off-diagonal coefficient by imposing a zero-pressure gradient on water, while pumping mercury through a sand-pack, and found coefficients smaller than 0.01. Similar experiments by Dullien and Dong [11] however resulted in much larger magnitudes than [10]. Dullien and Dong obtained non-symmetric coefficients as well, and were speculating about variability in interface configurations from one experiment to the other (wetting phase gradient as opposed to non-wetting phase gradient). It is clear from their data that there is also a saturation gradient within the sample in these experiments, and the saturation configurations were not the same in the two sets of experiments that they conducted. Several authors have speculated on the form of the diagonal coefficients, some suggesting that the off-diagonal coefficients are unequal (see e.g. [12]). The symmetry of the coefficients has been demonstrated by Auriault [13], who correctly argues that the off-diagonal coefficients are negligible if the interface is nearly rigid with zero velocities. Others such as Lasseux et al. [14] derive a reciprocal form based on volume averaging and closure conditions through essentially Auriault’s [13] method, but were unable to specify a self-consistent order of magnitude for the off-diagonal coefficients. For a viscosity ratio of unity, Rakotomalala et al. [15], computed off-diagonal coefficients that were about 1% compared to the diagonal ones. The network modeling of Goode and Ramakrishnan [4] relied first on a finite element analysis within a single pore to compute a duct conductance matrix, based on which it was shown that the off-diagonal coefficients were equal.2 However both duct conductance matrix coefficients and the effective network permeability matrix were seen to depend on both the saturation and the viscosity ratio of the flowing fluids. The coefficients were computed with stationary interfaces, and therefore a pressure perturbation was imposed on one of the fluids to compute the matrix of mobility coefficients. The general conclusion may be written compactly for quasi-static two-phase flow as
kkrc ðSw ; MÞ rpn ; lw þ ln kkrc ðSw ; MÞ kk ðS ; MÞ ¼ rpw rn w rpn ; lw þ ln ln
vw ¼
ð4Þ
where it was understood that the mobilities kij were the ratio of the effective permeabilities to viscosities; no discussion of how the viscosities appeared in the off-diagonal terms was given by Rose. Invoking the Onsager reciprocity principle, he asserted that knw ¼ kwn . A number of papers have speculated on the notion of a generalized mobility matrix for two-phase flow. For example, deGennes [6] wrote a form identical to Eqs. (3) and (4), with saturation-dependent mobilities. No specific viscosity factor appeared in his formulation. Bourbiaux and Kalaydjian [7] attempted to measure the offdiagonal coefficients by conducting a cocurrent displacement, followed by a counter-current imbibition. Since the gradients in the two phase-pressures are of opposite sign in counter-current flow, and are more or less matched in co-current flow, the measured profiles or water-cut in the two experiments would be indicative of the magnitude of the coefficients knw . Symmetry of the coefficients was assumed for processing the data, and a characteristic peak in knw was obtained. Given the viscosity ratio of about ten and the implementation of the boundary conditions, it is difficult to determine whether such an indirect measurement is an artifact or not. For example, no special effort was taken to eliminate capillary end-effects [8,9]. The consequences of end-effects are quite different for counter-current and co-current flow. Furthermore, experiments were performed in natural media (clayey sandstone), and although these were stated to be fairly homogeneous, displacement profiles indicated moderate heterogeneities, the effect of which on the interpretation of data is unclear. Zarcone
393
vn
kkrw ðSw ; MÞ
lw
rpw
ð5Þ ð6Þ
where krc is the off-diagonal or cross-coefficient, krw and krn are the wetting and the non wetting phase relative permeabilities, and M is the viscosity ratio equal to ln =lw . Note that in our formulation, the relative permeabilities and the cross-coefficients are functions of saturation and viscosity ratio, and is different from those proposed by Rose [5] and deGennes [6], where only explicit dependence on saturation alone is assumed. The dependence of the wetting fluid relative permeability (krw ) on M is however rather weak and may be ignored for practical purposes. The key conclusion of Goode and Ramakrishnan [4] was that a nonzero krc implies a dependence on M for all of the three coefficients, whereas the converse is not necessarily true. For example, when the wetting phase is disconnected, krn may depend on M, but krc may be negligible since the motion near the interfaces do not appreciably contribute to the superficial velocity. Also, the form chosen in [4] for the viscosity dependence of the off-diagonal mobility is also symmetric. This form also gives comparable magnitudes for krc with respect to the viscosity ratio. Clearly, the experimental results have varied, and while some inferences have been indirect, others have used methods where the velocities are measured, while maintaining a zero pressure gradient in one of the phases. The latter have been performed in sand-packs and consistent results with regard to the 2 Contrary to the statements given by Li et al. [16] that [4] implied invalidity of reciprocity.
394
T.S. Ramakrishnan, P.A. Goode / Journal of Colloid and Interface Science 449 (2015) 392–398
magnitude and importance of krc have not emerged. It should however be noted that experiments conducted within the domain of validity of the extended forms of Darcy’s law should satisfy symmetry.
2. Experimental concept Departing from previously published literature, the experimental apparatus discussed in this paper is designed to infer the off-diagonal coefficients in relation to the relative permeabilities directly. Inference is not based by comparing the results to a numerical model and regression. Rather than measure small velocities induced by drag, we measure a pressure drop to negate the small velocity induced by drag. Thus the objective of the experimental technique is to provide an unambiguous indication of the existence and the magnitude of krc . The exercise involves conducting a ‘‘null’’ experiment, i.e., measuring a wetting-phase pressure drop, when jDpn j=pc 1, for v w ¼ 0. At steady-state, any non-zero pressure drop, in a direction opposite to that of the non-wetting phase, implies a positive krc . The essential steps in the experiment are as follows. Bring the system to capillary pressure equilibrium by imposing a fixed pressure in the non-wetting phase over that of the wetting phase. Perturb the equilibrated system by a small Dpn , so that the saturation is virtually unaffected. Obviously Dpn has to be kept small compared to pc in order to infer krc at a given saturation. Force the wetting phase velocity to be zero. Measure Dpw =Dpn at steady-state. With regard to the choice of fluids, for many clean natural media and porous ceramics, water or a low-salinity aqueous solution is an acceptable wetting fluid. For the non-wetting fluid, we prefer to choose one that will give us a resolvable response without disturbing capillary pressure equilibrium. Besides requiring that the pressure in the non-wetting phase be small in relation to the capillary pressure, we need the wetting phase pressure increase
due to a nonzero krc to be resolved easily. The pressure drop in the wetting phase for the null experiment is
Dpw ¼
krc ðSw ; MÞ 1 ðDpn Þ: krw ðSw ; MÞ 1 þ M
ð7Þ
Now, theoretical network calculations [4] show that krc monotonically increases with M. However krc =ð1 þ MÞ decreases with viscosity ratio. Furthermore, since krw shows little variation with M, it is clear that for any saturation, the largest counter pressure gradient in the wetting phase is obtained when M is the smallest practical value, suggesting that a suitable non-wetting fluid would be nitrogen (used here) or any other inert gas. For the wetting-fluid, we used low-salinity water, measuring about 1 Xm resistivity. Therefore, water is used synonymously with the wetting fluid in the rest of the paper. 3. The apparatus The experimental set-up is illustrated in Fig. 1. It is designed to be vacuum-filled by a wetting fluid to 100% saturation, and then brought to capillary-pressure equilibrium. An arbitrary pressure drop in the gas-phase may also be imposed, and by choice the wetting phase may be allowed to flow out of the core or kept stagnant. The gas-phase pressures as well as differential gas and wetting-phase pressures are measured along with the wettingphase saturation. The operational details are given below. A porous sample of about 38 mm (1.5 in.) in diameter with a length up to 200 mm may be placed in the core-holder. The particular sample we used was 65 mm in length. The core was from Cleveland Quarry, commonly known as Berea 400 sandstone, and has a nominal permeability of 0:4 lm2 ( 400 mD). The sample was chosen for being a relatively clean naturally occurring sandstone. The core is sealed on its lateral surface with a rubber sleeve on the external side of which a compressed nitrogen pressure of about 650 kPa is applied. The transducers labeled Dpg is for measuring the drop in the gas phase (non-wetting), and has a range of ±34:5 kPa (±5 psi). The transducer is Sensotec, Model Z-882-12. Since M 0:01 and is small compared to unity, the
20cc pipette for saturation measurement
Water Line Gas Line
20cc pipette for fillup
pw
V6
V5
R2
V3 R1
V9 N 2 gas in V8
V7
pg
pg
V4
V2
Core V1
Fig. 1. The experimental apparatus for the null-flow measurement.
To vacuum
T.S. Ramakrishnan, P.A. Goode / Journal of Colloid and Interface Science 449 (2015) 392–398
395
Fig. 2. The expanded assembly that illustrates wetting and non-wetting phase inlet and outlet through the hydrophilic membrane.
magnitude of krc =krw approximately determines the measured jDpw =Dpg j. In drainage, for the four-cusp model of [4], at Sw 0:6, we expect jDpw j to be about 20 times smaller than jDpg j. A differential transducer with a range of ±3:45 kPa (±0:5 psi, Sensotec, Model Z-882-28) was therefore chosen. Each of these gauges has an accuracy specification ±0:25% of full scale, repeatability within 0.05% and hysteresis error within 0.10%. The nonlinearity component is 0.15%, and is easily taken into account. Since our transducers are used only unidirectionally for pressure difference measurement, and are subject to only a fraction of the entire range, the hysteresis errors are expected to be substantially smaller than the specification. For completeness, we have assumed two different error criteria of 0.25% and 0.10% of full-scale for indicating error bars. The ±34:5 kPa transducer is calibrated against a secondary standard traceable to NIST. For the ±3:45 kPa transducer, we used an inclined tube reference manometer with a fluid whose specific gravity was known to three digits (Dwyer Red Gage Fluid). The transducers are full-bridge strain gauges excited at 225 Hz carrier frequency, and a 10 Hz Bessel filter is used to condition the output. The amplifier and signal conditioning itself was precision calibrated to 1 part in 10,000. For the purpose of measuring and controlling phase pressures, the porous-sample or the rock core is sandwiched by two strongly hydrophilic porous plates, 6.35 mm thick. The plates have a sufficiently small pore size to prevent gas-phase intrusion of N2 for all capillary pressures of interest for the Berea sample. The ‘‘semi-permeable’’ plate’s permeability range is 0:0001 lm2 to 0:0003 lm2 and its pore size for a percolating path is sufficiently small for gas to break through water only at about 175 kPa (25 psi).3 The plate is made by CoorsTek and has a designation of P-12AC. Four evenly spaced holes are drilled through the porous plate and are aligned with holes in the nylon end-plug through which gas directly flows into or out of the core. The wetting phase can access 3 The stated bubble pressure is about 80 psi. Our experiments in water-dodecane indicated the conservative number to be closer to 30 psi. Adjusted for surface tension in fresh samples of dodecane this amounts to about 50 psi for air–water systems. A safety factor of two is used in the quoted number.
the core only through the porous plate. The central port in the nylon end-plug feeds the wetting fluid to the semi-permeable plate. A rubber gasket pressed between the nylon end-plug and the porous semi-permeable plate isolates the wetting and non-wetting fluids. This is illustrated in the expanded assembly of Fig. 2. 3.1. Operating procedure Prior to the experiment, lines to atmosphere from the coreholder past V5 and V6 are filled with water and a pipette connected for saturation measurement. The position of the interface in the pipette is noted prior to any water displacement from the core. Before commencing the experiment, the entire assembly is first helium leak-tested. With the exception of the two tanks upstream of V8, the system is then evacuated. Core seal pressure is initiated, and valves V1 and V7 are closed. V3 is then opened to fill the core and water lines with water. We estimate the core filling time by assuming Darcy’s law within the wetting phase, the velocity of which determines the frontal movement. The fill-up time T f is given by
Tf ¼
L2 lw 1 ; 2/ k p0
ð8Þ
where L is the length of the sample, / is the porosity, and p0 is the atmospheric pressure driving the wetting fluid. Gas pressure sufficiently higher than p0 is set upstream of V9 pressure and then V9 is opened to allow nitrogen gas to bubble into the two 1L chambers. The chambers, which are connected in series and partially filled with water, are used to saturate the intruding gas. This minimizes water evaporation into the gas phase within the rock. Once the core is filled i.e., after a few T f , V8 and V7 opened, and the regulator R1 is set to be slightly higher than p0 so as to displace water from the gas lines between V7 and the core, and V4 and the core. The pressure in the gas-phase past the regulator should be low to prevent gas-phase intrusion into the core. After a few minutes, valve V3 may then be shut and a step in capillary pressure may be made. The capillary pressure is controlled by the regulator R1 and measured by the pressure gauge
pg . When a change in pc is imposed, valves V5 and V6 are opened and water is permitted to leave the core. Core saturation is determined by measuring the volume of water which flows into the measuring pipette. No discernible movement in the interface position is indicative of reaching equilibrium. It is only necessary to asymptotically reach equilibrium, since the interpretation of the results is unaffected by small deviations away from it. For the gas to leak at a small rate, a gas gradient is now created by turning the metering valve V2. Since this valve may not have enough resolution to ensure a very small rate so as to have a pressure drop much smaller than the capillary pressure, we also simultaneously adjust the regulator R2 for flow bypass. Adjusting R2 gives us control over the magnitude of the gas phase pressure drop across the core. Since the lines containing R2 and the core are in parallel, the gas phase pressure difference across the core is that across R2. This pressure difference is measured by the transducer labeled Dpg . Since the change in saturation is negligible with the small imposed gradient in the gas-phase, no noticeable movement should occur in the pipette for measuring saturation. Valves V5 and V6 may be closed; there is no other water entry/exit line. At steady-state, for the wetting phase no-flow boundary conditions, the velocity v w ¼ 0. We allow for sufficient time to elapse for stabilization of ðDpg =Dpw ). Since the transducers have a dead volume of a few mL, a deflection of the transducer diaphragm is needed to measure the signal, and therefore a tiny volume of water needs to flow to the higher pressure side for an equilibrated pressure difference to be established. Following the equilibrated measurement of Dpg to Dpw at one saturation, a new saturation is achieved by closing V2, and opening V5 and V6, and adjusting R1 to a higher pc and therefore a lower Sw . The procedure is repeated to infer krc =krw ratio. According to the conventional extended form of Darcy’s law, one would expect the pressure drop in the wetting phase to be zero, when the phase velocity is zero. A non-zero Dpw indicates inadequacy of this form, and furthermore, a positive pressure drop in the wetting phase would constitute a failure of the extended Darcy’s law and the one with the off-diagonal coefficients. As explained previously, the magnitude of Dpw is reflective of the ratio krc =krw , and since M 1, we may write
Off-diagonal to diagonal (wetting phase) mobility
T.S. Ramakrishnan, P.A. Goode / Journal of Colloid and Interface Science 449 (2015) 392–398
Viscosity Ratio, M 0.01 0.1 1 10 100 1000 10000
1.5
1.0
0.5
0.0 0.2
0.4
0.6
0.8
1.0
Wetting phase saturation Fig. 3. Theoretical calculations for the ratio of the off-diagonal to diagonal wetting phase mobility for various M values. Note that the characteristic peak at very low saturations.
0.12
Water pressure increase ( pw, kPa)
396
0.10
Sw
0.54
0.08
0.06 line with 0.153 slope
0.04
0.02
0.00 0.0
0.2
0.4
0.6
0.8
Gas phase pressure drop (- pg, kPa)
ð9Þ
Fig. 4. Proportional wetting-phase pressure increase with respect to gas phase pressure drop at Sw 0:54. The best slope gives Dpw =Dpg ¼ 0:153 with an intercept of zero. There are error bars for both x and y axis based on the 0.10% and 0.25% of full-scale of the respective transducers. The best-fit suggests that the error bars exaggerate the inaccuracy of the measurement.
The ratio of off-diagonal mobility to the direct coefficient krc =½krw ð1 þ MÞ as computed by Goode and Ramakrishnan [4] is shown in Fig. 3. The curve is plotted for a viscosity ratio ranging from M ¼ 0:01 to M = 10,000. Since the viscosity of N2 at experimental conditions is approximately 0:0175 mPa s, M ¼ 0:01 is representative of the gas–water system. As M ! 0, there is little sensitivity to M, and therefore the precise value of it is inconsequential. For Sw > 0:6, the ratio decreases below 1%, and the response is not expected to be resolvable. Therefore only one measurement, at about Sw ¼ 0:75, was performed. As expected, for Sw ¼ 0:75, the response was below the resolution of the transducer and the instrument amplifier, indicating that krc ð0:75; 0Þ krw ð0:75; 0Þ for the Berea sample. Following the measurement at Sw ¼ 0:75; pc was increased, equilibrating at a saturation of Sw ¼ 0:54. At this saturation, since a nonzero krc was clearly evident, further tests were conducted to test the veracity of the data. As per Eq. (9), any increase in Dpg should lead to a proportional increase in Dpw , as long as Dpg pc . Measurements at four different Dpg were conducted,
and for each, the equilibrated Dpw is shown in Fig. 4. Numerical values of these are also shown in Table 1. As seen in Fig. 4, at an Sw of 0.54, the approximate ratio of krc to krw is 0.153. Given the minute pressure differences imposed and measured, it is remarkable that the data are consistent with very little scatter. For a saturation of Sw ¼ 0:54; pg pw was about 10.34 kPa. The elevated gas pressure leads to a gradual dissolution of gas into water. Thus, water within the porous medium has a higher N2 chemical potential compared to the water saturating the porous semi-permeable plate or the lines connected to the differential transducer. Within the porous plate, gas remains in the liquid phase, because the entry capillary pressure within the plate is much higher than that of the Berea sandstone. However, through diffusion, upon reaching a sufficient concentration of nitrogen in the tubing water where its equilibrium pressure in the gas phase would be greater than the water pressure, nitrogen bubbles would spontaneously appear. We call this as gas break-through. At the point of gas-breakthrough, the experiment is terminated, the core is cleaned and reassembled to restart the process.
krc ðSw ; 0Þ Dp w: krw ðSw ; 0Þ Dpg 4. Results
397
Table 1 Saturation, non-wetting and wetting phase pressure drops and mobility ratio. Pressures and error bars are in kPa; Eðpg Þ and eðpg Þ are 0.25% and 0.10% transducer errors. The error bars for the mobility ratio are shown in Fig. 6 for both 0.25% and 0.10% errors in differential pressure. The error shown for the mobility ratio in the table is based on 0.10% error in pressure, with the upper and lower values in parenthesis. Sw
Dpg
Eðpg Þ
eðpg Þ
Dpw
Eðpw Þ
eðpw Þ
krc =½krw ð1 þ MÞ
0.75
0.490
0.086
0.034
0.000
0.0086
0.0034
0.54
0.317
0.086
0.034
0.050
0.0086
0.0034
0.668
0.086
0.034
0.104
0.0086
0.0034
0.520
0.086
0.034
0.078
0.0086
0.0034
0.415
0.086
0.034
0.062
0.0086
0.0034
0.000 (0.000, 0.008) 0.157 (0.132, 0.189) 0.156 (0.143, 0.170) 0.150 (0.134, 0.168) 0.149 (0.130, 0.172) 0.160 (0.141, 0.180) 0.387 (0.354, 0.445) 0.374 (0.346, 0.409) 0.738 (0.653, 0.817)
0.53
0.459
0.086
0.034
0.073
0.0086
0.0034
0.44
0.379
0.086
0.034
0.150
0.0086
0.0034
0.525
0.086
0.034
0.197
0.0086
0.0034
0.352
0.086
0.034
0.256
0.0086
0.0034
0.40
Off-diagonal mobility/Wetting phase mobility
T.S. Ramakrishnan, P.A. Goode / Journal of Colloid and Interface Science 449 (2015) 392–398
1.5
1.0
0.5
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Water saturation Fig. 6. The ratio of the off-diagonal mobility to the wetting phase mobility at various saturations is plotted alongside the results of the numerical network model for a qualitative comparison. The thicker error bars with wide caps are with 0.10% error in the transducer. The thinner error bars with narrow caps are with 0.25% error in differential pressures. The data are also presented in numerical form in Table 1 along with the errors based on transducer specfications.
0.5
0.16
M = 0.01, Network calculation Experiment, gas-water system
0.5
0.4 Sw = 0.44
0.14 0.4
0.2
0.06
R a tio
0.08
0.3 0.3
Ratio - pg pw
0.2
0.2
0.1
P h a se p re ssu re d ro p , k P a
0.3 Ratio - pg pw
Phase pressure drop, kPa
0.10
Ratio
0.4
Sw = 0.53
0.12
0.1
0.04 0.1 0.02
0.0 0.00
0
0.0 2
4
6
8
10
Time, hr Fig. 5. Non-wetting and wetting phase pressure drops for v w ¼ 0 at a saturation of Sw ¼ 0:53. The ratio is Dpw =Dpg .
20
25
Fig. 7. Non-wetting and wetting phase pressure drops for stationary wetting phase at a saturation of Sw ¼ 0:44. The ratio is Dpw =Dpg .
0.5 0.7 Sw = 0.40
0.6
0.4
0.5 0.3 0.4 0.2
0.3
Ratio - pg pw
0.2 0.1
0.1
P h a se p re ssu re d ro p s, k P a
Following reassembly, to check for repeatability, we attempted to go back to a saturation of Sw ¼ 0:54 and obtain a repeat point. For the same gas pressure over the wetting phase pressure, a saturation of Sw ¼ 0:53 was actually achieved. The steady-state ratio of Dpw =Dpg was 0.16, in excellent agreement with the previously acquired data at Sw ¼ 0:54. We also show the time dependence of the pressure drops in water and gas in Fig. 5. Stabilization in the ratio of the pressures occurs in a few hours at Sw ¼ 0:53. Subsequent to the repeat point, we increased the gas pressure and reached a saturation of 0.44. Again two different Dpg were imposed in sequence, each resulting in nearly the same ratio of the phase pressure drops. The lowest saturation that we could reach was 0.4. This corresponded to a capillary pressure of about 34.5 kPa (5 psi). Above this pressure, gas break-through occurred before Dpw stabilization, and therefore it was impossible to measure krc . All of the stabilized pressure drop data are summarized in Table 1. The computed off-diagonal to diagonal mobility ratio for
15
Time, hr
10
R a tio
0
5
0.0 0
5
10
15
20
25
30
35
Time, hr Fig. 8. Non-wetting and wetting phase pressure drops for v w ¼ 0 at a saturation of Sw ¼ 0:40. The ratio is Dpw =Dpg .
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T.S. Ramakrishnan, P.A. Goode / Journal of Colloid and Interface Science 449 (2015) 392–398
the wetting phase is presented in Fig. 6. The continuous curve is obtained from network modeling of 4-cusp pores [4] and has been included purely for comparative purposes; it is certainly not suggested that the experimental data should quantitatively agree with the results of the network modeling. The approach to the stabilized pressure drop for Sw ¼ 0:44 and Sw ¼ 0:40 are given in Figs. 7 and 8. The data indicate that the off-diagonal coefficient in two-phase flow is consistent with momentum transfer at the fluid interface. Momentum transfer due to the presence of the wetting fluid in corners in the pores intruded by the non-wetting phase is the primary reason for nonzero elements in the transport matrix. The measured magnitudes are in qualitative agreement with the theoretical calculations. A point worth discussing is the contribution of krc to the flow of reservoir fluids. If one assumes that in large scale reservoir flow rpw rpn , then below about Sw of 0.5, we see that a significant fraction of the water flow is due to the off-diagonal coefficient. But given that krw itself is small compared to krn for small Sw , the off-diagonal contribution, while being significant compared to the diagonal one, is nevertheless small in relation to the nonwetting phase flow magnitude. The non-wetting gas phase flow’s diagonal term is also significantly larger than the off-diagonal one, both because krc is small compared to krn at low Sw , and lw is much larger than gas-phase viscosity. Thus, unless one is interested in remedying small water flow in gas production, the off-diagonal terms do not impact overall flow rate in co-current flow. 5. Summary We have presented a null method to measure off-diagonal transport coefficients for two-phase flow in porous media. At capillary equilibrium, a perturbation in one of the phase pressures may be imposed to enable a small pressure gradient, while keeping the second phase stationary. The resulting positive gradient in the direction of flow of the first phase is a measure of the off-diagonal
coefficient. The technique is powerful because a positive pressure gradient in the stationary phase is a direct indication of the inadequacy of the presently practiced two-phased extended Darcy formulation. For conditions of gas–water flow, our experimental data are consistent with the expected momentum transfer between the wetting phase in films and the non-wetting phase. The data indicate that in gas–water flow, for intermediate saturations, one may not neglect the drag induced by the gas phase on the more viscous wetting fluid. Eqs. (3) and (4) with nonzero krc is necessary. Acknowledgment We thank Schlumberger for allowing us to perform this work and permitting us to publish the material. References [1] R.D. Wyckoff, H.G. Botset, Physics 7 (1936) 325–345. [2] M. Muskat, R.D. Wyckoff, H.G. Botset, M.W. Meres, Trans. AIME 123 (1937) 69– 96. [3] M.C. Leverett, Trans. AIME 132 (1939) 149–171. [4] P. Goode, T.S. Ramakrishnan, AIChE J. 39 (7) (1993) 1124–1134. [5] W. Rose, in: Fundamentals of Transport Phenomena in Porous Media, 1972. [6] P.G. deGennes, Phys. Chem. Hydrodyn. 4 (1983) 175–185. [7] B.J. Bourbiaux, F.J. Kalaydjian, in: 63rd Annual Technical Conference and Exhibition of SPE, 1988. [8] R. Collins, Flow of fluids: through porous materials, Reinhold chemical engineering series, Reinhold Pub. Copr., 1961. [9] C. Marle, Multiphase Flow in Porous Media, Éditions technip, 1981. [10] C. Zarcone, R. Lenormand, C. R. l’Academie. Sci., Ser. II 318 (11) (1994) 1429– 1435. [11] F.A.L. Dullien, M. Dong, Trans. Porous Media 25 (1) (1996) 97–120. [12] D. Avraam, A. Payatakes, in: Multiphase Flow in Porous Media, Springer, 1995, pp. 135–168. [13] J.-L. Auriault, Trans. Porous Media 2 (1) (1987) 45–64. [14] D. Lasseux, M. Quintard, S. Whitaker, Trans. Porous Media 24 (2) (1996) 107– 137. [15] N. Rakotomalala, D. Salin, Y. Yortsos, Appl. Sci. Res. 55 (2) (1995) 155–169. [16] H. Li, C. Pan, C.T. Miller, Phys. Rev. E 72 (2) (2005) 026705.
