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Non-equilibrium and Nonlinear Effects in Water-in-Oil Emulsion Flows in Porous Media† Mars M. Khasanov,‡ Guzel T. Bulgakova,*,§ Aleksey G. Telin,‡ and Alfir T. Akhmetov|| ‡
Rosneft, 26/1, Sofiyskaya Embankment, 1, GSP-8, 117997 Moscow, Russia Ufa State Aviation Technical University, 12, Karl Marx Street, 450000 Ufa, Russia Institute of Mechanics of Ufa Branch of Russian Academy of Sciences (RAS), 71, Oktyabrya Prospect, 450054, Bashkir State University, 32, Zaki Validi Street, 450007 Ufa, Russia
)
§
ABSTRACT: The nonlinear behavior of emulsions in hydrodynamics is mostly pronounced when such emulsions flow in microchannels in experiments involving a permanent pressure gradient at the channel. It has been discovered that, despite the permanent pressure gradient, the flow of the water-in-oil emulsion almost completely ceases with time. This effect is known as dynamic blocking. It becomes pronounced when the emulsions flow through a cylindrical microchannel, in the case of a plane flow in the elements of a fracture, and when the emulsion flows through a 3D capillary structure, such as a core. The sizes of the water microdrops in the emulsion are an order of magnitude smaller than the cross-section sizes of the microchannels. However, the structure that is formed of such microdrops blocks not only individual microchannels but also the entire microchannel system, including cores. The physical nature of the dynamic blocking of emulsions is connected to the deformations of the emulsion microdrops as the pressure gradient increases and to the friction among these microdrops. The friction between emulsion microdrops brings about non-equilibrium and nonlinear effects in the filtration flow. On the basis of these ideas, this work describes a mathematical model of two-phase filtration through a porous medium. The results of the mathematical model satisfactorily agree with the experimental data.
’ INTRODUCTION An emulsion is a mixture of two or more immiscible liquids. In an emulsion, one liquid (the dispersed phase) is dispersed in the other (the continuous phase). An emulsion is two immiscible liquids mixed together (e.g., by shaking) with small droplets of one liquid dispersed (separated and distributed throughout the space) in the other liquid. This dispersion is usually not stable, and the droplets will “clump” together over time and form two layers. Because of the immiscibility, the emulsion is classified according to the chemical nature of the liquids, such as oil-inwater (oil is the dispersed phase, and water is the continuous phase) or water-in-oil (water is the dispersed phase, and oil is the continuous phase). These classical types of emulsions can be stabilized against coalescence (i.e., preventing the droplets from clumping together) by the presence of surfactant molecules. Part of the molecular structures of these surfactant molecules is soluble in water, and the other part is soluble in oil-like solvents. Water-in-oil emulsions are widely used in various oil-production processes.1,2 At the same time, no clear-cut theories exist today that explain the behavior of emulsions as they move through porous media. Using emulsions to equalize the profiles of the well water intake capacities requires thorough investigations into the flow of emulsions in the porous structure and studies of the mechanisms involved in the processes of transforming disperse and dispersion phases under the flow filtration. Results from core-flood experiments using well-characterized silicon-carbide packs provide preliminary evidence of aqueous †
This paper should be considered as part of the 11th International Conference on Petroleum Phase Behavior and Fouling special section. r 2011 American Chemical Society
droplet adsorption as the main retention mechanism in porous media. This finding was expected considering the average droplet size of 0.3 mm. The mother formulation of the water-in-oil emulsion is a concentrate, containing 80 wt % of the water phase and 8 wt % of the active scale inhibitor. The product can be diluted to 2% water phase by the droplet size of the emulsion.3 The thermodynamic basis for capillary pressure in porous media is discussed elsewhere.4 The authors employ microscopic mass- and momentum-balance equations for phases and interfaces to develop an understanding of the capillary pressure at the microscale. In particular, the presence of interfaces and their distribution within a multiphase system are shown to be essential in describing the state of the system. A thermodynamic approach to the definition of capillary pressure provides a theoretically sound alternative to the definition of capillary pressure as simple hysteretic-function saturation. The pore-scale flow mechanisms and the relative phase permeabilities (RPPs) during steady-state two-phase flow in a glass model pore network were studied experimentally for the case of strong wettability. The capillary number, the fluid-flowrate ratio, and the viscosity ratio were changed systematically, while all other parameters were kept constant. The flow mechanisms at the micro- and macroscopic scales were examined visually and video-recorded. As in the case of intermediate wettability, the pore-scale flow mechanisms include many strongly nonlinear Received: August 10, 2010 Revised: January 26, 2011 Published: February 15, 2011 1173
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Energy & Fuels phenomena, including breakup, coalescence, stranding, and mobilization, when observed over a broad range of the system parameters.5 Such microscopically irreversible phenomena cause macroscopic nonlinearity and irreversibility, which make an Onsager-type theory inappropriate for this class of flows. The main effects of strong wettability are changes in the domains of the system parameters where the various flow regimes are observed and an increase in the RPPs, whereas the qualitative aspects of the flow remain the same.5 Steady-state two-phase flows in porous media were studied experimentally using a model pore network of the chamber-andthroat type etched in glass. The size of the network was sufficient to make end effects negligible. The capillary number, the flowrate ratio, and the viscosity ratio were changed systematically within a range of practical interest, whereas the wettability, the coalescence factor, and the geometrical and topological parameters of the porous medium were kept constant. Optical observations and macroscopic measurements were used to determine the flow regimes and to calculate the corresponding RPP and fractional flow. Four main flow regimes were observed and video-recorded: large-ganglion dynamics, small-ganglion dynamics, drop-traffic flow, and connected-pathway flow. Qualitative mechanistic explanations for these experimental results are proposed. The conventional RPP and the fractional flow of water are found to be strong functions of not only the water saturation but also the capillary number and viscosity ratio (with the wettability, coalescence factor, and all other parameters kept constant). These results imply that a fundamental reconsideration of fractional-flow theory is warranted.6 A series of papers7-9 describe the mechanism of foam-layer adhesion and slip relative to each other. In a shear flow between plates in foams, bubbles are deformed and the foam layers shift relative to each other. This process causes squeezing of a dispersion medium from the interbubble space until “black films” are formed. The thicknesses of these “black films” can reach of 3-5 nm. They are thermodynamically stable, and their stability is not time-limited because a potential-energy barrier must be overcome for film destruction. As a result, the foam structure is locally transformed into a “hardened” condition, and the authors suggest that this effect should also occur in concentrated emulsions. The above mechanism is used to explain the current yield point in foams and emulsions. If the stress falls below a critical value, the foam and emulsion structure maintains its elasticity. However, if the stress is above a critical value, it moves like a liquid. Concerning the emulsions studied in our experiments, as judged from the measured rheological characteristics, the critical stress is negligible compared to the stresses accompanying the dynamic blocking. The mentioned studies suggest that the dynamic blocking mechanism is based on the friction between water droplets while squeezing the interface fluid. The goal of this work is to determine the conditions for forming and destroying emulsions as water-in-oil systems seep through porous structures. Techniques involving visualization of multiphase flows2 are currently considered to be the most informative ways of investigating microprocesses taking place as the emulsion flows through porous structures. Therefore, in addition to conventional experiments involving viscometers and cores, techniques were used that enable the visualization of the processes taking place in the viscometer and in the porous formation elements, i.e., capillaries, fractures, and the porous structure.3-6 The visualizations of multiphase flows allow for the refinement and development of concepts of the nature of oil
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displacement under various physical and chemical conditions and the mechanism of oil retention and extraction. The experimental research has established that the rheological behavior of water-in-oil emulsions in a porous medium is nonNewtonian.10 This phenomenon is attributed to structural transformations in emulsions. Within the context of the phenomenological approach, a mathematical model is considered that simulates the process of water-in-oil displacement from the porous medium, taking into account the emulsion state of the fluids. This paper proposes to consider non-equilibrium effects related to changes of the rheological properties in the fluid emulsion systems.
’ FILTRATION STUDIES USING LINEAR FORMATION MODELS Filtration studies were performed on singular water-saturated cores measuring 5 cm in diameter and 5.9 cm in length. The initial stage consisted of determining the permeability for water. A slug of inverted water-in-oil emulsion with a volume of 0.1 Vp was injected, followed by water until the pressure gradient stabilized. Next, the second slug of 0.3 Vp of the emulsion was injected, followed by water. If necessary, a third slug of the emulsion was injected, having a volume of 0.5 Vp, followed by water until the pressure gradient stabilized. A constant flow of fluid at the input to the formation model was maintained throughout the experiment. Here, Vp is the void volume of the core. The experiments yielded the filtration characteristics. In this experiment, the resistance factor R is defined as the ratio of the pressure drop when filtering an agent (emulsion or polymer) ΔPR to the pressure drop when filtering water ΔPW before the agent injection. The residual resistance is the ratio of the pressure drop when filtering water ΔPWR after the agent injection to the pressure drop when filtering water ΔPW before the agent injection. R ΔPR ΔPW , Rs ¼ ΔPW ΔPW As shown in Figure 1a, in the first experiment, an injection of 0.1 Vp of the emulsion slug into a porous medium with permeability, K0 = 0.307 μm2 and porosity = 0.18 resulted in a considerable growth of the residual resistance factor, Rs = 10.1. A subsequent 0.3 Vp injection of the emulsion brings about a still greater growth of the residual resistance factor, Rs = 20.1. For comparison, the second experiment involved injecting slugs of a cross-linked polymer composition (CPC) based on Sedipur-brand polyacrylamide (0.2%) and chromium acetate (0.015%) into a porous medium with permeability, K0 = 0.304 μm2 and porosity = 0.18 (Figure 1b). The resistance factor after injecting the first slug was 56.4, whereas the residual resistance factor was Rs = 39.0. The second slug of the polymer composition increased the residual factor to 83.7 and the residual resistance factor to Rs = 51.4. The injection of the polymer composition used for comparison yields provides results that qualitatively correspond to those of the emulsion injections under comparable conditions. This similarity is clearly visible when comparing panels a and b of Figure 1. It follows from the experiment that the water-in-oil emulsion possesses residual resistance that allows its use in diverter technologies. To better understand the cause of these surprising results, a series of experiments was conducted using the simplest inverted dispersions: a highly concentrated water-in-oil emulsion obtained by mixing reservoir water and oil (non-stabilized) and a similar emulsion stabilized with the artificial emulsifier Neftenol-NZ from Himeco-GANG (stabilized). Both emulsions are statically highly stable. An artificial stabilizing emulsifier is a surfactant such as Neftenol-NZ. When the emulsion is prepared, this reagent is added to oil such that it is 4% of the total emulsion.
R ¼
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Figure 1. Variation of the pressure drop on the core as (a) the emulsion slugs are successively injected with the integral flow held constant and (b) the polymer composition slugs are successively injected with the integral flow held constant. In both cases, the concentration of the disperse phase (water microdrop) was 75%. The diameters of the water microdrops in the stabilized emulsion were under 1 μm, whereas those in the non-stabilized emulsion were under 5 μm. Investigating the oil-displacement processes using visualization methods enables the refinement and development of new concepts of the nature of oil displacement under various physical and chemical conditions and the mechanism of oil extraction and retention. We used a 2D dismountable transparent micromodel of a porous structure whose pore pattern reflects the cross-section of a real-life core, a slot-like model of a fracture, a Hele-Shaw cell, and model of a single pore, a glass capillary pipe. Hereafter, we will use the terms “stabilized emulsion” and “non-stabilized emulsion”.
’ STUDIES OF EMULSION FLOW IN THE FRACTURE MODEL CONSISTING OF A HELE-SHAW SLOT-LIKE CELL AND IN THE POROUS-STRUCTURE MICROMODEL The water-in-oil emulsion flow was obtained by feeding fluids under pressure to the intake of the Hele-Shaw cell with the pressure at the outlet corresponding to the atmospheric pressure. The cell was formed in a 35 μm gap between the two thick cylindrical plates made of optical-grade glass, which were housed in a steel holder. The gap was defined by the thickness of the foil placed between the plates. A rectangular window of dimensions 2 4 cm was cut in the foil. The movement occurs within the volume formed by the plates and the foil. Two holes were drilled through the upper glass plate, one serving to supply the water-in-oil emulsion and the other serving to channel it to the measuring instrument. The flow characteristics were measured by electronic scales connected to a
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Figure 2. Time dependence of the volume of the water-in-oil emulsion flowing through the Hele-Shaw cell (2 4 cm, 35 μm gap) at a constant pressure drop of ΔP = 0.1 MPa. The photographs show images of the flow structure at various times: (a) non-stabilized emulsion and (b) stabilized emulsion.
computer. The constant pressure drop on the cell was maintained at 0.2 MPa. Under the plane flow in thin-gap conditions, the non-stabilized water-in-oil emulsion changes its structure during the initial flow period (2 min). This change involves the coalescence of water microdrops, drop formation, and further coalescence into progressively larger drops that move at greater speeds, leaving the continuous phase behind. As the water-in-oil emulsion is injected further, the flow structure changes substantially (Figures 2a and 3a). The most surprising finding is that the measurements of the flow characteristics reveal a gradual blocking of the cell. The structure being observed also “freezes” and does not reveal any visible changes. With a permanent pressure gradient, the threadlike structure survives for quite a long time (more than 10 days). The stabilized water-in-oil emulsion also displays some degree of destruction (Figures 2b and 3b). The emerging drops of the water phase move significantly faster than those of the continuous phase. Water drops, forming from the coalescence of emulsion droplets, move faster than the ambient emulsion because the wall friction for water is much lower than that for the emulsion, and the field of pressure forces is almost uniform. As the injection progresses, flow paths appear, which testifies to the laminarity of the flow. Later on, the flow tends to become chaotic and slows substantially. After an hour, according to the scale readings, a complete lockup takes place and the flow pattern itself also “freezes”. Nevertheless, the water-in-oil emulsion inside the cell substantially transforms after several days, although the blocking persists. 1175
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Figure 3. Magnified image of the flow structure of the emulsion in the Hele-Shaw cell at various times: (a) non-stabilized emulsion and (b) stabilized emulsion.
Figure 4. Dismountable micromodel of a porous medium: (a) glass plate with porous-medium channels, (b) second plate with input and outlet holes, (c) mounting of plates on optical contact, (d) fully assembled model, with the front of injected water visible.
As shown above, both emulsion types show a considerable slowing of the flow and its complete block (if one relies on the readings of the scales), although the pressure gradient was kept constant. The time dependence of the volume of liquid that has flown through the fracture model at a constant pressure gradient is shown in panels a and b of Figure 2 for the non-stabilized and stabilized water-in-oil emulsions, respectively. The path of the
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Figure 5. Volume of the passed fluid versus time for the emulsion flowing through a cylindrical tube (Ø = 100 μm; L = 4 cm; ΔP = 0.2 MPa): (a) stabilized emulsion and (b) non-stabilized emulsion.
curve displays a “lockup” of the model as time progresses. A more detailed microscope-assisted scrutiny of the flow revealed the presence of a microflow, whose value was 4 orders of magnitude smaller than that of the initial flow. The discovered effect was therefore called the dynamic blocking effect. A study of the water-in-oil emulsion flow in the microchannel complex structure, representing a porous medium section, was carried out using a porous structure micromodel. As a complicated capillary structure, we used a system of microchannels, modeling the pore pattern in a polished section of an oil-bearing core prepared by the selective etching. This structure is conventionally referred to as a micromodel.11 A specific feature of our micromodel was the ability to disassemble it using glass plates with high-density parallel surfaces placed on the optical contact. The etching depth in this particular case is 15 μm, and the dimensions of the etched area are 2 4 cm (Figure 4). The statistical properties of the micromodel were not studied (the pore size in the plan is 6-100 μm). The emulsion flow in the adopted micromodel at a constant pressure gradient also exhibited dynamic blocking. The supply of the emulsion to the micromodel leads to some phase separation. The viscosity of the carrier phase (oil) is significantly smaller that of the water-in-oil emulsion. At large pressure gradients, a certain fraction of oil is filtered between water microdrops and outflows first. In the emulsion passing via channels of a complicated configuration, water microdrops exhibit coalescence and an aqueous phase (water) separates and flows at a velocity that is significantly greater than the velocity of the emulsion proper. During this phenomenon, the separated water drops continue to grow and merge. The blocking behavior in a capillary structure, observed in the micromodel, and the blocking-curve behavior are similar to the 1176
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process in the Hele-Shaw cell. For the porous structure, the flow rate with a high-pressure gradient is an order of magnitude smaller compared to the Hele-Shaw cell.
’ CAPILLARY FLOW OF INVERT WATER-IN-OIL EMULSIONS As a check on the blocking effect in the capillary microchannels, we conducted a series of experiments in glass capillaries. The relationships between the emulsion volume passed through the capillary and time are presented in panels a and b of Figure 5. In the experiment, we used 4 cm long capillary tubes cut from one long capillary that had been preliminarily cleaned with alcohol and water. The capillary diameter was 100 μm, and its pressure drop was 0.2 MPa. In comparison to the case of the fracture model (the Hele-Shaw cell) and the capillary structure (the micromodel), the transition into the blocked state was accomplished in a much more vigorous manner, although the capillary diameter was much larger than the fracture-model gap (100 μm) and the channel depth (4 cm) in the micromodel. It can be seen in panels a and b of Figure 5 that this time (the transition into the blocked state) is approximately 2 min. A fairly considerable spread was observed in terms of time, and the volume of the fluid passed until the blocking condition was reached, although the flow value in the transitional state was nearly equal. This substantial difference from the results obtained on 2D models can be explained by the fact that the capillary flow progresses at a constant rate for quite a long time until the blocking, whereas in 2D models, the volume flow rate starts to change from the very beginning of the experiment. In capillaries, the transition from constant flow to blocking takes place quite abruptly, and the time required to accomplish this transition does not exceed half a minute. A more detailed microscope-assisted scrutiny of the blocked system reveals that a slow creeping flow takes place in the blocked state whose volume rate is 4 orders less than the flow rate in the transitional state. Therefore, in the capillary case, the blocking effect can be considered to be dynamic blocking. The effect reveals that the liquid flow completely stops (as can be seen from the scale readings), despite a constant pressure drop on the capillary with an averaged pressure gradient of up to 8 MPa/m. The stabilized water-in-oil emulsion movement before reaching the blocking condition occurs at a nearly constant flow, as demonstrated by a series of curves in Figure 5a. Apparently, this phenomenon is caused by its high dynamic stability. A certain scattering is observed in the passed emulsion volume and the blocking time, which is explained by emulsion interactions with the capillary walls and a complex mechanism of dynamic blocking. The non-stabilized emulsion movement takes place with a noticeable flow-rate scattering (Figure 5b), which is conditioned by its low dynamic stability. The scattering of volumes and times before the blocking is not less than that in the case of a stabilized water-in-oil emulsion. The curves shown on the graphs correspond to experiments conducted under identical conditions. Their difference is caused by the complexity of the structure-formation process from water droplets. The beginning of the process has a probabilistic nature. In these experiments, the stabilized emulsion flow rates before blocking are almost equal (Figure 5a) and the emulsion is dynamically stable. The flow rates are different for the nonstabilized emulsion (Figure 5b). The emulsion is partially destroyed in the capillary flow and is dynamically unstable.
Figure 6. Shear stress versus shear-strain rate (solid line, forward stroke; dashed line, return stroke): (a) stabilized emulsion and (b) non-stabilized emulsion.
Extensive research was dedicated to investigating the rheological properties of water-in-oil emulsions.10 Nevertheless, to mathematically model emulsion flows, we performed a study of the rheological properties of the emulsions under study. As is common, the rheological characteristics were measured using a rotational viscometer using the cone-plate system. Its principal advantage is the immense value of the shear-deformation strain rate throughout the sample volume. The rheology of the disperse system is quite complicated. The behaviors of the stabilized and non-stabilized emulsions in the gap between the cone and plate differ. The measured values of the shear stress with a gradual increase of the shear-strain rate up to a certain value (forward stroke) and a subsequent decrease (return stroke) align along two different curves (Figure 6). Because of structure-forming processes, the viscosity of the stabilized emulsion increases as time progresses (i.e., it possesses the property of rheopexy), whereas the viscosity of the nonstabilized emulsion decreases (partially because of the separation of the water phase) (i.e., it possesses the property of thixotropy). Thus, these two emulsion types display different dynamics in shear flows. The dependence of the shear-strain rate γ_ obtained for the shear stress τ (Figure 6) is sufficiently well approximated by the model of a power-law fluid τ = Kγ_ n, where n is the nonNewtonian behavior index and K is the consistency index of the fluid.11 In accordance with the obtained values of K and n for the forward and return strokes, the time dependencies were constructed from the fluid volume that had passed through the capillary. These dependencies are presented as straight lines in Figure 6. 1177
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Energy & Fuels The results of this experiment allow for the determination of the parameters of the approximation function τ = Kγ_ n. For the stabilized emulsion, K = 0.115 and n = 1.02 on the forward stroke and K = 1.53 and n = 0.668 on the return stroke. For the nonstabilized emulsion, K = 2.42 and n = 0.664 on the forward stroke and K = 0.45 and n = 0.83 on the return stroke.
’ DISCUSSION The emulsion flow in capillary structures and microchannels of various configurations reveals a remarkable, quite unexpected effect called dynamic blocking, whereby the emulsion flow through a microchannel ceases with time despite the presence of a continuous pressure gradient. The term “dynamic blocking” reflects the fact that, despite an apparent flow decay, the flow can nevertheless be observed on a microscopic scale at a much lower (by 3-4 orders of magnitude) flow rate and with a significantly modified microflow structure. It should be noted that, in all cases of the dynamic blocking effect, the size of water microdrops was significantly (by more than an order of magnitude) smaller than the characteristic transverse size of the channel. The effect of dynamic blocking was discovered during the flow of concentrated inverse water-in-oil emulsions in a flat microchannel representing a Hele-Shaw cell with lateral dimensions of 4 2 cm2 and a 17 μm wide gap between thick optical glass plates. During the emulsion flow in this flat microchannel, the dynamic blocking effect was most clearly manifested (Figures 2 and 3). The experiments were performed with stabilized emulsions with rheopexic properties and non-stabilized emulsions, which are thixotropic. In both cases, the concentration of the aqueous phase was 75%, and the water microdrop size was 0.3-1 μm in the rheopexic emulsion and 1-2 μm in the thixotropic emulsion. The flow of the stabilized emulsion was accompanied by the formation of flow tubes, followed by the chaotization of the flow structure and the formation of isobaric lines with subsequent flow blocking (Figure 3b). In the case of a nonstabilized emulsion, the aqueous phase was intensively separated as a result of the coalescence of water microdrops, after which the separated aqueous phase acquired a rivulet structure and the flow was blocked (Figure 3a). Questions naturally arise as to how the dynamic-blocking manifestations depend upon the gap width and the pressure difference and which factor accounts for the flow blocking (including the role of the capillary input and cell volume). To check for a universal character of the observed phenomenon and to vary the main parameters, we studied the flow of emulsion in the simpler system of an axisymmetric cylindrical channel and then proceeded to a complicated system of microchannels. Dynamic blocking was also observed in cylindrical capillaries. However, there are certain limits for the manifestation of the blocking effect depending upon the capillary diameter. For example, no dynamic blocking took place in a capillary with a diameter of 600 μm, while the capillaries with diameters of 250, 100, and 40 μm exhibited this effect. A detailed examination of the flow structure along the entire channel length under a microscope showed that, simultaneously with a decrease in the flow velocity, water drops appeared in the channel along its entire length. The dimensions of these drops are comparable to the capillary diameter. The flow velocity decreases by 2-3 orders of magnitude, as compared to the initial value. The moving becomes creep, piston. The flow velocity keeps decreasing further until it drops by another 1-2 orders of magnitude but
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very slowly by 1 or 2 days. At this time, the emulsion is subjected to partial degradation with the separation of the aqueous and carrier (oil) phases in the microchannel and the flow velocity becomes negligibly small (0.2 μm/s). One could try to explain the observed behavior by the effect of surface forces using an analogy with foams. A 4 cm long capillary accommodates approximately 200 menisci between water drops. There is no difference between the angles at the oncoming and trailing fronts, a difference of which could determine the resistance to the drop motion. Nevertheless, even a rough estimation from the above using the Laplace formula with the surface tension at the boundary of water and the carrier phase assumed to be 5 mN/ m, the total pressure difference amounts to 40 kPa, which is an order of magnitude smaller than the value for dynamic blocking. The case of a complicated capillary structure was studied using a system of microchannels that model the pattern of pores in a polished section of an oil-bearing test core as prepared by the selective-etching technique. This structure is conventionally referred to as a micromodel. The emulsion flow in the adopted micromodel at a constant pressure difference also exhibits dynamic blocking. The emulsion supply to the micromodel leads to some phase separation. The viscosity of a carrier phase (oil) is significantly smaller to that of the emulsion. At large pressure gradients, a certain fraction of oil is filtered between water microdrops and outflows first. In the emulsion passing via channels of complicated configuration, water microdrops exhibit coalescence and an aqueous phase (water) separates and flows at a velocity that is significantly greater than the velocity of the emulsion proper. During this occurrence, the separated water drops continue to grow and merge. An analogous pattern is observed in a separate capillary, but there it takes place upon the onset of blocking. Numerous experiments with mechanical action upon the capillary input zone have shown that the main factor in dynamic blocking is a structure at the capillary input, which is formed in the tube supplying emulsion to the capillary and has a length not exceeding 500 μm. The formation of this structure is probably related to the action of friction (drag) forces between water microdrops, with 3 nm long surfactant molecules occurring on the water and oriented perpendicular to its surface. With these shells of surfactant molecules, water microdrops resemble hedgehogs. The physical nature of the dynamic blocking effect is related to the interaction between nanodimensional shells consisting of surfactant molecules that surround water microdrops and is expressed by the friction between shells and the deformation of microdrops. A microdrop structure that is formed as a result of these interactions near the microchannel input zone begins to slow the flow, which leads to an increase of the pressure gradient in this zone. This event results in the deformation of microdrops, the passage from their contact point to contact surfaces, and the transformation of spherical contact structures into polyhedra forming a low-permeable framework. Thus, in addition to the traditional hydrodynamic properties of liquid-liquid dispersions, their flow in microchannels is also characterized by the effect of dynamic blocking that is manifested in a broad range of pressure gradients (from 1 MPa/m to 3 GPa/m). The dynamic blocking mechanism caused by the friction arising between water droplets while squeezing an interphase fluid is sufficiently described elsewhere.7-9 The friction causes nonlinear and unbalanced effects in two-phase flow. The dynamic blocking effect was observed when filtering the water emulsion at a constant pressure gradient. The pressure-gradient 1178
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Energy & Fuels variations are observed in the case of the two-phase flow of a nonstabilized water-in-oil emulsion in a porous medium at a constant flow rate. We assume that the pressure gradient variations for two-phase filtration are caused by the following: the non-Newtonian properties of the water-in-oil emulsion, the effect of flow blocking, and the lag time (relaxation time) in the process of structural transformations in two-phase systems. Experiments show that, when oil is displaced by water from a porous medium, the displaced phase moves not as an independent phase but is segregated by microflows of a displacing phase, replacing water in some channels after it pushes the oil out. In addition, all oils contain surfactants, and oil displacement from the porous medium with aqueous solutions is accompanied by both the dispersion and coalescence of droplets of both liquids, i.e., with the formation of a water-in-oil emulsion. The dispersion of oil, contacting water in the porous medium, takes place almost instantly.12 Thus, in the pore space of real reservoirs, we have a flow of the non-stabilized water-in-oil emulsion.12 As mentioned above, the non-stabilized emulsions are considered to be non-Newtonian fluids. The anomaly of the emulsion viscosity is caused by the deformation of dispersed particles while increasing the applied stress. The emulsions in the porous medium are characterized by thixotropic properties. As a reminder, a thixotropic environment is an environment with unsteady rheological properties, in which the structure is gradually destroyed under deformation at a constant shear rate, leading to a decrease in the effective viscosity with time. The characteristic time of structural reconstruction is comparable to the characteristic time of stress changes. This finding leads to the fact that the relation between stress and strain becomes non-equilibrium. It takes some time to establish an equilibrium stress, corresponding to a specific shear-rate value. The emulsion can be simplistically described as a suspension of particles whose size is comparable to the pore size. Particles move together with the liquid through various channels, being exposed to deformation and friction. This phenomenon leads to the dynamic blocking of pore channels. Because the flow in the porous medium takes place at a constant flow rate, the pressure begins to rise, leading to the destruction of the structure and a decrease of the pressure. After the relaxation time is reached, the fluid returns to its original state and begins to flow and the process repeats. This qualitative picture together with general observations on non-equilibrium effects in porous media13-15 allow for the proposal of a simple qualitative theory on the dynamics of pressure behavior. The relationship of pressure gradient versus time obtained during laboratory investigations of samples of a porous medium are significantly different from the theoretical curves calculated using the Muskat-Leverett16 models of two-phase seepage. The current macroscopic theories of two-phase flow in porous media are based on the extended Darcy’s law and an algebraic relationship between the capillary pressure and saturation. Both of these equations have been challenged in recent years, primarily on the basis of theoretical works using a thermodynamic approach, which have led to new governing equations for two-phase flow in porous media. Many theoretical and experimental studies have shown that this assumption is invalid and that the pressure difference between the two fluids is not only equal to the capillary pressure but is also related to the variation of saturation with time in the domain. This phenomenon is referred to as the nonequilibrium capillarity effect.13-15,17-19
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The authors have shown that the extended model can properly capture physical processes, such as capillary hysteresis. The results illustrate that the non-equilibrium capillarity coefficient (the relaxation time) is a function of saturation and the viscosity ratio. However, much remains to be done regarding finding ways of studying various terms in these equations and gaining insight into the interplay of various effects. Our model considers the non-equilibrium effects of two-phase filtration by the introduction of non-equilibrium RPPs. When a two-phase mixture flows through a porous medium at a constant flow rate, pressure-gradient variations are observed. It is shown below that these oscillations are associated with nonequilibrium and nonlinear effects. Non-equilibrium effects as a result of processes involving the redistribution of the phases in pores have been considered. In this paper, we proposed to consider the non-equilibrium effects associated with a change in the rheological properties in emulsified media. Here, the twophase fluid is in the form of an emulsion, the particles of which possess viscoelastic properties. During the motion through the porous channels, which is accompanied by particle deformation, there is a change in the seepage resistance of the flow as a result of the restructuring of the emulsion with a relaxation time that is characteristic for the given system. When these concepts are taken into account, a mathematical model of the process in which oil is displaced from a sample of a porous medium by water is then considered. Laboratory experiments show that, when the pressure gradient is increased, the resistance to seepage in the flow of an emulsion is reduced and increases when the pressure gradient is decreased. In this sense, the rheological behavior of an emulsion is of a nonNewtonian nature (the effective viscosity decreases with increasing applied stress). The reasons for this behavior are associated with structural rearrangements in emulsions. However, not all of its special features have been fully investigated. We will therefore confine ourselves to a phenomenological approach to the description of these phenomena and the specification of functions of the RPPs of water and oil, accounting for their nonequilibrium nature. We will write the kinetic equations for the non-equilibrium RPP of water (the displacing phase), n1(s), and oil (the displaced phase), n2(s), in the form ∂ni ðsÞ ¼ ki ðsÞψðqÞ ni ðsÞ þ τ1 ∂t ∂p q ¼ i ¼ 1, 2 ∂x
ð1Þ
where τ1 is the restructuring time, k1(s) and k2(s) are the equilibrium RPPs of water and oil, respectively, s is the saturation of the displacing phase, and p is the pressure. To simplify the analysis, we neglect the jump in capillary pressure for all phases. The introduction of the function ψ(q) enables the description of the non-Newtonian properties of the water-in-oil emulsion. The function ψ(q) is defined as ψ = f(q, sgn((∂q)/(∂t))) ( 1 - expð - bqÞ, ∂p=∂t > 0 ψ¼ 0:01½expðbqÞ - 1, ∂p=∂t < 0 where b is a parameter that affects the curve shape. This parameter is selected to obtain the function 0 < ψ < 1. A possible presentation of this function is shown in Figure 7. 1179
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Figure 7. Graphical presentation of the function describing the nonNewtonian properties of the emulsion.
The values of this function for the same absolute values of the pressure gradient, determined for a decreasing and an increasing pressure gradient q, are not the same. That is, the phase permeabilities exhibit a hysteresis (Figure 7). The equations of motion and continuity for the displacing and displaced phases are written in the form VB1 ¼ -
kn1 grad p, μ1
VB2 þ τ2
∂ VB2 kn2 ¼ grad p ð2Þ ∂t μ2
∂p ∂s þ m þ div VB1 ¼ 0, ∂t ∂t ∂s ∂p ð1 - sÞβ2 - m þ div VB2 ¼ 0 ∂t ∂t
sβ1
ð3Þ
where μ1 and μ2 are the viscosities of the displacing and displaced B2 are the seepage rates of the phases, τ2 is the phases, V B1 and V relaxation time, β1* and β2* are the coefficients of elastic compliance of the phases, k is the absolute permeability of the porous medium, and m is the porosity. For generality, the equation of motion of the displaced phase is taken in the relaxation form.20 This consideration is particularly necessary for short models, in which the process is determined not by transmissibility but by the restructuring time. Equations 1-3 are used to describe the process of the immiscible displacement from a sample of a porous medium where a constant flow rate of the displacing fluid is maintained in the inlet cross-section and the initial pressure in the model of the porous medium is maintained at the outlet. The initial and boundary conditions, which close the system of equations, therefore have the form n1 ðx, 0Þ ¼ 0, n2 ðx, 0Þ ¼ F2 - const, sðx, 0Þ ¼ sc , sð0, tÞ ¼ sT , V ð0, tÞ ¼ V0 , pðL, tÞ ¼ p0
pðx, 0Þ ¼ p0 ,
where sc and sT are the initial and final (limiting) sat s(x, 0) = sc, saturation of the displacing phase, F2 is the value of the RPP of the displacing fluid, p0 is the initial pressure in the model of the stratum when s = sc, V = V1 þ V2 is the overall seepage rate, V0 is the seepage rate in the inlet cross-section of the model of a porous medium, and L is the length of the model. For a numerical solution, the initial equations were written in dimensionless variables x p qL Vi x ¼ , p ¼ , q ¼ , Vi ¼ , L p0 p0 V0 V0 t V0 τ1 V0 τ 2 , τ1 ¼ , τ2 ¼ t ¼ mL mL mL
V ¼
V , V0
In the text below, we do not use dashes. The system of eqs 1-3 was solved numerically to investigate the qualitative features of the proposed model of two-phase
Figure 8. Experimental and calculated curves of pressure drop versus time.
seepage. Equations 1 and 2 were approximated by special finitedifference schemes, with an exponential adjustment that accounts for the existence of small parameters accompanying the higher derivatives.21 The equations were integrated numerically using the well-known implicit-pressure explicit-saturation scheme. To ensure the stability and necessary accuracy of the difference schemes, a spatial step of the variable h = 0.01 and a time step l = 0.0001 were selected. The numerical simulation [of oil displacement by water using the reservoir (core) model] was applied for the BS1-5 zone under the conditions for the Prirazlomnyi reservoir (L = 0.1647 m; m = 0.17; k = 0.021 pm; F2 = 0.58; sc = 0.285; sT = 0.718; V0 = 232.32 m/year; and μm2; μ0 = μ1/ μ2 = 0.27). The experimental results allow for the determination of the relative permeabilities as the following functions: s - sc 5=2 sT - s 3=2 ; k2 ðsÞ ¼ 0:58 k1 ðsÞ ¼ 0:138 sT - sc sT - sc These functions were used in the numerical calculations. The results of the experiments with reservoir fluids and natural rocks demonstrate that the values of principal parameters characterizing the displacement process of immiscible liquids at various stages are controlled by not only the phase properties but also the degree of interaction between the displaced and displacing fluids between the fluids and the rock, the pore space geometry, and the construction of the walls of pore channels. Processes in a complex system, such as a porous medium saturated by residual (or injected) water and oil, being a multicomponent system and most often in a colloidal state, can take place with characteristic relaxation times, controlled by the structure, composition, and condition of a specific system.17-19 In our case, the relaxation times τ1 and τ2 are defined from the functional minimization criteria I = ∑i(Δpiexp - Δpicalc)2 f min, where Δpiexp and Δpicalc are the experimental and calculated pressure gradients, respectively. The functional minimum is achieved at τ1 = 0.025 and τ2 = 0.1. This level of dimensionless time corresponds to 3 103 s. The pressure drop, Δp(t) = p(0, t) - p(1, t) = p(0, t) - p0, as a function of time t is represented by the solid curve in Figure 8. The experimental dependence of the pressure drop on time is shown by the dashed curve. The result of the simulation is shown by a solid line. It can be seen that pressure drop variations take place in a system with nonlinear properties. The elastic properties of the stratum system amplify the pulsation in Δp(t), which 1180
dx.doi.org/10.1021/ef101053k |Energy Fuels 2011, 25, 1173–1181
Energy & Fuels are observed until the saturation throughout the whole length of the model attains a limit value of sT. When τ1 = 0, there are no variations in Δp. The same effect is observed when ψ(q) = 1; i.e., there is no hysteresis in the flow character for non-Newtonian media. This result means that the existence of a lag in the structural rearrangement processes of rheologically complex media leads to the occurrence of pressure drop variations. Calculations were carried out for various values of τ1, τ2, μ0, k, and V0. The variation of these quantities leads to a change in the oscillation amplitude and frequency. The described phenomenological model enables the interpretation of the pressure-gradient variations observed in experiments on the displacement of oil by water from a sample of a porous medium. The model describes the nonlinear properties of non-stabilized water-in-oil emulsions and allows for consideration of the relaxation time when determining the permeabilities for immiscible displacement processes.
’ SUMMARY It was established that a water-in-oil emulsion, depending upon the presence of an artificial stabilizing emulsifier, is described by various structural transformations of hydrodynamic flows. The flow of non-stabilized water-in-oil emulsions is accompanied by emulsion restructuring in all investigated flow types. The emulsion stabilized with an artificial emulsifier is dynamically stable. In the case of planar flow at high shear-deformation rates, negligible restructuring is observed. Dependent upon the presence of a stabilizing emulsifier, the water-in-oil emulsion may belong to different types of media: rheopectic (with the emulsifier) or thixotropic (without the emulsifier). Both of these emulsion types have a common element: when they flow in reservoir models, they experience the dynamic blocking. The physical nature of emulsion dynamic blocking is explained by friction among microdrops and their deformation as the pressure gradient increases. When visualization techniques were employed in the viscometer and in the experiments involving the elements of an oilbearing reservoir, such as a fracture, porous structure, and capillary, the authors explored the mechanism of an inverted system transformation in these elements and the new effect of dynamic blocking, which forms a basis for flow-diverting technologies. Stabilized water-in-oil emulsions can be used as waterproofing compositions, enabling one to efficiently control the water flows in well production without changing the structure of the pore space in the near-wellbore zone.
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’ REFERENCES (1) Sj€oblom, J. Emulsions and Emulsion Stability, 2nd ed.; CRC Press: New York, 2006. (2) Orlov, G. A.; Kendis, M. S.; Glushchenko, V. N. Use of Invert Emulsions in Oil Production; Nedra: Moscow, Russia, 1991. (3) Romero, C.; Bazin, B.; Zaitoun, A. SPE Prod. Oper. 2007, 22 (2), 191–201. (4) Hassanizadeh, S. M.; Gray, W. G. Water Resour. Res. 1993, 29 (10), 3389–3405. (5) Avraam, D. G.; Payatakes, A. C. Ind. Eng. Chem. Res. 1999, 38 (3), 778–786. (6) Avraam, D. G.; Payatakes, A. C. J. Fluid Mech. 1995, 293, 207–236. (7) Denkov, N. D.; Tcholakova, S.; Golemanov, K.; Ananthapadmanabhan, K. P.; Lips, A. Phys. Rev. Lett. 2008, 100, No. 138301. (8) Denkov, N. D.; Tcholakova, S.; Golemanov, K.; Lips, A. Phys. Rev. Lett. 2009, 102, No. 118302. (9) Denkov, N. D.; Tcholakova, S.; Golemanov, K.; Ananthapadmanabhan, K. P.; Lips, A. Soft Matter 2009, 5, 3389–3408. (10) Schramm, G. A. Practical Approach to Rheology and Rheometry; Cebrueder Haake GmbH: Karlsruhe, Germany, 1994. (11) Davis, J. A.; Jones, S. C. J. Pet. Technol. 1968, 2, 1413. (12) Rebinder, P. A. Selected Works. Surface Phenomena in Dispersed Systems. Physical-Chemical Mehanics; Nauka: Moscow, Russia, 1979. (13) Barenblatt, G. I.; Patzek, T. W. SPE J. 2003, 8 (4), 409–416. (14) Joekar-Niasar, V.; Hassanizadeh, S. M.; Dahle, H. K. J. Fluid Mech. 2010, 655, 38–71. (15) Joekar-Niasar, V.; Majid Hassanizadeh, S. Int. J. Multiphase Flow 2011, 37, 198–214. (16) Muskat, M.; Meres, M. W. Physics 1936, 7, 346–363. (17) Aker, E. K.; Maloy, K. J.; Hansen, A. Phys. Rev. B: Condens. Matter Mater. Phys. 1998, 58, 2217–2226. (18) Bravo, M. C.; Araujo, M.; Lago, M. E. Phys. A 2007, 375, 1–17. (19) Dahle, H.; Celia, M.; Hassanizadeh, S. M. Transp. Porous Media 2005, 58 (1-2), 5–22. (20) Barenblatt, G. I.; Mamedov, Yu. G.; Mirzadjanzadeh, A. Kh.; Shvetsov, I. A. Fluid Dyn. 1973, 8 (5), 742–748. (21) Doolan, E. P.; Miller, J. J.; Schilders, W. H. A Uniform Numerical Methods for Problems with Initial and Boundary Layers; Boole Press: Dublin, Ireland, 1980.
’ NOTE ADDED AFTER ASAP PUBLICATION The Abstract in the article published February 15, 2011 has been revised. In addition, the units for k for the Prirazlomnyi reservoir were revised in the Discussion section. The correct version was reposted March 3, 2011.
’ AUTHOR INFORMATION Corresponding Author
*Fax: þ7-347-272-81-69. E-mail:
[email protected].
’ ACKNOWLEDGMENT This study was supported by the Ministry of Education and Science of the Russian Federation, the Presidential Program for Support of Leading Scientific Schools in Russia (Projects 4381.2010.1 and 65497.2010.9), and the Russian Foundation for Basic Research (Project 08_01_97032_r_povolzh’e.a). 1181
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