Simulations reveal mechanisms of seismic waves for EOR

The pore-scale effects of seismic stimulation on two-phase flow are numerically modeled in random two-dimensional grain-pack geometries.  

Steven R. Pride, Lawrence Berkeley National Laboratory; Eirik G. Flekkøy and Olav Aursjø, University of Oslo 

Seismic stimulation aims to enhance oil production by sending seismic waves across a reservoir to liberate immobile patches of oil. For seismic amplitudes above a well-defined (analytically expressed) dimensionless criterion, the force perturbation associated with the waves can indeed liberate oil trapped on capillary barriers and get it flowing again under the background pressure gradient. Subsequent coalescence of the freed oil droplets acts to further enhance oil movement because longer bubbles more efficiently overcome capillary barriers than do shorter bubbles. Poroelasticity theory defines the effective force that a seismic wave adds to the background fluid-pressure gradient, while the lattice-Boltzmann model in two dimensions is used to perform the pore-scale numerical simulations. The dimensionless numbers (groups of material and force parameters) involved in seismic stimulation are carefully defined so that the numerical simulations can be applied to field-scale conditions. Using the analytical criteria defined by the authors,* there is a significant range of reservoir conditions over which seismic stimulation can be expected to enhance oil production.

The objective of this article is to simulate numerically the effect of a passing seismic wave on the pore-scale two-phase flow. We use the lattice-Boltzmann model to perform the simulations, and poroelasticity theory to define the effective force that the seismic wave adds to the background fluid-pressure gradient. The simulations are presently done in 2D on cells that are typically 10 x 10 grains in size; however, extension to 3D and to larger systems is straightforward. Over a significant parameter range, stimulation is observed to mobilize oil by causing coalescence of smaller droplets into bigger ones that flow more easily.

POROELASTICITY OF SEISMIC STIMULATION

We imagine an oil reservoir being produced through the use of injection and extraction wells. The oil production in the field, through time, has fallen to low levels; water is flowing, but much of the oil became effectively trapped when the waterflood passed through. Such remaining oil patches (so-called “ganglia”) may be present at economically significant volume fractions; e.g., 30% or more of the pore space in a reservoir might commonly be occupied by such trapped oil. We focus on some arbitrarily chosen small region within the reservoir that we call a “simulation” or “flow cell.” The distribution of wells in the field is creating a net fluid pressure gradient F_o across this flow cell; however, any further details of the production wells are not required in what follows.

An oil ganglion of downstream length h (typically much larger than grain sizes) becomes stuck when the downstream pressure drop hFo along the bubble is just balanced by a capillary pressure increase where  is the oil/water surface tension, and where R_down and R_up denote the radii of curvature of the furthest downstream and furthest upstream meniscii that bound the stuck bubble of oil. Throughout this article, water is taken as wetting the solid grains and oil as non-wetting. For the oil bubble to be trapped, either the downstream meniscus has a radius of curvature (multiplied by the cosine of the contact angle) that is greater than the radius of the pore-throat constriction through which it is trying to pass, or the upstream meniscus has a radius of curvature (multiplied by the cosine of the contact angle) smaller than the main pore into which it is trying to enter. The goal of the seismic stimulation is to displace these meniscii enough that they may pass through their respective capillary barriers, and begin flowing again under the background force F_o. Subsequent linking up with other bubbles to make longer and more efficiently transported bubbles is shown by means of numerical simulations to be a very important effect.*

As shown in Fig. 1, there is assumed to be a seismic source in a stimulation well located a distance r from a given flow cell under study. By perturbing the fluid pressure in this borehole over a certain depth range, we aim to perturb the flow in the distant cell with the goal of enhancing the oil flux across that cell. Fluid exchanges between the stimulation well and the surrounding reservoir play no important role in enhancing the oil flow. It is the seismic waves that are overwhelmingly responsible for any stimulation effect. Any oil ganglia that are marginally trapped (on the verge of moving by F_o alone) are susceptible to seismic stimulation.

Fig. 1. A fluid-pressure perturbation ∆P is applied over a certain depth range of a stimulation well with the goal of stimulating flow at points in the reservoir a distance r away. The seismic waves are able to deliver significant perturbation to a flow cell; however, the diffusional penetration from the borehole is negligible.

PARAMETERIZATION AND PROTOCOL

In all the present simulations, the background force F_o is applied from left to right (the flow direction), while both the top and bottom and left and right flow boundaries are periodic.

Some characteristic snapshots corresponding to the various stages of the simulation protocol used are given in Fig. 2, while the associated average Darcy velocity of the oil during each stage is shown in Fig. 3. The porous material in this demonstration example has porosity  = 0.5 and permeability k = 0.5 (in lattice-Boltzmann units). Furthermore, the fraction of pore space occupied by oil is 50% (with water in the other 50%).

Fig. 2. Snapshots during the four stages of the seismic-stimulation simulation, with fluid flow from left to right. Solid discs are gray, oil is light gray and the water is black. a) The initial phase separation occurs. b) Background force Fo has caused the droplets to get stuck, and there is no flow of oil. c) The stimulation is turned on (Fa > 0), and the droplets coalesce. d) A new steady state emerges as Fa = 0. There is now a flow through the system-spanning droplets.

Fig. 3. The spatially averaged Darcy velocity in the system as a function of simulation time (in units of the time step) corresponding to the geometry and flow in Fig. 2. Figure 2’s distinct time intervals (a–d) are separated by the vertical dashed lines. The thin curve is the instantaneous velocity, while the thick curve is a running average over a time window of ∆t = 1,000. In all graphs, units are lattice-Boltzmann units.

The simulation protocol is the same for each production run presented in this article and is broken into the four time intervals denoted in both Figs. 2 and 3 and defined as follows:

a. The oil and water are allowed to spontaneously separate from an initial homogeneous distribution with all applied forcing set to zero.

b. The background force F_o is then applied uniformly to both oil and water, initially causing some flow but often resulting (depending on both the value of F_o and the oil-volume fraction) in the oil becoming trapped on capillary barriers with an associated large decrease in the average Darcy velocity in the system.

c. The seismic stimulation is then applied, such that F_a (the amplitude of any time-harmonic force created by the fluid-pressure perturbations applied in the stimulation borehole) is greater than zero.

d. The seismic stimulation is turned off with a new steady flow state emerging under the influence of F_o alone.

Only three seismic-wave periods of stimulation (corresponding to roughly 0.1 s of stimulation in the field) are ever applied in our simulations. If, as in field applications, the stimulation is applied for many millions of wave periods, the effect on the total volume of produced oil will necessarily increase beyond what we have determined here.

The main pore-scale effect of applying the seismic stimulation is to mobilize the stuck oil droplets, which allows them to coalesce and form longer bubbles. Longer bubbles have a greater applied pressure drop along them, which allows them to more easily overcome the capillary barriers they encounter. In Fig. 3, it is seen that the oil was stuck (barely moving) prior to the application of the stimulation. In the final steady state that emerges once the stimulation is turned off, there is observed a system-spanning steady stream of oil created by the wave-induced coalescence of the oil droplets.

Stimulation can easily create such system-spanning streams when the oil occupies roughly 40–60% of the pore space. As the oil volume fraction increases, there becomes enough oil that the background force alone can create a system-spanning stream and the oil never becomes trapped. As the oil volume fraction decreases, the likelihood of creating such a single system-spanning stream, even with stimulation-induced mobilization and coalescence, is progressively reduced. In order to keep the oil droplets moving at such lower oil volume fractions, it is necessary to repeatedly apply the stimulation. In real, 3D, non-periodic systems, even if a system-spanning oil stream develops, it will potentially break up due to the Rayleigh instability, which is the 3D effect responsible for the pinching off of droplets at a slowly dripping faucet. The smaller bubbles so created will become trapped again on capillary barriers, in which case repeated stimulation will again be required.

RESULTS FOR SIX MATERIALS

We next focus on the oil production occurring in six porous materials, Fig. 4. The permeabilities and porosities of these six materials are shown in the legend of Fig. 5 (and subsequent figures). In all the examples presented here, the oil occupies one-third of the pore space and water occupies the other two-thirds. We conservatively chose this oil-volume fraction so that system-spanning oil streams were not created.

Fig. 4. The six different porous media studied at N = 128 resolution. Light gray = solid grains; black = wetting water; dark grey = non-wetting oil; white = boundary points surrounding the oil patches. All snapshots here are taken just after oil-water separation has occurred and before macroscopic forcing. The porosities and permeabilities (lattice-Boltzmann units) for the six materials shown are given in the legend in Fig. 7.

Fig. 5. The specific oil volume Voil as a function of the dimensionless stimulation number S (the ratio of Fo to the capillary pressure gradient opposing oil movement) for the six porous media. When S > 1, oil bubbles tend to move by Fo alone; when S < 1, they tend to be stuck.

For each of the six different porous materials studied, Fig. 5 shows the corresponding total oil production during an entire production run (the entire 18,000-time-step simulation) when no seismic stimulation is applied. The plotted quantity is the specific oil volume that we define as:

where j_oil is the local volume flux of oil (from left to right),  is the spatial average of this flux over the entire simulation (flow) cell, and the time integral runs over the entire simulation time of 18,000 time steps. In steady state, the specific oil volume V_oil multiplied by the height of the simulation cell will converge to the volume of oil that crosses a surface perpendicular to the flow (the produced oil volume).

Figure 5 shows that for all six media studied, the dimensionless “stimulation number” S ≈ 1 corresponds to the threshold background force F_o above which the oil flows by the background force alone without becoming trapped. The roughly linear dependence V_oil ~ S in Fig. 5 once S > 1 is simply a statement of the emergent Darcy’s law. In order to determine S at each level of applied force F_o, it is necessary to numerically measure the average length h of the oil ganglia found in each system.

The same experiments are next performed with the inclusion of three cycles of seismic stimulation between time steps 9,000 and 12,000 with F_a / F_o = 1. The result is shown in Fig. 6 for two of the six materials; however, similar results hold for all six. The stimulation is seen to enhance the total oil produced during the run even though the seismically coalesced bubbles are not able to form system-spanning streams. If stimulation was applied for more than just three time-harmonic cycles, the enhancement would have been greater.

Fig. 6. Voil that results both with three time periods of stimulation applied (open symbols) and without any stimulation applied (solid symbols) for two of the six materials. Similar results hold for all six materials studied.

A better way to see the effect of stimulation on the oil production is to monitor (as in Fig. 3) the average Darcy flow throughout the system  as a function of time. Note that  corresponds to the average flux of oil volume across each vertical slice of a system and is thus equivalent to the average rate of oil production throughout the system. Figure 7 shows  for the six systems at each time step of a production run with an additional 1,000-time-step running average applied in order to see the net effect of each stimulation cycle. The average production rate histories without stimulation applied are shown with solid symbols, and those with the three cycles of stimulation are shown with open symbols. Although the oil production drops back to almost zero after stimulation is turned off, the net oil production is greatly enhanced during stimulation. Again, when the oil does not form system-spanning streams (as in this example), stimulation must be applied continuously to maintain enhanced production.

Fig. 7. The average oil flux joil in each of the six systems with an additional 1,000-time-step running average applied over the entire production run. Solid symbols are with no stimulation applied. Open symbols are when three cycles of stimulation are applied between time steps 9,000 and 12,000 with Fa = Fo. Significant enhancement in the oil production rate is observed for all six materials during stimulation.

Mobilization and coalescence of trapped bubbles may only occur if the oil bubbles are moved a significant distance in a wave period compared to the pore length. To quantify this condition, we carry out simulations with different ratios (T) of the pore-length scale ℓ to the fluid displacement due to the acoustic oscillations. This suggests that when T > 1, stimulation ceases to have an effect. In Fig. 8, total produced oil is plotted as a function of T, and indeed a critical T is observed around the value of 1.

Fig. 8. Voil as a function of the dimensionless frequency T when S ≈ 1⁄2. At the lower frequencies corresponding to T < 1, there enhanced oil production is seen because the stimulation has enough time in each cycle to push the trapped meniscii through their bounding pore throats.

The finite oil production for T > 1 is due both to the initial displacement that takes place before stimulation is applied (Figs. 3a and 3b) and to the fact that not all oil becomes completely trapped in the steady state prior to stimulation.

In Fig. 9, the specific volume  is plotted as a function of F_a / F_o. In these simulations F_o was kept constant at the value indicated in the figure, while F_a was increased. For all six materials, the value of 1/S − 1 was between ½ and 1. Thus, the stimulation criterion of F_a / F_o > 1/S − 1 is seen to be at least roughly satisfied. This is perhaps more clearly seen in Fig. 10, in which the simulation cell’s spatially averaged oil flux  is time-averaged over the three cycles of the applied stimulation and plotted as a function of F_a / F_o. If the bubbles remain trapped, they do not contribute to this averaged velocity, so the steady increase in the average oil velocity with increasing F_a / F_o is a demonstration that the stimulation has a strong mobilization effect while active. Again, there is a finite oil velocity even when F_a= 0 because not all of the oil is completely trapped at the level of background force F_o used.

Fig. 9. The total volume displacement of oil as a function of Fa / Fo when 0 < 1/S − 1 < 1 for each of the six materials.

Fig. 10. The oil velocity averaged over the three cycles of applied stimulation when T = ¼ and with Fo fixed to the values in the legend. In all six cases, 0 < 1/S − 1 < 1, so these results are consistent with the stimulation criterion that the oil becomes mobilized when Fa / Fo > 1/S − 1. If trapped oil was not liberated in each cycle, the average velocity would not increase with increasing values of Fa.

CONCLUSIONS

The numerical simulations performed in this study strongly support the conclusion that seismic stimulation will mobilize trapped oil, thus increasing oil production, when certain criteria are met, among them:

• The seismic wave has to push on the oil bubble with enough force that the downstream meniscus has its radius of curvature reduced to the point that it can get through the pore throat constriction that is blocking its downstream progress.

• In a cycle of the time-harmonic stimulation, the meniscus has enough time to advance through the constriction before the seismic force changes direction and begins to push the meniscus upstream.

These two conditions may be achieved by using sufficiently large stimulation amplitudes and sufficiently small stimulation frequencies. Interestingly, when the stimulation force is modeled as that due to a seismic wave, and if seismic strain is independent of frequency, we predict that Fa increases linearly with frequency, while T becomes independent of frequency. Of course, the imposed seismic strain from different seismic sources working at different frequencies need not be the same. Furthermore, in the field, the strain of a wave is always reduced at higher frequencies by seismic attenuation. So there are many practical tradeoffs to consider when choosing which source to work with and at what frequency.

Further numerical studies should be performed that 1) take the simulations from two dimensions to three; 2) work with more grains in each flow cell; 3) apply the stimulation for many hundreds (or more) of seismic wave periods; and 4) use alternative ways to simulate the way the oil becomes trapped in the system in the first place. As an example of this last point, one may wish to start with a large volume fraction of oil in the system, and perform an imbibition experiment until the advancing water front forms a percolating backbone. The oil that remains may be only marginally stuck and thus more susceptible to the effects of seismic stimulation.

There is no intrinsic difficulty in performing the lattice-Boltzmann simulations in 3D. It is mainly a matter of applying a larger computational effort. We conjecture that 3D simulations will reveal that the greater number of junctions and oil branchings that can occur in three dimensions will result in a larger number of marginally stuck fingers of oil. On the other hand, the Rayleigh instability (an effect confined to 3D) will tend to break up larger ganglia into smaller ones, in particular when the solid is wetted by the water as in our simulations. In this case, the stimulation will need to be applied repeatedly in order to continuously remobilize and coalesce the oil droplets. Lastly, in 3D the porosities can be reduced to those in real rocks because the pore space remains connected across a sample even as porosity approaches zero.   

  *  Analytical criteria are defined in Pride, S. R., Flekkøy, E. G. and O. Aursjø, “Seismic stimulation for enhanced oil recovery,” Geophysics, 73. No. 5, September–October 2008. This article excerpts that paper with permission from the Society of Exploration Geophysicists. Find the original article at http://dx.doi.org/10.1190/1.2968090. 

THE AUTHORS
	Steve Pride obtained his doctoral degree in geophysics from Texas A&M University in 1991.  After two years as a post-doctoral student at MIT, he accepted a job as a professor at the University of Paris in France in 1993, then moved to the University of Rennes in 1998. In 2003, he took joint positions as a staff scientist at the Lawrence Berkeley National Laboratory and an adjunct professor at the University of California at Berkeley. In addition to his research and teaching, he also owns and runs a winery in the Napa Valley.
	Eirik G. Flekkoy received his PhD degree in physics at the University of Oslo in 1993. He has worked as a research scientist in physics and geophysics at the Massachusetts Institute of Technology, the City of Paris High School of Industrial Physics and Chemistry (ESPCI) and the University of Honolulu. He became a full professor at the University of Oslo in 2001. Presently he works part time in the small Norwegian oil company PetroMarker, where he develops methods for electromagnetic prospecting.
	Olav Aursjø has a master’s degree in theoretical particle physics and general relativity. He is currently working on his PhD in physics at the University of Oslo, where Dr. Eirik Grude Flekkøy is his supervisor. The current focus of his research is on theoretical study and simulation of two-phase flow in porous media using a lattice-Boltzmann method for immiscible fluids. He can be contacted at olav.aursjo@fys.uio.no.