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  What Is Relative Permeability?

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- This talk is on relative permeability in porous media. In the first parts of the talk, I'll give a brief review of Darcy's law as a lead-in to the definition of relative permeability. I'll then show some examples of what relative permeability looks like in two phase systems. such as oil-water mixtures and oil-gas systems. I'll then follow up with a description of the mechanism that gives rise to things like residual oil saturation on the microscopic level. I'll finally follow up with a issues slide that discusses some of the issues that one needs to be concerned about when looking at relative permeability data and using a reservoir simulation. Before I discuss, with the concept of relative permeability, I'd like to introduce and give a brief review of Darcy's law. Since the extension of Darcy's law from single phased to multi phased flow, gives rise to the concept of relative permeability. Henry Darcy, a French engineer in the 1800s, performed a series of experiments where he took a vertical metal cylinder and packed it with different types of sand and flowed water through it, and measured the flow rate as a function of the difference in head between the entrance and the exit of the fluid. What he found was the Darcy's law effectively which we see in equation one, which says that the flow rate, the volumetric flow rate of the fluid is directly proportional to the pressure drop between the entrance and the exit of the fluid times the cross-sectional area flow times the density of the fluid divided by its viscosity. The proportionality constant was deemed the permeability of the porous media. If we assume that the flow is horizontal, then the gravitational terms in Darcy's law drop out, and we get the more familiar equation in terms of pressure drop, rather than potential drop. This equation is derived under the following assumptions: that we have single phase flow and that we're at a laminar flow regime, which means essentially slow fluid flow with no inertial effects. What if we had multiple phases flowing through the rock? Such as an oil water mixture? Can we generalize Darcy's law to the case of multi phase flow, so that we can predict the flow rate of any given phase in terms of its associated pressure drop? Let us assume, for simplicity's sake, that we're talking about horizontal flow. Then what we do is we rewrite Darcy's law in terms of a fluid phase. As we show on equation four, we define the flow rate, the volumetric flow rate of an ith phase as proportional to its associated pressure drop, times the density of that phase divided by its viscosity, times the cross-sectional area, times an effective permeability of the ith flowing fluid. Experimentally, the effective permeability of a given phase, is found to be a function of the rock type and the saturation of the fluid phase in question. The saturation of a given phase is simply the volume fraction of that fluid phase in the porous media. The relative permeability is then defined as the effective permeability of the given phase, divided by the absolute permeability of the rock to single phase flow. One way to determine the relative permeability of a system experimentally, is to perform a two phase flow experiment on core plugs that are taken from a full core, from a well. Usually, what happens is core plugs are drilled out of the full core, these are usually one inch in diameter and one inch in length, and I stack a setup of four to six of these and then, these are put into core holder and a two phase flow mixture is flown through the core. In the next slide, I show an example of this, a very simple schematic of this where we have an oil-water mixture that's going into the core holder that's holding the core. We establish a flow with a certain pressure drop and we, then, make measurements when the pressure drop basically stabilizes. And, if the pressure drop stabilizes we assume that, then, the saturation's within, the core aren't changing as a function of time. The mixture that exits the core holder goes through a separation procedure and then, we can determine the flow rates of each of the associated phases. The saturation in the core holder is determined usually by something like a CT scanner, where the scan is taken and then processed to determine the saturation is function of the image from the CT scan. Whether the two fluid phases in the pore space are miscible or not, has a pronounced impact on the relative permeability. What I mean by miscible is the ability to mix in all proportions, without the presence of an interface between the two fluid phases. In the next slide, I show an example of this, where we see the relative permeability curves, for two fluid phases where they're totally miscible and there's no interfacial effects. An example of this would be sea water, or sailing water and fresh water flowing in the pore space. What you see is that the relative permeability of any given phase is a linear function of its saturation and then it starts off at zero and increases from there. That is, the minute the fluid phase is present in the pore space, it is mobile. This behavior's also seen in a fractured flow where the presence of interfacial effects is dramatically reduced. You also see this in surfactant floods where the interfacial between the two fluids is very,very low. In the next slide, I show examples of two phase relative permeability curves, obtained for systems that do have interfacial effects. Specifically, a two phase oil and water system and a two phase oil and gas system. What you see is that these curves are very different than what you saw in the previous slide, and that they have curvature. What this curvature means is that the, for a given pressure drop, the total flow rate from the system will be less than would be expected or calculated from the flow rates from the single phases, if they're just waited by their saturations. Thus, the two phases interfere with the flow of each other. Also, there is a region, or there are regions of saturation where the relative permeability of a phase is zero. And what this means is, over those regions of saturation, there's no flow for that phase. There are two points on the curve, on each of these curves that are very important, that we need to understand. On the left side, for the oil water system they are the irreducible water saturation and the residual oil saturation to water flood. The irreducible water saturation is that point where if we were to flood the core, a core filled with water with oil, that we would come to a point where essentially the water wouldn't flow anymore but there would be a saturation in the core that wouldn't be changing. And that's the irreducible water saturation. Correspondingly, if the core was filled with oil and we were flooding it with water, displacing the oil with water, that there'd come a point where the oil saturation wouldn't change anymore, and that's the residual oil saturation to water flood. And the oil basically wouldn't flow, and the saturation would be constant at that value. We see the same thing for the oil gas system, where we have irreducible liquid saturation and this is a point where the gas is displacing the liquid, essentially, the liquid becomes immobile. And this liquid is a combination of oil and water. On the other side, we see a critical gas saturation and this is the saturation of gas that needs to be established before the gas can actually move. If we have core where we're dropping the pressure of reservoir, we're dropping the pressure, that gas will start coming out a solution at the bubble point, but it won't become mobile until the gas saturation builds up to a critical gas saturation value where we get connected phase for the gas and then it becomes immobile. Experimentally, we find that relative permeability is not a unique function of the saturation. And we refer to this as relative permeability hysteresis. And this simply reflects the fact that the relative permeability depends on the saturation history of the core, not on the saturation. In the next slide, I show examples of this where the curves show the relative permeability of the two phases under imbibition, that is when the phase is increasing, and saturation or drainage, where it's decreasing in saturation. And, generally, see here that for the water phase the relative permeability is greater under imbibition rather than drainage. Interfacial effects determine the residual saturations. That is, they determine the area of saturation where the relative permeability is equal to zero. In the next slide, I show an example of what goes on in a core flood, on a microscopic level, the pore level. And this is an idealized example where we have cross-sections of an idealized rock, where we're flooding it with water, and the water saturation is increasing. So the grains of the rock are given by the spheres and the open spaces between the grains are the pores and the pore throats. And we have two examples here, one for a water-wet rock and one for an oil wet-rock. What do I mean by water-wet and oil-wet? Water-wet means that the water, preferentially, adheres to the surface of the grains. Oil-wet means, conversely, that the oil preferentially adheres to the grains. So, we see here, in the top series going from the left to the right, is where the initial saturation of water is very low and it's only around the grains of the rock. And that is, we increase the water saturation, the layer around the grains thickens, connects up into a continuous phase, and then basically displaces the oil out of the rock. Finally,what we see is, we have these isolating globules of oil, that are located in some of the pores in the rock. And this is the residual oil saturation. Why do we have this residual oil saturation? The interfacial tension effects on these globules make it such that the pressure drop that's causing the flow cannot displace these isolated individual globules through the small pore throats. Conversely on the oil-wet system, we have a situation where you can see the light blue in the mass of green, which is the isolated water globules, and the water, as it's being injected, basically, links up these globules and allows them to be displaced. So, the residual oil saturation in an oil-wet rock is, basically, a thin film around the grains. Finally, there are two issues that I'd like to bring to your attention, that you need to consider when evaluating relative permeability data and using it, for maybe, potentially reservoir simulation. The first is that laboratory measurements on two phase flow might not be representative of the actual behavior in the porous media. And this can come about because of the choice of fluids that are used in the experiments, and the actual laboratory protocol. And a good description of this is given by Dake. The second is the issue of scale. Relative premeability is required information in reservoir simulation of two phase flow, and the resolution of the laboratory data is vastly different than usually occurs when we're doing reservoir simulation. The laboratory data has a scale on the order of inches to maybe a foot, while the reservoir simulation has a scale order of tens of feet to hundreds of feet, is associated with the size of a grid block. Thus, if we want to take this laboratory information and use it in reservoir simulation, we need to perform a scale-up procedure, that essentially accounts for things like the gravity effects that could be occuring on the size of the grid block, and also, issues of sub grid block size inhomogeneities that could be present, that were not taken into account in the simulation. And these can have a pronounced impact on the flow of the two phase fluids