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  5. Primary Imbibition Saturation-Height Modeling

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- [German] So now, I'm gonna go over the primary imbibition part. So before going into the imbibition part dyadically, I'm gonna talk a little bit about one of the endpoints of this modeling, which is the trapped gas saturation. But I'm gonna recap a little bit and talk about why we are doing this. So remember that the lack of sealing capacity or the tilt in the trap can cause imbibition of water, again, into what was previously saturated with gas. So the point at which imbibition starts is called initial water saturation, and it's indicated here. And this point can equal your, if it, it reduces the water saturation or could be a higher water saturation if you are in the transition zone. So our modeling approach assumes that reservoirs were fully charged with hydrocarbons, and then they went through imbibition from the irreducible water saturation condition. We are always, in this model, are starting from here. So the mercury extrusion tests are commonly used for modeling imbibition from scar date. So we understand that the not extruded or trapped mercury might not be an accurate representation of in situ pressure the gas saturation and perhaps a saturation at low capillary pressures. For rocks with significant amount of micro-porosity, the mercury usually exceeds the gas trapping due to the absence of something called rapid-gas diffusion. So this mechanism takes place at gas-liquid interfaces in small spore with high capillary pressures. So in this plot right here, in the upper right, you see, in the x-axis, the maximum trapped gas saturation interpreted from counter-current imbibition tests, and you have, in the y-axis, the trapped gas saturation interpreted from mercury extrusion. You see that, how this trapped gas saturation from mercury extrusion consistently overestimate your trapped gas in these micro-porous Almond core samples. So it's almost twice the amount of trapped gas when interpreted from mercury extrusion. So in the case of a counter-current imbibition test, the samples, as you know, the samples are fully dry with air. And then, they are submerged in toluene. So this, the air, starts kind of bubbling, and the remaining is considered a residual gas. Since this test start with wetting phase saturation or toluene of zero, this saturation needs to be taken to irreducible. And for that, we are using an average of the two models that we have available, which are the Land and the Jerauld. So in other words, this test in the lab starts here, but you need to correct it by either the initial water saturation or the place where your imbibition starts. So for this, we are using, again, an average of these two models. So now, how we are gonna combine this extrusion mercury with this maximum, with this trapped gas saturation, it's not even maximum, trapped gas saturation. So the modeling consists of combining the mercury extrusion, which is, the original data is shown in red, in here. I hope that you can see the cursor. So we are combining this, only this part of the curve, which is, in this case, something higher than 10 psi capillary pressure, with the trapped gas from the counter-current imbibition, which is shown in this yellow point here. In this particular example, we are considering mercury, we are considering that the mercury extrusion is not accurate below 15 psi, over here, and therefore it's disregarded. So we are going to see, in the next slide, how we do that in practical terms. So this slide, I kind of show the modeling approach for generating this extrusion, capillary pressure. So what we are doing is to take the Brooks-Corey water saturation model for drainage, and we're modifying it to characterize primary imbibition. So the wetting phase saturation, which is here, is linearly rescaled between the boundaries of the experimental data, and we are generating a parameter called normalized water saturation, which is right here. And these normalized parameters has this form, this equation. Subsequently, experimental data are fitted using the following hyperbolic equation, which is shown here. So in this hyperbolic equation, you have two main parameters, which are A, which is the capillary pressure of the trapped gas saturation, and in this case, for convenience, we are using, we are setting this parameter to one, and the other parameter is B, which is the curvature of the function that we are generating. Then, we are recombining these two parameters, and we are generating this function for reproducing water saturation for imbibition. So this is the function that we are going to be using. And again, in the same way that we did it for primary drainage, we are correlating porosity, permeability, and square root of K over Phi against the parameters involved in this equation. And we are taking the irreducible water saturation from the drainage model, and what you have here is a correlation between permeability and the curvature of the function over parameter B. And then, in here, you see the correlation between permeability and the trapped gas saturation. So this is how we built our primary imbibition curves.