The Cause and Devilish Effect of Heterogeneity
- Hi, I'm Wayne Narr, and I've been talking about heterogeneity in naturally fractured reservoirs. In previous modules of this series, I've talked about first, evidence for heterogeneity, and by heterogeneity, what I mean is well-to-well variability in productivity, and I'm describing that as productivity heterogeneity in naturally fractured reservoirs. So we looked at occurrences of heterogeneity, and then, I've gone through and talked about the causes of heterogeneity, and there are two major categories of causes. The first that I talked about is sampling issues. That is, the way that we go about sampling a subsurface fracture system, usually with individual wells, introduces extreme heterogeneity in productivity. In part, because we're almost completely unable to sample the representative properties of a subsurface fracture system. This is because the basis for our sampling of the subsurface fracture system is probabilistic, and therefore hit-or-miss on whether we're going to intersect the fracture in any interval in the subsurface. And the second part of the heterogeneity comes from geological variability in the fracture system itself. We also looked at the variability introduced by variability in the geology natural fracture systems. And we did that by looking at examples in outcrop analogs, and reviewing, discussing at least, what impact this variability would have on production heterogeneity in naturally fractured reservoirs. Now, this is the final module, and I'm going to talk about heterogeneity based on some modeling of naturally fractured reservoirs. The first approach that I'd like to talk about is the effect of medium modeling approach. And the basis of this approach is shown here. What we're looking at are two different simulation grid cells, that might be extracted from a much larger model. So these grid cells, for a typical petroleum reservoir, might be on the order of 100 meters by 100 meters, in map dimension, and maybe something greater than ten meters thick, and, as you can see, they each have their own fracture system, in the grid cell. The approach used here with this single effective medium model is to take these individual fracture sets in each grid cell and upscale them, in order to homogenize the fractures. When we upscale them, we blend them with the matrix permeability, and for each grid cell, then describe a permeability tensor. So we can go about this, building a discrete fracture system, on a cell-by-cell basis, and calculating the permeability tensors for each grid cell. First, we generate the discrete fractures in the cells, the size and orientation distributions of those fractures are based on both subsurface data and analogs in outcrop. Secondly, the location within the grid cells for fractures are chosen stochastically. Therefore, even if we have identical properties in two different grid cells, the resulting permeability tensor is likely to be different. After generating those fractures, we upscale them, and make them the basis for the permeability tensors, blending the fractures together, and blending the fractures with the matrix as well. So, a result of this modeling is shown in the graph here, and let me just say a few words about the graph. What we're looking at is maximum effective permeability as a function of fracture density. So, by maximum effective permeability, what I'm referring to is the long axis of the permeability tenser. The matrix permeability is indicated by the colors of the individual dots. Each dot on this plot, or small triangle on the plot, represents a single grid cell. So you see, this is a rather large model, with many thousands of grid cells in it. Where fracture density is very low, in the lower left portion of the plot, down near the origin, what we see is the grid cells show a range of permeability values that are close to the matrix permeability. That is, they range from being extremely low, to low permeability. From red to blue. But, where fracture density is very low, the main control on permeability is the matrix. So down in this region, we have no, or very few, through-going fractures, matrix dominant permeability, and you see that as we're increasing fracture density, and still have low fracture density, there's an order when you increase in permeability, that we see. However, at some point, at some fracture density value, that is, the points on the plot sometimes jump, and jump markedly. So whereas the increase that we're seeing down here by the dashed line is a factor of maybe five to ten increase in permeability, where we see these jumps in points values, they're jumping by many orders of magnitude. What's controlling that are through-going, or through-connected fractures, here. So, down in the matrix dominant area, the flow is restricted to the matrix. That is, some of the flow has to go through the matrix. Whereas, when we get up into this upper, more chaotic region of results, we have fractures that cross-cut the entire grid cell, and they're what determine the permeability, up here. The other aspect of the result here, is that where fracture dominant flow is occurring, the organization of points is highly disordered, and there's a scant relationship with matrix permeability. And what we're seeing, or what's happened here, is illustrated by the small grid cell map, in the upper right portion of the diagram. So this represents a single grid cell, and it has three fractures located in that grid cell. And if I were to do a fluid-flow experiment through this grid cell, you see I'd have flow in the fractures, perhaps, but there would have to be flow occurring through the matrix as well. But if I just have a slightly different organization of those fractures, same fracture density, same description of the fractures in all ways, except for the location of one of the fractures, I have now created a through channel for fracture flow. And fluids can move much more efficiently across this grid cell. Hence, the permeability of this grid cell is quite a bit higher. This linking up of fractures to allow a direct connection from one side of a grid cell to another is referred to as percolation. And usually we say, when we see this linkage established, that the cell has crossed the percolation threshold. But because the process is completely stochastic, at least in the location and size distribution of the fractures we put in the grid cell, or the sampling of the size distribution, is what I should say, in the grid cell, we can get quite a bit of variability in the pathways that we establish across grid cells. Some can be efficient, some are much less efficient, and this results in extreme variability in permeability, on a grid cell by grid cell basis. And the result is this rather chaotic or highly disordered result of maximum permeability, on a grid cell by grid cell basis. And this will result in high variability in a flow simulation model, for example, as well. And it's probably a pretty good mimic for the variability and basis for high heterogeneity that we see in actual naturally fractured reservoirs. We just considered one type of model, an effective medium model in which we used a fracture system as the basis for computing effective permeability of grid cells. I'd like to look now at a different type of model, a discrete fracture network model, and consider the fracture organization that we see there, and we'll look at how subtle or small scale differences in a fracture property can result in a rather major difference in the character of the connected fracture network within a model. So, let's look here, first, this is a discrete fracture model representation, that you're looking at. There are 2000 fractures here. It's a single fracture set, by that, I mean, well, constrained group of fractures so far as orientation is concerned. There's a length-height aperture distribution that defines the fractures, and you can see, there's one well, that light red feature on the upper surface of the model is a well, and we're going to look at the intersection of the well with fractures in the model. So, here's a model with 2000 fractures, here's the identical model, statistically identical model, but with 3000 fractures. The orientations, which I already mentioned, are shown here in the lower hemisphere equal area net, these are poles to fracture planes, and what they tell us is the fractures dip steeply and are oriented more or less North-South, with some variability in that orientation. This shows the intersection of our well with fractures in the model. So we've got this model with fractures in it, then we drill a well, and we're looking at the intersections, and the well happens to intersect six fractures in the model. In the lower model, with 3000 fractures, we did the same experiment, and here, the well intersected seven fractures. Now, if we look at the number of connected fractures, that is, fractures that we intersect with the well, and the fractures that are connected to those fractures, we see that in the upper model, there are 34 connected fractures, and there's a fracture area, something times ten to the sixth square meters. But compare the result in doing the same analysis in the lower model with 3000 fractures, and you see our seven fractures have over a thousand connected fractures, and the fracture surface area is now greater than ten to the eighth square meters. So, a major difference between these two models, even though the difference in terms of number of fractures is modest. So, fracture density is one point five times higher in the lower model. The number of connected fractures is 34 times greater, and the connected fracture surface area is 32 times greater. What we see here, similar to the effective medium model approach, here we have a percolation threshold, and I think that we would all agree, we're pretty much across the percolation threshold, as we've gotten into the lower model. The plot that you're looking at here, labeled with the percolation threshold labels, is showing the surface area of the largest cluster in the model as a function of fracture density. In other words, as fracture density goes up, the surface area of the largest cluster goes up, and it's certainly a very non-linear result. Network connectivity, any kind of network, is strongly non-linear, and in this case, it's a non-linear function of fracture density. We see a moderate increase in fracture density has led to strong increase in network connectivity, and what that's really telling us is that there's a high sensitivity to small changes, and that leads to heterogeneity, and thinking of the impact that such a response would have in a reservoir, you can see that as we move from one area to another within the reservoir, such modest changes in fracture density are going to result, similarly, in great heterogeneity in productivity within a reservoir. This slide shows the results of a full-field flow-simulation model in a naturally fractured reservoir, in which we were trying to do a history match. This is a single-porosity model, we used an effective medium approach, so we've built these small fracture systems on a cell-by-cell basis, much as I described several slides ago, and then up-scaled those permeability, up-scaled the fractures in order to calculate effective permeability tensors for each grid cell. And what we're looking at here is a comparison of two wells, Well A and Well B, and specifically we're looking at static wellbore pressure, as a function of time, and we see that, from the black data points, the pressure goes down as time passes. So when we tried to model this, our initial model shows lower static wellbore pressure in both cases than the data indicates. So, we had to change the permeability in order to more closely match the data with our modeling approach, and in order to maintain higher pressure, that means we have to increase the permeability. We're making a global change to the fracture description here, in other words, we're trying not to go in and just put a little bullseye around each of these wells and force a history match in that way. We're trying to step back, make a global change in the properties and see if we can better model the results of the well. So you see, for Well A, the red dash curve is iteration number one, and in order to get this, what we have done, was gone in and globally, we raised the permeability, and the way we raised the permeability was by adjusting the fracture length of fractures in a model. Since fracture length is basically an unknown parameter, we can't get it from any subsurface data, we have to take it from an outcrop analog. Therefore, its actual value in the subsurface is, I was going to say poorly known, it's really unknown, so we feel comfortable making an adjustment to that parameter within the model. At any rate, we adjusted, made the fractures longer, and you see in Well A, Iteration 1 the pressure went higher, in fact, it goes too high. Look at the result in Well B, and that is the pressure field, or pressure response drops, does exactly the opposite from what it did in Well A. So what we're seeing is a strong effect of heterogeneity here, and that is when we change our parameter, we don't really know what the outcome's going to be, because there's so much stochastic basis for that parameter. So what direction do we go now? We decided, based on a number of wells in our data set that, indeed, we went too far with changing the permeability. So we made another adjustment, we trimmed back the size of the fractures a bit, but still greater than the initial model, and you see Iteration 2, the blue-dashed line on Well A, comes in closer to the actual data points. And likewise, so Iteration 2 in Well A, the response has been a decrease in pressure, but in Well B, as we move from Iteration 1 to Iteration 2, the static wellbore pressure profile goes up, it approaches that initial model, again. So it has moved in the opposite direction from what it did in Well A. The result is, or the understanding of this, is that it's very difficult to predict, when placing fractures at random, and have properties that are randomly sampled, to describe the fractures, it's very difficult to predict what the resulting organization and percolation properties through the fracturing network are going to be. So, there's a real limit to what we can predict in a naturally fracturing reservoir. And, really, heterogeneity has steered our predictions in unexpected directions.