- Good Day. I'm Rocky Detomo, representing Omoted Geophysical Consulting, LLC., and I welcome you to this series of Geophysical Online Lectures. Today's topic is an introduction to the Mathematical Description of Complex Physical Phenomena, specifically as applied to Rock Physics. Today's outline will briefly cover our motivation for this lecture, some simple examples, specifically that of composite rock densities. Then, we'll discuss the art of choosing an appropriate functional form suitablefor physical descriptions, and show an example of amore complex relationship between rock densitiesand rock velocities. Finally, there will be a fewshort words on considerations in building an integrated rock property description. Most geoscientists find themselves in a situation where they're trying to understand different rock property measurements from many different wells, and understand the relationship between these properties. A solid mathematical description relating these rock properties canprovide significant insight as to the physical processes that drive their values and their variability, and help us in understanding how and why rock properties change laterally and vertically. This can then provide a physical basis for making rock property predictions away from well control, which can be critical in exploration and production success. Some rock properties arerelatively straight forward to understand and to describe, such as the density of a rock, which is simply a volumetric average of its component masses, as described in equation one. Note we must explicitly take into account not only the rock component volume and masses, but also that of the fluid that fills the pore spaces. For a typical clastic, 3-component system, made up of sand, shale, and brine, this can simply expressed asI indicated in equation two, where the ratio of sand to shale is taken into account. Other rock properties are more complex to describe, such as rock velocity, since this rock property is the result of chemical and physical processes of chemical and physical processes that have taken place overlong periods of geologic time. Therefore, the velocity ofa rock cannot necessarily be calculated directly fromits contributing components, although if these historical processes can be understood and can be estimated, some prediction and/or bounds on these values can often be made. can often be made. In addition to relating rock properties of a particular rock to each other, geoscientist often want to describe how these rock properties are changing over geologic time. Over geologic-scale time intervals, rocks will undergo pressure changes, such as compaction with burial depth and over-pressures, thermal changes with depth and proximity to thermal conduits orsalt, and chemical changes, such as shale, carbonateor volcanic diagenesis or quartz cementation, et cetera. Most commonly, these effects over time are attempted to be understood by relating how the properties of a rock of similar composition vary as a function of burial depth. Therefore, it is common to see plots such as figure one, which are assembled from many well log measurements of compositionally similar materials. The implicit assumptionis that these rocks have undergone a common, monotonic thermal burial history. It is then most commonto fit these data to some mathematical function, such as the second order polynomial, such as that which is displayed. such as that which is displayed. However, there is often absolutely no physical basis for such a choice in functional form, and these choices often display behavior outside of the input data's range that are completely unphysical. In addition, the spread in the data are often attributed to natural variability, and are simply carried along as an error bar or as a fitting uncertainty. However, if we examine the data more closely, and we account for the asymptotic values for a shale's density both at the surface and at great depth, we might rather choose an exponential fit for the functional form instead, allowing us to examine the remaining variability in terms of other rock properties and burial history. For instance, classification of these shale densities in terms of their pressure regime gives the plot in figure two, where some of the variability that we saw before is clearly associated with the shale's pressure history. Close examination of the other log data rock properties allow one to further characterize much of the remaining shale density's variability as a result of the shale's relative silt content. The result is that this more holistic description then gives us insights into what is driving the vertical and lateral viability in reflectivity that results from these changes in rock properties and thus provides us a clue as to what such variability that we observe means. as to what such variability that we observe means. Recognizing and describing the impact of pressure on shale densities carries over to a more consistent description of other rock properties. Figure three shows the resultant description of shale velocities versus shale densities in this same region. Here, the variation inthe exponential fit's fit's parameterization depends on a simple way with pressure. I hope this brief introduction will help you in your development ofrock property models. You have seen that a consistent description of rock's properties with respect to its burial history is fundamental to understanding seismic reflectivity, and that rock property models need to be more closely tied to their associated basin model. A geoscientist will seek to use the rock property data to better understand the physical processes that have taken place that have driven the variability in rock properties that they observed laterally and vertically in the earth, and use this in their interpretation. Thank you have a nice day.