Scale Dependence of Governing Laws in Earth Materials
Abstract
Earth materials are heterogeneous, not only in their pore-scale microgeometry, but also in their ensuing physical properties (porosity. elastic moduli, strength, hydraulic permeability, electrical conductivity). These properties are usually non-uniform at all scales of measurement: for example, a "point in space" sampled with geophysical remote sensing represents a composite of many finer-scale elements that might be studied in the lab. The governing laws (or "transforms") relating geophysical measurements to desired physical properties are most often derived at the fine-scale using controlled physical measurements or digital simulations. It is not obvious that these same transforms remain valid at the coarser field scale. For example, volumetric properties (porosity, fluid content, composition) upscale arithmetically, while transport (permeability, electrical conductivity) and mechanical properties (stiffness, strength, material creep) require non-linear upscaling operations. While upscaling methods for individual physical properties are well developed, upscaling methods for property cross-relations are not. Therefore, critical questions are: Do lab-derived transforms remain valid at the coarser field scale? If a lab-derived physical transform holds at the coarser field scale, then why? When will this agreement break down? When physical transforms are not scale-invariant, how do we modify them in order to make predictions accurate to within mission specifications? Do we modify the equation or do we modify the smoothed property? Our pilot studies to date indicate agreement of fine-scale and coarse scale transforms for (a) relations between the elastic-wave velocities and porosity, mineralogy, and pore fluid, obtained from borehole data (cm scale) and applied to surface seismic data (10 m scale) and (b) relations between porosity and permeability, elastic moduli, and electrical conductivity from pore-scale (sub-mm scale) computational experiments applied to lab and borehole data. Our scientific objective in this project is to understand the fundamental reasons behind the observed scale-independence of governing laws and, by so doing, reveal where this behavior holds and where it breaks down. The effort will be based on two sources of data: digital and physical. Digital data include high -resolution CT-scans of rock samples and granular dynamics simulations of sediment compaction. Physical data include lab measurements and data measured in hydrocarbon boreholes worldwide. The relations we will focus on include elastic wave velocity versus porosity, mineralogy, and sediment texture (cementation); frequency/scale dependence of the pore fluid effects on the velocity; and permeability and electrical conductivity versus porosity. We will upscale the variables forming the elemental trends by existing mathematical methods, analytical (e.g.. elastic averaging for layered porous media with fluids and elastic bounds) and numerical (e.g. Darcy and electrical flow simulation and granular dynamics). Relations between the upscaled properties will be then compared to those between the elemental properties. These systematic math- and physics-based case studies will lead us to revealing at least some of the reasons behind scale-stationarity of governing laws or absence thereof.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Sep 11, 2018
- Source ID
- W911NF1810008
Entities
People
- Gary M. Mavko
Organizations
- Army Contracting Command
- Stanford University
- United States Army