Elasticity of High-porosity Sandstones: Theory for Two North Sea Datasets

We have analyzed two laboratory datasets obtained on high-porosity rock samples from the North Sea. The velocities observed are unusual in that they seem to disagree with some simple models based on porosity. On the other hand, the rocks are unusually poorly-cemented (for laboratory studies, at least), and we investigate the likelihood that this is the cause of the disagreement. One set of rocks, from the Oseberg field, is made of slightly cemented quartz sands. We find that we can model their dry-rock velocities using a cementation theory where the grains mechanically interact through cement at the grain boundaries. This model does not allow for pressure-dependence. The other set of rocks, from the Troll field, is almost completely uncemented. The grains are held together by the applied confining pressure. In this case, a lower bound for the velocities can be found by using the Hertz-Mindlin contact theory (interaction of uncemented spheres) to predict velocities at a critical porosity, combined with the modified HashinStrikman lower bound for other porosities. This model, which allows for pressure-dependence, also predicts fairly large Poisson's ratios for saturated rocks, such as those observed in the measurements. The usefulness of these theories may be in estimating the nature of cement in rocks from measurements such as sonic logs. The theories could help indicate sand strength in poorly-consolidated formations and predict the likelihood of sand production. Both theoretical methods have analytical expressions and are ready for practical use. INTRODUCTION Exploring seismic-velocity-to-porosity transformations in various lithologies has been one of the important research areas in rock physics. The main application of this knowledge is to predict porosity from sonic logs and from seismic. Often the elastic moduli of rock are used instead of velocities. Among them are the compressionalwave modulus (or the M -modulus) and the shear modulus ( G ). They can be expressed through Vp , Vs , and density ρ as M = ρVp 2 , G = ρVs 2 . It has been established (Nur et al., 1991) that in consolidated sandstones elastic moduli depend approximately linearly on porosity. The straight line connects two end members: one has zero porosity and a modulus close to that of the solid phase, another has "critical" porosity and (for dry rock) a modulus close to zero. The critical porosity value for sandstones is between 0.36 and 0.4. Two groups of highporosity sandstones from the Oseberg and the Troll fields defy this simple rule (Figure 1). They represent two different trends where the modulus changes with changing porosity not as dramatically as it does in consolidated sandstones. Our goal is to understand the physical principles that govern velocity-porosity relations in unconsolidated sandstones and to develop a predictive theory. The two laboratory datasets have been obtained on high-porosity rock samples from the North Sea. The first set is from the Oseberg field (Strandenes, 1991). Some of these samples are shales and/or have strong intrinsic anisotropy. We select a subset that includes structurally and acoustically isotropic quartz sands with porosities between 0.13 and 0.32. Quartz volumetric content varies from 0.6 to almost 1.0. The thin section images reveal slight quartz cementation among the grains. Two other dominant minerals here are mica (up to 0.15 volume fraction) and clay (up to 0.25 volume fraction).