The PFC code requires spheres for its calculations. Because individual grains have very different shapes, multiple spheres are grouped together to form realistic grains. (Images courtesy of Numerical Rocks)

When reservoirs are depleted, they change. We know this from observations of surface subsidence and reservoir compaction, from associated micro-seismicity, and from collapsing well casings. Time-lapse seismic permits us to see where changes occur and thereby position new wells in safer locations.

While petrophysics traditionally has contributed to a static description of the reservoir, the concept of a dynamic reservoir also requires a dynamic petrophysical characterization. When reservoirs are depleted, the stresses change. So petrophysics has to be brought under stress.

This implies, of course, that cores should be tested in the laboratory for all petrophysical properties such as permeability, porosity, resistivity, nuclear magnetic resonance (NMR) parameters, compressibility, and wave velocities under controlled conditions of stress and pore pressure. Although feasible, this is not likely to happen: Cost and time constraints represent obvious limits. Furthermore, cores may not always be available, and core alteration may be so severe that reliable measurements are not possible.

Numerical laboratory

One solution which still honors the structure and composition of the rock at grain and pore scale is a “numerical laboratory,” where the 3-D microstructure is represented digitally and rock properties are computed according to the physical laws controlling characteristics such as fluid flow and electrical charges.

Numerical Rocks, based in Trondheim, Norway, has established such a laboratory — “e-Core” — that can predict formation properties from input data extracted from thin sections (a thin section is a 30-micron polished slab of the rock). The information extracted from the thin section is the grain size distribution, various cements, and the amount of clay in the pore space. These parameters are fed into the computer, and a 3-D digital rock model is constructed by mimicking nature’s way of creating sandstones — sedimentation, compaction, and diagenesis (i.e., cementation upon deeper burial). Once the numerical (or virtual) rock has been generated, petrophysical properties such as permeability, formation factor, and NMR relaxation can be determined by numerically solving the governing differential equations directly on the virtual rock.

An alternative to the reconstructive rock modeling is to use a 3-D Micro-CT image directly. These numerical rock models are, of course, static in the sense that the initial image is not deformed by the action of stresses.

Particle flow code

SINTEF Petroleum Research in Trondheim has established a Joint Industry Project (JIP) called “Petro-physics under stress — core applications,” where discrete particle modeling is used to model deformation and failure of rocks based on the same input data used by the e-Core technology. This kind of modeling, which builds on the discrete element method (DEM) pioneered by Peter A. Cundall and his colleagues at Itasca in Minneapolis, permits computation of complex and dynamic rock behavior based on simple contact laws controlling deformation and failure of bonded contacts between grains within the rock. The commercially available code for doing this is called “Particle Flow Code” (PFC). This code predicts in a qualitative manner aspects of rock behavior such as formation of deformation bands during rock failure, stress memory (Kaiser effect), core alteration during coring, borehole failure and sand production, elastic wave propagation, and long-term deformation (“creep”). To make quantitatively correct predictions, however, the micro-structure and the grain contact behavior have to be fully honored.

The basic code (in 3-D) uses spheres as building blocks. In order to mimic realistically grain shapes, individual sand grains are built from spheres so that they have the same mechanical properties as quartz (Figure 1). The composite grains interact through intergranular contact laws, requiring stiffness and strength parameters to be defined.

Obviously, a large number of microscopic parameters are required to describe dynamic rock behavior. Experimental calibration of the model is therefore a necessary part of the development of the numerical laboratory. In the calibration, samples of artificially cemented glass beads as well as various sandstones are used (Holt et al., 2005).

Experiments include rock mechanical and petrophysical measurements under various stress conditions. In addition, simple experiments that can be used on small pieces of field core material are included. For example, the intergranular cement bond strength parameters can be related to the forces required to remove individual grains in a core scratch experiment (Figure 2).

As part of the JIP, SINTEF and Numerical Rocks are working closely together to combine the two techniques described above. Three-D Micro-CT images are taken from a selected rock sample. These images are treated to partition the individual grain particles and are described with small spheres. The initial sample image will then be deformed according to the DEM under influence of increasing stresses, and the resulting particle model is converted to obtain the grain and pore structure required for e-Core. The workflow is illustrated in Figure 3. This final grain and pore structure will be fed to e-Core for a staggered computation of petrophysical properties as a result of the applied stress.

Application of the method on a real sandstone rock sample has demonstrated its potential. Stress-strain curves show a good match, and a realistic decrease of porosity and permeability has been observed. A wide variation of sandstones with different properties has been selected to test and calibrate the technique further.