**5. Digital rock physics**

Digital rock physics is based on the concept "image and compute," image rock at the pore scale (**Figure 19**) and then simulate in the computer various processes in such an image to arrive at a desired rock property. These simulations include viscous fluid flow to arrive at hydraulic permeability, electrical charge flow to arrive at electrical resistivity, as well as elastic deformation to arrive at the elastic moduli and velocities.

The advantage of such digital approach is that the same sample can be reused multiple times, unlike in physical experiments where a sample is altered after every test; the sample can be digitally altered by, e.g., introducing digenetic cementation, which is hardly possible in physical experiments, as well as subsampling of a digital volume to investigate how various rock properties vary within the volume and how relations between rock properties depend on the spatial scale of investigation.

**Figure 19.**

*Segmented digital images of loose sand (porosity about 30%), sandstone (porosity about 20%), and carbonate (porosity about 15%) showing the mineral matrix and pores. The images are a few mm across.*

Although the aforementioned concept is simple, its implementation is not. First, the imaging has to be conducted at the appropriate scale and resolution to reveal the salient features of natural rock relevant to the process under examination. Second, the image has to be segmented to separate minerals from pores and segregate various minerals within the solid matrix, as well as fluid phases inside the pores. Third, powerful computational engines have to be utilized and verified to simulate processes relevant to the physical experiment.

In spite of these complexities, during the last decade, DRP has emerged as a powerful technique complementing (if not replacing) physical testing, mostly due to the recent advances in imaging hardware and image processing and computational software, the latter combined with steadily improving computational power. Not only DRP has become a novel research tool in academia and national labs, but is has also been adopted by leading oil and service companies.

There is one more inherent feature of DRP that needs to be accounted for. Porescale rock images are only a few mm in size, and the higher the resolution needed to revel the salient features, the smaller the field of view. At the same time, these computational results have to be relevant at much larger spatial scales of feet for wireline measurement interpretations in the well or tens and hundreds of feet in seismic prospecting. Even such basic property as porosity may be different if measured on an inch-sized sample an on mm-sized fragment of the same sample.

One way out of this conundrum is instead of directly comparing data points generated by different methods of measurement, compare trends formed by such data points, such as permeability versus porosity trends. Dvorkin et al. [18] show that such trends are often hidden inside a very small digital sample and can be derived by subsampling it. Moreover, these computational trends often match

**Figure 20.** *Illustration of the subsampling approach.*

**Figure 21.**

*Permeability versus porosity in Fontainebleau sandstone. Left: Laboratory data matched with a Kozeny-Carman theoretical curve. Right: Multiple permeability versus porosity data points computed from a few digital Fontainebleau samples and subsamples thereof (adopted from Dvorkin et al. [18]).*

**Figure 22.** *Formation factor versus porosity computed on carbonate cuttings. The curves are from Archie's equation with the cementation exponent m 2.0, 2.5, and 3.0 (bottom to top) (adopted from Dvorkin et al. [18]).*

relevant physical trends and/or theoretical rock physics transforms, hence validating computational results and making them relevant at much coarser spatial scales.

The approach is to subsample a digital volume into 23 , 3<sup>3</sup> , or 43 subvolumes (**Figure 20**) and then compute the desired property pairs (e.g., porosity and permeability) on each of these subvolumes. Very often, the property pairs thus computed form a meaningful trend supported by physical measurements and/or theories (see examples in **Figures 21**–**23**). We can call this subsampling approach "to see the rock in a grain of sand."

These results open ways to a meaningful utilization of DRP in research and industry. Publications related to DRP are many and the number is growing. We refer the reader to Kameda and Dvorkin [19], Dvorkin et al. [20], Dvorkin and Derzhi [21], and Andra et al. [22, 23].

### **Figure 23.**

Vp *versus porosity for Fontainebleau sandstone as computed from a few digital samples and subsamples thereof (squares). Gray circles are from laboratory measurements of dry samples. The curve is from the stiff-sand model (adopted from Dvorkin et al. [18]).*
