IEEE Signal Processing - March 2018 - 126

I, is computed. Ohm's law then is used to compute the electrical
conductivity of the sample v sample = I/V. Conductivity is often
presented in terms of the formation factor
F = v fluid .
v sample

(9)

Elasticity
The finite-element method is a well-established method for
numerically computing elastic properties, such as the bulk and
shear modulus. An integral form of the linear elastic equation is
used to discretize the finite element mesh. The segmented CT
scan is used as the simulation mesh in which each voxel is used as
a finite element [21], [22]. Localized voxel-based material properties, such as the bulk and shear modulus, are supplied to the
simulation algorithm. Composite properties are then computed by
applying a strain and computing appropriate stress averages. The
relationship between the strain field, x, and the strain field e is
given by Hook's law: x = Ce.

Challenges and recent innovations
The traditional DRP workflow presented suffers from several
challenges. For example, availability of relevant rock samples
is limited, and physical rock properties can vary significantly
between relatively proximate samples. Heterogeneity within a
rock sample or between different rock samples makes the interpretation and validation of computed rock properties a challenging process. Furthermore, pressure and temperature conditions in
reservoirs are different from laboratory environments. In the next
sections, we present several proposals that can better handle limitations in the traditional DRP workflow.

Stochastic reconstruction
Core rock samples are used in simulating physical rock properties. Due to the high variation in the properties of rocks in hydrocarbon reservoirs, a large volume of samples is needed to build
an accurate model of the reservoir. Using stochastic simulations,
researchers can reconstruct a variety of rock samples that resemble variations of a given reference sample and satisfy desired constraints. In addition, the ability to simulate 3-D samples is valuable

(a)

in situations in which acquiring 3-D scans is time-consuming or
not easily attainable [24].
Reconstruction algorithms employ multiple point statistics of
the given reference image to generate simulated data. The neighborhood-based conditional probability distribution [25], Markov
chain Monte Carlo simulations [26], and the cross-correlation
function [27], are examples of methods that have been proposed
to solve the reconstruction problem.
In [25], the original image to be reconstructed is divided into
small patches. Next, a search over the entire image or data available is used to compute the probability of occurrence for each
patch. Computed probabilities are used to generate the simulated images. This approach is computationally expensive and
the suggested implementation fails in cases where the data is
highly heterogeneous because the method ignores patches with
few occurrences.
An alternate reconstruction method that is based on Markov
chains was proposed in [26]. The method performs a raster scan
where two pixel values are generated at each step. The update of
two pixel values was found to generate more realistic simulations.
The updated values depend on 2-D/3-D neighborhoods of previously generated intensity values. Transition probabilities are generated from available training data.
A common technique used in generating simulated images
is to divide the training image into blocks and fill the simulated
image with random realizations from these blocks. The problem
with this approach is that it does not preserve continuity between
blocks. The cross-correlation-based method (CCSIM) [27] is
a patch-based algorithm that employs a one-dimensional raster
scan to generate reconstructed data. The algorithm divides the
training and simulated images into overlapping blocks. Figure 7
graphically illustrates the progress of the reconstruction process.
The algorithm fills each block in the reconstructed image with a
block selected from the training image that satisfies a threshold
on the cross-correlation function between the overlap region and
the selected block. If the threshold is satisfied by more than one
candidate, one of these candidates is chosen randomly. The algorithm iterates until all image blocks are filled. If no blocks are
found to match the threshold, the block is split into four sections,

(b)

(c)

Figure 7. The CCSIM for porous media reconstruction computes cross-correlation between the overlap region and candidate images: (a) the original image and
(b) raster scan used to generate the new image. Extracted portions are the (c) overlap regions. (Simulated image used with permission from [23].)

126

IEEE Signal Processing Magazine

March 2018

Table of Contents for the Digital Edition of IEEE Signal Processing - March 2018