Estimation of properties for saturated rocks

Estimating the elastic properties of saturated rocks is the final stage in rock physics modelling within fmu-pem. Several models are available. Historically, PEM workflows for clastic rocks separated the task into (1) estimating dry frame properties and (2) applying fluid substitution. fmu-pem is extended to carbonate rocks, and the inclusion-based T-Matrix model performs both steps in a single calculation; for consistency the workflow ensures both dry and saturated responses are produced for every selected model. For regression-based and patchy or friable cement models, the dry frame and saturation steps remain sequential. Both steps are executed automatically when either model is selected, using Gassmann fluid substitution for the saturation step (fluid substitution).

As fluid saturation and pore pressure change during production, it is necessary to calculate the properties of saturated rocks for all selected time steps in the reservoir simulator.

Model calibration

This section applies to internal use of fmu-pem within Equinor.

It is recommended to calibrate rock physics models in RokDoc. Both RokDoc-plugins and fmu-pem use the same rock physics libraries, so parameters determined in RokDoc can be applied directly in fmu-pem. Calibration is performed using available well log data, and RokDoc’s interactive features and graphics assist users in selecting suitable parameters for each case.

Regression models

Where a reasonable trend exists between porosity and elastic properties—either Vp and Vs, or bulk and shear modulus (K and Mu)—the modelling of dry rock properties for each lithology can be simplified using a polynomial function:

$$V_p=a_1+a_2\varphi +a_3{\varphi }^2+a_4{\varphi }^3+...+a_n{\varphi }^{(}n-1)$$

In the current version of fmu-pem, regression models are limited to two phases/minerals: sand and shale. Users may choose regression models for Vp and Vs, or for K and Mu. Density is typically determined by volume fractions and the mineral density of each fraction, but it can also be estimated using a polynomial function. The polynomial coefficients must be provided by the user for each regression model. The degree of the polynomial is determined by the number of coefficients; for example, [a_1, a_2] produces a first-degree polynomial. Pressure sensitivity is not directly included in the regression model. A separate pressure sensitivity model for dry rocks is applied prior to the fluid substitution or saturation step.

Patchy cement model and friable model

Patchy cement and friable models are contact theory models for clastic rocks, typically used for sandstones. Both describe the rock as a collection of grains in contact. If grain contacts are cemented, the framework becomes stiffer and the friable model is no longer valid. A fully cemented model, where all grain contacts are cemented, is theoretically insensitive to pressure changes. In most sandstones with hydrocarbon accumulations, some sensitivity to pressure changes remains, even if in situ temperature is high enough for quartz dissolution and precipitation. In such cases, the patchy cement model is appropriate, covering scenarios where some grain contacts are cemented and others are not. Calibration to observed logs determines the degree of cementation and, consequently, the sensitivity to pressure changes. Further documentation is available in the references below.

[1] Mavko, G., Mukerji, T., & Dvorkin, J. (2020). The Rock Physics Handbook (3rd ed.). Cambridge University Press.

[2] Avseth, P., Mukerji, T., & Mavko, G. (2005). Quantitative Seismic Interpretation. Cambridge University Press.

[3] Avseth, P. & Skjei, N. (). Rock physics modeling of static and dynamic reservoir properties - A heuristic approach for cemented sandstone reservoirs. The Leading Edge, January 2011.

Parameters for patchy cement model

Parameters for the friable model are a subset of those used in the patchy cement model. Most parameters have default values, which may not be suitable for all cases. The example below shows default values. In the fmu-pem user interface, each parameter is described.

cement_fraction: 0.04  # must be lower than the upper bound cement fraction (0.1)
critical_porosity: 0.4  # porosity when the sand grains fall out of suspension
shear_reduction: 0.5  # parameter that affects the tangential friction between grains
coordination_number_function: PorBased  # coordination number is the number of grain contacts per grain, assumed
                                        # to be inversely correlated to porosity

Two parameters are not accessible: upper_bound_cement_fraction and lower_bound_effective_pressure. The upper bound for cement fraction ensures the model remains within a sensible range. The constant cement model, part of the patchy cement model, is only valid within certain limits; high cement fractions yield erroneous results. For highly cemented sandstones, an inclusion model such as T-Matrix is more appropriate. The lower bound for effective pressure is based on experience from partially cemented cases, but is not a numerical constraint, unlike the upper bound for cement fraction.

T-Matrix model

The T‑Matrix model is an inclusion-based effective medium approach. The rock framework is treated as a homogeneous elastic background, while pores, fractures and softer patches are represented as ellipsoidal inclusions embedded in this background. This formulation is appropriate for rocks with frameworks stiffer than partially cemented sandstones (e.g. many carbonates), and has also been applied to unconventional reservoirs (e.g. organic-rich shales) and some tight sandstones. The current implementation of T-Matrix does not include any pressure sensitivity. An additional pressure sensitivity step is added as post-processing.

Key concepts:

• Geometry: Each inclusion is idealised as an ellipsoid defined by a single aspect ratio (shortest axis / longest axis). A spherical pore has aspect ratio 1.0; a thin fracture may have an aspect ratio ≪ 1 (e.g. 1e-4).
• Compliance effect: Lower aspect ratios (flatter inclusions) increase the compliance (soften the aggregate response) more than higher aspect ratios.
• Practical limits: The model becomes unreliable if a large fraction of total porosity is assigned extremely flat (very low aspect ratio) inclusions. In practice, most porosity is assigned aspect ratios ≥ 0.5, with a controlled fraction at lower values to capture fractures or crack-like pores.
• Parameter richness: Required inputs include background (frame) elastic moduli, inclusion aspect ratios, inclusion volume fractions, and inclusion (or pore fluid) elastic properties. Some parameters cannot be constrained from conventional well logs alone and may need core, CT scan, petrographic or laboratory measurements.
• Optimisation workflow: In RokDoc-plugins two usage contexts are distinguished:
  • Exploration (EXP): Sparse constraints; more parameters solved via optimisation.
  • Appraisal / production (PETEC): Better mineralogical, saturation and fluid property control; fewer free parameters. Resulting optimised parameter files can be supplied directly to fmu-pem.

In practice:

• Select parameters that can be fixed from data (e.g. mineral frame moduli, fluid properties). This is common to all saturated rock models.
• Assign geologically plausible aspect ratio populations (fractures vs equant pores).
• Use optimisation to solve for poorly constrained fractions or aspect ratios within reasonable bounds.
• Leave truly insensitive parameters at default values after sensitivity screening.

Limitations:

• Assumes dilute or moderately interacting inclusions; extreme crack densities reduce accuracy.
• Single aspect ratio per inclusion population simplifies reality and may under-represent multimodal pore systems.
• Strong anisotropy from aligned fractures is only approximated unless explicit orientational distributions are incorporated.

Literature

The theory for T-Matrix can be found in the papers and in the references therein:

[1] Agersborg, R., Jakobsen, M., Ruud, B.O. and Johansen, T. A. 2007. Effects of pore fluid pressure on the seismic response of a fractured carbonate reservoir. Stud. Geophys. Geod., 51, 89-118.

[2] Agersborg, R., Johansen, T. A. and Ruud, B.O. 2008. Modelling reflection signatures of pore fluids and dual porosity in carbonate reservoirs. Journal of Seismic Exploration, 17(1), 63-83.

[3] Agersborg, R., Johansen, T. A., Jakobsen, M., Sothcott, J. and Best, A. 2008. Effect of fluids and dual-pores systems on pressure-dependent velocities and attenuation in carbonates, Geophysics, 73, No. 5, N35-N47.

[4] Agersborg, R., Johansen, T. A., and Jakobsen, M. 2009. Velocity variations in carbonate rocks due to dual porosity and wave-induced fluid flow. Geophysical Prospecting, 57, 81-98.

All of the papers and a extended explanations of the involved equations can be found in Agersborg (2007), phd thesis: Agersborg (2007), PhD thesis