Getting the right acoustic and elastic parameters
for a rock is crucial for accurate modelling, but choosing
geologically realistic parameters is not easy.
If the porosity is doubled how will that affect
Vp?
If the pores are filled with fizzy water rather than an oil/gas
mixture will Vs go up or down?
What effect will variations in the clay/sand ratio have? Etc.
The Rock Physics suite is a set of three small utilities, matProp, poreProp and rockProp, that between them can help answer such questions. matProp and poreProp take a simple lithological description of the reservoir and surrounding rocks and, using first principles, compute realistic estimates of the acoustic and elastic properties for the solid and fluid components of a rock. Once a matrix of acoustic and elastic properties has been built rockProp can be used to combine them to get geologically sensible values of the P- and S-wave velocities, Vp and Vs, for input to modelling tools such as tune3D or massp, leading to better and more geologically sound synthetic seismic data.
The first step in the modelling process involves setting the rock matrix properties and calculating the resulting properties (bulk and shear moduli, together with density) of the dry rock. This step is handled by matProp.
The physical properties of various minerals are known, so by specifying a mineral assemblage – either estimated from gamma logs or other downhole sources such as mud logs, or inferred from the seismic velocities and a knowledge of the general geology – the user is effectively specifying the properties of the matrix. (Here, the term matrix means the material composing the non-pore parts of the rock-plus-pore space assemblage. The rock + empty pore space is referred to as dry rock or bulk rock.)
matProp allows the user to define a lithology as a mixture of common components (clay, sand, limestone etc.) and estimates the relevant physical properties of the matrix, the bulk modulus κm, the shear modulus μm and the density ρm, from a weighted average of the individual components.
Given the properties of the matrix and a porosity φ one can see the density of the dry rock is simply:
ρb = ρm · (1 - φ)
The empirical relations:
κb =
κm· (1 - φ)[3/(1-φ)]
μb =
μm
· (1 - φ)[3/(1-φ)]
can be used to estimate the bulk and shear moduli of the dry rock. (Here and in the following discussion a b subscript refers to the bulk or dry rock parameter and the m subscript to the matrix material. An f subscript will be used to indicate a value for the pore fluids.)
Working out the matrix properties is a relatively straightforward process, for all that it’s somewhat ad hoc, relying as it does on empirical formulae. And the user input, the mineralogical composition, is likely to be fairly well founded.
Estimating the pore fluid properties is the exact reverse. The underlying physical theory is sound and well known, but the composition, and even more so the temperature and pressure conditions, of the pore fluids is almost always going to involve much guesswork unless there is exceptionally good information from wells.
The pore fluids can be divided into their three phases, gas (typically a mixture of various hydrocarbons, carbon dioxide and water vapour), oil (with or without dissolved gases) and brine. Batzle and Wang found (Seismic Properties of Pore Fluids, Geophysics vol. 57 no. 11 (November 1992), pp. 1396–1408) that the adiabatic bulk modulus of the gas phase could be approximated by:
| γ · P | |||
| κgas ≈ | —————————————————— | ||
| [1 - (P/Z)·(∂Z/∂P)]T |
where P is the pressure and T the temperature of the gas. The term Z is a function of the gas density (itself, of course, dependent on gas composition), pseudo-reduced pressure, Ppr, and pseudo-reduced temperature, Tpr. γ is a simple function of Ppr.
The value of the bulk modulus increases with increasing pressure and decreases with increasing temperature. It is also critically dependent on composition, so choosing the correct parameters is critical (Batzle and Wang, ibid).
Dealing with oils is slightly easier. Given that the user has some idea of the type of oil to be expected (approximate API, whether the oil has dissolved gas (“live” oil) or not (“dead” or black oil), etc.) the bulk modulus is:
κoil = V2oil·ρoil
The techniques for estimating the acoustic velocity V2oil and density ρoil depend on the type of oil, its density, the temperature and pressure and, for a live oil, the gas–oil ratio.
Finally the density and acoustic velocity in brine can be estimated (though only approximately it is true) using empirical formulae involving temperature, pressure and salinity (Batzle and Wang, ibid). The density and bulk modulus of the pore fluid system, ρf and κf, is then estimated by taking a weighted average of the components’ individual properties.
Once the physical properties of the rock matrix and pore fluids have been estimated, the bulk modulus of the wet rock is estimated using Gassman’s equation:
| κf · (1 - κb/κm)2 | ||||
| κs = | κb + | ——————————————————— | ||
| [φ + (κf/κm)·(1-φ) - (κb/κm)] |
where κs is the bulk modulus of the saturated rock, κb the bulk modulus of the bulk (dry) rock, κm that of the rock matrix material (both calculated by matProp) and κf is the bulk modulus of the average pore fluid (calculated by poreProp).
The shear modulus of the saturated rock, μs, is of course the same as for the dry rock, μb, and the density of the saturated rock is given by:
ρs = ρm · (1 - φ) + ρf · φ
Finally, having estimated the physical parameters of the rocks from a basic lithological description geologically sensible P- and S-wave velocities, Vp and Vs, are calculated from the standard equations:
Vp =
√[(κs/ρs) +
(4·μs/3·ρs)]
Vs =
√[μs/ρs]