Rock Physics Modelling

Rock physics is an essential part of any seismic reservoir characterisation project. The rock physics may be as simple as establishing empirical relationships between rock properties or as complex as poroelastic numerical modelling. The sophistication of the modelling will depend on the objectives and the quality and availability of data. Typical objectives of rock physics modelling studies based on well log data are: 1. Quality control of the measured elastic logs: density, p-sonic and s-sonic. Demanding a consistent and meaningful rock physics model can often indicate anomalous and poor quality measurements in the elastic logs that need correction or mitigation. 2. Quality control of petrophysical interpretation. It has been suggested that empirical relationships are more reasonable to use with well log data as it is hard to establish a consistent numerical rock physics model to explain the information from various wells even though they are drilled close by and in the same formations. However, the root of these problems may lie in inconsistencies in the petrophysical interpretations. Insisting on a consistent rock physics model can often highlight inconsistencies within the petrophysical interpretations. 3. Synthesis of elastic logs where missing or the quality is poor. Once a strong relationship has been established between the petrophysical interpretation (or other well log data) and the elastic properties, prediction of elastic logs can be made. In particular the synthesis of shear information is often required due to the lack of measured data in many wells. 4. Correcting logs for invasion effects and dispersion. Many logs measure the properties of the invaded zone. To establish correct relationships between seismic data and log data, the latter must be corrected for the effects of invasion. In addition sonic tools measure the rock properties at a different frequency and scale to seismic data. Compensation for these differences can be applied through rock physics modelling. 5. Scenario modelling. Once a rock physics model has been established it is possible to change the petrophysical properties of the rock to understand the impact on the elastic properties. For example, changing the porosity would help to understand what might be observed when a reservoir changes depth. Changing the clay content would predict the response of the shaling out of a reservoir. The most common scenario modelling, though, is fluid substitution. 6. Establishing a framework for interpretation of seismically derived elastic properties. Deriving and understanding the relationships between the elastic properties of the rocks and their petrophysical properties enables the interpretation of seismically derived elastic models in terms of petrophysical properties and provides an understanding of the limitations of any such interpretation.

Normally the well log data will form the major source of information. Core information is required to calibrate the rock physics models, but a great deal can be achieved even in the absence of core data. The degree to which the well log data need to be modelled will depend on how the elastic logs are to be used in the seismic reservoir characterisation. Typical uses are: 1. Wavelet estimation. Deriving effective wavelets for inversion of seismic data to elastic properties 2. Low frequency model building. Low frequency models are required to provide information below the seismic bandwidth in order to generate absolute models of rock properties from seismic inversion. Low frequency models are also used for time to depth conversion and pore pressure prediction, for example. 3. Calibration of seismically derived velocities. Seismic velocities can provide additional control on the low frequency model building. As such they should be consistent with and therefore calibrated to the well log data. 4. Time to depth conversion. In the absence of checkshot data and often even when checkshot data are available, tying the well to the seismic through matching of synthetics with the seismic data is used to establish a time-depth relationship. 5. Hard data in geostatistical modelling. Well log data can be set as hard data in geostatistical modelling ensuring that all geostatistical realisations exactly honour the well log data.

Clearly the detail of the log data required for geostatistical modelling is higher than for low frequency modelling. What is also important to note is that the quality of the well log data needs to be high even in the non-reservoir intervals, that is, over the entire time or depth interval of the study. Quite often the quality of the well log data in the non-reservoir intervals is lower than in the reservoir interval due to poorer borehole conditions, especially in older wells. This makes rock physics modelling all the more important. However, establishing a good rock physics model for the non-reservoir intervals is often more complicated than for the reservoir interval for various reasons: 1. Lack of a reliable, consistent, detailed or, sometimes, meaningful petrophysical interpretation in the non-reservoir. 2. Lack of knowledge of or uncertainty in the elastic properties of clay minerals. 3. Anisotropy of clay minerals.

The difficulty with clays forms one of the many obstacles to rock physics modelling. These obstacles will be discussed further when describing the numerical rock physics modelling.

The rock physics approach ultimately should show consistency with all the input data and petrophysical and geological understanding of the rocks. It should show consistency between the wells and it should be able to explain the seismic data. It is important to establish this consistency from the petrophysical interpretation all the way to the synthetic to seismic well tie. This may result in an iterative process whereby not only the rock physics model but also the petrophysical model needs to be updated to improve the overall consistency. Consequently rock physics modelling is the focus of the integration of data and disciplines and is therefore extremely important.

Rock physics modelling

There are three fundamental properties that are modelled during rock physics modelling for seismic reservoir characterisation: density, p-velocity and s-velocity. Other properties such as Poisson’s ratio, compressional to shear ratio, slowness, impedances, Lamé’s parameters are all derived from the three fundamentals.
In numerical rock physics modelling the rock is modelled as a mixture of minerals and fluids. Therefore it is important to highlight the difference between clay and shale. Shale is the term applied to a rock type that is composed predominantly of clay minerals. For many techniques it is difficult to work with volume of shale as an input to rock physics modelling, although it is quite often preferentially derived during petrophysical interpretation. The volume of clay and ideally the volumes of the different clay minerals present are preferred.

Density

Given the volume of minerals and fluids present in the rock and their respective densities, the density of the whole rock can be found unambiguously:

[pic],

where [pic] is the relative volume and[pic] is the density of the ith component. The volumes of the components are estimated during the petrophysical interpretation. The densities of the components are often known, or reasonable values assumed in the case of minerals and can be measured from samples or calculated from empirical equations in the case of fluids. For calculation of fluid properties, additional information is required such as the brine salinity, the API of the oil and the specific gravity of the gas, as well as the in situ conditions of temperature and pressure. The main concerns with the density modelling are 1. The quality of the density measurement. The density tool is a contact tool. In boreholes affected by washouts the tool does not always remain in contact with the borehole wall and the density estimated is consequently incorrect, generally lower than actual. Washouts tend to occur in softer formations, typically shales. This may not have an impact on formation evaluation of the reservoir but has a significant impact on seismic reservoir characterisation. The density log is often relied upon heavily in the petrophysical interpretation. Poor quality density data can often lead to poor estimation of component volumes. 2. The accuracy of the volumes. The mineral and fluid volumes are derived during the petrophysical analysis. Volume uncertainties can be as high as [pic]2 p.u. for porosity and [pic]10% for mineral volumes and [pic]10% for saturation even when log quality is reasonable. 3. The uncertainty in the component densities. Not all mineral densities are known with the same degree of confidence. A given clay mineral may have various forms that have different densities. 4. Identification of all the components. Minor minerals with extremely high densities such as pyrite can have a significant impact on the density measurement, for example. If these minor minerals are missed or ignored in the petrophysical interpretation then the volume estimates of the other components will be wrong. In addition, in many cases only a total volume of clay is estimated when various clay minerals with different densities are present throughout the interval. 5. Drilling effects. The area around the borehole can be affected by the drilling process and the presence of drilling fluids: stress release, invasion, hydration of swelling clays. The density tool can only measure a short distance into the formation and so will be affected by these disturbances. That is the true formation density is not measured directly and must be estimated from the measurements made.

Velocity

The compressional and shear velocities are modelled simultaneously. Modelling of the velocities is not as straightforward as for density. Even if the volumes of the components and their elastic properties are known accurately then it is only possible to establish an upper and lower limit to the velocities of the whole rock and if one of the components is a fluid then the lower limit is almost useless. The reason for this is that the density of the rock is independent of how the components are arranged whereas the structure of the rock has a strong influence on the velocities. This may not be significant if the properties of the components are similar, but fluids have extremely different properties than minerals, in particular the shear velocity is zero. The additional information on the structure of the rock required to model the velocities can be derived from 1. Core analysis. Thin section analysis, say, can establish the distribution of clay as dispersed, structural or laminar. The degree of cementation and the grain contacts can be understood. 2. Petrophysical analysis. Cross-plots of neutron porosity and density porosity or the Thomas-Steiber analysis may also indicate the distribution of clay. 3. Rock physics modelling. The assumption of a particular structural model may explain the data more readily than others. Cross-plots of velocity or other derivatives versus petrophysical properties such as porosity can indicate a particular structural model as determined from previous rock physics or theoretical studies.

The main concerns with velocity modelling are:

1. The quality of the measured data The most consistent and reliable rock physics models are produced when the well log data are of high quality. Poor quality data leads to ambiguity and uncertainty. 2. The availability of sufficient shear data It is possible to build a variety of rock physics models that match the acoustic properties of the rocks within an acceptable tolerance but give very different shear properties. Without sufficient good quality shear data the potential for poor prediction of shear is quite high. 3. The accuracy of the petrophysical analysis Porosity, saturation and volume of clay have strong influences on elastic properties. Consistent and accurate estimates are required to help constrain the rock physics models. 4. The uncertainty in the component velocities The elastic properties of some minerals are well known. However, the properties of others, in particular the clay minerals, are not. In addition, clay minerals are often intrinsically anisotropic and their effective properties will depend on their orientation to the propagation of the elastic waves. 5. The uncertainty in the structure of the rock The large range of possible values for a given set of volumes and properties requires detailed knowledge of the structure to provide accurate results. This information is not always readily available. 6. The large variety of rock physics modelling techniques The number of rock physics modelling techniques and approaches is daunting. Each has its limitations. Choosing an inappropriate technique can lead to severe problems. 7. Invasion and other drilling effects Well log data typically do not provide information on the virgin properties of the formation due to invasion and other disturbances at the borehole wall. Knowledge of the invasion profile and how it influences the velocity measurements is not easy to obtain. 8. Dispersion A model that works at the log scale and for logging frequencies may not work for seismic scales and frequencies. 9. The availability of core measurements. Without core it is not possible to calibrate the petrophysics or rock physics models. We rely entirely on producing a good match to the well log data. The availability of core allows some constraints to be applied to the type of rock physics models to use. However, core data introduces additional problems caused by sample size and scale of measurements.

Fluids

Fluids are most often included in the rock model through the Gassmann equations. The Gassmann equations assume that the pressure is equalised throughout all the fluids present within the rock. Therefore the effective properties of the mixture of fluids should be calculated from the iso-stress equation:

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where [pic]is the bulk modulus of the fluid mixture and [pic]is the bulk modulus of the ith component. In the case of a mixture of brine and gas this equation would become:

[pic],

where [pic] is the water saturation and [pic]and [pic]are the bulk moduli of the brine and gas respectively. However, it has been established that under some circumstances the pressure may not be equalised. For example, when the distribution of fluids is not homogenous and the pressure pulse is too quick or the permeabilities are too low for equilibration to be established between the patches of rock with different fluid mixes. In such cases the Gassmann equations are not strictly applicable. They can, however, still be used and effective properties of the fluid mix used instead of the equations above. Brie proposed a formula to calculate the effective properties for a range of cases:

[pic],

where e varies from 1, where the fluids are completely isolated from each other, to infinity, where the fluids are completely mixed. The heterogeneity of the fluid mixture is scale dependent and what may be heterogeneous at logging scales and frequencies may be homogeneous at seismic scales and frequencies. That is we may choose a particular value of e when modelling the well log data and a different value when producing logs for synthetic to seismic correlation.