Th-04-07 3D Interpolation and Extrapolation of Sparse Well Data Using Rock Physics Principles - Applications to Anisotropic VMB R. Bachrach* (Schlumberger WesternGeco) & K. Osypov (Schlumberger WesternGeco) SUMMARY Mapping sparse well data into 3D volumes is a challenging problem. Specificall as seismic velocity model buildings (VMB) requires 3D velocity and anisotropy volumes, and as most anisotropy estimates are calibrated to borehole data, generating 3D anisotropic models relies heavily on extrapolating sparse well data. This spatial extrapolation problem is ill posed and many subjective decisions are made during model building. New advancements in basin modeling and analysis enable us to model temperature and compaction history using simple physical principles. These fields can be used as auxiliary fields to solve spatial interpolation problems of sparse well log data using the concept of interpolation in the rock physics domain. The principle suggests that sparse well log measurements in the physical (x,z) 3D space may still adequately sample the rock physics space of temperature, porosit and ective stress to allow proper reconstruction of the anisotropic velocity field in a manner that is consistent with the diagenetic process. We present the basic concepts and give a synthetic example to support this direct link between basin modeling and anisotropic velocity model building.
Introduction Mapping sparse well data into 3D volumes is a challenging problem. Specificall as seismic anisotropic velocity model buildings requires 3D volumes, and as most anisotropy estimates are calibrated to borehole data, generating 3D anisotropic models relies heavily on extrapolating sparse well data. This spatial extrapolation problem is ill posed and many subjective decisions are made during model building. Recent advances in rock physics modeling of anisotropy showed that as rock physics models depend on rock porosit temperature, and ective stress (among other factors), it is possible to construct a rock model to predict anisotropy in a basin (Draege et al. 2006; Bachrach 2011). The rock model parameters are porosit temperature, and ective stress. Basin modeling, and specificall the modeling of temperature, stress, and compaction has become a quantitative tool that allows one to generate a calibrated temperature, porosit and ective stress fields from basic principles (Hantchel and Kauerauf 2009). Temperature modeling using Fourier heat transfer and stress using a finite element allows one to quantitatively derive temperature, stress, and porosity associated with a certain basin. These fields should be quantitatively integrated into the velocity model building process to improve our ability to resolve the subsurface and constrain tomography. In this work, we show how to use these basin modeling attributes in the automatic extrapolation of sparse well data in 3D space. Specificall it is shown that 3D extrapolation in the physical (x,z) space can be achieved using interpolation in the space of the rock physics parameters. Below, we present the basic theory and assumptions as well as a synthetic example for mapping anisotropic parameters in space using this new method. Theory 1. Rock physics fields and elastic representation of sedimentary rock The elastic stiffness tensor C of a rock governs the seismic wave speed, and is related to the state of compaction, diagenesis and stress, and burial history. Thus, one can conceptualize a relation where C is expressed as a function of porosity, temperature T and ective stress, and time t, C C (,, T, t). (1) We assume that a basin is characterized by a specific compaction history that can be expressed by a large deformation relation. For a fixed temperature, one can define a relationship between ective stress and porosity (Bachrach 2012), while in general, we define compaction as a relation between temperature, ective stress, and porosit which can be expressed as: (, T, t). (2) It is clear from equations (1) and (2) that the porosit stress, and temperature are not independent and are correlated. From a geological point of view, we can define this correlation as basin history. From a rock physics point of view, it shows that the ability to model or estimate porosit temperature, and stress through time will enable us to define the seismic wave velocities. 2. Extrapolation into the physical space using rock physics fields Anisotropic velocity fields can be represented as the distribution of the elastic stiffness in the physical (x,z) space, i.e., ( x, z). Thermal fields and stress / porosity fields can be also modeled or C
derived in 3D physical space and be represented as T ( x, z), ( x, z), ( x, z). As the model is defined at the current time (or the time of seismic acquisition), there is no time dependency associated with these fields. If we can estimate and model the porosit stress, and temperature in the physical (x,z) space and have some knowledge of C (,, T ), then we can extrapolate the elastic stiffness and with it the anisotropy in the physical (x,z). The difference between the physical space and the rock parameter space is that the rock model parameter space, (,, T ), is made out of correlated fields (i.e., stress and porosity and temperature and porosity are not independent). Thus, it is claimed that the 3D extrapolation of well log data is a better posed problem when rock physics parameter fields are considered. This will be shown in the synthetic example below. Single well extrapolated into 3D volume using thermal and compaction modeling of shales: Synthetic example In Figure 1 (left), a 2D basin model, consisting of temperature, porosit and ective stress fields for a Gulf of Mexico (GoM) basin is presented (here, simple modeling of constant thermal gradient is used to generate a simple thermal field that follows the water bottom). The porosity and ective stress are modeled using a simple compaction model for shales. More details about different compaction models and strategies to model compaction are given by Bachrach (2012). It is important to note that such models can be derived using more complex approaches (e.g., 3D basin modeling approach (Hantchel (2005)). In Figure 1 (center) the velocity and anisotropy fields associated with the basin are presented. Here, the seismic anisotropy is modeled using the GoM-calibrated compacting shale model presented by Bachrach (2011). In the right side of Figure 1, a plot of the velocit epsilon, and delta is presented as a single trace. It will be assumed here that this information can be measured locally in the well. We will define this single trace as the well data. In Figure 2 (right), we present the expected porosity ective stress temperature relations for the basin. Note the correlation between the rock model parameters. A plot of the deterministic relation is accompanied by some reasonable uncertainty that one may expect in the relations. Here, we assume a very conservative error of 10 C in temperature, 10% error in porosit and 10 MPa in ective stress. It is important to note that components of the rock model parameter space are not independent. This fact is well understood because it is clear that high temperature is associated with the deeper portion of the basin where the porosity is low and the ective stress is high. This is a key observation; porosit ective stress, and temperature are not independent along the basin. We will now use this observation to prescribe a recipe to map single well information into a 3D volume with the help of the temperature, porosit and ective stress fields. In Figure 2 (left), a plot of the well data (single trace from Figure 1) in the rock parameter space (porosit temperature, and ective stress) is presented. Note that a single well in (x,z) space samples a single trajectory in the (,, T ) space. However, the trajectory samples the rock model parameter space much better than the spatial (x,z) space. In other words, if the anisotropy is related to the porosit ective stress, and temperature, a single well samples better the rock model parameter space than the physical (x,z). Interpolation of well data in rock model parameter space In Figure 3 (left) we present mapping of the single well observation into the two-dimensional image. The points that have similar temperature, porosit and stress are all given an anisotropy value according to our assumption (,, T ) and (,, T ). Note that a single well does not provide information about the whole image; however, there are many points in space that have similar porosit temperature, and ective stress. This is because the high stress, high temperature, and low porosity are all related to the deeper portion of the basin, while shallow sediments are much more
likely to have low temperature, low stress, and high porosity (,, T ). This shows that the support volume associated with anisotropy measurements in the rock model parameter space is larger than the volume of influence a single well has in the physical (x,z) space. To complete our spatial interpolation method, we now perform simple spatial interpolation to derive an estimate of the anisotropy in space. This is presented in Figure 3 (right). Comparing Figure 3 (right) to Figure 1 (middle) shows that the anisotropy parameters are reconstructed such that the main features and trends are available. Temperature (deg C) Vp (m/s) Vp (m/s) Porosity Effective Stress (MPa) Figure 1 Left: Temperature, porosit and ective stress. middle: Vp, and modeled for the basin. right: A single well is being measured along the dotted line. Summary and conclusions Porosit temperature, and ective stress are natural rock model parameters, and can be modeled in space. These fields can be used for extrapolating borehole data in the physical space. A two-step extrapolation procedure is presented where the input is the temperature, porosit and ective stress fields, which can be modeled /estimated using various techniques varying from basin modeling to simplified thermal gradient and 1D compaction analysis and modeling. Using these fields, the first step is to project the borehole data into (,, T ) using the modeled field. The second step is to interpolate in the physical (x,z) space, the projected data to insure spatial continuity. Although, in this example, we present a very simple linear interpolation method, we have also tested various geostatistical techniques that can improve the image. We note that the extrapolated parameters are consistent with the compaction and diagenetic processes expected in the basin. This method provides a direct link between basin modeling and seismic velocity model estimation / building.
Porosity-Effective stress (Temperature colorbar),t),t) Figure 2 Left: Compaction model: porosity-ective stress relations (model in black and expected ranges as scatter point color coded by temperature). Right: Anisotropy from a single well in the rock parameter space of porosit temperature, and ective stress. Top: Thomsen s epsilon. Bottom: Thomsen s delta. Figure 3 Anisotropy from a single well data: Left: and derived by interpolating the anisotropy using temperature, porosit and ective stress. Right: Simple spatial interpolation of estimated parameters yields a 2D image closer to the original data. References Bachrach, R. [2011] Elastic and resistivity anisotropy of shale during compaction and diagenesis: Joint ective medium modeling and field observations. Geophysics, E175-E186. Bachrach, R., Osypov, K., Nichols, D., Yank, Y. and Woddward, M. [2012] Application of deterministic and stochastic rock physics modeling to anisotropic velocity model building. Geophysical Prospecting, in Press. Dræge, A., Morten, J. and Johansen, T.A. [2006] Rock physics modeling of shale digenesis. Petroleum Geoscience, 12, 49-57. Hantchel, T. and Kauerauf, A. [2009] Fundamentals of petroleum system modeling. Springer.