Methodology for Variogram Interpretation and Modeling for Improved Reservoir Characterization

The variogram is a critical input to geostatistical studies. It is the most widely used tool to investigate and model spatial variability of lithofacies, porosity, and other petrophysical properties. In addition, 90% of geostatistical reservoir characterization studies use variogram-based geostatistical modeling methods. Furthermore, the variogram reflects our understanding of the geometry and continuity of reservoir properties and can have an important effect on predicted flow behavior and consequent reservoir management decisions. Yet, the practice of variogram interpretation and modeling is poorly documented and unexperienced practitioners find themselves at lost when required to provide a reliable variogram model. This often results in wrong variogram models used in subsequent geostatistical studies. Our approach is a two-step procedure similar to that used in modern well test interpretation, that is, model identification followed by parameter estimation. The total variance of the phenomenon under study is divided into variance regions. The behavior of each variance region is shown to follow clearly defined behaviors reflecting well-understood geological features. Establishing a mathematically consistent and geologically interpretable variogram model is straightforward after model recognition. The proposed methodology for variogram interpretation and modeling provides a better, more rigorous, quantification of spatial variability, which leads to improved flow models and management decisions. The theoretical background of our methodology will be presented. A number of case studies are then shown to illustrate the practical importance of the methodology. Introduction The variogram has been widely used to quantify the spatial variability of spatial phenomena for many years; however, calculation and interpretation principles have advanced slowly. This is particularly true in the petroleum industry due to the limited number of well data and limited resources to undertake detailed geostatistical studies. Time and budget constraints force reservoir modelers to proceed directly to facies and petrophysical property modeling almost immediately. The preliminary steps of variogram calculation, interpretation, and modeling are often performed hastily or even skipped altogether. This practice should be reversed and much more attention should be devoted to establish a robust model of spatial variability (variogram) before proceeding with building numerical reservoir models. The reservoir modeler can significantly influence the appearance and flow behavior of the final model through the variogram model. Reservoir modeling proceeds sequentially. Large-scale bounding surfaces are modeled, then facies, and then petrophysical properties such as porosity and permeability. The variogram is needed for stochastic modeling of surfaces and petrophysical properties; facies modeling, however, may be performed with object-based techniques, which do not require the use of a variogram. The importance and relevance of object-based methods do not diminish the importance of the variogram for a large fraction of reservoir modeling algorithms. The counterpart of the variogram in object-based methods is the size and shape specifications of the geological objects. We limit our consideration to variograms for geologic surfaces, facies indicator variables, and continuous petrophysical properties. Geostatistical model-building algorithms such as sequential Gaussian simulation, sequential indicator simulation, and truncated Gaussian simulation take an input variogram model and create a 3-D model constrained to local data and the variogram model. The variogram has an extremely important role to play in the appearance and flow behavior of 3-D models due to the sparse data available for petroleum reservoir characterization. The seemingly large volume of seismic data is at a large scale and must be supplemented by the variogram to control smaller scale variations. SPE 56654 Methodology for Variogram Interpretation and Modeling for Improved Reservoir Characterization E. Gringarten, SPE, Landmark Graphics Corp., and C. V. Deutsch, SPE, University of Alberta 2 E. GRINGARTEN, C. V. DEUTSCH SPE 56654 Thorough variogram interpretation and modeling are important prerequisites to 3-D model building. The practice of variogram modeling and the principle of the Linear Model of Regionalization have been covered in many text (e.g. Refs. 1-4). However, none have presented a strict and rigorous methodology to easily and systematically produce a licit and consistent 3D-variogram model. We present a methodology of variogram interpretation and modeling whereby the variance is divided into a number of components and explained over different length scales in different directions. The paper defines the variogram and its idealized behavior, and shows a small flow simulation study to illustrate the importance of the variogram in geostatistical operations. A number of real variograms are shown with a consistent interpretation. The Variogram The variogram has been defined in many books and technical papers. For completeness, however, we recall the definition of the variogram and related statistics. Consider a stationary random function Y with known mean m and variance σ. The mean and variance are independent of location, that is, m(u) = m and σ(u) = σ for all location vectors u in the reservoir. Often there are areal and vertical trends in the mean m, which are handled by a deterministic modeling of the trend and working with a residual from the locally variable mean. The variogram is defined as: ( ) ( ) ( ) [ ] { } 2 2 h u u h + − = Y Y E γ (1) In words, the variogram is the expected squared difference between two data values separated by a distance vector h. The semivariogram γ(h) is one half of the variogram 2 γ(h). To avoid excessive jargon we simply refer to the variogram, except where mathematical rigor requires a precise definition. The variogram is a measure of variability; it increases as samples become more dissimilar. The covariance is a statistical measure that is used to measure correlation (it is a measure of similarity): ( ) ( ) ( ) [ ] { } 2 m Y Y E C − + ⋅ = h u u h (2) By definition, the covariance at h=0, C(0), is the variance σ . The covariance C(h) is 0.0 when the values h-apart are not linearly correlated. Expanding the square in equation (1) leads to the following relation between the semi-variogram and covariance: ( ) ( ) ( ) ( ) h h h h γ γ − = − = ) 0 ( ) 0 ( C C or C C (3) This relation depends on the model decision that the mean and variance are constant and independent of location. These relations are the foundation for variogram interpretation. That is, (1) the “sill” of the variogram is the variance, which is the variogram value that corresponds to zero correlation, (2) the correlation between Y(u) and Y(u+h) is positive when the variogram value is less than the sill, and (3) the correlation between Y(u) and Y(u+h) is negative when the variogram exceeds the sill. This is illustrated in Fig. 1, which shows three h-scatterplots corresponding to three lags on a typical semivariogram. Geostatistical modeling generally uses the variogram instead of the covariance for purely historical reasons. A single variogram point γ(h) for a particular distance and direction h is straightforward to interpret and understand. Practical difficulties arise from the fact that we must simultaneously consider many lag vectors h, that is, many distances and directions. The variogram is a measure of “geological variability” versus distance. The “geologic variability” is quite different in the vertical and horizontal directions; there is typically much greater spatial correlation in the horizontal plane. However, the well-known principle in geology known as Walther’s law entails that the vertical and horizontal variograms are dependent. There are often common features such as short scale variability (nugget effect) and variogram shape. Although there are similarities between the vertical and horizontal variogram, we often face confounding geologic features such as areal and vertical trends, cyclicity, and stratigraphic continuity across the areal extent of the reservoir. Understanding Variogram Behavior The link between geological variations in petrophysical properties and observed variogram behavior must be understood for reliable variogram interpretation and modeling. Figs. 2(a)-(c) show three geologic images and corresponding semivariograms in the vertical and horizontal directions for each image. In practice, we do not have an exhaustive image of the reservoir and the variogram behavior must be interpreted and related to geological principals from directional variograms. The primary variogram behaviors are: Randomness or lack of spatial correlation: certain geological variations appear to have no spatial correlation. These random variations are the result of deterministic depositional processes. At some scales, however, the processes are highly non-linear and chaotic, leading to variations that have no spatial correlation structure. Typically, only a small portion of the variability is explained by random behavior. For historical reasons, this type of variogram behavior is called the nugget effect. Decreasing spatial correlation with distance: most depositional processes impart spatial correlation to facies, porosity, and other petrophysical properties. The magnitude of spatial correlation decreases with separation distance until a distance at which no spatial correlation exists, the range of correlation. In all real depositional cases the length-scale or range of correlation depends on direction, that is, the vertical range of correlation is much less than the horizontal range due to the larger lateral distance of deposition. Although the correlation range depends on distance, the nature of the decrease in correlation is often the same in different directions. The reasons for this similarity are the same reasons that underlie Walther’s Law. This type of variogram behavior is called geometric anisotropy. SPE 56654 METHODOLOGY FOR VARIOGRAM INTERPRETATION AND MODELING FOR IMPROVED RESERVOIR CHARACTERIZATION 3 Geologic trends: virtually all geological processes impart a trend in the petrophysical property distribution, for example, fining or coarsening upward or the systematic decrease in reservoir quality from proximal to distal portions of the depositional system. Such trends can cause the variogram to show a negative correlation at large distances. In a fining upward sedimentary package, the high porosity at the base of the unit is negatively correlated with low porosity at the top. The large-scale negative correlation indicative of a geologic trend show up as a variogram that increases beyond the sill variance σ. As we will see later, it may be appropriate to remove systematic trends prior to geostatistical modeling. Areal trends have an influence on the vertical variogram, that is, the vertical variogram will not encounter the full variability of the petrophysical property. There will be positive correlation (variogram γ(h) below the sill variance σ) for large distances in the vertical direction. This type of behavior is called zonal anisotropy. A schematic illustration of this is given in Fig. 3. Stratigraphic layering: there are often stratigraphic layer-like features or vertical trends that persist over the entire areal extent of the reservoir. These features lead to positive correlation (variogram γ(h) below the sill variance σ) for large horizontal distances. Although large-scale geologic layers are handled explicitly in the modeling; there can exist layering and features at a smaller scale than cannot be handled conveniently by deterministic interpretation. This type of variogram behavior is also called zonal anisotropy because it is manifested in a directional (horizontal) variogram that does not reach the expected sill variance. Geologic cyclicity: geological phenomenon often occur repetitively over geologic time leading to repetitive or cyclic variations in the facies and petrophysical properties. This imparts a cyclic behavior to the variogram, that is, the variogram will show positive correlation going to negative correlation at the length scale of the geologic cycles going to positive correlation and so on. These cyclic variations often dampen out over large distances, as the size or length scale of the geologic cycles is not perfectly regular. For historical reasons, this is sometimes referred to as a hole-effect. Real variograms almost always reflect a combination of these different variogram behaviors. Considering the three images and their associated variograms presented on Fig. 4, we see evidence of all the behaviors mentioned above: nugget effect most pronounced on the top image, geometric anisotropy and zonal anisotropy on all, a trend on the middle one, and cyclicity most pronounced on the bottom image. The top image is an example of migrating ripples in a man-made eolian sandstone (from the U.S. Wind Tunnel Laboratory), the central image is an example of convoluted and deformed laminations from a fluvial environment. The original core photograph was taken from page 131 of Ref. 5. The bottom image is a real example of large-scale cross laminations from a deltaic environment. The original photograph was copied from page 162 of Ref. 5. The intent of this paper is to present a systematic procedure for variogram interpretation and modeling of real geological features. Requirement for a 3-D Variogram Model All directional variograms must be considered simultaneously to understand the variogram behavior. The experimental variogram points are not used directly in subsequent geostatistical steps; a parametric variogram model is fitted to the experimental points. A detailed methodology for this fitting is a central theme of this paper. There are a number of reasons why experimental variograms must be modeled: The variogram function γ(h) is required for all distance and direction vectors h within the search neighborhood of subsequent geostatistical calculations; however, we only calculate the variogram for specific distance lags and directions (often, only in the principle directions of continuity). There is a need to interpolate the variogram function for h values where too few experimental data pairs are available. In particular, the variogram is often calculated in the horizontal and vertical directions, but geostatistical simulation programs require the variogram in off-diagonal directions where the distance vector simultaneously contains contributions from the horizontal and vertical directions. There is also a need to introduce geological information regarding anisotropy, trends, sampling errors and so on in the model of spatial correlation. As much as possible, we need to filter artifacts of data spacing and data collection practices and make the variogram represent the true geological variability. Finally, we must have a variogram measure γ(h) for all distance and direction vectors h that has the mathematical property of positive definiteness, that is, we must be able to use the variogram, or its covariance counterpart, in kriging and stochastic simulation. A positive definite model ensures that the kriging equations can be solved and that the kriging variance is positive, in other words, a positive definite variogram is a legitimate measure of distance. For these reasons, geostatisticians have fit variograms with specific known positive definite functions like the spherical, exponential, Gaussian, and hole effect variogram models. It should be mentioned that any positive definite variogram function could be used, including tabulated variogram or covariance values. The use of any arbitrary function or nonparametric table of variogram values would require a check to ensure positive definiteness. In general, the result will not be positive definite and some iterative procedure would be required to adjust the values until the requirement for positive definiteness is met. With a “correct” variogram interpretation, the use of traditional parametric models is not limiting. In fact, the traditional parametric models permit all geological information to be accounted for and realistic variogram behavior to be fit. Moreover, the use of traditional variogram models allows straightforward transfer to existing geostatistical simulation codes. 4 E. GRINGARTEN, C. V. DEUTSCH SPE 56654 Importance of the Variogram in Geostatistics The variogram is used by most geostatistical mapping and modeling algorithms. Object-based facies models and certain iterative algorithms, such as simulated annealing, do not use variogram. Conservatively, 80% of all geostatistical reservoirmodeling studies involve the use of the variogram for building one or all of the facies, porosity, and permeability models. All Gaussian simulation algorithms including sequential Gaussian simulation and p-field simulation, indicator simulation methods such as sequential indicator simulation and the Markov-Bayes algorithm, and most implementations of simulated annealing-based algorithms rely on the variogram. Technical papers presented at the SPE annual meetings present a glimpse at current practice in the petroleum industry. In 1997, there were 76 papers related to the reservoir, one third concerned geostatistical techniques, and 90% of these involved modeling procedures that required a variogram model, yet very little emphasis was made on the importance and impact of variogram modeling. The 1998 report contains 73 papers related to formation evaluation of which 18% considered geostatistical techniques. 70% of those required the use of a variogram, the rest dealt with fracture/fault modeling and one with channel modeling, but note that for completeness, a reservoir model very often requires porosity and permeability simulations which would entail the use of a variogram. Not only is the variogram used extensively, it has a great effect on predictions of flow behavior. The flow character of a model with a very short range of correlation is quite different from a model with a long range of correlation. Occasionally there are enough well data to control the appearance and flow behavior of the numerical models; however, these cases are infrequent and of lesser importance that the common case of sparse well control. The available well data are too widely spaced to provide effective control on the numerical model. Seismic and historical production data provide large scale spatial constraints. The variogram provides the only effective control on the resulting numerical models. The lack of data, which makes the variogram important, also makes it difficult to calculate, interpret, and model a reliable variogram. Practitioners have been aware of this problem for some time with no satisfactory solution. Analogue data available from better-drilled fields, outcrop studies, or geological process modeling provide valuable input; however, this data must be merged with field-specific calculations to be useful. A 3-D variogram model is often assembled with a combination of field data and analogue information for horizontal correlation ranges. Variogram modeling is important and the “details” often have a crucial impact on prediction of future reservoir performance. In particular, the treatment of zonal anisotropy is important. Recall that zonal anisotropy is where the variogram “sill” appears different in different directions. Consider a 2D cross-section through an oil reservoir with a gas cap. We will take a horizontal well in the oil leg and perform flow simulation to assess gas breakthrough. Furthermore, we will consider that there are significant areal variations in average porosity and, hence, permeability. These areal variations translate to a zonal anisotropy where the vertical variogram reaches a lower sill than the horizontal direction. Case A will be the correctly modeled zonal anisotropy and Case B will consist of mistakenly treating the zonal anisotropy as a geometric anisotropy where only the range is different in the vertical and horizontal directions. Examples of the resulting permeability fields are shown in Figs. 5(a) and (b) for Cases A and B respectively, note the stronger areal trends visible on Fig. 5(a). The VIP flow simulator was used to model the performance of the horizontal well in terms of oil production and timing of gas breakthrough. Figs. 5(c) and (d) show oil saturation distributions after 100 time steps for the permeability fields of Figs. 5(a) and (b) respectively. To avoid interpreting an artifact of one realization, multiple (100) stochastic simulations are considered. The distributions of the time needed to reach a GOR of 0.6 (just after breakthrough) are shown in Figs. 5(e) and (f) for Cases A and B respectively. The spread in responses for Case A is much greater than for Case B, the variance is 4 times greater. The mean of the responses is different by 10 days. Even through this simple illustrative example, one can see the impact the variogram model may have on consequent reservoir management decisions. There are other bad practices in variogram interpretation and usage. For example, a systematic vertical or horizontal trend in sometimes included in the variogram as a long range structure; however, it should be handled explicitly by working with data residuals or considering a form of kriging that explicitly accounts for the trend. In such case, we require the variogram of the data residuals in place of the original Z variogram, see later section on Removing the Trend. Variogram modeling is more than just picking vertical and horizontal ranges. Care must be taken to account for zonal anisotropy, trends, and cyclic geologic variations. We develop a systematic procedure that promotes easy interpretation and