REN R 690 Lab Geostatistics Lab

Transcription

1 REN R 690 Lab Geostatistics Lab The objective of this lab is to try out some basic geostatistical tools with R. Geostatistics is used primarily in the resource and environmental areas for estimation, uncertainty quantification and integrating data with difference volumetric supports, precisions and sources. This lab will introduce you to variograms and kriging for spatial data using the R package gstat. These concepts will be introduced using a 2D data set from a West Texas oil deposit. This is a nice small data set consisting of 62 wells from a carbonate-siltstone reservoir in West Texas. It includes measurements of porosity (void fraction) and permeability (measure of how easily fluid flows through the rock). The X and Y coordinates in the data set correspond to Easting and Northing values (in ft), respectively. We are interested in mapping porosity for this reservoir since this directly correlates to the oil in place. If you have time at the end, you can try the same procedure with permeability since this directly correlates with our ability to extract the oil. Even better, you could apply this procedure to some of your own spatial data! 1. Getting Started Download the dataset 2dwelldata.csv. Load this data set into R and check that it imported correctly. welldata = read.csv("2dwelldata.csv") fix(welldata) attach(welldata) Let s have a look at a map of the well locations. We can plot a simple map of where the wells were drilled. plot(y~x, xlab="easting",ylab="northing") This isn t very interesting though. So we could colour this map by porosity. The porosity varies between about 4 and 12%. To make a plot coloured by porosity where the high porosity rock is coloured red and the low porosity rock is coloured yellow, a set of possible commands would be: mycol=seq(12,4,-0.01) mycol=heat.colors(length(mycol))[rank(mycol)] porcol=round((porosity-4.0)*100.) porcol=mycol[porcol] plot(y~x,pch=16,col=porcol,xlab="easting", ylab="northing") Jared Deutsch

2 Northing You can colour your map anyway you wish. A map of porosity colored with this scheme: Easting The high porosity rock is all in the East of the area and concentrated in the North-East of the reservoir now the cluster of wells in the North-East makes sense. A large number of wells were drilled in the high porosity region. 2. Variograms Download and install the package gstat. This is an R package with basic geostatistics functionality. Install then load the package in R. install.packages('gstat') library(gstat) This also loads the sp package which is a spatial data frame package. To run the gstat library with this data, the data needs to be coerced into a spatial data frame. This is done by first assigning the coordinates as a set of spatial points and then adding on the porosity data. R code to do this: xyspatial=spatialpoints(cbind(x,y)) porspatial=data.frame(porosity) spatialdata=spatialpointsdataframe(xyspatial,porspatial) Check the spatial data to make sure that it assembled the data correctly. The first few rows should look like: > spatialdata Jared Deutsch

3 coordinates Porosity 1 (9856, 5652) (9767, 4271) (8062, 9321) A variogram can be calculated with gstat using the command: porvario=variogram(porosity~1,spatialdata) This is an omnidirectional (isotropic) variogram. This means that we are assuming that the spatial variability is the same in all directions. This is fine for this exercise although we can tell by the map that this is not the case! The reservoir is much more continuous in the North-South direction than the East-West direction. A variogram plot should always include a line with the sill. The sill is the point at which there is no spatial correlation. This value corresponds to the data variance. Figure from University of Alberta MIN E 310 Lecture Notes We can now calculate the variogram sill (variance) and plot the variogram. One quick note: the terms variogram and semivariogram are used interchangeably. Technically the semivariogram (what we are calculating) is the variogram value divided by 2, but since we always divide by 2 to calculate the semivariogram they are used interchangeably. porsill=var(porosity) plot(porvario$dist,porvario$gamma,xlim=c(0,8000),ylim=c(0,porsill+1),xlab="distance (ft)",ylab="semivariogram") abline(h = porsill) text(6000,porsill+0.2,paste("sill =",round(porsill,3))) Jared Deutsch

4 Semivariogram The abline command with h tells R to plot a horizontal line at the sill value. We are adding text with the sill value above; the paste command tells R to concatenate the strings. Your plot should look like: Sill = Distance (ft) Right now we know experimental variogram values at a few specific distances but we need to model this variogram so that we know the variogram values at all distances. This means that we need to determine: The nugget effect The variogram shape The variogram contribution The variogram range For simple cases like this, it is a good idea to pick the nugget effect yourself based on your knowledge of the variable. Recall that the nugget effect can be thought of as the y-axis intercept. In this case, the nugget effect looks pretty low. We might estimate a nugget effect of about 0.3. There are a number of permissible variogram models the reason we need to pick a defined variogram model rather than fitting the curve with any function is that the calculated covariance matrix must be positive definite. With the defined variogram models (spherical, exponential, Gaussian) this is always true. If we were to use another function this might not be the case. Here we could choose the spherical variogram model. Jared Deutsch

5 The variogram contribution should sum with the nugget effect to the data variance (sill). Since we picked a nugget effect of 0.3, the variogram contribution would be = If you picked a different nugget effect then adjust your variogram contribution accordingly. The variogram range is the point at which the variogram reaches the sill. It looks like this occurs at around 8000 ft, but we can use the variogram fitting function in the gstat package to help us pick the range. The variogram range is determined by: porvm=fit.variogram(porvario, model = vgm(3.379, "Sph", 8000, 0.3), fit.sills=false) You can see the values we determined in the variogram model function is our guess for the range. This function will try and fit the range using 8000 as a starting point. We can look at the calculated range by: > porvm model psill range 1 Nug Sph So it fit a range of 8769 ft. This means that our variogram model equation is zero for a distance of zero (no variability at zero distance!) and is the nugget effect plus our spherical variogram model equation for distances larger than 0: ( ) { [ ( ) ( ) ] We can now plot the variogram model and the experimental points together to check if our fit is reasonable. There is a built in plotting capability in gstat, but the plots aren t that pretty so we can do this ourselves given the above function for the spherical variogram model. If you still have your variogram plot open you can add a plot of the model with: curve( *(1.5*(x/8769)-0.5*(x/8769)^3),add=true) You should now have a reasonable variogram model that looks like: Jared Deutsch

6 Semivariogram Sill = Distance (ft) 3. Kriging We can now use our variogram model to estimate porosity over the entire area by kriging. To do this we need a list of locations at which we are going to estimate. We can do this by creating a regular grid which paves the area. Look back at the area quickly. The area is could be summarized as spanning Easting (X) values of 0 to ft and Northing (Y) values of 0 to ft. We could consider estimating a grid where the cells were 250 ft by 250 ft. This would mean we would have 42 cells each in the X and Y directions. The procedure for generating this grid is then: gt = GridTopology(cellcentre.offset=c(125,125), cellsize=c(250,250), cells.dim=c(42,42)) grd=spatialgrid(gt) The generated grid can be checked: > summary(grd) Object of class SpatialGrid Coordinates: min max [1,] [2,] Is projected: NA proj4string : [NA] Jared Deutsch

7 Grid attributes: cellcentre.offset cellsize cells.dim We will use simple kriging. This means that we have to provide a mean (the normal procedure for doing this is declustering, but we will use a naïve mean here). The mean is: > mean(porosity) [1] To krige we provide the variable we are kriging, porosity data, the grid of points to estimate, the variogram model and the mean (beta parameter here). The results can be plotted using the spatial plotting utility. krigedpor=krige(formula=porosity~1, spatialdata, grd, model=vgm(3.379, "Sph", 8769, 0.3), beta=8.402) spplot(krigedpor["var1.pred"], cuts=length(mycol), col.regions=mycol) Kriging is exact so it reproduces the data points exactly. We should have a look and check that this is the case. To do this we first extract the kriged estimates: krigedpoints=cbind(coordinates(grd),krigedpor$var1.pred) Jared Deutsch

8 Northing We can use the same level plot from the lattice library to plot this (we did this in an earlier lab). Install and load the lattice library if you need to. install.packages('lattice') library(lattice) We can make a level plot using the same colors as before: levelplot(krigedpoints[,3]~krigedpoints[,1]*krigedpoints[,2], cuts=length(mycol), col.regions=mycol,xlab="easting", ylab="northing",groups=1) To add the points, we can use the lattice trellis method: trellis.focus("panel", 1, 1, highlight=false) panel.xyplot(x,y,pch=21,cex=1.3,fill=porcol) trellis.unfocus() Easting We see a transition from low to high porosity values moving to the East and all the data values are reproduced. You should see that in the far South East corner where there is very little data, the estimated values tend towards the mean. Now if you have time you could try the same procedure with the permeability data and see if you can estimate permeability over the area. Jared Deutsch