Well-log interpolation

One of the commonly used methods for velocity inversion is the iterative method. To start the inversion process, one needs to have an initial velocity model. The updated velocity model is obtained by perturbing the previous velocity model in a way. To control the stability of the inversion process, one can set up a constraint to penalize large perturbations in the the new velocity model from a predetermined model. Usually a constant velocity model or a smooth velocity model extracted from data is used for both the initial model and the predetermined model. Actually, it would be nice that these models are constructed from some independent and hard data, such as sonic log measurements from wells that are located along the seismic survey lines.

To create a 2-D velocity model from sonic logs, one needs to do interpolation. Wells are often drilled with large separating distances. Data collected with such a large sample interval are often believed to be seriously aliased. Is it possible to interpolate this kind of data? The answer is positive if we are interested in having a smooth 2-D velocity model. Because smooth sonic logs are non-Gaussian, they can be interpolated with a non-linear method.

The next question one may ask is how the interpolation should be done. My idea is that this kind of data should be interpolated along its contour lines because data samples along these trajectories have constant values. Based on this idea, I developed a two-step algorithm to interpolate this kind of data. First, I use the known well logs to predict the contour lines of data. Then I interpolate data along these contour lines. Predicting contour lines for such sparsely sampled data is an underdetermined problem. I use a model to parameterize the contour lines. If only two well logs are available, the contour lines are straight lines. If three or more well logs are available, then the contour lines are modeled by cubic-spline functions. These models are parameterized by the depth positions of the contour lines at the locations of the known wells. Once we find these depth positions, we can generate the whole set of contour lines.

Suppose a contour has a depth position z at well 1 and a depth position $z+\Delta z$ at well 2. Then the log sample u₁(z) at well 1 should be equal to the log sample $u_2(z+\Delta z)$ because they are on the same contour line. Therefore, we can estimate $\Delta z$by minimizing the difference between u₁(z) and $u_2(z+\Delta z)$.Obviously, $\Delta z$ is a function of z. We can estimate the function $\Delta z(z)$ through the constrained non-linear optimization as we did for dip estimation. The constraint for this problem is that the contour lines do not cross each other. When three or more well logs are available, $\Delta z(z)$ is estimated for each pair of well logs. From these functions, I generate all contour lines and interpolate data along these contour lines.

I used a synthetic model to test my interpolation algorithm. The top panel of Figure shows a synthetic slowness model. The model has fairly complicated subsurface structure. The interval slowness within each layer is constant. The middle panel of the figure shows seven slowness profiles that simulates the well logs measured from the synthetic model and with a large separating distance. The bottom panel of Figure displays the slowness model interpolated from those slowness profiles. The interfaces of the original model are plotted on the same panel. The algorithm does a very good job. The interpolated model is very close to the original model except at the pinch-out.

 

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Figure 4 A synthetic example. Top: a slowness model. Middle: 7 well logs. Bottom: the interpolated slowness model.

Now I show some field data examples. Figure a displays three smoothed sonic logs. The distances between wells are several kilometers. Figure b shows the result of interpolation. The interpolated model is smooth. It has a well-defined high velocity layer at depth of 3 km and a well-defined low velocity layer at the depth of 4 km. Unfortunately, I am not able to check the accuracy of this result.

The well logs shown in Figure are collected from two wells between which a cross-well experiment was conducted. The middle panel of this figure displays the interpolated slowness model. The dashed lines on this panel are the geological structures interpreted from an independent source (Harris et al., 1990). It is clear that the two results agree fairly well.

 

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Figure 5 Interpolation of sonic logs: (a) two sonic logs; (b) interpolated slowness model

 

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Figure 6 Interpolation of a sonic log. Left: well log A. Right: well log B. Middle: interpolated slowness model.