Missing log data estimation

Building on our previous work (Bader et al., 2018), we estimate a complete sonic log using all other sonic logs in the dataset and compare it against the actual sonic log from the well. These results are also compared against a conventional approach for estimating missing sonic logs. We then extend the approach to honor true well log values for well logs that have incomplete or partial well logs.

There are several potential sources of information that can be used to constrain the estimation of missing log data: (1) the same well log type at other well locations, (2) other well logs within the same well, and (3) the seismic data. We focus on using other well logs of the same type in our estimation of a missing log. Generally, we include information from all other wells in our estimation:

$$\begin{bmatrix}\boldsymbol{W_1} \\ \boldsymbol{W_2} \\ \vdots \\ ... ...ol{\hat{l_2}} \\ \vdots \\ \boldsymbol{W_N}\boldsymbol{\hat{l_N}} \end{bmatrix}$$ (4)

where our estimated log, $\boldsymbol{\tilde{l}}$, is a weighted function of well logs from different wells denoted by the subscript $k$. If we simplify the prediction to one unknown log and one known log, Equation 4 simplifies to the following linear relationship:

$$W_k(z) \tilde{l}(z) \approx W_k(z) \hat{l_k}( S_k(z) ),$$ (5)

where $W_k(z)$ weights the specific value used to estimate the missing log value, $\tilde{l}(z)$, from available well log, $\hat{l_k}($$S_k(z)$$)$. To estimate a missing log at each depth sample, we must first remove structural and stratagraphic variations between the well logs by correlating the well logs to common geologic time using function $S_k(z)$, based on the shifts estimated from LSIM. The correlation is done by selecting a well log type that is available in all wells, for example, the gamma ray log. We then select one reference gamma ray log and estimate the function $S_k(z)$, that aligns all remaining gamma ray logs to the reference. $S_k(z)$ is applied to the remaining well logs to align all well logs (density, velocity, etc.) to constant geologic time.

We design the weight, $W_k(z)$, in Equation 5, as a product of two factors: the distance between the unknown and available well logs and the caliper value at that depth, which measures the size of the borehole at each depth. We assume that the borehole is drilled to be a specific diameter and deviations, measured by the caliper, from this anticipated borehole size likely indicates an inaccurate log measurement. Although many environmental factors may impact the well log data, for simplicity we weight the log values in our inversion based on caliper information. Thus, $W_k(z)$ can be expressed as:

$$W_k(z) = \phi(\vert x - x_k\vert)*C_k( S_k(z) ),$$ (6)

where $\phi(\vert x - x_k\vert)$ is a radial basis function, $x_k$ is the well location, $x$ is the well with a missing log, and $C_k$ is inversely proportional to the deviation between the expected and actual caliper value at each depth.

There are several different radial basis functions (Powell, 1985). We chose to implement the inverse mulitquadratic radial function

$$\phi(\vert x - x_k\vert) = \dfrac{1}{\sqrt{1 + (\epsilon \vert x - x_k\vert)^2}} \epsilon>0$$ (7)

which gives a larger weight to a well closer to the unknown well as compared to a well further away.

Returning to our original linear relationship, Equation 4, the estimated log, $\tilde{l}$, is a function of available well logs weighted by each well's distance and caliper log. By solving the least-squares problem in Equation 4, we can predict a new `pseudo well log' at each depth as follows:

$$\tilde{l(z)} = \frac{\sum\limits_{k=1}^N W_k^2(z) \hat{l_k}(\text{\new{$S_k(z)$}\old{$p_k(z)$}})}{\sum\limits_{k=1}^N W_k^2(z)}$$ (8)