Sequential Value of Information for Subsurface Exploration Drilling

Abstract

Quantitative methods are needed for systematic decision-making during exploration for subsurface resources, but few methods exist that fully incorporate the interaction of geological, operational, and financial conditions. The sequential nature of planning where to drill for subsurface exploration is not commonly addressed by conventional techniques in a quantitative fashion, despite its foundational relevance to hypothesis testing. Value of information (VOI) can incorporate various aspects of subsurface exploration decision-making as well as sequence. Here, we use VOI to determine the optimal sequence and placement of exploration boreholes when varying conditions such as target resource volume and drilling cost. Using VOI, we show that the optimal placement and selection of exploration boreholes change when planning to drill one borehole at a time compared to planning to drill two boreholes sequentially. A formulation and tutorial explanation of VOI for sequential decision situations are shown using a synthetic case. We demonstrate a test case using data from a real metal deposit.

Appendices

Appendix 1

Conditioning the Prior Model with Data

The prior model consists of a set of $L$ discrete realizations (${{\varvec{m}}}^{\mathcal{l}}$), drawn from the same geological scenario. These prior realizations are unconditional and need to be conditioned to drilling observations (hard data). Scheidt et al. (2018) showed that if the conditioning problem can be expressed a linear inverse problem (Eq. 1), then there exist techniques such as the Bayes-linear-Gauss equation that can efficiently solve the conditioning problem. We start with a forward model of the data:

$${\varvec{d}} = G{\varvec{m}}\;\;{\text{where }}\;\;{\varvec{m}}\;\;{\text{is}}\;\;{\text{the}}\;{\text{disctete}}\;{\text{model}}\;{\text{and}}\;G\;\;{\text{is}}\;{\text{a}}\;\;{\text{linear}}\;\;{\text{operator}}$$

(25)

The linear operator $G$ is a matrix containing as many columns as grid cells in the model. The number of rows in $G$ is the number of hard data points (Fig. 

Figure 14

Linear operator G

14). The values in each cell of $G$ are whether the cell contains hard data—not the values of the hard data itself.

The high-dimensionality of the model ${\varvec{m}}$ necessitates the use of a dimensionality reduction technique. Any technique can be used in this approach, such as principal component analysis (PCA), or discrete cosine transform (DCT). These dimensionality reduction techniques express the model ${\varvec{m}}$ as:

$${\varvec{m}} = \varphi \alpha$$

where φ is a matrix containing the set of basis functions and α, a matrix of coefficients.

The linear relationship from Eq. 25 becomes:

$${\varvec{d}} = G_{2} \alpha \;\;{\text{with}}\;\;G_{2} = G_{\varphi }$$

Assuming that $\mathrm{\alpha }$ are normally distributed, Bayes-linear-Gauss equation can be applied to do the conditioning. If $\mathrm{\alpha }\sim N(\mu ,\Sigma )$, then ${\mathrm{\alpha }}_{\mathrm{post}}=\mathrm{\alpha }|{\varvec{d}}\sim N(\overline{\mu },\Sigma )$ with:

• $\overline{\mu } = \mu + \Sigma G_{2}^{^{\prime}} \left( {G_{2} \Sigma G_{2}^{^{\prime}} } \right)^{ - 1} \left( {{\varvec{d}}_{obs} - G_{2} \mu } \right)$, with ${{\varvec{d}}}_{obs}$ the observed drilling data

• $\overline{\Sigma } = \Sigma - \Sigma G_{2}^{^{\prime}} \left( {G_{2} \Sigma G_{2}^{^{\prime}} } \right)^{ - 1} G_{2} \Sigma$

A set of posterior coefficients α_post are then sampled using the above multi-variate normal distribution ($\bar{\mu },\;\bar{\Sigma }$). “Pre-posterior” realizations are then obtained by mapping back the posterior coefficients into the original space (by inverting the linear dimensionality reduction technique, PCA, DCT, etc.). Next, a threshold is applied to the pre-posterior realizations to generate the final, discrete posterior models:

$${\varvec{m}}_{{{\text{post}}}} = {\varvec{m}}_{{{\text{prepost}}}} > 0.5$$

This suite of discrete posterior models is then used as an input to the value of information calculation.

Appendix 2

Conditioning the Prior Model with Data with Ensemble Smoother

Ensemble smoother (van Leeuwen and Evensen, 1996; Emerick and Reynolds, 2013) is a data assimilation technique that uses an ensemble of models to perform model updating. In the case of linear forward modeling (${\varvec{d}}=G{\varvec{m}}$) and under the Gaussian assumption: ${\varvec{m}}\sim \mathrm{ N}({{\varvec{\mu}}}_{m},{\mathrm{C}}_{\mathrm{m}})$, the ensemble smoother updates each model of the prior ensemble using the following equation:

$${\varvec{m}}_{{{\text{pos}}}}^{\ell } = {\varvec{m}}^{\ell } { + }C_{{{\text{md}}}} *C_{{{\text{dd}}}}^{ - 1} \left( {{\varvec{d}}_{{{\text{obs}}}} - {\varvec{d}}^{\ell } } \right)$$

where ${\mathrm{C}}_{\mathrm{md}}$ represents the cross-covariance matrix between ${\varvec{m}}$ and ${\varvec{d}}$, and ${C}_{dd}$ the covariance matrix of ${\varvec{d}}$. Here, ${{\varvec{d}}}_{obs}$ is the observed data, and ${{\varvec{d}}}^{\mathcal{l}}$ is sampled from the prior model ${{\varvec{m}}}^{\mathcal{l}}.$

In the case of discrete models, additional prior transformations are needed to better meet the Gaussian assumption. One such idea is to apply a signed distance transform (SDT, Osher and Fedkiw, 2002; Hakim-Elahi and Jafarpour, 2017). The SDT is defined at grid cell each location by the signed distance to the boundary of the closest object $d(x)$:

$$SDT\left( x \right) = \left\{ {\begin{array}{*{20}c} { - d\left( x \right)} & {{\text{if }}x{\text{ is in the object}}} \\ {{ }d\left( x \right)} & {{\text{ if }}x{\text{ is outside the object}}} \\ 0 & {{\text{ if }}x{\text{ is at the boundary}}} \\ \end{array} } \right.$$

Because the SDT transformation is not linear, the linear forward model no longer applies (${\varvec{d}}\ne G*SDT({\varvec{m}}))$, hence we propose an additional sigmoid transformation making the relationship between data and transformed models $Sig\left(SDT\left({\varvec{m}}\right)\right)$ closer to linear (see Fig. 

Figure 15

(top) Example of a prior model (left) and its transformations: $SDT\left({\varvec{m}}\right)$ (middle) and $Sig\left(SDT\left({\varvec{m}}\right)\right)$ (right) with their relation to the data and the relation between $SDT\left({\varvec{m}}\right)$ and ${\varvec{d}}$ and $SDT\left({\varvec{m}}\right)$ and $Sig\left(SDT\left({\varvec{m}}\right)\right)$, respectively

15):

$$Sig\left( x \right) = { }\frac{1}{{1 + {\text{exp}}\left( { - {\text{x}}} \right)}}$$

The ensemble smoother is then applied on the non-linear transformation $Sig\left(SDT\left({\varvec{m}}\right)\right)$ Because these transformations are bijective, they can be undone to obtain the discrete ore body model.

Here, we demonstrate the necessity of the signed distance transform and sigmoid function for preparing the prior models before input to the ensemble smoother. In Figure 

Figure 16

A comparison of one realization of the prior model (top left) with a realization of the posterior model without using any transform (top right), only using the signed distance transform (bottom left), and using both the sigmoid and signed distance transforms (bottom right). The prior model was conditioned on one outcome of Borehole 61 for this demonstration

16, we show posterior models when the transforms are not performed to the posterior model using fully transformed data. When neither transform is performed, the posterior model closely matches the borehole data, but the posterior model does not appear realistic due to the outer “shell” of ore lithology which isn’t fully updated from the prior model. If only the signed distance transform is performed, the posterior model does not match the borehole data as closely, and appears over smoothed. When using both the sigmoid and distance transforms, the borehole data is closely matched and the posterior model maintains features from the prior model without over smoothing.

Table 5 shows the VOI of single boreholes when performing both the sigmoid and distance transformations, neither transformation, and the signed distance transform only. When performing neither transform, VOI is higher than the fully transformed case for all boreholes except BH3, where the VOI is equal in both cases. As shown in Figure 16, this is due to the improper updating of the prior model which leads to a much greater estimate of cells containing ore. Nevertheless, the ranking of boreholes BH61, BH21, and BH3 is preserved between both cases. When performing only the signed distance transform, the VOI for all boreholes is zero. This low VOI is due to the over smoothing shown in Figure 16.