Conditioning Truncated Pluri-Gaussian Models to Facies Observations in Ensemble-Kalman-Based Data Assimilation

Abstract

The truncated pluri-Gaussian model is a powerful tool for representing realistic spatial distributions of facies in reservoir characterization. It is suitable for generating stochastic three-dimensional facies realizations with complex vertical and lateral relationships such as are observed in algal mound behavior. Truncated pluri-Gaussian realizations account for anisotropies and relative proportions of the facies. Despite their advantages, truncated pluri-Gaussian models have not been extensively used in data assimilation techniques such as ensemble-Kalman-based algorithms. One of the major limitations encountered in the existing implementations is the difficulty of preserving facies observations at well locations through the data assimilation procedure arising even for weakly correlated data. In this work, the problem of maintaining consistency of realizations with facies is solved by merging the data assimilation algorithm (Levenberg–Marquardt ensemble randomized maximum likelihood) with an interior-point method suitable of inequality constraints. The iterative ensemble smoother is effective at assimilating highly nonlinear production data, while the interior-point method takes into account the inequality constraints on the Gaussian model variables during the data assimilation. The formulation uses an objective function that includes a model-mismatch term, a data-mismatch term and a boundary penalization function. The method allows for its approximate version neglecting model-mismatch. The algorithm was tested on a three-dimensional synthetic reservoir mimicking the algal mounds shapes of an outcrop in the Paradox Basin of Utah and includes a large number of strongly correlated facies data and significant change of petrophysical properties for different facies. It resulted in good decrease in the data-mismatch while preserving the mound structure realism and variability in the ensemble.

Appendices

Appendix A: Barrier Function Derivatives

To compute the model update at each iteration, one needs to evaluate the derivatives and the Hessian of $\hat{S}(m)$ [Eq. (3)]

$$\begin{aligned} \Bigl (\frac{\partial ^2 \hat{S}}{\partial \mathbf {m}^2}\Bigr )_{\mathbf {m}_n} = \Bigl (\frac{\partial ^2 S}{\partial \mathbf {m}^2}\Bigr )_{\mathbf {m}_n} + t \Bigl (\frac{\partial ^2 f_\mathrm{b}}{\partial \mathbf {m}^2}\Bigr )_{\mathbf {m}_n}. \end{aligned}$$

Thus the first and the second order partial derivatives of $f_\mathrm{b}(m)$ [Eq. (5)] are developed as

$$\begin{aligned} \Biggr (\Bigl (\frac{\partial ^2 f_\mathrm{b}}{\partial \mathbf {m}^2} \Bigr )_{\mathbf {m}_n} \Biggr )_{\alpha \beta }&= \Bigl ( \frac{\partial ^2 f_\mathrm{b}}{\partial (m)_{\alpha } \partial (m)_{\beta }} \Bigr )_{\mathbf {m}_n}, \\ \Bigl (\frac{\partial f_\mathrm{b}}{\partial (m)_{\alpha }}\Bigr )_{\mathbf {m}_n}&= - \frac{1}{c_{\alpha }(\mathbf {m}_n)} \Bigl ( \frac{\partial c_{\alpha }}{\partial (m)_{\alpha }}\Bigr )_{\mathbf {m}_n}, \\ \Bigl (\frac{\partial ^2 f_\mathrm{b}}{(\partial (m)_{\alpha })^2}\Bigr )_{\mathbf {m}_n}&= \frac{1}{c_{\alpha }^2(\mathbf {m}_n)} \Bigl ( \frac{\partial c_{\alpha }}{\partial (m)_{\alpha }}\Bigr )^2_{\mathbf {m}_n} - \frac{1}{c_{\alpha }(\mathbf {m}_n)} \Bigl (\frac{\partial ^2 c_{\alpha }}{(\partial (m)_{\alpha })^2}\Bigr )_{\mathbf {m}_n}. \end{aligned}$$

If $c_\alpha (m) = (m_{\text {max,}\alpha } - (m)_{\alpha })( (m)_{\alpha } - m_{\text {min,}\alpha })$, then

$$\begin{aligned} \Bigl (\frac{\partial ^2 f_\mathrm{b}}{(\partial (m)_{\alpha })^2}\Bigr )_{\mathbf {m}_n}&= \frac{((m_{\text {max,}\alpha }-(m_n)_\alpha )-((m_n)_\alpha -m_{\text {min,}\alpha }) )^2}{(m_{\text {max,}\alpha } - (m_n)_\alpha )^2((m_n)_\alpha - m_{\text {min,}\alpha })^2}\nonumber \\&\quad \quad - \frac{-2}{(m_{\text {max,}\alpha } - (m_n)_\alpha )((m_n)_\alpha - m_{\text {min,}\alpha })} \nonumber \\&= \frac{1}{(m_{\text {max,}\alpha } - (m_n)_\alpha )^2} +\frac{1}{((m_n)_\alpha - m_{\text {min,}\alpha })^2}. \end{aligned}$$

For the two linear forms of $c_\alpha (m)$

$$\begin{aligned} \Bigl (\frac{\partial ^2 f_\mathrm{b}}{(\partial (m)_{\alpha })^2}\Bigr )_{\mathbf {m}_n} = \frac{1}{c^2_\alpha (\mathbf {m}_n)}. \end{aligned}$$

Note that $\frac{\partial ^2 f_\mathrm{b}}{\partial \mathbf {m}^2} ({\mathbf {m}_n})$ is a diagonal matrix, because for $\alpha \ne \beta$

$$\begin{aligned} \Bigl (\frac{\partial ^2 f_\mathrm{b}}{\partial (m)_{\alpha } \partial (m)_{\beta }}\Bigr )_{\mathbf {m}_n} = 0. \end{aligned}$$

Appendix B: Update Direction Implementation

The formula for the model update [Eq. (14)] with the interior-point penalty can be derived as follows

$$\begin{aligned} \widehat{\delta m_l}&= - \Delta m_l^e [(1+\lambda _l)I_e + {\Delta d_l^e }^\mathrm{T} \Delta d_l^e + t {\Delta b_l^e}^\mathrm{T} \Delta b_l^e]^{-1} {\Delta m_l^e}^\mathrm{T}\\&\times [ C_\mathrm{M}^{-1} (m_l - m_j^{\text {pr}}) + G_l^\mathrm{T} C_\mathrm{D}^{-1} (g(m_l)- d_{\text {obs,}j}) + t \nabla _{m_l}f_\mathrm{b}(m_l) ]\\&= -\Delta m_l^e [(1+\lambda _l)I_e + {\Delta d_l^e }^\mathrm{T} \Delta d_l^e + t{\Delta b_l^e}^\mathrm{T} \Delta b_l^e]^{-1} {\Delta m_l^e}^\mathrm{T} C_\mathrm{M}^{-1} (m_l - m_j^{\text {pr}})\\&- \Delta m_l^e [(1+\lambda _l)I_e + {\Delta d_l^e }^\mathrm{T} \Delta d_l^e + t {\Delta b_l^e}^\mathrm{T} \Delta b_l^e]^{-1} {\Delta m_l^e}^\mathrm{T} ({\Delta m_l^e}^\mathrm{T})^{-1} \\&\times {\Delta d_l^e}^\mathrm{T} C_\mathrm{D}^{1/2} C_\mathrm{D}^{-1}(g(m_l)- d_{\text {obs,}j}) - t \Delta m_l^e [(1+\lambda _l)I_e + {\Delta d_l^e }^\mathrm{T} \Delta d_l^e \\&+ t{\Delta b_l^e}^\mathrm{T} \Delta b_l^e]^{-1} {\Delta m_l^e}^\mathrm{T} \nabla _{m_l}f_\mathrm{b}(m_l)=- \Delta m_l^e [(1+\lambda _l)I_e + {\Delta d_l^e }^\mathrm{T} \Delta d_l^e\\&+ t {\Delta b_l^e}^\mathrm{T} \Delta b_l^e]^{-1} {\Delta m_l^e}^\mathrm{T} U_{m_0}^{p_{m_0}} (W_{m_0}^{p_{m_0}})^{-2} { U_{m_0}^{p_{m_0}}}^\mathrm{T} (m_l - m_j^{\text {pr}})\\&- \Delta m_l^e [(1+\lambda _l)I_e + {\Delta d_l^e }^\mathrm{T} \Delta d_l^e + t {\Delta b_l^e}^\mathrm{T} \Delta b_l^e]^{-1} {\Delta d_l^e}^\mathrm{T}C_\mathrm{D}^{-1/2} (g(m_l)- d_{\text {obs,}j})\\&- t \Delta m_l^e [(1+\lambda _l)I_e + {\Delta d_l^e }^\mathrm{T} \Delta d_l^e + t {\Delta b_l^e}^\mathrm{T} \Delta b_l^e]^{-1} {\Delta m_l^e}^\mathrm{T} \nabla _{m_l}f_\mathrm{b}(m_l). \end{aligned}$$