A New Data-Space Inversion Procedure for Efficient Uncertainty Quantification in Subsurface Flow Problems

Abstract

Uncertainty quantification for subsurface flow problems is typically accomplished through model-based inversion procedures in which multiple posterior (history-matched) geological models are generated and used for flow predictions. These procedures can be demanding computationally, however, and it is not always straightforward to maintain geological realism in the resulting history-matched models. In some applications, it is the flow predictions themselves (and the uncertainty associated with these predictions), rather than the posterior geological models, that are of primary interest. This is the motivation for the data-space inversion (DSI) procedure developed in this paper. In the DSI approach, an ensemble of prior model realizations, honoring prior geostatistical information and hard data at wells, are generated and then (flow) simulated. The resulting production data are assembled into data vectors that represent prior ‘realizations’ in the data space. Pattern-based mapping operations and principal component analysis are applied to transform non-Gaussian data variables into lower-dimensional variables that are closer to multivariate Gaussian. The data-space inversion is posed within a Bayesian framework, and a data-space randomized maximum likelihood method is introduced to sample the conditional distribution of data variables given observed data. Extensive numerical results are presented for two example cases involving oil–water flow in a bimodal channelized system and oil–water–gas flow in a Gaussian permeability system. For both cases, DSI results for uncertainty quantification (e.g., P10, P50, P90 posterior predictions) are compared with those obtained from a strict rejection sampling (RS) procedure. Close agreement between the DSI and RS results is consistently achieved, even when the (synthetic) true data to be matched fall near the edge of the prior distribution. Computational savings using DSI are very substantial in that RS requires $O(10^5$–$10^6)$ flow simulations, in contrast to 500 for DSI, for the cases considered.

Appendix: Detailed Mapping Operations

In this Appendix, the general pattern-based mapping operations applied in this study are described. For generality and simplicity of notation, we describe the mapping operations for an ensemble of time-series functions $y_i(t)~(i=1,2,\dots , N_{{\text {r}}})$. Note that the data values in $({\mathbf {d}}_{{\text {full}}})_i$, discussed in Sect. 3.1, are simply the values of these time-series functions at different time steps.

Fig. 20

Illustration of mapping operations. a Original functions, b functions after mapping

It is assumed that the functions can all be separated into the same number of stages, with the same general behavior within each stage. For example, the functions in Fig. 20a can be separated into four stages: constant, decline, constant and decline. The times separating the different stages are referred to as transition times. The goal is then to map these functions to those shown in Fig. 20b, in which the corresponding transition ‘times’ for all functions are the same.

The overall starting and ending times, $t_{{\text {s}}}$ and $t_{{\text {e}}}$, are assumed to be the same for all functions. A total of $M+1$ stages are then identified ($M=3$ for the cases shown in Fig. 20a), and the corresponding transition times are designated $t_i^j~(i=1,2,\dots ,N_{{\text {r}}}; j=1,2,\dots , M)$. Defining $t_i^0=t_s$ and $t_i^{M+1}=t_e$, the mapped functions are then constructed as

$$\begin{aligned} \hat{y}_i(\tau )=y_i\left( t_i^j+(t_i^{j+1}-t_i^j)\frac{\tau -\tau ^j}{\tau ^{j+1}-\tau ^j}\right) , \quad \tau ^j \le \tau \le \tau ^{j+1}, \ \ 0 \le j \le M, \end{aligned}$$

(30)

where $\hat{y}_i(\tau )$ denotes the mapped function for member i in the ensemble, and $\tau ^j$ is the transition ‘time’ for all mapped functions (note $\tau ^j$ is the same for all functions). The values of $\tau ^j~(j=0,1,\dots ,M+1)$ must be predefined. In this paper, a particular transition ‘time’ is defined as the mean of the corresponding transition times for all of the original functions, i.e.,

$$\begin{aligned} \tau ^j = \sum _{i=1}^{N_{{\text {r}}}}\frac{t_i^j}{N_{{\text {r}}}} , \quad 0 \le j \le M + 1. \end{aligned}$$

(31)

Figure 20b shows the mapped functions corresponding to the original functions in Fig. 20a, with transition ‘times’ defined by Eq. (31).

The forward mapping operation for $y_i(t)$ is expressed as

$$\begin{aligned} {\mathscr {F}}: y_i(t) \rightarrow \hat{y}_i(\tau ),~t_i^1,~t_i^2,\dots ,t_i^M. \end{aligned}$$

(32)

The transition times $t_i^1$ to $t_i^{M}$ must also be included since they are needed for the backward mapping. Note that the values in the mapped data vectors $\widehat{{\mathbf {d}}}_i$, introduced in Sect. 3.1, are simply the values of $\hat{y}_i(\tau )$ at different ‘time’ steps, plus the identified transition times.

Once we construct a predicted mapped function, denoted by $\hat{y}_{{\text {p}}}(\tau )$, and its associated (predicted) transition times $t_{{\text {p}}}^1, t_{{\text {p}}}^2, \dots , t_{{\text {p}}}^M$, the backward mapping is given by

$$\begin{aligned} {\mathscr {F}}^{-1}: \hat{y}_{{\text {p}}}(\tau ),~t_{{\text {p}}}^1,~t_{{\text {p}}}^2,\dots ,t_{{\text {p}}}^M \rightarrow y_{{\text {p}}}(t), \end{aligned}$$

(33)

where

$$\begin{aligned} y_{{\text {p}}}(t)=\hat{y}_{{\text {p}}}\left( \tau ^j+(\tau ^{j+1}-\tau ^j)\frac{t-t_{{\text {p}}}^{j}}{t_{{\text {p}}}^{j+1}-t_{{\text {p}}}^{j}}\right) , \quad t_{{\text {p}}}^{j} \le t \le t_{{\text {p}}}^{j+1}, \ \ 0 \le j \le M. \end{aligned}$$

(34)

Here $\tau ^j$ are the pre-computed values from Eq. (31), and $t_{{\text {p}}}^0$ and $t_{{\text {p}}}^{M+1}$ are the known start and end simulation times. This completes the description of the forward and backward mapping operations used to ‘align’ the transitions between the various stages.

An alternate mapping approach, based on the use of histogram transformations, was introduced in Sun et al. (2016). That procedure is applicable when the production data do not display clear patterns, as is the case when wells are abruptly shut and opened at different times. Because the histogram transformations are applied independently for each data variable, the nonlinear correlations between data variables (as shown in Fig. 3e) might not be as effectively mitigated with this treatment. In future work, it will be of interest to compare results with the two approaches for systems that display clearly identifiable stages.