Uncertainty quantification of overpressure buildup through inverse modeling of compaction processes in sedimentary basins

Abstract

This study illustrates a procedure conducive to a preliminary risk analysis of overpressure development in sedimentary basins characterized by alternating depositional events of sandstone and shale layers. The approach rests on two key elements: (1) forward modeling of fluid flow and compaction, and (2) application of a model-complexity reduction technique based on a generalized polynomial chaos expansion (gPCE). The forward model considers a one-dimensional vertical compaction processes. The gPCE model is then used in an inverse modeling context to obtain efficient model parameter estimation and uncertainty quantification. The methodology is applied to two field settings considered in previous literature works, i.e. the Venture Field (Scotian Shelf, Canada) and the Navarin Basin (Bering Sea, Alaska, USA), relying on available porosity and pressure information for model calibration. It is found that the best result is obtained when porosity and pressure data are considered jointly in the model calibration procedure. Uncertainty propagation from unknown input parameters to model outputs, such as pore pressure vertical distribution, is investigated and quantified. This modeling strategy enables one to quantify the relative importance of key phenomena governing the feedback between sediment compaction and fluid flow processes and driving the buildup of fluid overpressure in stratified sedimentary basins characterized by the presence of low-permeability layers. The results here illustrated (1) allow for diagnosis of the critical role played by the parameters of quantitative formulations linking porosity and permeability in compacted shales and (2) provide an explicit and detailed quantification of the effects of their uncertainty in field settings.

Résumé

Cette étude illustre une procédure conduisant à une analyse préliminaire du risque de développement de surpression dans un bassin sédimentaire caractérisé par l’alternance de niveaux de grès et de schiste argileux. L’approche repose sur deux éléments clef : (1) une modélisation préalable des écoulements et de la compaction et (2) une application d’une technique de réduction de la complexité du modèle basée sur un développement généralisé du polynôme du chaos (gPCE). Le modèle développé considère des processus de compaction verticale unidimensionnelle. Le modèle gPCE est ensuite utilisé en situation de modélisation inverse pour obtenir une estimation fiable des paramètres du modèle et une quantification de l’incertitude. La méthodologie est appliquée à deux cas de terrain décrits dans des publications antérieures, le Venture Field (Scotian Shelf, Canada), et le Basin de Navarin (Mer de Bering, Alaska, USA), reposant sur des données de porosité et de pression disponibles pour le calibrage du modèle. Le meilleur résultat est obtenu lorsque les données de porosité et de pression sont considérées simultanément dans la procédure de calibrage du modèle. La propagation de l’incertitude des paramètres d’entrée non connus sur les données de sortie du modèle, telles que la distribution verticale de la pression dans les pores, est étudiée et quantifiée. Cette stratégie de modélisation permet de quantifier l’importance relative du phénomène clef régissant l’interaction entre la compaction du sédiment et les processus d’écoulements, et conduisant à un excès de surpression hydraulique dans un bassin sédimentaire stratifié caractérisé par la présence de niveaux de faible perméabilité. Les résultats illustrés ici (1) permettent le diagnostic du rôle critique joué par les paramètres des équations reliant la porosité et la perméabilité dans des schistes argileux compactés et (2) fournissent une quantification explicite et détaillée des effets de leur incertitude pour des conditions naturelles de terrain.

RESUMEN

Este estudio ilustra un procedimiento apropiada para el análisis preliminar del riesgo de desarrollo de sobrepresión en cuencas sedimentarias que se caracterizan por la alternancia de eventos de deposición de capas de areniscas y lutitas. El enfoque se basa en dos elementos fundamentales: (1) el modelado directo del flujo del fluido y la compactación, y (2) la aplicación de una técnica de reducción de la complejidad del modelo basado en una expansión polinómica generalizada del caos (gPCE). El modelo directo considera procesos unidimensionales de compactación vertical. El modelo gPCE se utiliza en un contexto de modelado inverso para obtener la estimación de parámetros de un modelo eficiente y la cuantificación de la incertidumbre. La metodología se aplica a dos escenarios de campo considerados en la literatura de trabajos anteriores, es decir, Venture Field (Scotian Shelf, Canada) y la Cuenca del Navarín (Mar de Bering, Alaska, EEUU) basándose en la información de la porosidad y la presión disponible para la calibración del modelo. Se encontró que se obtiene un mejor resultado cuando los datos de porosidad y de presión se consideran conjuntamente en el procedimiento de calibración del modelo. Se investigaron y cuantificaron la incertidumbre de propagación de los parámetros desconocidos de entrada en los resultados del modelo, y la distribución vertical de la presión poral. Esta estrategia de modelado permite cuantificar la importancia relativa de los fenómenos claves que regulan la retroalimentación entre los procesos de flujo de fluido en la compactación de los sedimentos y la conducción y acumulación del exceso de presión de fluido en cuencas sedimentarias estratificadas que se caracterizan por la presencia de capas de baja permeabilidad. Los resultados ilustran aquí por (1) permitir el diagnóstico del papel fundamental desempeñado por los parámetros en las formulaciones cuantitativas que enlazan la porosidad y la permeabilidad en las lutitas compactadas y (2) proporcionar una cuantificación explícita y detallada de los efectos de la incertidumbre en la configuración de campo.

摘要

本研究详细阐明了砂岩和页岩层交替沉积的沉积盆地中超压发展的整个过程,这个过程有利于对超压发展进行初步风险分析。该方法依赖于两个关键要素:(1)流体流动和压实的正演模拟,及(2)基于概化多项式混沌膨胀(gPCE)的模型-复杂度还原技术。正演模型考虑了一维垂直压实过程。然后概化多项式混沌膨胀(gPCE)模型用于反演模拟环境中,以获取有效的模型参数估算结果和不确定性量化结果。该方法应用于过去文献中考虑到的两个野外情况下,即(加拿大Scotian大陆架)的Venture牧场及(美国阿拉斯加州白令海)的Navarin盆地,依靠可现有的孔隙率和压力信息进行模型校准。发现在模型校准程序中,综合考虑孔隙率和压力就能获得最好的结果。调查和量化了模型输出中未知输入参数诸如孔隙压力垂直分布导致的不确定性传播。这个模拟策略能使人们量化关键现象的相关重要性,这些关键现象控制着沉积压实和液体流动过程之间的反馈及推进存在低渗透性地层的分层沉积盆地中液体超压的逐渐积累。这里描述的的结果:(1)能够诊断连接压实页岩中孔隙率和渗透性的定量公式参数所化发挥的关键作用以及 (2)提供野外背景下不确定性影响的明确和详细的量化结果。

Resumo

Este estudo ilustra um procedimento útil para uma análise preliminar do risco de sobrepressão desenvolvido em bacias sedimentares caracterizadas por eventos alternados de deposição de camadas de arenitos e xisto. A abordagem apoia-se em dois elementos chave: (1) modelagem direta do fluxo de fluido e compactação; e (2) aplicação de uma técnica de redução de complexidade do modelo baseado em uma expansão polinomial generalizada do caos (qPCE). A modelagem direta considera um processo de compactação vertical unidimensional. O modelo qPCE é então usado em um contexto de modelagem inversa para obter um modelo eficiente de estimativa de parâmetros e quantificação de incertezas. A metodologia é aplicada em duas configurações de campo consideradas em obras de literatura anteriores, por exemplo, o Campo Venture (plataforma Escocesa, Canadá) e a Bacia Navarin (Mar de Bering, Alasca, EUA), contando com informações de porosidade e pressão disponíveis para calibração do modelo. Verificou-se que o melhor resultado é obtido quando os dados de porosidade e pressão são considerados em conjunto no processo de calibração do modelo. Propagação de incerteza para entrada de parâmetros desconhecidos para modelar os resultados, tais como a distribuição vertical da pressão dos poros, é investigada e quantificada. Esta estratégia possibilita uma modelagem para quantificar a importância relativa dos fenômenos chave que regulam a retroalimentação entre compactação do sedimento e o processo de fluxos de fluido e levam ao acúmulo de fluido em sobrepressão em bacias sedimentares estratificadas caracterizadas pela presença de camadas pouco permeáveis. Os resultados aqui ilustrados (1) permitem diagnósticos do papel crítico desempenhado pelos parâmetros de formulação quantitativos que ligam a porosidade e permeabilidade em xistos compactados e (2) fornecem uma quantificação explicita e detalhada dos efeitos da sua incerteza em configurações de campo.

Appendices

Appendix 1

Model reduction technique

Modeling overpressure development and porosity variations in sedimentary basins is subject to two major challenges: (1) the complexity of the processes involved, and (2) the high number of parameters which are difficult (if not impossible) to measure and/or are characterized by a strong variability. A complete compaction model is characterized by a system of partial differential equations which are fully coupled and strongly non-linear, the solution of which is associated with a significant computational cost. The numerical model includes a large number of parameters which are typically uncertain.

Reduction of the computational cost by taking into account parameter uncertainty can be accomplished by introducing a surrogate model of the full system model described by Eqs. (1)–(8). This surrogate model is expressed in the form of a polynomial approximation and is solely a function of the uncertain model parameters. For example, obtaining a single solution of the full model for the Navarin Basin is associated with a computational time of 83 s, the evaluation of the porosity profile by the surrogate model requiring only 3 s. Model calibration through parameter estimation requires performing multiple evaluations of the model outputs. Therefore, considering the surrogate (gPCE) model allows reducing computational costs especially during the inverse modeling phase: for example, the CPU time required for model inversion through the gPCE relying on an Intel(R) Core(TM) 2 CPU 6600 @ 2.40 GHz 2.31 GHz (RAM 2 GB, operational system 32 bit) is approximately 0.4 h when both porosity and pressure data are used for model calibration, 11.5 h being needed for model calibration through numerical solution of the full model (Eqs. 1–8). The model reduction technique adopted is briefly illustrated in the following, additional details being presented by Formaggia et al. (2013).

A set of N _Par unknown parameters are considered, each described by a uniform probability distribution within a corresponding interval of variability. These parameters are collected in a vector p. A parameter space Γ = [p ^min, p ^max] can then be defined and the output of the full compaction model can be expressed as a function of the unknown parameters only, by means of a generic function f(p) : Γ → R. The generalized polynomial chaos expansion (gPCE, e.g. Le Maitre and Knio 2010) recasts f(p) as a linear combination of a finite number W of multivariate Legendre polynomials ψ _i (p) multiplied by real numbers α _i (i = 1, …, W), termed as gPCE coefficients:

$$ f\left(\mathbf{p}\right)\approx {\displaystyle \sum_{i=1}^W{\alpha}_i\;{\psi}_i\left(\mathbf{p}\right)} $$

(16)

This enables one to recast the problem in terms of a linear combination of polynomials that can be evaluated very efficiently for any given value p ∈ Γ. Legendre polynomials are here employed because they are orthonormal with respect to the uniform probability distribution considered for the uncertain parameters. Parameter distributions different from the uniform are treated upon replacing the Legendre polynomials with an appropriate family of orthonormal polynomials (e.g. Le Maitre and Knio 2010; Xiu and Karniakidis 2002).

In this work, the evaluation of the gPCE coefficients α _i is performed by relying on a sparse grid sampling technique, following the procedure proposed by Formaggia et al. (2013) and employed by Porta et al. (2014). A gPCE approximation of order 2 is selected as an appropriate surrogate model in both cases analyzed. Such a gPCE model is built upon relying on a sparse grid sampling with 255 and 303 collocation points (corresponding to the number of full model runs which are needed to be performed), respectively for the Venture Field and for the Navarin Basin.

The ability of gPCE to approximate the porosity and pressure outputs is assessed through a blind test. The latter is performed by randomly selecting 30 sets of input parameters within the parameter space Γ (see Table 3) and then calculating the resulting porosity and pressure profiles by (1) the full model described by Eqs. (1)–(8) and (2) the gPCE-based surrogate model. Figure 9 depicts the results of an example through which one can gauge the quality of the agreement between the outputs of the gPCE-based surrogate model and the complete forward numerical model. Figure 9a,b depicts scatter plots of gPCE- and full model-based porosity and pressure values at locations corresponding to four selected depths (−2,284, −4,932, −5,038 and −5,251 m) at the final simulation time for the Venture Field. The corresponding depiction for the Navarin Basin is shown in Fig. 9c,d (at depths −577, −2,016, −2,387 and −3,971 m). These results show that the gPCE provides an approximation of the full model results of sufficient quality for the purpose of model calibration.

Fig. 9

Scatter plots of a,c porosity and b,d pressure values for Venture Field (a–b) and Navarin Basin (c–d), obtained through the employed gPCE approximation and the full model. Results are associated with 30 sets of input parameters randomly selected in the parameter space and are representative of four selected vertical locations at the final simulation time for each test case

Appendix 2

Global sensitivity analysis

Sensitivity analysis of a model is key to understanding and quantifying the relative importance of the variations of input parameters on the model target outputs (Saltelli et al. 2006). Global sensitivity analysis (GSA) typically investigates the influence of the uncertainty of input parameters on the model output variance. In this work, a variance-based method, also called ANOVA (ANalysis of VAriance) technique, is employed according to which the variance of the model output is decomposed into the sum of terms embedding the contributions of each input parameter as well as effect of mutual interactions between parameters through the so-called Sobol indices. These indices are particularly useful to study uncertainty propagation in nonlinear models (Sobol 2001, 2005). Following Sudret (2007) and Crestaux et al. (2009), the ANOVA is performed upon relying on the gPCE of model outputs (e.g. porosity and pressure distributions). The Sobol indices can be analytically evaluated through a proper manipulation of the gPCE coefficients, leading to a considerable reduction of the computational cost. To this end, it is possible to recast Eq. (16) as:

$$ f\left(\mathbf{p}\right)\approx {\alpha}_{\mathbf{0}}{\psi}_{\mathbf{0}}+{\displaystyle \sum_{n=1}^N{\displaystyle \sum_{\mathbf{i}\in {\mathcal{P}}_n}{\alpha}_{\mathbf{i}}{\psi}_{\mathbf{i}}\left(\mathbf{p}\right)}}+{\displaystyle \sum_{n=1}^N{\displaystyle \sum_{m=n}^N{\displaystyle \sum_{\mathbf{i}\in {\mathcal{P}}_{n,m}}{\alpha}_{\mathbf{i}}{\psi}_{\mathbf{i}}\left(\mathbf{p}\right)}+\dots }} $$

(17)

where i is a multi-index vector which collects the order of each polynomial ψ _i with respect to each parameter p _n . Let Λ be the set of multi-indices i included in the expansion (17). ${\mathcal{P}}_n\subset \varLambda$ denotes the subset of Λ containing all of the multi-indices i such that only the n-th component is non-zero, and ${\mathcal{P}}_{n,m}\subset \varLambda$ denote the subset of Λ containing the all of the multi-indices i such that only the n-th and the m-th components are non-zero. Note that:

$$ \mathbb{E}\left[f\right]={\alpha}_{\mathbf{0}},\kern2em \mathbb{V}\left[f\right]={\displaystyle \sum_{\mathbf{i}\in {\mathbb{N}}^N}{\alpha}_{\mathbf{i}}^2}-{\alpha}_{\mathbf{0}}^2\approx {\displaystyle \sum_{\mathbf{i}\in \varLambda }{\alpha}_{\mathbf{i}}^2}-{\alpha}_{\mathbf{0}}^2 $$

(18)

where $\mathbb{E}\left[f\right]$ and $\mathbb{V}\left[f\right]$ are the mean and the variance of f(p) (Sudret 2007). The quantities:

$$ {S}_n={\displaystyle \sum_{\mathbf{i}\in {\mathcal{P}}_n}\frac{\alpha_{\mathbf{i}}^2}{\mathbb{V}\left[f\right]}},\kern3em {S}_{n,m}={\displaystyle \sum_{\mathbf{i}\in {\mathcal{P}}_{n,m}}\frac{\alpha_{\mathbf{i}}^2}{\mathbb{V}\left[f\right]}}, $$

(19)

are approximations of the so-called Sobol indices that can be used to perform a GSA of the system outputs with respect to the input parameters (Sobol 2001, 2005; Sudret 2007). The quantity S _n collects all contributions to the total variation of f(p) which are only due to the n-th parameter, p _n , and is termed the principal Sobol index of the n-th parameter. As such, S _n provides a quantitative measure of the actual need for an accurate estimate of p _n to reduce the uncertainty associated with the prediction of f(p). The total Sobol index S ^T _n is defined as:

$$ {S}_n^{\mathrm{T}}={S}_n+{\displaystyle \sum_{k\ne n}{S}_{n,k}}+{\displaystyle \sum_{k,j\ne n}{S}_{k,j,n}}+\dots $$

(20)

The first-order indices, S _i , quantify the influence of the uncertainty of each parameter on the output variance, while the high-order indices consider the jointly effect of the input parameters uncertainties. The total Sobol indices S ^T _n are also considered. These describe the influence of each parameter considered not only alone but also in combination with all of the other uncertain parameters.

The Sobol indices are considered as useful indicators to characterize the sensitivity of a model with respect to its input parameters mainly because their information content is not limited by the existence of any kind of linear or monotonic trend describing the functional relationship linking the model outputs to system parameters (Saltelli et al. 2006).