[Paper Review] Estimating nonlinearities in twophase flow in porous media

Jianfeng Zhang, Guy Chavent+1 more|arXiv (Cornell University)|Apr 9, 2009

Reservoir Engineering and Simulation Methods24 references3 citations

TL;DR

This paper presents a geometric nonlinear analysis framework for estimating relative permeability and capillary pressure functions in one-dimensional two-phase flow through porous media, using inverse modeling of laboratory experiments. By combining linear and nonlinear stability analysis with Hessian-based uncertainty quantification, it demonstrates that the nonlinear least-squares problem is Q-well posed and provides sharp confidence regions for estimated parameters, with uncertainty decreasing as measurement density increases.

ABSTRACT

In order to analyze numerically inverse problems several techniques based on linear and nonlinear stability analysis are presented. These techniques are illustrated on the problem of estimating mobilities and capillary pressure in one-dimensional two-phase displacements in porous media that are performed in laboratories. This is an example of the problem of estimating nonlinear coefficients in a system of nonlinear partial differential equations.

Motivation & Objective

• To develop a robust numerical framework for estimating nonlinear coefficients—specifically relative permeability and capillary pressure—under laboratory conditions in porous media.
• To address the inverse problem of parameter estimation in one-dimensional two-phase flow governed by nonlinear partial differential equations.
• To quantify uncertainty in estimated parameters using both uniform and directional first-order stability estimates.
• To demonstrate the practical applicability of nonlinear stability analysis in real experimental settings with limited measurements.
• To show that increasing measurement density improves parameter stability and reduces uncertainty bounds.

Proposed method

• Formulates two-phase flow using a global pressure model with Darcy’s law, incorporating saturation-dependent mobility and capillary pressure functions.
• Reformulates the inverse problem as a nonlinear least-squares minimization problem to estimate saturation-dependent coefficients.
• Applies linear stability analysis to assess well-posedness and define neighborhoods of stability around the estimated parameters.
• Employs Hessian-based geometric analysis to compute confidence regions and uncertainty bounds for estimated parameters.
• Uses edgehog extremal solutions to compute confidence intervals and validate the stability of the solution.
• Implements numerical validation using synthetic data with 30 and 90 saturation measurements to compare uncertainty bounds under different measurement densities.

Experimental results

Research questions

• RQ1 How can nonlinear stability analysis be used to assess the well-posedness of inverse problems in two-phase flow through porous media?
• RQ2 What is the impact of measurement density on the stability and uncertainty of estimated relative permeability and capillary pressure functions?
• RQ3 How do uniform and directional first-order uncertainty estimates compare in terms of sharpness and practical relevance?
• RQ4 Can geometric nonlinear analysis tools reliably quantify parameter uncertainty in the absence of full statistical modeling?
• RQ5 What conditions ensure that the nonlinear least-squares problem for parameter estimation is Q-well posed?

Key findings

• The nonlinear least-squares problem is Q-well posed when measurement errors are bounded: for 30 measurements, error must be ≤1.11×10⁻³ in norm; for 90 measurements, ≤1.7×10⁻².
• With a per-measurement saturation error of 7.3×10⁻⁵, the uniform uncertainty estimate on the parameter vector is ∥δp∥ ≤ 0.0191 for 30 measurements and ≤0.0027 for 90 measurements.
• The directional first-order uncertainty estimate (inequality 20) produces significantly smaller uncertainty domains than the uniform estimate (inequality 18), confirming its sharper bound.
• Uncertainty domains are independent of the optimal parameter value p₀, but their size depends on measurement density and error magnitude.
• Increasing the number of measurements from 30 to 90 reduces the parameter uncertainty by a factor of approximately 7, demonstrating improved stability.
• The Hessian-based geometric analysis enables reliable construction of confidence regions without requiring full Bayesian inference or strong statistical assumptions.