Comparison of Two Mathematical Models for 3D Groundwater Flow: Block-Centered Heads and Edge-Based Stream Functions

Abstract

Traditionally, groundwater flow models, as well as oil reservoir models, are based on the block-centered finite difference method. Well-known models based on this approach are MODFLOW (groundwater) and ECLIPSE (oil and gas). Such models are well proven and robust; their underlying principles are well understood by hydrologists and petroleum reservoir engineers. Nevertheless, the desire to improve the block-centered finite difference paradigm has always been alive, for instance, to be able to apply deformed grid blocks, or to model anisotropy that is not aligned along the coordinate axes. This article introduces the edge-based stream function as a potential alternative to the paradigmatic model, not only to mitigate the above mentioned limitations, but especially for its promise to inverse modeling. Computer programs have been developed for the discrete analog equations of the stream function method and the conventional method. The two methods are tested by using synthetic forward modeling problems of uniform and radial flow. The theoretical formulation and the numerical results show that the two methods are algebraically equivalent and yield the same flux output. However, for rectangular grid blocks and anisotropy aligned along the coordinate axes, the block-centered method is shown to be computationally more efficient than the edge-based stream function method. The major advantage of the stream function method is that it is linear in the resistivities, proving it an ideal candidate for direct inverse modeling. Moreover, any arbitrary specification of stream functions yields a solution that satisfies the mass balance.

A Velocity-Oriented Approach for Modflow

“…To be precise, a more reliable determination of relatively small vertical velocity components could also be obtained by a stream function method, because such a method solves the water balance equation (Eq. 3) exactly, even if the iterations to solve the system of linear equations are terminated too early [for details of the three-dimensional stream function method, see Mohammed et al (2009)]. However, since the stream function method is mathematically equivalent with the earlierdiscussed block-centered method (like MODFLOW) and mixed-hybrid method, we propose here the more accurate method-the VOA.…”

“…(ii) Application of the three-dimensional stream function method, a method that honors the water balance exactly, as presented by (Zijl and Nawalany 2004;Mohammed et al 2009). …”

“…However, numerical experiments by Kuppen (1988) show that, even when allowing for smoothness, good results can be obtained for practical problems. The advantage of research direction (ii) is its greater flexibility in handling heterogeneity (discontinuities in any direction), while its disadvantage is the relatively great programming effort, because standard software for the three-dimensional stream functions does not exist, and we should also take into account that the stream function method is mathematically equivalent to the block-centered finite difference method (Mohammed et al 2009). As a consequence, if the system of linear algebraic equations of the block-centered finite difference method is solved exactly (i.e., with sufficient iterations to solve the water balance exactly), the accuracy of the stream function method is the same as the accuracy of the block-centered finite difference method (MODFLOW), while the MODFLOW-based VOA is much more accurate.…”

Hydraulic fracturing and the environment: risk assessment for groundwater contamination from well casing failure

“…6-14) (Schlumberger 2011). Aside from modeling oil and gas reservoirs, this simulator has applications in groundwater modeling (Mohammed et al 2009) and environmental assessment and remediation (Zhou and Arthur 1994).…”