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

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## 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.

## Keywords

Discrete analogs Block-centered method 3D stream function Edge-based vector potential## List of Symbols

(Vectors are denoted by letters with right-pointing arrow above their names. Matrices and arrays are denoted as upper case symbols)

*A*_{f}Surface area of grid block face f (

*L*^{2})- D
*N*_{V}×*N*_{F}matrix relating volumes to faces, discrete differential operator corresponding to div (–)*h*_{i}Hydraulic head gradient in the

*i*direction (–)- \({\vec{h}}\)
Head gradient vector (–)

- H
Column array of dimension

*N*_{F}representing head differences including those caused by boundary conditions (L)*k*_{ij}Component of hydraulic conductivity tensor along the

*i*and*j*directions (L/T)- K
*N*_{V}×*N*_{V}matrix containing hydraulic conductivities and grid metrics (*L*^{2}/T)- \({\vec{\vec{k}}}\)
Hydraulic conductivity tensor (L/T)

*N*_{BN},*N*_{BF},*N*_{BE}Number of boundary nodes, boundary faces, and boundary edges (–)

*N*_{X},*N*_{Y},*N*_{Z}Number of grid blocks in the

*x*,*y*, and*z*directions (–)*N*_{V},*N*_{F},*N*_{E}Number of grid volumes, faces, and edges (–)

*q*_{i}Flux density in the

*i*direction (*L*^{3}/*T*/*L*^{2})- \({\vec{q}}\)
Flux density vector (

*L*^{3}/*T*/*L*^{2})- Q
Column array of dimension

*N*_{F}representing normal components of flux densities \({\vec{q}}\) integrated over faces (*L*^{3}/T)- R
*N*_{F}×*N*_{E}matrix relating faces to edges, discrete differential operator corresponding to curl or rot (–)- \({\dot {s}_0}\)
Storage per unit volume (1/T)

- \({\dot {{S}}_{\rm o}}\)
Column array of dimension

*N*_{V}representing storage (*L*^{3}/T)*x*_{i}Cartesian coordinate in the

*i*direction (L)

## Greek Symbols

- \({\phi}\)
Hydraulic head (L)

- Φ
Column array of dimension

*N*_{V}representing heads at the center of grid volumes (L)*γ*_{ij}Component of resistivity tensor along the

*i*and*j*directions (T/L)- \({\vec{\vec{\gamma}}}\)
Resistivity tensor (T/L)

- Γ
*N*_{V}×*N*_{V}matrix containing resistivities and grid metrics (*T*/*L*^{2})- Π
Column array of dimension

*N*_{F}representing oriented boundary heads, nonzero only on boundary faces (L)*θ*Angular coordinate (rad)

- \({\vec{\psi}}\)
Vector potential (

*L*^{3}/T)- Ψ
Column array of dimension

*N*_{E}representing 3D stream function, i.e., vector potentials integrated along edges (*L*^{3}/T)

## Additional Symbols

- ×
Cross product

- ·
Dot product

- ∇
= (∂/∂

*x*, ∂/∂*y*, ∂/∂*z*), del (nabla) operator in 3D Cartesian coordinate system (1/L)- ∇·
Divergence, \({\nabla \cdot \vec{a}=(\partial a_x /\partial x+\partial a_y /\partial y+\partial a_z /\partial z)}\)

- ∇×
Curl, \({\nabla \times \vec{a}=\left( {\partial a_z /\partial \tilde{y}-\partial a_y /\partial z,\partial a_x /\partial \tilde{z}-\partial a_z /\partial x,\partial a_y /\partial \tilde{x}-\partial a_x /\partial y}\right)}\)

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