Documentation for the “XT3D” option in the Node Property Flow (NPF) Package of MODFLOW 6

Techniques and Methods 6-A56
By: , and 



This report describes the “XT3D” option in the Node Property Flow (NPF) Package of MODFLOW 6. The XT3D option extends the capabilities of MODFLOW by enabling simulation of fully three-dimensional anisotropy on regular or irregular grids in a way that properly takes into account the full, three-dimensional conductivity tensor. It can also improve the accuracy of groundwater-flow simulations in cases in which the model grid violates certain geometric requirements. Three example problems demonstrate the use of the XT3D option to simulate groundwater flow on irregular grids and through three-dimensional porous media with anisotropic hydraulic conductivity.

Conceptually, the XT3D method of estimating flow between two MODFLOW 6 model cells can be viewed in terms of three main mathematical steps: construction of head-gradient estimates by interpolation; construction of fluid-flux estimates by application of the full, three-dimensional form of Darcy’s Law, in which the conductivity tensor can be heterogeneous and anisotropic; and construction of the flow expression by enforcement of continuity of flow across the cell interface. The resulting XT3D flow expression, which relates the flow across the cell interface to the values of heads computed at neighboring nodes, is the sum of terms in which conductance-like coefficients multiply head differences, as in the conductance-based flow expression the NPF Package uses by default. However, the XT3D flow expression contains terms that involve “neighbors of neighbors” of the two cells for which the flow is being calculated. These additional terms have no analog in the conductance-based formulation. When assembled into matrix form, the XT3D formulation results in a larger stencil than the conductance-based formulation; that is, each row of the coefficient matrix generally contains more nonzero elements. The “RHS” suboption can be used to avoid expanding the stencil by placing the additional terms on the right-hand side of the matrix equation and evaluating them at the previous iteration or time step.

The XT3D option can be an alternative to the Ghost-Node Correction (GNC) Package. However, the XT3D formulation is typically more computationally intensive than the conductance-based formulation the NPF Package uses by default, either with or without ghost nodes. Before deciding whether to use the GNC Package or XT3D option for production runs, the user should consider whether the conductance-based formulation alone can provide acceptable accuracy for the particular problem being solved.

Suggested Citation

Provost, A.M., Langevin, C.D., and Hughes, J.D., 2017, Documentation for the “XT3D” option in the Node Property Flow (NPF) Package of MODFLOW 6: U.S. Geological Survey Techniques and Methods, book 6, chap. A56, 40 p.,

ISSN: 2328-7055 (online)

Table of Contents

  • Abstract 
  • Introduction
  • Mathematical Model of Groundwater Flow
  • Conceptual Basis for the XT3D Method
  • XT3D Flow Expression
  • XT3D Control-Volume Finite-Difference (CVFD) Equation
  • XT3D Formulation of the CVFD Equation for Solution
  • Newton-Raphson Formulation of the XT3D CVFD Equation for Solution 
  • Correction Applied by the Horizontal Flow Barrier Package
  • Features of the XT3D Option
  • Applicability and Limitations of the XT3D Option
  • Example Problems
  • References Cited 
  • Appendix A. List of Symbols
  • Appendix B. XT3D Calculations on a Rectangular Grid
Publication type Report
Publication Subtype USGS Numbered Series
Title Documentation for the “XT3D” option in the Node Property Flow (NPF) Package of MODFLOW 6
Series title Techniques and Methods
Series number 6-A56
DOI 10.3133/tm6A56
Year Published 2017
Language English
Publisher U.S. Geological Survey
Publisher location Reston, VA
Contributing office(s) Office of Ground Water
Description vi, 27 p.
Larger Work Type Report
Larger Work Subtype USGS Numbered Series
Larger Work Title Section A: Groundwater in Book 6 Modeling techniques
Public Comments This report is Chapter 56 of Section A: Groundwater in Book 6 Modeling techniques.
Online Only (Y/N) Y
Additional Online Files (Y/N) N
Google Analytic Metrics Metrics page
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