Drained rock volume around hydraulic fractures in porous media: planar fractures versus fractal networks

This study revisits the mathematical equations for diffusive mass transport in 1D, 2D and 3D space and highlights a widespread misconception about the meaning of the regular and cumulative probability of random-walk solutions for diffusive mass transport. Next, the regular probability solution for molecular diffusion is applied to pressure diffusion in porous media. The pressure drop (by fluid extraction) or increase (by fluid injection) due to the production system may start with a simple pressure step function. The pressure perturbation imposed by the step function (representing the engineering intervention) will instantaneously diffuse into the reservoir at a rate that is controlled by the hydraulic diffusivity. Traditionally, the advance of the pressure transient in porous media such as geological reservoirs is modeled by two distinct approaches: (1) scalar equations for well performance testing that do not attempt to solve for the spatial change or the position of the pressure transient without reference to a well rate; (2) advanced reservoir models based on numerical solution methods. The Gaussian pressure transient solution method presented in this study can compute the spatial pressure depletion in the reservoir at arbitrary times and is based on analytical expressions that give spatial resolution without gridding-meaning solutions that have infinite resolution. The Gaussian solution is efficient for quantifying the advance of the pressure transient and associated pressure depletion around single wells, multiple wells and hydraulic fractures. This work lays the basis for the development of advanced reservoir simulations based on the superposition of analytical pressure transient solutions.

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Flow in Fractured Porous Media Modeled in Closed-Form: Augmentation of Prior Solution and Side-Stepping Inconvenient Branch Cut Locations

Fluids ◽

10.3390/fluids5020051 ◽

2020 ◽

Vol 5 (2) ◽

pp. 51 ◽

Cited By ~ 1

Author(s):

Ruud Weijermars ◽

Aadi Khanal

Keyword(s):

Porous Media ◽

Complex Analysis ◽

Fractured Porous Media ◽

Flow In Porous Media ◽

Far Field ◽

Natural Fractures ◽

Resolution Model ◽

Drained Rock Volume ◽

Cam Model ◽

Branch Cut

Carefully chosen complex variable formulations can solve flow in fractured porous media. Such a calculus approach is attractive, because the gridless method allows for fast, high-resolution model results. Previously developed complex potentials to describe flow in porous media with discrete heterogeneities such as natural fractures can be modified to expand the accuracy of the solution range. The prior solution became increasingly inaccurate for flows with fractures oriented at larger angles with respect to the far-field flow. The modified solution, presented here, based on complex analysis methods (CAM), removes the limitation of the earlier solution. Benefits of the CAM model are (1) infinite resolution, and (2) speed of use, as no gridding is required. Being gridless and meshless, the CAM model is computationally faster than integration methods based on solutions across discrete volumes. However, branch cut effects may occur in impractical locations due to mathematical singularities. This paper demonstrates how the augmented formulation corrects physically unfeasible refraction of streamlines across high-permeability bands (natural fractures) oriented at high angles with respect to a far-field flow. The current solution is an important repair. An application shows how a drained rock volume in hydraulically fractured hydrocarbon wells will be affected by the presence of natural fractures.

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Lattice Boltzmann Simulation of Non-Darcy Flow In Stochastically Generated 2D Porous Media Geometries

SPE Journal ◽

10.2118/146689-pa ◽

2013 ◽

Vol 18 (01) ◽

pp. 12-26 ◽

Cited By ~ 27

Author(s):

M.S.. S. Newman ◽

X.. Yin

Keyword(s):

Porous Media ◽

Lattice Boltzmann ◽

Darcy’S Law ◽

Contact Length ◽

Darcy's Law ◽

Hydraulic Fractures ◽

Large Set ◽

Pressure Drops ◽

Darcy Flows ◽

2D Porous Media

Summary It is important to consider the additional pressure drops associated with non-Darcy flows in the near-wellbore region of conventional gas reservoirs and in propped hydraulic fractures. These pressure drops are usually described by the Forchheimer equation, in which the deviation from the Darcy's law is proportional to the inertial resistance factor (β-factor). While the β-factor is regarded as a property of porous media, detailed study on the effect of pore geometry has not been performed. This study characterized the effect of geometry on the flow transition and the β-factor using lattice Boltzmann simulations and stochastically constructed 2D porous media models. The effect of geometry was identified from a large set of data within a porosity range of 8–35%. It was observed that the contrast between pore throat and pore body triggers an early transition to non-Darcy flows. Following a quick transition where the correction to the Darcy's law was cubic in velocity, the flows entered the Forchheimer regime. The β-factor increased with decreasing porosity or an increasing level of heterogeneity. Inspection of flow patterns revealed both steady vortices and onset of unsteady motions in the Forchheimer regime. The latter correlated well with published points-of-transition. In developing a dimensionally consistent correlation for the β-factor, we show that it is necessary to include two distinctive characteristic lengths to account for the effect of pore-scale heterogeneity. This finding reflects the fact that it is the contrast between pore bodies and throats that dictates the flow properties of many porous media. In this study, we used the square root of the permeability and the fluid-solid contact length as the two characteristic lengths.

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