Research Collection

Master Thesis

Nonlocal formulation for flow in porous media

Author(s): Delgoshaie, Amir Hossein

Publication Date: 2014

Permanent Link: https://doi.org/10.3929/ethz-a-010262375

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ETH Library

Nonlocal Formulation for Flow in PorousMedia

Amir Hossein Delgoshaie

Mechanical Engineering

Master’s thesis FS 2014

Institute of Fluid dynamicsETH Zurich

Professor Patrick Jenny

Professor Hamdi Tchelepi

Acknowledgments

First of all, I would like to thank ETH Zurich for funding my master’s studies throughthe ETH Excellence Scholarship program. This support allowed me to learn so manynew things and have a wonderful experience over the past two years.I want to thank my supervisor Prof. Patrick Jenny for his guidance and supportthroughout my master’s studies. It has been an honor and a great experience to bea part of his group. Also I wish to thank Prof. Hamdi Tchelepi for hosting me atStanford University, and all the time that he spent guiding me for this thesis.I would like to thank Dr. Mohammad Karimi-Fard and Dr. Bradley Mallison forstimulating discussions.Last but not least, I would like to thank my parents for supporting me through mylife, giving me a better vision and encouraging me to continue my education.

iii

Abstract

A general nonlocal model for flow in porous media is proposed. The model assumesthat each point in the medium is connected to a nonlocal region around that point,and that the flow rate is a function of the nonlocal pressure differences. A method isproposed for discretizing a homogeneous nonlocal transmissibility kernel in multipledimensions. The approach is demonstrated using a homogeneous anisotropic porousmedium. The results are compared with a reference Darcy solution and it is shownthat the model converges to the standard two-point solution upon refinement. In caseswhere the principal axis of anisotropy is not aligned with the grid, the method isshown to yield a monotone solution by ensuring that the resulting coefficient matrixis an M-matrix.

iv

Contents

Contents

1 Introduction and literature review 1

2 Nonlocal transport theory 32.1 Darcy’s law as a special case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Peridynamic formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Flux calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Modeling Anisotropy 73.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Comparison with the local model . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Numerical convergence studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.4.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4.2 Example 1: wells aligned with the main axis of permeability . . . . . . . . 11

3.4.2.1 Reference solution . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4.2.2 Nonlocal solution with the transmissibility kernel aligned with

the grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4.2.3 Nonlocal solution with the transmissibility kernel rotated with

respect to the grid . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4.3 Example 2: wells tilted with respect to the main axis of permeability . . . 24

3.4.3.1 Reference solution . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4.3.2 Nonlocal solution with the transmissibility kernel aligned with

the grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4.3.3 Nonlocal solution with the transmissibility kernel rotated with

respect to the grid . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Possible ways for finding the transmissibility kernel 36

5 Final remarks 38

Bibliography 39

v

Chapter 1. Introduction and literature review

1 Introduction and literature review

Darcy’s law, the constitutive equation for flow in porous media, is represented as a flux that isproportional to the local gradient of pressure [1]. This type of gradient diffusion model is alsoused for modeling other physical phenomena. Examples include Fick’s law and Fourier’s law fortransport of mass and heat, respectively. These models all use local information, i.e., gradientat a point, to compute a flux. However, the nature of flow in porous media suggests that theflux at each point might depend on a neighborhood around that point, and considering only thelocal gradient at the point may not be enough to describe the flow accurately.

Natural porous media can be thought of as grains of sand with void space between them thatprovide microchannels for fluids to move. The distribution of these microchannels may be ho-mogeneous, anisotropic or both. In Darcy’s law, this translates into having a permeability thatis a function of position, or a tensor permeability [1]. In addition to these grains one can imaginetubes with impermeable walls with different length distributed in the porous medium. In orderto have flow in these tubes there should be a pressure difference between the two ends of thetube, separated with a finite distance. A local pressure gradient at either end of the tube will notinduce flow in the microchannel. If the length of the microchannels present is small compared tothe overall size of the porous medium, Darcy’s law can correctly capture the flow [2]. However,flow in the presence of such channels cannot generally be modeled with Darcy’s Law.

Figure 1.1: Illustration of a non-Darcy medium [2]

1

The need for nonlocal modeling is not limited to flow in porous media. A similar idea has mo-tivated research in different fields of mechanical engineering in order to derive nonlocal modelsfor phenomena, for which diffusion-type equations have been used [3], [4], [5].

The first field for which this idea has been developed is solid mechanics. The difficulty of mod-eling cracks using the partial differential equations that are used in the classical theory of solidmechanics was the main motivation for using nonlocal models [3]. The peridynamic theory pro-posed by Silling et al. [3] aims to avoid spatial derivatives by formulating the problem usingintegral equations. In this theory, the mathematical description of the solid mechanics is reformu-lated such that the equations hold anywhere in a domain, even in the presence of discontinuities.

Nonlocal models have also been used for modeling heat transfer. Similar to solid mechanics,using peridynamic theory for heat transfer was motivated by cases where some form of discon-tinuity forms and evolves in the conducting body due to material failure [4]. In the work byBobaru et al. [4], [5] a method has been proposed for arriving at a peridynamic formulation us-ing the classical heat transfer model in domains with homogeneous conductivity. In their work,several test cases have been solved both with the classical model and the peridynamic theory.The results from the peridynamic theory are shown to agree well with the results obtained fromFourier’s law.

A nonlocal model for flow in porous media was proposed by Sen et al. [2]. In their work itwas shown that the proposed integral formulation reduces to Darcy’s law as a special local case.They also suggest a method to determine the transmissibility kernel (the centerpiece of the in-tegral formulation) in a general setting. However, there is no example to show how this methodperforms in comparison with predictions from Darcy’s law.

Peridynamics has been recently introduced for modeling flow in porous media. Katiyar et al. [6]generalized the work of Bobaru et al. [4] to propose a method for constructing a peridynamicformulation for anisotropic diffusion in heterogeneous material with application to flow in porousmedia. They also compared the results from the peridynamic representation with classical the-ory for a quarter five-spot pattern. In their work, they also propose a method for handlingpoint sources. Despite the general form of their formulation, the comparisons do not includeanisotropy, and heterogeneity is considered only in the simple case of having an impermeableblock in a homogeneous permeable medium.

Although the potential of nonlocal models for handling anisotropy has been mentioned in pre-vious research [6], there has been very little work on the subject. The purpose of this report isto investigate the potential of nonlocal models for anisotropic systems and show the advantagesof this model in comparison with classical Darcy’s law.

2

Chapter 2. Nonlocal transport theory

2 Nonlocal transport theory

Consider a porous medium in which each pore can have connections to many other pores. Theconnections can span a wide range of length scales, and the conductivity properties may bedifferent for the different connections. We introduce a nonlocal formulation in one dimension.The equation for volume flow rate q(x) is proposed in terms of the pressure, p(x), as [2]

q(x) =

∫ +∞

−∞T (x, x′)[p(x)− p(x′)] dx′, (2.1)

where T (x, x′) is the transmissibility kernel, which represents the strength of the connectionbetween points x and x′. It is a property that includes the effects of channel length and cross-sectional area of the connection between x and x′, as well as the viscosity of the fluid [2]. Thisequation can be viewed as relating the volume flow rate to the pressure field. Since the strengthof the connection between two point is the same for flow in either direction between the twopoints, T (x, x′) is symmetric. That is,

T (x, x′) = T (x′, x). (2.2)

The extension of Eq. 2.1 to higher dimensions is straight forward. It can be shown that

q(x) =

∫ΩT (x,x′)[p(x)− p(x′)] dV (x′), (2.3)

where Ω is a two- or three-dimensional control volume, and T (x,x′) = T (x′,x).

2.1 Darcy’s law as a special case

One way of showing that 2.1 is consistent with Darcy’s law is to show that both equations becomeidentical in the limit of L << H, where L and H are the largest connection length and the size ofa finite-volume cell, respectively [8]. For simplicity, it is assumed that T (x, x+ r) = T (x, x− r).For a control volume around xI , integrating Eq. 2.1 leads to∫ xI+H/2

xI−H/2

(∫ +L

−LT (x, x′)[p(x)− p(x′)] dx′

)dx =

∫ xI+H/2

xI−H/2q(x) dx. (2.4)

3

2.1. Darcy’s law as a special case

Substitution of the Taylor expansion p(x′) = p(x) +∑∞

m=1∆x′m

m!∂mp∂xm

∣∣∣∣x

with ∆x = x′ − x leads

to

−∫ xI+H/2

xI−H/2

(∫ +L

−LT (x, x′)∆x′

∂p

∂x

∣∣∣∣x

+T (x, x′)∆x′2

2

∂2p

∂x2

∣∣∣∣x

dx′)dx

−∞∑m=3

∫ xI+H/2

xI−H/2

(∫ +L

−L

T (x, x′)∆x′m

m!

∂mp

∂xm

∣∣∣∣x

dx′)dx =

∫ xI+H/2

xI−H/2q(x) dx. (2.5)

Neglecting the terms of order ∆x′4 and higher in the expansion of pressure, one obtains

−∫ xI+H/2

xI−H/2

k/µ︷ ︸︸ ︷( ∫ +L

−L

T (x, x′)∆x′2

2dx′) ∂2p

∂x2

∣∣∣∣x

dx =

∫ xI+H/2

xI−H/2q(x) dx. (2.6)

This shows that the solution governed by Darcy’s law, i.e., by

−kµ

∂2p

∂x2= q(x) with

k

µ=

∫ xr

xl

T (x, x′)(x− x′)2

2dx′, (2.7)

and

∫ xr

xl

T (x, x′)[p(x)− p(x′)] dx′ = q(x),

are consistent.

4

2.2. Peridynamic formulation

2.2 Peridynamic formulation

The integration in Eq. 2.1 can also be performed in a neighborhood surrounding the point x.The peridynamic formulation originally introduced by Silling et al. [7], and further developedby Bobaru et al. [4] and Katiyar et al. [6], uses the bounded form of this integral equation.In peridynamics, this neighborhood is called the horizon of a point. Although, the form of therepresentative equation for volume flow rate is the same in the peridynamic theory proposedby Bobaru et. al [4], they arrive at this equation using different assumptions. Here we followthe assumptions used by Bobaru et al. to write the peridynamic equations for flow in porousmedia. Porous media is considered as material points that have associated porosity and volume.Connections exist between each material point and the neighbors in the horizon of that point.Let i, j be unit vectors along the x-axis and y-axis, respectively, and let θ be the angle betweenj and the bond direction. The peridynamic flux corresponding to a bond is introduced in [4]as:

q = K(x,x′)p(x)− p(x′)

‖x− x′‖e, (2.8)

where K(x,x′) is the conductivity of the bond between points x and x′, and e is the unit vectoralong the vector x−x′. In their model, for a two-dimensional case the flux associated with thebond between x and x′ is

q = Kx(x,x′)p(x)− p(x′)

‖x− x′‖cos(θ)i +Ky(x,x

′)p(x)− p(x′)

‖x− x′‖sin(θ)j, (2.9)

where Kx(x,x′) and Ky(x,x′) are the conductivities of the bond in the x and y directions. For

a case with K(x,x′) = Kx(x,x′) = Ky(x,x′), the conservation of mass for a single bond leads

to

∂(ρφ)bond∂t

(x− x′).e = K(x,x′)p(x)− p(x′)

(x− x′).e, (2.10)

where φ is the porosity of the bond. Dividing both sides of Eq. 2.10 by ‖x−x′‖ and integratingover the horizon of point x, which we denote Hx, one obtains

∫Hx

∂(ρφ)bond∂t

dAx′ =

∫Hx

K(x,x′)p(x)− p(x′)

‖x− x′‖2dAx′ . (2.11)

It is assumed that the following relation is valid between the material properties at point x andthe properties of the bonds connecting to that point:

1

Ax

∫Hx

∂(ρφ)bond∂t

dAx′ = (ρφ)x, (2.12)

5

2.3. Flux calculation

where Ax is the area of the horizon at point x. The micro-conductivity of the bond between xand x′ is defined as

k(x,x′) = K(x,x′)/Ax. (2.13)

Equation 2.11 then becomes

∂(ρφ)x∂t

=

∫Hx

k(x,x′)p(x)− p(x′)

‖x− x′‖2dAx′ . (2.14)

If a source is present at x, then Eq. 2.14 becomes

∂(ρφ)x∂t

=

∫Hx

k(x,x′)p(x)− p(x′)

‖x− x′‖2dAx′ +Q(x). (2.15)

It can be seen that Eq. 2.15 has the same form as the general nonlocal formulation Eq. 2.1, andthe two equations become identical for a steady state problem by choosing

T (x,x′) =k(x,x′)

‖x− x′‖2. (2.16)

2.3 Flux calculation

The flux through a single bond is given by Eq. 2.10. To find the flux into a point at locationx, the fluxes from all points in Hx with a pressure higher than p(x) are considered. We denotethis part of the horizon by H+. The flux into a point can be written as:

f(x) =

∫H+

k(x,x′)p(x′)− p(x)

‖x− x′‖ex′dAx′ . (2.17)

Using Eq. 2.16, the flux can be also written as

f(x) =

∫H+

T (x,x′) (x− x′) [p(x′)− p(x)]dV (x′). (2.18)

6

Chapter 3. Modeling Anisotropy

3 Modeling Anisotropy

3.1 Motivation

Accurate representation of anisotropic permeability is one of the challenging aspects of reservoirsimulation. Traditional reservoir simulation models yield a correct discretization on a nonorthog-onal grid, only if the grid directions are aligned with the principal directions of the permeabilitytensor [10]. Multipoint Flux Approximation (MPFA) methods have been developed to givea correct discretization of the flow equations for general non-orthogonal grids, as well as forgeneral orientation of the principal directions of the permeability tensor [11]. However thereare limitations on the numerical stability of these methods, and they do not always producea monotone solution [12]. Monotonicity is guaranteed if the discrete approximation yields anM-matrix. In a linear system of the form Au = q, A is an M-matrix if and only if the entriesaij satisfy [13]:

aii > 0, ∀i,aij ≤ 0, ∀i, j, i 6= j∑

i diaij ≥ 0, ∀j(3.1)

with strict inequality for at least one column and some positive vector d. These conditionsguarantee that

A−1 ≥ O, (3.2)

where O is the zero matrix. If the pressure on the domain boundary is zero, no positive pressuresource can induce a negative pressure inside the domain. An M-matrix will always satisfy thisconstraint, but being an M-matrix is not a necessary condition for satisfying the inequality3.2. MPFA methods often lead to matrices that are not an M-matrix. These methods yielda monotone solution conditionally, and they fail to satisfy the maximum principle for stronganisotropy or grid skewness [12]. On a cartesian grid, if we denote the angle between the principleaxis of permeability and the x-direction by φ, then for a given anisotropic permeability, MPFAmethods violate the maximum principle for some φ less than 45o [12].

In this section, we show that by representing anisotropy in the nonlocal formulation, one canassure that the coefficients matrix is an M-matrix, and that the solution is monotone for anyvalue of φ.

7

3.2. Framework

3.2 Framework

To model anisotropy, we need a transmissibility kernel, T (x, x′), that has a major and minoraxis of connectivity. Many functions can be used for this purpose. One of the simplest optionsis the HeavisideLambda function

T (x,x′) = HeavisideLambda(X,Y ), (3.3)

X =1

Lx

((x− x′) cos(θ) + (y − y′) sin(θ)

), (3.4)

Y =1

Ly

((x′ − x) sin(θ) + (y − y′) cos(θ)

), (3.5)

where x =

(xy

)and x′ =

(x′

y′

). Lx and Ly represent the connection length in the principle

directions of T , and θ is the angle between the principle axis and the reference frame. Fig. 3.1shows the HeavisideLambda function for θ = 0, Lx = 4, Ly = 2.

-4

-2

0

2

4-4

-2

0

2

4

0.0

0.5

1.0

(a)

-4 -2 0 2 4

-2

-1

0

1

2

0.1 0.3 0.5 0.7 0.9

(b)

Figure 3.1: (a) HeavisideLambda function for θ = 0 ; (b) contour plot of (a)

In order to solve the integral equation 2.3, we employ a Galerkin method [14]. A uniform 2Dmesh with grid spacing, δ, is employed. We consider the approximate pressure field p′(x)

p′(x) =

N∑J=0

pJΦJ(x, y). (3.6)

8

3.2. Framework

The basis function ΦJ(x, y) can have different forms. Here, a top-hat function [14] is used.

ΦJ(x, y) =

1 if |x− xJ | < δ

2 and |y − yJ | < δ2

0 otherwise.(3.7)

Substitution of p′(x) into Eq. 2.3 multiplication with the weighting function ΨI(x) = ΦI(x),and integration over Ω leads to the balance equation [8]:

∫Ω

ΦI(x′′)

∫ΩT (x′′,x′)[

N∑J=0

pJΦJ(x′′)−N∑J=0

pJΦJ(x′)]dV x′dV x′′ =

∫Ω

ΦI(x′′)q(x′′)dV (x′′),

(3.8)

from which a linear system can be constructed as Ap = Q with

AI,J =

∫Ω

ΦI(x′′)

∫ΩT (x′′,x′)[ΦJ(x′′)− ΦJ(x′)]dV x′dV x′′, (3.9)

and

QI =

∫Ω

ΦI(x′′)q(x′′)dV (x′′). (3.10)

For top-hat basis functions and i 6= j, Eq. 3.9 simplifies to

AI,J = −∫

Ω

∫Ω

ΦI(x′′)ΦJ(x′)T (x′′,x′)dV x′dV x′′. (3.11)

In order to guarantee conservation of mass, AI,I is calculated from

AI,I =∑I 6=J

AI,J . (3.12)

Provided that T (x′,x) is positive for every pair of x and x′ and top-hat basis functions are used,the coefficients obtained from 3.11 will be negative. This ensures that the resulting coefficientmatrix will be an M-matrix by construction, which ensures that the approximate solution ismonotone.

9

3.3. Comparison with the local model

3.3 Comparison with the local model

We show that the nonlocal model yields the same result for the cases where using a two pointflux approximation (TPFA) is correct. To construct a relation between the nonlocal model andDarcy’s law, we assume that the flux calculated by both models is the same when the pressurefield is changing linearly in one direction and is constant in other directions.

Eq. 2.18 can be written as

f(x) =

∫H+

T.(x− x′)[p(x′)− p(x)]dV (x′)

= i

∫H+

Tp(x′)− p(x)

‖x′ − x‖‖x′ − x‖2cos(φ)dA(x′) + j

∫H+

Tp(x′)− p(x)

‖x′ − x‖‖x′ − x‖2sin(φ)dA(x′)

= i

∫H+

Tp(x′)− p(x)

x′ − x‖x′−x‖2cos2(φ)dA(x′) + j

∫H+

Tp(x′)− p(x)

y′ − y‖x′−x‖2sin2(φ)dA(x′)

= i

∫H+

Tp(x′)− p(x)

x′ − x(x′ − x)2dA(x′) + j

∫H+

Tp(x′)− p(x)

y′ − y(y′ − y)2dA(x′). (3.13)

Here, φ is the angle between x′ − x and positive x direction and T (x,x′) is written as T forbrevity. Now let us assume that the pressure field varies linearly in the x direction and isconstant in the y direction, then Eq. 3.13 simplifies to

f(x) =∂p

∂xi

∫H+

T (x′ − x)2dA(x′). (3.14)

The integral along the j direction is zero because for the given linear pressure profile the inte-grand is an odd function integrated over a symmetric domain. Equating this flux with the fluxobtained from the local formulation one obtains

∫H+

T (x′ − x)2dA(x′) = kx. (3.15)

Using a similar a similar assumption in the y direction one obtains

∫H+

T (y′ − y)2dA(x′) = ky. (3.16)

Equations 3.15, 3.16 can be used to compute the local permeability tensor corresponding to anonlocal permeability kernel.

10

3.4. Numerical convergence studies

3.4 Numerical convergence studies

In this section, the solutions for the nonlocal formulation are demonstrated for a simple exampleof flow in a homogeneous anisotropic medium. The convergence test is performed by comparingthe nonlocal solution to a reference TPFA solution. Both the reference and nonlocal solutionsare obtained numerically.

3.4.1 Problem setup

A circular domain with diameter D = 17 is considered. In all of the considered cases, the com-putational nodes are points on a cartesian grid with δx = δy = δ located inside the circle. All ofthe cases in this chapter have one injection and one production well at a fixed distance d = 10from each other, and their locations are symmetric with respect to the domain center. In orderto have the same problem definition, the area of the injection and the injected flux are chosenequal for all cases. The source terms are distributed inside a patch with unit area. There is noflow over the boundaries of the domain.

To illustrate the performance of the nonlocal model, a HeavisideLambda function is used. Thekernel parameters are chosen as Lx = 0.4 and Ly = 0.2. The coefficient matrix is constructedfrom Eq. 3.9 using numerical integration. The fluxes are calculated from Eq. 2.17 using amid-point numerical integration

qx|xi ≈∑p

T (xi,xp) (xp − xi)[p(xp)− p(xi)]dA(xp), (3.17)

qy|xi ≈∑p

T (xi,xp) (yp − yi)[p(xp)− p(xi)]dA(xp). (3.18)

In this section, the fluxes in the x and y direction are denoted as qx and qy.

Two cases are considered. In the first case, the line connecting the wells is aligned with themain axis of anisotropy. In the second case, it is tilted 20o clockwise with respect to the mainaxis of anisotropy. For each case, the corresponding local TPFA solution is obtained as a firststep. The nonlocal solution is obtained for a case where the main axis of the kernel is alignedwith the grid and for a case where it is tilted with respect to the grid.

3.4.2 Example 1: wells aligned with the main axis of permeability

In this example, the main direction of the permeability kernel is aligned with the line connectingthe wells. The results are obtained for the case where this line is aligned with the x-direction,

11

3.4. Numerical convergence studies

and for the case where it makes a 45o angle with the x-direction. A reference TPFA solution isobtained next.

3.4.2.1 Reference solution

Using equations 3.15 and 3.16, the local principal permeability values corresponding to theconsidered kernel are kx = 10.7 and ky = 2.7. The problem is solved using TPFA for variouslevels of refinement. The finest grid is chosen as the reference solution based on the behavior ofthe convergence curves. For the convergence studies, the finest TPFA solution is interpolatedon each grid and the maximum norm

Lmax = ‖Pi − pi‖, (3.19)

and the mean-square norm

L2 =[ n∑i=1

(Pi − pi)2Ai

]1/2(3.20)

are used. Here, Pi is the reference value, and pi is the numerical solution at node i, and Aiis the area of the control volume associated with node i. Table 3.1 and Fig. 3.2 summarizethe convergence results for the TPFA solution. The reference TPFA solution is shown is Fig.3.3.

Table 3.1: Maximum and mean-square norms for TPFA solutions for example 1

n δ Lmax L2

848 0.5 0.0155 0.00783,512 0.25 0.0093 0.004614,296 0.125 0.0076 0.003557,640 0.0625 0.0074 0.003390,224 0.005 0.0074 0.0032130,064 0.0417 0.0073 0.0032

3.4.2.2 Nonlocal solution with the transmissibility kernel aligned with the grid

In this section, the nonlocal solution is obtained for the case where the principal axis ofthe transmissibility kernel is aligned with the x-direction. The solution is obtained for δ =0.1, 0.0625, 0.05, 0.0385. The result are compared to the reference TPFA solution. Table 3.2 and

12

3.4. Numerical convergence studies

0 0.1 0.2 0.3 0.4 0.53

3.5

4

4.5

5

5.5

6

6.5

7

7.5

8x 10

−3

δ

LM

ax p

(a)

0 0.1 0.2 0.3 0.4 0.50.007

0.008

0.009

0.01

0.011

0.012

0.013

0.014

0.015

0.016

δ

L2 p

(b)

Figure 3.2: Convergence of the maximum (a) and mean-square (b) residual norm for TPFA solutionfor example 1

Figure 3.3: Reference TPFA pressure solution for example 1

13

3.4. Numerical convergence studies

Fig. 3.4 show the convergence results. The nonlocal solution for δ = 0.0385 is shown in Fig.3.5.

Table 3.2: Maximum and mean-square norms for the nonlocal pressure solution with the kernelaligned with the grid for example 1

n δ Lmax L2

22,416 0.1 0.0102 0.040457,640 0.0625 0.0040 0.014690,224 0.05 0.0025 0.0082152,712 0.0385 0.0012 0.0033

To visually compare this solution with the TPFA solution, the pressure on the line connectingthe wells and the flux in the x-direction at x = 0 are shown in Fig. 3.6. The nonlocal modelcaptures the same behavior for the pressure solution near the wells, and it also predicts themaximum and minimum pressure values correctly. The flux profile at x=0 is also matching wellbetween the two solutions. Convergence of qx at x = 0 is shown in Fig. 3.7.

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.11

2

3

4

5

6

7

8

9

10

11x 10

−3

δ

LM

ax p

(a)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

δ

L2 p

(b)

Figure 3.4: Convergence of the maximum (a) and mean-square (b) norms for the nonlocal pressuresolution with the kernel aligned with the grid for example 1

The fluxes calculated with the finest nonlocal model are compared with the reference TPFAsolution in figures 3.8 and 3.9. As it can be seen from the contours, the flux field is quite similaraway from the wells, but near the sources the nonlocal formulation produces a slightly differentfield. The reason for this difference lies in the assumption used for finding the local permeability

14

3.4. Numerical convergence studies

Figure 3.5: Fine nonlocal pressure solution for the case with the transmissibility kernel aligned withthe grid with δ = 0.0385 for example 1

0 2 4 6 8 10

−0.1

−0.05

0

0.05

0.1

Distance from the left well

Pre

ssure

TPFA

Nonlocal

(a)

−10 −5 0 5 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

y

qx

TPFA

Nonlocal

(b)

Figure 3.6: Comparison of the fine nonlocal solution and the TPFA solution with the kernel alignedwith the grid for example 1: (a) pressure solution between the two wells , (b) qx atx = 0

15

3.4. Numerical convergence studies

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.5

1

1.5

δ

LM

ax q

x

(a)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1

2

3

4

5

6

7

8

9

10

δ

L2 q

x

(b)

Figure 3.7: Convergence of the maximum (a) and mean-square (b) norms for qx at x = 0 forexample 1 with θ = 0

values. Since the pressure field near the wells is highly nonlinear, equations 3.15 and 3.16 donot hold in that region.

Fig. 3.10 and 3.11 show the convergence behavior of the nonlocal fluxes with respect to thereference TPFA solution. In these plots, a circular neighborhood with radius 1.5 around the wellsis not considered for calculating the residuals. Including the wells shows a similar convergencebehavior, but since the maximum error occurs near the well, the maximum residual values arehigher in that case.

16

3.4. Numerical convergence studies

−5 −4 −3 −2 −1 0 1 2 3 4 5−1.5

−1

−0.5

0

0.5

1

1.5

0.4

0.4

0.3

0.3

0.3

0.3

0.2

0.2

0.2

0.2

0.2

0.2

0.1

8

0.1

8

0.18

0.1

8

0.1

8

0.18

0.1

6

0.16

0.16

0.16 0.16

0.1

6

0.16

0.1

6

0.1

4

0.14

0.14

0.140.1

4

0.14

0.14

0.14

0.1

4

0.13

0.130.13

0.1

3

0.1

3

0.130.13

0.13

0.1

3

0.12

0.12 0.12

0.1

2

0.1

2

0.120.12

0.12

0.1

2

(a)

−5 −4 −3 −2 −1 0 1 2 3 4 5−1.5

−1

−0.5

0

0.5

1

1.5

0.4

0.4

0.3

0.3

0.3

0.3

0.3

0.3

0.2

0.2

0.2

0.2

0.2

0.2

0.1

8

0.18

0.18

0.1

8

0.1

8

0.1

8

0.1

8

0.1

8

0.1

6

0.16

0.16

0.1

6

0.16

0.1

6

0.16

0.1

6

0.14

0.14

0.14

0.1

4

0.1

4

0.14

0.14

0.1

4

0.13 0.130.13

0.1

30.13

0.130.13

0.1

3

0.1

2

0.12 0.12

0.1

2

0.1

2

0.120.12

0.12

0.1

2

(b)

Figure 3.8: Comparison of the reference solution with the fine nonlocal solution:(a) tpfa qx, (b) nonlocal qx

17

3.4. Numerical convergence studies

−10 −8 −6 −4 −2 0 2 4 6 8 10

−6

−4

−2

0

2

4

6

0.08

0.08

0.05

0.05

0.03

0.03

0.03

0.0

3

0.03

0.01

0.0

1

0.01

0.01

0.01

−0.01

−0.0

1

−0.01

−0.01

−0.03

−0.03

−0.03

−0.0

3

−0.05

−0.05

−0.08

−0.08

(a)

−10 −8 −6 −4 −2 0 2 4 6 8 10

−6

−4

−2

0

2

4

6

0.08

0.08

0.05

0.05

0.03

0.03

0.03

0.03

0.01

0.0

1

0.01

0.01

0.0

1

0.01−0.01

−0.0

1

−0.01

−0.01

−0.0

1

−0.01−0.03

−0.03

−0.03

−0.03−0.05

−0.05

−0.08

−0.08

(b)

Figure 3.9: Comparison of the reference solution with the fine nonlocal solution:(a) tpfa qy, (b) nonlocal qy

18

3.4. Numerical convergence studies

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.5

1

1.5

2

2.5

δ

LM

ax q

x

(a)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1

2

3

4

5

6

7

8

9

10

δL

2 q

x(b)

Figure 3.10: Convergence of the maximum (a) and mean-square (b) norms for qx with the kernelaligned with the grid for example 1.

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

δ

LM

ax q

y

(a)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.5

1

1.5

2

2.5

3

3.5

4

δ

L2 q

y

(b)

Figure 3.11: Convergence of the maximum (a) and meas-square (b) norms for qy with the kernelaligned with the grid for example 1.

19

3.4. Numerical convergence studies

3.4.2.3 Nonlocal solution with the transmissibility kernel rotated with respect to thegrid

In this example the wells are placed such that the line connecting them makes an angle θ = 45o

with the x axis. The permeability kernel is also rotated in the same way and the solution isobtained for δ = 0.1, 0.0625, 0.05, 0.0385. The solution is compared with the base tpfa solutionby a −θo rotation. Fig 3.12 and table 3.3 show the convergence behavior for this test case.

Table 3.3: Maximum and mean-square norms of the pressure solution for the nonlocal model withθ = 45 for example 1

n δ Lmax L2

22,416 0.1 0.0123 0.043157,640 0.0625 0.0056 0.013190,224 0.05 0.0040 0.0075152,712 0.0385 0.0027 0.0036

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.12

4

6

8

10

12

14x 10

−3

δ

LM

ax p

(a)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

δ

L2 p

(b)

Figure 3.12: Convergence of the maximum (a) and mean-square (b) norms for the nonlocal pressuresolution with θ = 45o for example 1

Fig. 3.14 indicates that the pressure solution along the line connecting the wells is matched wellby both the aligned nonlocal solution and the nonlocal solution with θ = 45o.

Similar to the previous example, qx is calculated at the line perpendicular to the line connectingthe wells. Fig 3.15 shows the convergence for this flux.

20

3.4. Numerical convergence studies

Figure 3.13: Fine nonlocal pressure solution δ = 0.0385, θ = 45o

0 2 4 6 8 10

−0.1

−0.05

0

0.05

0.1

Distance from the left well

Pre

ssu

re

TPFA

Nonlocal θ = 0o

Nonlocal θ = 45o

Figure 3.14: Pressure solution along the line connecting the wells for example 1

21

3.4. Numerical convergence studies

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

δ

LM

ax q

x

(a)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

δ

L2 q

x

(b)

Figure 3.15: Convergence of the maximum (a) and mean-square (b) norms for qx at x = 0 forexample 1 with θ = 45o

Fig. 3.16 and 3.17 show the convergence behavior of nonlocal fluxes with respect to the referenceTPFA solution. In the rotated case, the fluxes along y = x and y = −x lines are calculated andcompared to the reference solution with a −450 rotation. In these plots a circular neighborhoodwith radius 1.7 around the wells is not considered for calculating the residuals. Since the wellpatches are rectangular, only the distance between the center of the well cell groups can be setequal to the reference solution and the position of the well cell groups are not exactly the samein the two cases. This is one source of error for this test case. The largest absolute value ofthe error occurs near the location of the wells at the boundary, and it does not diminish uponrefinement. In the convergence plots presented here, an outer ring with thickness ∆ = 0.1 is notconsidered when calculating the residuals. The results show that the flux field is converging inall other parts of the domain. Excluding this ring only affects the convergence of the maximumresidual and the second norm converges considering every point in the domain.

The coefficient matrix obtained from Eq. 3.9 is an M-matrix for any θ; therefore, the pressuresolution obtained with this method always satisfies the discrete maximum principle, Eq. 3.2.The classical way of solving a case with a general full matrix permeability is an MPFA method[11]. However, for the given values of kx and ky the inverse of the MPFA coefficient matrix hasnegative entries [12]; hence, it does not satisfy Eq. 3.2. For a mild anisotropy ratio like the onein this test case, it is hard to visually illustrate the non-monotonicity caused by these negativevalues. However, having these entries in A−1 leads to spurious oscillations in the pressuresolution [12].

22

3.4. Numerical convergence studies

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.0155

0.016

0.0165

0.017

0.0175

0.018

0.0185

δ

LM

ax q

x

(a)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

δL

2 q

x(b)

Figure 3.16: Convergence of the maximum (a) and mean-square (b) norms for nonlocal qx withθ = 45o for example 1

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.02

0.0205

0.021

0.0215

0.022

0.0225

0.023

0.0235

0.024

δ

LM

ax q

y

(a)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

δ

L2 q

y

(b)

Figure 3.17: Convergence of the maximum (a) and mean-square (b) norms for nonlocal qy withθ = 45o for example 1

23

3.4. Numerical convergence studies

3.4.3 Example 2: wells tilted with respect to the main axis of permeability

In the previous example the main axis of permeability is aligned with the shortest path betweenthe wells and therefore the flow is dominated by this path. In order to test the nonlocal modelin a more general setting, in this example the line connecting the wells is rotated 20o clockwisewith respect to the main axis of permeability.

3.4.3.1 Reference solution

The problem is solved using a TPFA for various levels of refinement. Table 3.4 and Fig. 3.18summarize the convergence results for the TPFA solution. The finest grid is chosen as thereference solution based on the behavior of the convergence curves. The reference TPFA solutionis shown in Fig. 3.19.

Table 3.4: Maximum and mean-square norms for TPFA pressure solutions for example 2

n δ Lmax L2

848 0.5 0.0154 0.04543,512 0.25 0.0064 0.026414,296 0.125 0.0051 0.019857,640 0.0625 0.0010 0.0027203,416 0.0333 0.0002 0.0004

0 0.1 0.2 0.3 0.4 0.50

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

δ

LM

ax p

(a)

0 0.1 0.2 0.3 0.4 0.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

δ

L2 p

(b)

Figure 3.18: Convergence of the maximum (a) and mean-square (b) norms for the TPFA pressuresolution for example 2

24

3.4. Numerical convergence studies

Figure 3.19: Reference TPFA pressure solution for example 2

3.4.3.2 Nonlocal solution with the transmissibility kernel aligned with the grid

In this section, the nonlocal solution is calculated for the case where the principal axis of thetransmissibility kernel is aligned with x-direction. The solution is obtained for δ = 0.1, 0.0625, 0.05,0.0385. The result are compared to the reference TPFA solution. Table 3.5 and Fig. 3.20 showthe convergence results. The nonlocal solution for δ = 0.0385 is shown in Fig. 3.21.

Table 3.5: Maximum and mean-square norms for the nonlocal pressure solution with θ = 0 forexample 2

n δ Lmax L2

22,416 0.1 0.0134 0.101057,640 0.0625 0.0071 0.044590,224 0.05 0.0042 0.0042152,712 0.0385 0.0035 0.0035

25

3.4. Numerical convergence studies

To visually compare this solution with the TPFA solution, The pressure on the line connectingthe wells, and flux in the x-direction at x = 0 are shown in Fig. 3.22. The nonlocal modelcaptures the same behavior for the pressure solution near the wells, and it also predicts themaximum and minimum pressure values correctly. The reference flux profile is matched wellaway from y = 0 line, but the maximum value of qx predicted by the nonlocal model is slightlyhigher that the TPFA reference value. Convergence or qx at x = 0 is shown in Fig. 3.23. One ofthe reasons for having a better match for the pressure solution compared with the flux field isthe difference of the accuracy of the respective integrations. The coefficients matrix is obtainedusing a high order numerical integration, whereas the fluxes are obtained from central numericalintegration.

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.12

4

6

8

10

12

14x 10

−3

δ

LM

ax p

(a)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

δ

L2 p

(b)

Figure 3.20: Convergence of the maximum (a) and mean-square (b) norms for the nonlocal pressuresolution with θ = 0 for example 2

The fluxes calculated with the finest nonlocal model are compared to the reference TPFA so-lution in figures 3.24 and 3.25. Similar to the previous example, the flux field is quite similaraway from the wells but near the sources the nonlocal model yields a different flux field. Thereason for this difference lies in the assumption used for finding the local permeability values.Since the pressure field near the wells is highly nonlinear, equations 3.15 and 3.16 do not holdin that region.

The convergence behavior of fluxes calculated with the nonlocal model with respect to thereference TPFA solution are presented in Fig. 3.26 and Fig. 3.27. In these plots, a circularneighborhood with radius 1.5 around the center of the wells is not considered for calculating theresiduals. Including the wells yields a similar convergence behavior, but the maximum residualvalues are higher in that case.

26

3.4. Numerical convergence studies

Figure 3.21: Fine nonlocal pressure solution with the transmissibility kernel aligned with the gridwith δ = 0.0385 for example 2

0 2 4 6 8 10

−0.1

−0.05

0

0.05

0.1

0.15

Distance from the left well

Pre

ssure

TPFA

Nonlocal

(a)

−10 −5 0 5 100

0.02

0.04

0.06

0.08

0.1

0.12

y

qx

TPFA

Nonlocal

(b)

Figure 3.22: Comparison of the fine nonlocal solution and the TPFA solution with the transmis-sibility kernel aligned with the grid for example 2: (a) pressure solution between thetwo wells , (b) qx at x = 0

27

3.4. Numerical convergence studies

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.2

0.4

0.6

0.8

1

1.2

1.4

δ

LM

ax q

x

(a)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1

2

3

4

5

6

7

8

9

10

δ

L2 q

x

(b)

Figure 3.23: Convergence of the maximum (a) and mean-square (b) norms for qx at x = 0 withθ = 0 for example 2

3.4.3.3 Nonlocal solution with the transmissibility kernel rotated with respect to thegrid

In this case, the well group cells are placed such that the line connecting the wells makes anangle θ = 65o with the x axis. The permeability kernel is rotated 45o so that the angle betweenthe main axis of permeability and the line connecting the wells remains 20o. The solution isobtained for δ = 0.1, 0.0625, 0.05, 0.0385 and it is compared with the base tpfa solution by a 45o

rotation. Fig 3.28 shows the convergence behavior for this test case.

Table 3.6: Maximum and mean-square norms for the nonlocal pressure solution with θ = 45 forexample 2

n δ Lmax L2

22,416 0.1 0.0151 0.097957,640 0.0625 0.0072 0.031490,224 0.05 0.0054 0.0143152,712 0.0385 0.0034 0.0058

Fig. 3.29 shows the pressure solution for δ = 0.0385. The pressure solution agrees well with theTPFA solution. The pressure solution along the line connecting the wells is compared betweenthe TPFA and the two nonlocal solutions in Fig. 3.30. It can be seen that this solution matchesboth the nonlocal solution from the previous section and the reference TPFA solution. qx is

28

3.4. Numerical convergence studies

−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5

0.3

0.3

0.2

0.2

0.18

0.1

80.18

0.18

0.16

0.16

0.16

0.16

0.14

0.14

0.14

0.14

0.13

0.13

0.13

0.1

3

0.12

0.1

2

0.12

0.12

0.1

2

0.12

0.09

0.09 0.09

0.09

0.090.09

0.0

5

0.05

0.05

0.0

5

0.0

5

0.05

0.05

0.0

5

0.0

4

0.04

0.04

0.04

0.0

4

0.0

4

0.04

0.04

0.0

4

(a)

−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5

0.3

0.3

0.2

0.2

0.2

0.2

0.18

0.18

0.18

0.18

0.16

0.16

0.16

0.16

0.14

0.14

0.14

0.14

0.13

0.1

3

0.13

0.13

0.1

3

0.13

0.12

0.12

0.12

0.12

0.12

0.12

0.090.09

0.09

0.0

9

0.09

0.09

0.0

5

0.05

0.05

0.05

0.0

5

0.05

0.05

0.0

5

0.0

4

0.04

0.04

0.0

4

0.0

4

0.04

0.04

0.04

0.0

4

(b)

Figure 3.24: Comparison of the reference solution with the fine nonlocal solution:(a) tpfa qx, (b) nonlocal qx

29

3.4. Numerical convergence studies

−10 −8 −6 −4 −2 0 2 4 6 8 10

−6

−4

−2

0

2

4

6

0.15

0.15

0.08

0.08

0.05

0.0

5

0.05

0.05

0.05

0.05

0.030.03

0.03

0.03

0.03 0.03

0.01

0.01

0.01

0.01

0.01

0.01

−0.01

−0.01

−0.

01

−0.01

−0.03

−0.03

−0.05

−0.05

−0.08

−0.08

(a)

−10 −8 −6 −4 −2 0 2 4 6 8 10

−6

−4

−2

0

2

4

6

0.15

0.15

0.08

0.08

0.05

0.0

5

0.050.05

0.0

5

0.05

0.03 0.03

0.03

0.0

3

0.03

0.03

0.0

3

0.01

0.01

0.01

0.01

0.01

0.01

−0.01

−0.01

−0.0

1

−0.01

−0.03

−0.03

−0.05

−0.05

−0.08

−0.08

(b)

Figure 3.25: Comparison of the reference solution with the fine nonlocal solution:(a) tpfa qy ,(b) nonlocal qy

30

3.4. Numerical convergence studies

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

δ

LM

ax q

x

(a)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

2

4

6

8

10

12

δL

2 q

x(b)

Figure 3.26: Convergence of the maximum (a) and mean-square (b) norms for nonlocal qx with thekernel aligned with the grid for example 2

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.5

1

1.5

2

2.5

δ

LM

ax q

y

(a)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1

2

3

4

5

6

δ

L2 q

y

(b)

Figure 3.27: Convergence of the maximum (a) and mean-square (b) norms for nonlocal qy with thekernel aligned with the grid for example 2

31

3.4. Numerical convergence studies

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.12

4

6

8

10

12

14

16x 10

−3

δ

LM

ax p

(a)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

δ

L2 p

(b)

Figure 3.28: Convergence of the maximum (a) and mean-square (b) error norms for the nonlocalpressure solution with θ = 45o for example 2

calculated at the line perpendicular to the main axis of anisotropy and is compared to the sameflux in the reference solution. Fig 3.31 shows the convergence for this flux.

Fig. 3.32 and Fig. 3.33 show the convergence behavior of nonlocal fluxes with respect to thereference TPFA solution. In the rotated case, the fluxes along y = x and y = −x lines arecalculated and compared with the reference solution with a −450 rotation. In these plots acircular neighborhood with radius 1.7 around the wells is not taken into account for calculatingthe residuals. Since the well patches are rectangular, only the distance between the center ofthe well cell groups can be set equal to the reference solution and the position of the well cellgroups are not the exactly the same in the two cases. The largest value for the maximum erroroccurs near the location of the wells at the boundary, and this error does not diminish uponrefinement. In the convergence plots presented here an outer ring with thickness ∆ = 0.3 is notconsidered for calculating the residuals. The results show that the flux field is converging in allother parts of the domain. Excluding this region only affects the convergence of the maximumerror, and the second norm of the error diminishes considering all points in the domain.

The results show that in a general case where the main axis of anisotropy is not aligned with theline connecting the wells, the nonlocal model is still able to match the reference TPFA solutionfor an arbitrary rotation of the permeability kernel with respect to the grid.

32

3.4. Numerical convergence studies

Figure 3.29: Fine nonlocal pressure solution with δ = 0.0385 and θ = 45o for example 2

0 2 4 6 8 10

−0.1

−0.05

0

0.05

0.1

0.15

Distance from the left well

Pre

ssure

TPFA

Nonlocal θ = 0o

Nonlocal θ = 45o

Figure 3.30: Pressure comparison for example 2 along the lines connecting the wells

33

3.4. Numerical convergence studies

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1

2

3

4

5

6

7x 10

−3

δ

LM

ax q

x

(a)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

δL

2 q

x

(b)

Figure 3.31: Convergence of the maximum (a) and mean-square (b) norms for qx at y = −x forexample 2 with θ = 45o

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.008

0.009

0.01

0.011

0.012

0.013

0.014

δ

LM

ax q

x

(a)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

δ

L2 q

x

(b)

Figure 3.32: Convergence of the maximum (a) and mean-square (b) norms for nonlocal qx withθ = 45o for example 2

34

3.4. Numerical convergence studies

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

δ

LM

ax q

y

(a)

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

δ

L2 q

y

(b)

Figure 3.33: Convergence of the maximum (a) and mean-square (b) norms for nonlocal qy withθ = 45o for example 2

35

Chapter 4. Possible ways for finding the transmissibility kernel

4 Possible ways for finding the transmissibility kernel

The first question that arises for generalizing the nonlocal formulation is how to find the trans-missibility kernel for a heterogeneous permeability field. Sen et al. [2] proposed a method basedon setting a system of linear equations with the discrete connectivities as the unknowns. Con-sidering a 1D domain of size l, an experiment should be set up in which pressure distributionalong the line and the resulting flow can be measured at every point. If the interval is dividedinto N equal parts of size δ = l/N the discrete form of Eq. 2.1 is

qi =

N+1∑j=1

Ti,j(pj − pi

), (4.1)

where Ti,j is the discrete equivalent of T (x, x′). Since the connection between any two nodeshas the same strength for flow traveling in either direction between the two points, it can beassumed that Ti,j is symmetric. Therefore, there are N(N −1)/2 independent unknowns in Ti,j ,and as many independent equations are needed to find them. Hence enough experiments shouldbe conducted to provide these independent equations. This method can be general, and it isnot limited to cases with a homogeneous transmissibility kernel. However, there is no exampleshowing the performance of this method. Preliminary trials of this method for homogeneousfields have showed that this method can be used as a way to construct discrete transmissibilitykernels from local solutions.

Exploiting the convolution-like form of Eq. 2.1 can provide another way for finding T . If oneassumes that the transmissibility kernel depends only on the distance between the two points,then

T (x, x′) = T (|x− x′|) = T (x− x′), (4.2)

which is a valid assumption for cases where the transmissibility kernel is the same everywherein the domain. Equation 2.1 becomes

∫ ∞−∞

T (x− x′)[p(x)− p(x′)] dx′ = q(x), (4.3)

which can be rearranged as

p(x)

∫ ∞−∞

T (x− x′) dx′ −

T ∗ p︷ ︸︸ ︷∫ ∞−∞

T (x− x′)p(x′) dx′ = q(x). (4.4)

36

Considering an experiment where p(x) and q(x) are known at every point, the transmissibilitykernel is the only unknown in Eq. 4.4. If the integral of T is considered as a constant, G, takingthe Fourier transform of both sides yields

GF(p)−F(T )F(p) = F(q). (4.5)

The kernel can be found from solving Eq. 4.5

T = F−1

GF(p)−F(q)

F(p)

. (4.6)

37

Chapter 5. Final remarks

5 Final remarks

In this work, a nonlocal formulation is used to model flow in porous media in multi-dimensionaldomains. A general method is proposed to discretize a given transmissibility kernel. A methodhas been suggested for comparing the nonlocal flux with the classical flux in problems withlinear pressure profile in anisotropic porous media. The resulting pressure and flux fields areshown to converge to the reference TPFA solution for mild anisotropy ratios. Possible waysto extend the nonlocal formulation to heterogeneous porous media have been proposed. Thisformulation can be used in the future for modeling porous media with an arbitrary distributionof permeability.

38

Bibliography

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[12] J.M. Nordbotten, Ivar Aavatsmark, Monotonicity conditions for control volume meth-ods on uniform parallelogram grids in homogeneous media, Computational Geosciences 9(2005) 61-72.

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