Simulating Fluid Flow In Vuggy Porous Media

Copyright by Dana Sue Brunson 2005

The Dissertation Committee for Dana Sue Brunson certifies that this is the approved version of the following dissertation:

Simulating Fluid Flow In Vuggy Porous Media

Committee:

Todd Arbogast, Supervisor Steve Bryant Oscar Gonzalez Richard Tsai Mary Wheeler


by Dana Sue Brunson, B.S., M.S.

DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT AUSTIN August 2005

Acknowledgments

My sincerest gratitude goes to my advisor, Todd Arbogast, for his unending patience, encouragement and guidance. There is no doubt that this could not have happened without his support and understanding. I am grateful to my husband, John Homer, whose contribution has been overwhelming. He has freed me from nearly all other tasks so that I may have the time to study and write. Of course, I must acknowledge our daughter, Raina, because it is for her that I persist in hopes of setting a good example. I thank my committee for their time and effort. In addition, I am grateful to each of the professors I have studied with and for their inspiration to go further. Nancy Lamm and her predecessor, Chris Marcin, in the Mathematics department were also indispensable to my success. I am thankful for all of the friends along the way who have supported, encouraged and entertained me all these years.

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Publication No. Dana Sue Brunson, Ph.D. The University of Texas at Austin, 2005 Supervisor: Todd Arbogast

We develop and analyze a mixed finite element method for the solution of an elliptic system modeling a porous medium with large cavities, called vugs. It consists of a second order elliptic (i.e., Darcy) equation on part of the domain coupled to a Stokes equation on the rest of the domain, and a slip boundary condition (due to Beavers, Joseph, and Saffman) on the interface between them. The tangential velocity is not continuous on the interface. We consider a vuggy porous medium with many small cavities throughout its extent, so the interface is not isolated. We use a certain conforming Stokes element on rectangles, slightly modified near the interface to account for the tangential discontinuity. This gives a mixed finite element method for the entire Darcy-Stokes system with a regular sparsity pattern that is easy to implement, no matter how complex the vug geometry may be. We prove optimal global first order L2 convergence of the velocity and pressure, as well as of the velocity gradient, in the Stokes domain. Numerical results verify these rates of convergence, and even suggest somewhat better convergence in v

certain situations. Finally, we present a lower dimensional space that uses Raviart-Thomas elements in the Darcy domain and uses our new modified elements near the interface in transition to the Stokes elements. We present two computational studies to illustrate and verify an homogenized macro-model of flow in a vuggy medium. And finally, we compare the effect of the Beavers-Joseph slip condition to using a no slip condition on the interface in a few simple examples.

vi

Table of Contents

Acknowledgments

iv

Abstract

v

List of Tables

x

List of Figures

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Chapter 1. Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4

Chapter 2. Pipe Creek Sample 2.1 Connectivity of the Vug System . . . . . . . . . . . . . . . . .

7 9

Chapter 3. The Model 3.1 Notation . . . . . . . . . . . . 3.2 The Regions . . . . . . . . . . 3.2.1 The Vuggy Region, Ωs . 3.2.2 The Porous Region, Ωd 3.2.3 The Interface, Γ . . . . 3.3 The Governing Equations . . . 3.4 Variational Form . . . . . . .

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14 14 15 15 16 17 18 19

Chapter 4. The Finite Element Space 4.1 The Standard Stokes Elements . . . . . . . . . . . . . . . . . . 4.2 The Modified Elements . . . . . . . . . . . . . . . . . . . . . .

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Chapter 5. A Priori Error Analysis 5.1 A π Operator for the Velocity . . . . . . . . . . . . . . . . . . 5.2 A Convergence Analysis . . . . . . . . . . . . . . . . . . . . .

29 29 34

Chapter 6. Computational Results 6.1 A Remark on the Solution Procedure 6.2 Some Simply Constructed Examples . 6.3 Some Smooth Examples with Corners 6.4 Some Simple Layered Examples . . . 6.5 A Lower Dimension Modification . . . 6.6 Pipe Creek Sample . . . . . . . . . .

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Chapter 7. Computational Results for the Homogenized Model 7.1 A Macro-Model and the Effective Permeability . . . . . . . . . 7.2 Examples of Effective Permeabilities . . . . . . . . . . . . . . . 7.2.1 A Layered Medium . . . . . . . . . . . . . . . . . . . . . 7.2.2 Flow through a Meandering Vug System . . . . . . . . . 7.2.3 Flow through a Constricted Vug System . . . . . . . . . 7.2.4 Flow through a Disconnected Vug System . . . . . . . . 7.3 A Simple Computational Test of the Macro-Model . . . . . . . 7.3.1 Fine-scale Results for Pressure Drop Flow . . . . . . . . 7.3.2 Effective Permeability of the Sample . . . . . . . . . . . 7.3.3 Darcy Computation using Effective Permeabilities . . . 7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52 53 54 55 57 59 59 60 61 62 63 65

Chapter 8. 8.1 8.2 8.3 8.4

Comparison of the Beavers-Joseph and No Slip Boundary Conditions 67 Analytic Example . . . . . . . . . . . . . . . . . . . . . . . . . 68 Varying the Rock Permeability . . . . . . . . . . . . . . . . . . 71 Varying the Vug Diameter . . . . . . . . . . . . . . . . . . . . 72 Varying the Plug between Vugs . . . . . . . . . . . . . . . . . 73

Chapter 9.

Conclusions

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viii

Bibliography

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Vita

83

ix

List of Tables

6.1 6.2

The numerical Test Cases 1–4. . . . . . . . . . . . . . . . . . . ˜ h for Observed convergence rates using the unmodified space V the errors Ep = p − ph and Eu = u − uh for test cases 1–4 on a uniform grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Observed convergence rates using the modified space Vh for the errors Ep = p − ph and Eu = u − uh for Test Cases 1–4 on a uniform grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Observed convergence rates using the modified space Vh for the errors Ep = p − ph and Eu = u − uh for Test Cases 1–4 on a randomly perturbed grid. . . . . . . . . . . . . . . . . . . . . . 6.5 The numerical Test Cases 5–6. . . . . . . . . . . . . . . . . . . 6.6 Observed convergence rates using the modified space Vh for the errors Ep = p − ph and Eu = u − uh for Test Cases 5–6. . . . 6.7 The numerical Test Cases 7–8. . . . . . . . . . . . . . . . . . . 6.8 Observed convergence rates using the modified space Vh for the errors Ep = p − ph and Eu = u − uh for test case 8. . . . . . ¯ h for the 6.9 Observed convergence rates using the modified space V errors Ep = p − ph and Eu = u − uh for Test Cases 1–6 and 8 on a uniform grid. . . . . . . . . . . . . . . . . . . . . . . . . . ¯ h for the 6.10 Observed convergence rates using the modified space V errors Ep = p − ph and Eu = u − uh for Test Cases 1–6 and 8 on a randomly perturbed grid. . . . . . . . . . . . . . . . . . . 7.1 7.2 7.3 7.4 7.5

Numerical convergence for the layered medium. The h = 0 results are the true analytical values from (7.11)–(7.12). . . . . Effect of varying the Beavers-Joseph coefficient α for the layered medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of varying the rock matrix permeability K for the layered medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of varying the vug aperture δ for the layered medium. . Effect of varying the plug between vugs for the disconnected medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

41 42 42 43 44 44 44 45

46 46 56 57 57 58 61

List of Figures

2.1 2.2 2.3

A close-up view of the Pipe Creek Reef. Note the lens cap at the lower left. . . . . . . . . . . . . . . . . . . . . . . . . . . . Cretaceous limestone with Caprinid fossils. . . . . . . . . . . . Two CT scan cross-sections of the Pipe Creek Reef sample. . .

7 8 9

3.1

Notation near the interface. . . . . . . . . . . . . . . . . . . .

15

4.1 4.2

The standard Stokes finite elements. . . . . . . . . . . . . . . The standard basis functions in the x-direction. The left picture is the edge basis function over two grid blocks. The right picture is the corner basis function over four grid blocks. . . . . . . . . The 7 patterns . . . . . . . . . . . . . . . . . . . . . . . . . . The 5 patterns (up to symmetries) around a corner point in the modified space Vh on ∂Ω. Here d represents a Darcy element, s is a Stokes element, unspecified is either, and a heavy line is a part of Γ. For x1 velocity components, a quarter circle in an element indicates that the corner basis function is present over the element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . The 7 elements (up to symmetries) for the x1 velocity components, on R = (0, a)×(0, b), in the modified space Vh . The quarter circles represent corner nodal values, and the edge crosses represent normal fluxes (or, equivalently, nodal values). . . . .

23

4.3 4.4

4.5

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

CT scan of slice 120 of Pipe Creek sample. Permeability of center quarter of slice 120. Clean pressure. . . . . . . . . . . . . . . . Muddy Pressure. . . . . . . . . . . . . . . Clean velocity and speed. . . . . . . . . . . Muddy speed and velocity. . . . . . . . . . CT scan of slice 180 of Pipe Creek sample. Permeability of center quarter of slice 180. Clean pressure. . . . . . . . . . . . . . . . xi

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6.10 Muddy Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Clean velocity and speed. . . . . . . . . . . . . . . . . . . . . . 6.12 Muddy speed and velocity. . . . . . . . . . . . . . . . . . . . . 7.1 7.2 7.3 7.4 7.5

8.1 8.2 8.3

The layered medium. The reference cell is on the left, the periodic medium is on the right. . . . . . . . . . . . . . . . . . . . The meandering vug medium. The reference cell is on the left, the periodic medium is on the right. . . . . . . . . . . . . . . . The constricted vug medium. The reference cell is on the left, the periodic medium is on the right. . . . . . . . . . . . . . . . The disconnected vug medium. The reference cell is on the left, the periodic medium is on the right. The plug width (in the x-direction) varies from 2 mm to 1 cm. . . . . . . . . . . . . . A synthetic vuggy rock with an interconnected vug system and an isolated vug. It is also divided into 2 × 2 and 3 × 3 coarse grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˜ 11 in the layered medium and meandering Plot of the effect on Φ vug medium as we increase the permeability of the rock. . . . ˜ for media with various rock permeabilities Plot of the effect on Φ as we change the vug constriction diameter. . . . . . . . . . . ˜ in the disconnected vuggy medium as Plot of the effect on Φ we vary the plug width. Rock permeabilities of 10md through 1000md are all represented by this one graph. . . . . . . . . .

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50 51 51 56 58 59 60 62 71 72

73

Chapter 1 Introduction

1.1

Motivation Many of the world’s aquifers and petroleum reservoirs contain small

cavities known as vugs, which are relatively small compared to the size of the domain itself but much larger than the usual intergranular pore space. Such vuggy inclusions are especially common in carbonate rocks, and formed from various dissolution processes including geochemical reactions and sedimentation around fossils. Clearly the spatial distribution of vugs resulting from these processes will be complex and irregularly interconnected. Although small, vugs can significantly increase both the effective porosity and permeability of the medium. Examples of vuggy media include the Cretaceous Edwards aquifer in Texas and the Columbia River Basin in the Northwestern US. There are both practical and theoretical problems for understanding fluid flow through such rocks, yet better flow models are of both economic and environmental interest in a variety of applications. For example, in petroleum production (more than half the world’s oil reserves are in carbonate rocks, many of which have vugs of the type described above), in obtaining adequate water supplies (groundwater accounts for about one third of public, agricul-

1

tural and industrial water use in the US, and many aquifers are in vuggy limestone), and to remediate subsurface contamination (such as nuclear operations on the Hanford site near the Columbia River). While there is basic understanding of many of the principles governing fluid flow over length scales smaller than the vug-size, the ultimate goal is to construct models that predict the effective flow properties of these types of rocks over hundreds of yards to several miles. This dissertation describes the development and analysis of a mixed finite element method for the solution of an elliptic system modeling a porous medium with vugs as described above. In particular, we consider the vugs to be ubiquitous throughout the medium, so the interface between rock and vug is not isolated. We are fortunate to have a sample of carbonate rock from the Glenn Rose outcrop at Pipe Creek, Texas, to serve as an example.

1.2

Overview Since only low Reynold’s number flow is to be expected, the system

is governed in the rock matrix Ωd ⊂ Ω by a second order elliptic equation

¯ d by representing Darcy’s law and mass conservation, and in the vugs Ωs = Ω\Ω the Stokes equation, with the Beavers-Joseph-Saffman boundary condition [8, 32] on the interface between the two regions Γ = ∂Ωd ∩ ∂Ωs . The system is difficult to approximate because the Darcy and Stokes solutions have very different regularity properties and, more importantly, the tangential velocity may be discontinuous on the Darcy-Stokes interface Γ. We limit our study to 2

Ω ⊂ R2 for simplicity, although it should extend to 3-D. Our solver precludes large problems, so 3-D is not yet feasible. Previous studies have developed numerical techniques appropriate to the case where the porous medium and the open (vuggy) region are well separated [18, 27, 33]. However, these techniques are not so readily adapted to the case of a vuggy medium, for which there is no distinct separation between the vuggy regions and the porous rock, i.e., to the case where the vugs and porous matrix are essentially intertwined everywhere. For example, Layton et al. [27] use a Lagrange multiplier on the interface Γ to both aid in the approximation of the Beavers-Joseph-Saffman boundary condition and to allow the use of existing Darcy and Stokes flow simulators in a domain-decomposition (Ωd and Ωs ) iterative technique. They also provide a very thorough error analysis of their underlying method and the equations in general. However, this approach is not so useful when Γ is large, since the size of the Lagrange multiplier space would preclude efficient solution. The approach taken here is to design a finite element that is appropriate for both the Stokes and Darcy regions of the domain. The rationale is that then an efficient code can be written with little regard to the nature of the underlying equations, i.e., whether an element lies in Ωs or Ωd . For simplicity, we use conforming elements. Since the Stokes equations are necessarily a saddle point system, we consider the second order elliptic Darcy equations in mixed form. Stokes approximation requires that the fluid velocities be approximated in (H 1 (Ω))2 , which is more restrictive than Darcy 3

simulation requires. Mixed methods approximate velocities only in the space H(div) = {u ∈ (L2 )2 : ∇ · u ∈ L2 } [14, 31]. This requires only the (weak) continuity of normal velocity components, whereas Stokes requires both normal and tangential components to be (weakly) continuous. We thereby restrict ourselves to using a finite element appropriate for Stokes simulation that simultaneously works well for Darcy simulation. There do not appear to be many candidates, but there is at least one, a low order space due to Fortin [17]. This space, and higher order generalizations, have been shown by Arbogast and Wheeler [5] to approximate well second order elliptic systems. To approximate the combined Darcy-Stokes system, since the tangential velocity may be discontinuous, this finite element space must be modified near Γ, but the modification is relatively minor and easily handled in a computer code.

1.3

Outline of Thesis The outline of the rest of the thesis follows. In Chapter 2, we describe

the Pipe Creek Sample and discuss the connectivity of the vug system. In Chapter 3, we state in detail the governing equations of the Darcy-Stokes system and a mixed variational form suitable for finite element approximation. In Chapter 4, we define our finite element spaces as a small modification of the Fortin spaces, and then the finite element method follows immediately. In §5.1, we present a projection operator for the velocity and examine its properties. This operator is used in §5.2, where we present an a priori error analysis. We show that the method approximates both the true velocity and pressure to the

4

optimal first order in the standard energy norm. That is, the Stokes velocity error is measured in the (H 1 (Ωs ))2 norm, the Darcy velocity error is measured in H(div; Ωd ), and the pressure error is measured in L2 (Ω). We present the results of several numerical experiments to verify these rates of convergence in Chapter 6. Since these test cases lead to very ill-conditioned linear systems, we also discuss our solution strategy. In §6.5, we consider a finite element space with lower total dimension, by replacing the Fortin elements by RaviartThomas [31] elements on Darcy elements away from the Darcy-Stokes interface. Finally, in §6.6 we show examples of the method applied to the Pipe Creek Sample. When the medium is large in extent, such as an aquifer or petroleum reservoir, it is not reasonable to solve the Darcy-Stokes system over the entire domain for several reasons. First, there is inadequate data regarding the geometry of the vugs over the many square kilometer areal extent of the domain. Secondly, at the centimeter resolution that would be required to resolve the vugs, the problem would be computationally intractable. Finally, the data generated would be much too detailed to be of use in engineering analyses, where only meter scale average flows would be used. Recently, Arbogast and Lehr [4] derived from the micro-scale model presented here a macroscopic model using the mathematical theory of homogenization [9, 20, 24, 34]. The theoretical prediction is that Darcy flow results at the macro-scale. What is important here is that an expression for the effective permeability is also derived. This expression involves the solution of 5

a Darcy-Stokes system on a representative cell involving the vug geometry (except that periodic boundary conditions must be imposed). The numerical method of this paper is suitable for solving this cell problem, and thereby allowing one to obtain effective permeabilities for vuggy media. In Chapter 7, we will demonstrate this and in Chapter 8 we study the significance of the Beavers-Joseph slip condition on effective permeability as compared to a no slip condition.

6

Chapter 2 Pipe Creek Sample

A sample of Cretaceous (Albian) limestone was collected from the Pipe Creek Reef, a caprinid buildup in the Glenn Rose Formation near the town of Pipe Creek, Texas [30]. Caprinids are a particular type of rudist, which are large shell-forming creatures that grew on the Cretaceous sea floor in sediment trapping reefs and back-reef sediment supported mounds (Figs. 2.1-2.2).

Figure 2.1: A close-up view of the Pipe Creek Reef. Note the lens cap at the lower left. The sample was broken from the outcrop and trimmed where necessary to

7

Caprinid Shells

Back-reef mounds Sediment supported

Reefs Sediment trapping

Figure 2.2: Cretaceous limestone with Caprinid fossils. fit within the CT scanner. It is irregular in shape, but vertically elongated with dimensions of approximately 36 cm (14 in) in height and 25 cm (10 in) in diameter. It has a volume of about a cubic foot and vugs up to 1 inch in diameter by 5 inches long. Empirical steady-state flow measurements of the sample indicate roughly a 100 Darcy effective permeability for a 10 cm long portion of the sample and 35 md for another 20 cm portion, indicating a scale-dependant flow system [38]. The sample was scanned with an industrial high-resolution X-ray CT scanner at The University of Texas Department of Geological Sciences. The Pipe Creek sample was scanned with 240 slices, each 1.5 mm thick covering a 278 mm by 278 mm area with 512 by 512 pixels, each pixel 0.543 mm by 0.543 mm square. Images from selected slices are shown in Fig. 2.3. The grayscale images

8

Partially Filled Vugs

Vugs

Slice 48

Slice 109

Figure 2.3: Two CT scan cross-sections of the Pipe Creek Reef sample. show relative X-ray attenuation, with relates closely to density. Lighter shades indicate more attenuating (denser) materials. These high energy scans have an approximately linear relationship between relative X-ray attenuation and density which can be calibrated from the known bulk density of the sample.

2.1

Connectivity of the Vug System As a simple means of exploring the nature of the heterogeneities in

the Pipe Creek sample, we initiated a series of fine-scale simulations assuming that Darcy’s law was applicable throughout the sample, in both matrix and in vugs. These simulations were to provide a baseline for comparing the multiphysics (Stokes’ and Darcy’s Laws) modeling, but they also yielded some unanticipated insights into the geometry and topology of the vugs and into traditional approaches to upscaling.

9

The basic idea is simple: we assign permeabilities to each voxel in the CT image of the sample, extract a convenient subset (rectangular prism) of the data, and compute steady-state flow for prescribed boundary conditions on that subset. The assignment procedure was applied to each voxel individually and consisted of two steps. First, a previously determined indicator function was consulted to determine whether the voxel was part of a vug or part of the matrix. If the former, it was assigned an arbitrary large value of permeability (e.g., 1000 Darcy). If the latter, its permeability was estimated from the voxel porosity using correlations for carbonate rocks [28]. The indicator function and the porosity of each voxel were inferred from the raw CT data [21]. Local permeabilities were assumed isotropic. The irregularity of the sample motivated the extraction of a computationally convenient subset of the data. The largest rectangular prism contained within the sample contained some 8 × 106 voxels. The voxels in the CT scan measured approximately 0.5 mm by 0.5 mm by 1.5 mm, where the long side is in the nominal z-direction (aligned with gravity). Because of the extreme contrasts in local permeabilities, convergence of the computation is slow. To facilitate exploration, preliminary studies were conducted on upscaled versions of this subsample. The first upscaling was applied in the lateral (x and y) directions to produce cubes 1.5 mm on a side. Thus 3 × 3 × 1 fine-grid clusters were replaced with single elements whose permeability was the one-third power-law average of the local permeabilities. The second level of upscaling was a traditional 2 × 2 × 2, again with a one-third power-law average, yielding 10

elements that were 3 mm on a side. After the second level of upscaling, the rectangular prism contained about 105 elements. A converged flow field could be obtained in reasonable wall-clock time (a few hours) on a linux cluster using the parallel single-phase flow code ParSSim [1]. Simulations were run with constant pressure boundary conditions on one pair of opposing faces of the rectangular prism, and no-flow boundary conditions on the remaining faces. The effective permeability in the direction of flow is the ratio of total flux at steady state to the imposed pressure drop. This choice of boundary conditions is likely to underestimate the true effective permeability if vugs intersect any of the no-flux faces. In comparison to the approximations implicit in treating the vugs as media obeying Darcy’s law, this error is likely not critical. Whether individual vugs are connected is of obvious importance to the effective permeability, but even after careful examination of the CT images it was not clear whether a continuous network of vugs spans the sample. By repeating the calculation of effective permeability with different values of the arbitrary permeability assigned to voxels within a vug, we can obtain a direct measure of the connectedness of the vug space. Using the approach described in the preceding two paragraphs, we found that once the effective permeability was significantly greater than the average matrix permeability, the effective permeability increased linearly with the nominal vug permeability. Thus it is clear that the vugs are connected, and also that obtaining a physically reasonable effective permeability by this means would require a physically reasonable vug permeability. That latter problem is not well-posed for cm-

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scale vugs, and underlines the need for a multiphysics model. In the approach described above, we carried out the upscaling calculations anew for each value of nominal vug permeability. That is, we assigned permeabilities to the voxels of the original CT scan, i.e., to the 0.5 × 0.5 × 1.5 mm3 elements and carried out the two levels of upscaling. A shortcut procedure is to carry out a global substitution of all values of nominal vug permeability only in the upscaled permeability field. This is physically incorrect (but instructive) because it does not change the permeability of elements that were upscaled from a mixture of vug and matrix elements. The only elements whose permeabilities are changed are those that were upscaled from 6 × 6 × 3 blocks of fine-scale elements, all of which were vugs. Thus only vugs that extend at least 3 mm in each direction are still recognizable as vugs in the upscaled permeability field. Increasing the arbitrary value of permeability of these vugs in the upscaled grid did not change the effective permeability. This implies that these coarse-scale vugs are not connected. We also carried out this short-cut procedure at the first level of coarsening, for which the grid contains about 106 elements. Only vugs extending at least 1.5 mm in all directions at the fine scale would still be recognized as vugs at the coarser scale. Increasing the arbitrary value of permeability of these coarser scale vugs did change the effective permeability. Thus the vugs are connected at this first level of upscaling, and therefore are also connected at the fine scale. The disconnection observed at the second level of upscaling implies that the vug connections are between 1.5 mm and 3 mm in size. Volumes 12

that size would be averaged with matrix permeability during the two levels of upscaling, and thus would have permeability less than the nominal vug permeability on the coarse grid. The connections are much smaller than the vugs themselves (2 to 10 cm), and this explains the difficulty of ascertaining connectivity in the CT images. These results also indicate that a successful upscaling method for highly heterogeneous rocks must preserve the topology of features like vugs. In these cases this requirement is more important than the question of which average is appropriate, which is the usual focus of attention. This conclusion would be applicable even if the rock did not require a multiphysics model. Simulations with different boundary conditions after the first level of upscaling showed some anisotropy, with the permeability in the z-direction about half the permeability in the x- or y-direction. Because the subsample on which the flow calculation is done is smaller in the x- and y-directions (12 cm) than the z-direction (22 cm), this apparent anisotropy may be scalerelated, rather than a feature of the rock. In §6.6 we show examples of the multiphysics model applied to slices of the Pipe Creek Sample.

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Chapter 3 The Model

3.1

Notation The vuggy medium will be denoted Ω ⊂ R2 and is composed of a porous

¯ d . The interface part, Ωd , and the vuggy region, Ωs . Note that Ωs = Ω \ Ω between the two regions is Γ = ∂Ωd ∩ ∂Ωs . In all of Ω, denote the fluid velocity by u and the pressure by p. We will often need to distinguish between these quantities on Ωs or Ωd , and especially their traces on Γ. Thus let us = u|Ωs ,

ud = u|Ωd ,

ps = p|Ωs ,

and pd = p|Ωd ,

where ud is the Darcy velocity or effective fluid velocity as opposed to the interstitial velocity that would be seen at the pore scale [7]. For reference, we also state here that we will let µ > 0 be the fluid viscosity, K ∈ L∞ (Ω) the uniformly positive permeability of the porous rock matrix, α > 0 the BeaversJoseph slip coefficient, q ∈ L2 (Ω) an external source or sink term (satisfying the compatibility condition that its average over Ω vanishes), and f ∈ (L2 (Ω))2 a term related to body forces such as gravity. Let ν be the outer unit normal to ∂Ω, and on Γ, let it be the outer unit normal to ∂Ωs . Let τ be a unit tangent to Γ, and let D be the symmetric gradient, i.e., D(ψ) is the matrix ¶ µ 1 ∂ψi ∂ψj . + 2 ∂xj ∂xi 14

Ωd

ν

µ ¡ ¡ @ R τ@

Ωs

Γ

Figure 3.1: Notation near the interface.

3.2

The Regions The first question that arises in the formulation of a mixed porous and

open flow is the proper mathematical form of the governing equations in each region 3.2.1

The Vuggy Region, Ωs The flow in an open domain is governed by the steady state Navier-

Stokes equations. Since we expect relatively low Reynold’s number flow, the Stokes’ equations should adequately model fluid flow in the vugs. Thus −2µ∇ · Du + ∇p = f,

(3.1)

∇ · u = q.

(3.2)

Note that because we do not assume a vanishing divergence, we must pose the first equation in terms of the symmetric gradient. For Stokes’ equations, we need two boundary conditions along all edges in this two-dimensional geometry. On the outer boundary, we will use a no slip condition. For the porous/open interface, Γ, we require continuity of normal 15

stress, which takes the form 2µν · Dus · ν = ps − pd . The second boundary condition is a specification of the shear stress and will be discussed in §3.2.3. The difficulty is that there are no shear stresses associated with Darcy’s law [33]. 3.2.2

The Porous Region, Ωd Since the experiments of H. Darcy in 1856 on packed sand columns, his

empirical law has been used to relate pressure and fluid velocity in a variety of porous media under a variety of subsurface conditions. Darcy’s law [7, 26, 29] gives the volumetric fluid flux u(x, t) (now called the Darcy velocity) as proportional to the gradient of the fluid pressure p: u=−

K ∇(p − ρgz), µ

where K is the (measured) permeability of the medium, µ is the viscosity of the fluid, ρ is its density, g is the gravitational constant, and z is the depth. Darcy’s law can be derived from first principles as well as through various averaging techniques [7, 20, 34, 37]. It is well known that Darcy’s law best describes low Reynold’s number flow through porous medium with relatively small porosity [33]. Only one boundary condition is necessary for Darcy’s law. We require no normal flow on the outer boundary and continuity of normal flow on the porous/open interface. 16

3.2.3

The Interface, Γ In some pioneering experiments in 1967, Beavers and Joseph [8] de-

termined experimentally that a free fluid in contact with a porous medium flows faster than a fluid in contact with a completely solid surface. That is, a small boundary layer develops near the interface within which the tangential velocity of the fluid does not vanish. Beavers and Joseph further proposed a slip boundary condition to model the effect of this boundary layer, in terms of a dimensionless parameter α that measures the quality of the interface. For unidirectional flow, this boundary condition takes the form ∂Us α = √ (Us − Ud ) , ∂y K where ∂/∂y is the normal derivative, Us and Ud are the tangential components of the Stokes and Darcy velocities, and K is the permeability of the porous medium. Saffman [32] justified this law theoretically, and showed that Ud could be dropped (see also [22, 23]—the model appears to be ill-posed with the Ud term!). Jones [25] extended this to multidimensional flows by realizing the velocity derivative in the unidirectional flow is closely related to the shear stress at the interface: 2ν · Dus · τ = −αK −1/2 us · τ.

(3.3)

An alternative model could be considered, in which Stokes flow is assumed within the entire vug-pore space. In that case, we would assume that the velocity vanishes on the pore boundaries in the rock matrix. However, 17

there would be small boundary layers within all the pores that would need to be resolved, as well as near the boundary Γ. This would be not only computationally challenging, but also quite unnecessary. The advantage of using the Beavers-Joseph-Saffman model is that we do not need to resolve these physically occurring boundary layers. Darcy’s Law averages their effect within the rock matrix, and the boundary condition (3.3) models their effect on Γ. Of course, the Beavers-Joseph-Saffman approximation will break down as the boundary layer thickness approaches the vug aperture, and boundary layers from opposite sides of the vug begin to overlap. However, we believe that this will not happen in natural vuggy media. The Beavers-Joseph experiments [8] involve vug gaps on the order of 0.34 mm to 3.4 mm. Since we will consider examples with vug gaps at least this large (see Chapter 2), we do not anticipate any difficulty in using the Beavers-Joseph-Saffman model.

3.3

The Governing Equations Gathering things together, we have

Vuggy region (Stokes equations) −2µ∇ · Du + ∇p = f

in Ωs ,

(3.4)

∇·u=q

in Ωs ,

(3.5)

18

Rock matrix (Darcy equations) µK −1 u + ∇p = f

in Ωd ,

(3.6)

∇·u=q

in Ωd ,

(3.7)

us · ν = u d · ν

on Γ ,

(3.8)

on Γ ,

(3.9)

on Γ ,

(3.10)

us = 0

on ∂Ω ∩ ∂Ωs ,

(3.11)

ud · ν = 0

on ∂Ω ∩ ∂Ωd .

(3.12)

Interface

2ν · Dus · τ = −αK −1/2 us · τ 2µν · Dus · ν = ps − pd Outer boundary

The interface conditions represent continuity of mass flux (3.8), the BeaversJoseph-Saffman condition on the tangential stress (3.9), and the continuity of normal stress (3.10).

3.4

Variational Form A suitable variational form is posed with the velocity u in the space

V = {v ∈ H(div; Ω) : vs = v|Ωs ∈ (H 1 (Ωs ))2 , v · ν = 0 on ∂Ω, v = 0 on ∂Ωs ∩ ∂Ω} , (3.13) 19

in which the outer boundary condition is imposed, and the pressure p in the space W = L2 (Ω)/R. Note that the vector valued functions in V have (weakly) continuous normal components on Γ [14], but that the tangential components need not agree. Let (·, ·) denote the L2 (Ω), (L2 (Ω))2 or (L2 (Ω))2×2 inner product or duality pairing, as appropriate. Also, (·, ·)` , ` = s, d, will be the same with Ω replaced by Ω` , and h·, ·i will be the L2 (Γ) inner product or duality pairing. To derive the variational form, for v ∈ V, the left side of (3.4) is manipulated as − 2µ(∇ · Du, v)s + (∇p, v)s = 2µ(Du, Dv)s − 2µhν · Dus , vi − (p, ∇ · v)s + hps , v · νi , and the first interface term on the right is broken into normal and tangential components and further manipulated as −2µhν · Dus , vi = −2µhν · Dus · ν, v · νi − 2µhν · Dus · τ, vs · τ i = −hps − pd , v · νi + µhαK −1/2 us · τ, vs · τ i , using the interface conditions (3.9)–(3.10). Thus (3.4) becomes 2µ(Du, Dv)s +µhαK −1/2 us ·τ, vs ·τ i−(p, ∇·v)s +hpd , v ·νi = (f, v)s . (3.14) Similarly, (3.6) is manipulated as µ(K −1 u, v)d + (∇p, v)d = µ(K −1 u, v)d − (p, ∇ · v)d − hpd , v · νi = (f, v)d , (3.15) 20

since ν points into Ωd . Thus, the entire system (3.4)–(3.12) for (u, p) ∈ V × W becomes, for test functions (v, w) ∈ V × W , 2µ(Du, Dv)s + µhαK −1/2 us · τ, vs · τ i + µ(K −1 u, v)d − (p, ∇ · v) = (f, v) , (∇ · u, w) = (q, w) .

(3.16) (3.17)

Note that (3.8) is implicit, and (3.11)–(3.12) are explicit, in the space V.

21

Chapter 4 The Finite Element Space

We are concerned with a domain for which the Darcy and Stokes regions are intertwined. So that an efficient code can be written with little regard for which region we are in, we use elements that are suitable for both Stokes and Darcy approximation. Stokes velocities must be approximated in H 1 (Ω)2 , while the Darcy velocity must only be in H(div). As mentioned in §1.2 and [2], we have chosen to use the elements first attributed to Fortin [17] for Stokes equations which also work well for Darcy flow [5]. To approximate the combined Darcy-Stokes system, we must modify the elements near the interface to allow for discontinuity of the tangential velocity, which will be done in §4.2.

4.1

The Standard Stokes Elements We begin by recalling the definition of the standard Stokes finite ele-

ment space that we use [5, 17]. The finite element itself is defined on a rectangle R = (0, a)×(0, b). On R, we approximate pressure as a constant and the velocity in the space Vh (R) = Q1,2 (R)×Q2,1 (R), where Qi,j (R) are the polynomials of degree i in x1 and degree j in x2 defined over R. The degrees of freedom

22

for v = (v1 , v2 ) ∈ Vh (R) are the 8 corner values vj (0, 0) ,

vj (a, 0) ,

vj (0, b) ,

and the 4 edge average normal fluxes Z 1 b − v1 (0, x2 ) dx2 , b 0 Z a 1 v2 (x1 , 0) dx1 , − a 0

vj (a, b) ,

j = 1, 2 ,

Z 1 b v1 (a, x2 ) dx2 , b 0 Z a 1 v2 (x1 , b) dx1 , a 0

(4.1)

(4.2)

as depicted in Figures 4.1 and 4.2. For purposes of implementation, the 4 edge degrees of freedom may be replaced by the nodal values at the midpoints of the edges; however, for easier definition of the space and for the analysis, we represent the degrees of freedom as above. ¢

× ¡

£

×(a − x1 )P2 (x2 ) + x1 P2 (x2 ) ¤

¢ ×

£

(b − x2 )P2 (x1 ) + x2 P2 (x1 )

× ¤ x2 -velocity ¡

x1 -velocity

Figure 4.1: The standard Stokes finite elements. We assume henceforth that both Ωs and Ωd are unions of rectangles. Let Th be a rectangular finite element partition of Ω, with h being the maximum element diameter, so that each element R is in either Ωs or Ωd . Then Γ is a union of edges of Th . We tacitly assume that the aspect ratio of the rectangles does not degenerate as h → 0. The standard Stokes finite element space is formed in the usual way. ˜ h = {v ∈ Let Wh ⊂ W consist of piecewise constant functions over Th . Let V

¯ : v|R ∈ Vh (R) for all R ∈ Th }, which is formed by piecing together V ∩ C(Ω) 23

1 0.5 0 −0.5 0

1.5 1 0.5 0 0

1

2

2

1 2 0

1 0

1

Figure 4.2: The standard basis functions in the x-direction. The left picture is the edge basis function over two grid blocks. The right picture is the corner basis function over four grid blocks. the elements by matching the degrees of freedom at corners and edges of the partition.

4.2

The Modified Elements ˜ h must be modified near the Darcy-Stokes interface The elements of V

Γ, since in general the tangential component of the velocity is not continuous there. We must remove some of the corner degrees of freedom to allow for a discontinuous tangential velocity component on Γ, and simultaneously reduce the size of the polynomial space (but we must not degrade its approximation properties). We want to do this as simply as possible, so that there are minimal changes to a finite element code using the unmodified elements everywhere. Around a corner point, exactly seven patterns arise, as depicted in Fig. 4.3. Modification is made in only three of the cases. For ease of exposition, we consider the case of horizontal, x1 -components only (x2 -components are

24

d ¤¡d £¢ d d 1

d

d £¢

d 2

s

d ¤¡s £¢ d s 3

s ¤¡ s d d 4

s ¤¡ d d s 5

s ¤¡s £¢ d s 6

s ¤¡s £¢ s s 7

Figure 4.3: The 7 patterns (up to symmetries) around a corner point in the modified space Vh . Here d represents a Darcy element, s is a Stokes element, and a heavy line is a part of Γ. For x1 velocity components, a quarter circle in an element indicates that the corner basis function is present over the element. The corner in Pattern 5 is called a checkerboard corner. The edge tick marks indicate which edges may be used to define the corner value via the ScottZhang operator. handled similarly). If two Darcy elements share a vertical edge e and a corner on Γ, we remove the corner nodal value (which disconnects the tangential velocity). We also reduce the polynomial space by one degree on e. Thus Pattern 2 has two modified elements on top, while Pattern 4 has its two lower elements modified. Pattern 5 of Fig. 4.3 has alternating Stokes and Darcy elements around the corner, so we call it a checkerboard corner. Such corner points are problematic, because the entire horizontal interface is part of Γ, which means that the x1 velocity of the solution on the top two elements is potentially discontinuous with that on the bottom. We need to break the continuity, so we remove the corner nodal value systematically on, say, the bottom Darcy and Stokes elements. For a corner point on ∂Ω, it is trivial to examine the possibilities, as depicted in Fig. 4.4. If the corner point is also a corner of the domain, there is only one element containing the corner point and no modification is needed.

25

¢

¡¢

£¢

1

2

s

d

¡¢ d

s

¢

d

3

4

s

5

Figure 4.4: The 5 patterns (up to symmetries) around a corner point in the modified space Vh on ∂Ω. Here d represents a Darcy element, s is a Stokes element, unspecified is either, and a heavy line is a part of Γ. For x1 velocity components, a quarter circle in an element indicates that the corner basis function is present over the element. Otherwise the corner point is part of two elements, and modification is required only if the elements are of different types, and then only to the Darcy element, and only to the component normal to ∂Ω (i.e., tangential to Γ). Thus only Pattern 5 in Fig. 4.4 is modified. ¢

× ¡

£

¢

×(a − x1 )P2 (x2 ) + x1 P2 (x2 ) ×

×(a − x1 )P1 (x2 ) + x1 P2 (x2 )

¤

¤

£

×

£

¢

×(a − x1 )P0 (x2 ) + x1 P2 (x2 ) ×

£

×(a − x1 )P1 (x2 ) + x1 P1 (x2 )

¤ ¢

×

×(a − x1 )P1 (x2 ) + x1 P1 (x2 ) ¤

£

×

×(a − x1 )P0 (x2 ) + x1 P1 (x2 ) ×

×(a − x1 )P0 (x2 ) + x1 P0 (x2 )

Figure 4.5: The 7 elements (up to symmetries) for the x1 velocity components, on R = (0, a) × (0, b), in the modified space Vh . The quarter circles represent corner nodal values, and the edge crosses represent normal fluxes (or, equivalently, nodal values).

26

We have a grand total of 16 types of elements for each velocity component, depending on which of the 4 corners are affected. Removing symmetries, we are left with the 7 distinct types depicted in Fig. 4.5. Each of these elements has degree 0, 1, or 2 on the left and right vertical edges. On R = (0, a) × (0, b), with Pj (ξ) denoting polynomials in ξ of degree up to j, the polynomial spaces for x1 -components are (a − x1 )Pi (x2 ) + x1 Pj (x2 ), i, j = 0, 1, 2 (and for the x2 -components the spaces are Pk (x1 )(b − x2 ) + P` (x1 )x2 , k, ` = 0, 1, 2). Note that in the most reduced case, the element becomes the lowest order space due to Raviart and Thomas [31]. Denote by Vh ⊂ V the modified finite element space. Note that no global basis function disappears (we always have an unmodified Stokes element near Γ). Of course, the finite element method is to find (uh , ph ) ∈ Vh × Wh such that 2µ(Duh , Dvh )s + µhαK −1/2 us,h · τ, vs,h · τ i + µ(K −1 uh , vh )d − (ph , ∇ · vh ) = (f, vh ) , (∇ · uh , wh ) = (q, wh )

(4.3) (4.4)

holds for test functions (vh , wh ) ∈ Vh × Wh . It is a trivial application of linear algebra to verify that there is a unique solution. In a computer code, in the matrix assembly routines, one needs to check the type of element (Stokes or Darcy) to decide which of two forms is used: 2µ(Duh , Dvh )s or µ(K −1 uh , vh )d . On a Darcy element, we need to check the nearest neighbors to decide on the local basis. When we are on a Stokes element 27

in contact with Γ, we also include the interface term µhαK −1/2 us,h · τ, vs,h · τ i. Moreover, when in contact with Γ, for x1 velocity components, we need to check the two lower corners to see if either or both are checkerboard corners (Pattern 5 of Fig. 4.3) and modify the local basis accordingly (we check, say, the two left-most corners for x2 velocity components). Since the number of global basis functions is determined a priori by the number of corner points and edges, the global matrix problem has a regular sparsity pattern. In closing this section, we note that for checkerboard corners, we could instead remove the corner nodal value altogether. However, two issues then need to be addressed. First, if a larger checkerboard pattern of Stokes and Darcy elements were to arise, so that we have an edge with checkerboard corners on both ends, then the polynomial space would drop to a constant along that edge. This would be insufficient to approximate the Stokes equations. We conclude that omitting checkerboard corner nodal values is acceptable only if such edges do not arise in the finite element partition. Second, if we indeed omit a nodal value, the sparsity structure of the matrix problem changes. Since checkerboard corner points will likely not occur in nature, it is not particularly important how we handle them. In our computer code, in fact, we ignored the issue and used the strategy that Pattern 5 of Fig. 4.3 is left unmodified, and only the other two Patterns 2 and 4 were treated in a special way.

28

Chapter 5 A Priori Error Analysis

5.1

A π Operator for the Velocity We now define an important operator that will be used heavily in the

next section to analyze the approximation error. For the unmodified space ˜ h , the operator has been defined in, e.g., [5, 19]. Let π ˜h V ˜ : (H 1 (Ω))2 → V denote this unmodified operator. We have the following properties [5], where PW : L2 (Ω) → Wh denotes the L2 projection operator and | · |j,ω is the usual H j (ω) Sobolev seminorm, |ψ|2j,ω

X Z ¯¯ ∂ α ψ(x) ¯¯2 ¯ ¯ = ¯ ∂xα ¯ dx , α,|α|=j

ω

where the sum is taken over all jth order derivatives (i.e., α runs over all multi-indices of order j). Later we will also need the usual H j (ω) Sobolev norm k · kj,ω , which is given by kψk2j,ω

=

j X k=0

|ψ|2k,ω .

We omit ω if ω = Ω, and write s or d in place of Ωs or Ωd . Lemma 5.1.1. Assume that v ∈ (H 1 (Ω))2 . There exists some constant C independent of h, so that the following hold. 29

(a) The linear operator π ˜ is bounded on (H 1 (Ω))2 independently of h. (b) For R ∈ Th and v ∈ (H r (R0 ))2 , r−j |˜ π v − v|j,R ≤ C|v|r,R0 hR ,

1 ≤ r ≤ 2, j = 0, 1 ,

where R0 is the union of R and its (up to) four nearest neighbor elements that share edges with R, and hR = diam(R). (c) For v ∈ (H r (Ω))2 , |˜ π v − v|j ≤ C|v|r hr−j ,

1 ≤ r ≤ 2, j = 0, 1 .

(d) PW ∇ · v = PW ∇ · π ˜ v. By definition [5], π ˜ is defined locally, element by element, as essentially the interpolant of the finite element degrees of freedom (4.1)–(4.2) (where the edge degrees of freedom must be understood as the average normal velocity component, not the midpoint nodal value). The only problem is that corner values are not defined for functions in H 1 (Ω), so the corner nodal values are set using the Scott-Zhang operator [35]. This operator defines corner point values through integration over an adjoining edge of the finite element partition. The above results are proved by using the Trace Theorem locally to relate edge integrals to area integrals. Thus in (b), R0 is larger than R and only needs to include for each corner point the element over which the area integral is computed. (We note in passing that π ˜ is actually a projection, because of the way the Scott-Zhang operator is defined.)

30

˜ h has been modiWe now modify π ˜ locally near Γ wherever the space V fied to form Vh . We must begin by making a specific choice regarding the use of the Scott-Zhang operator. We apply the operator to x1 - and x2 -components of the velocity independently. Since tangential velocity components may be discontinuous across Γ, we need to define point values only using edges on which the given velocity component is continuous. Thus we require that horizontal, x1 -component point values be defined only on vertical edges, and similarly x2 -components use only horizontal edges. Moreover, we insist that the edge chosen for the integration borders both a Darcy and a Stokes element, so the Trace Theorem analysis can be taken entirely in either Ωd or Ωs , as needed. The only exception is when this is not possible, in which case we take the edge that borders two Stokes elements. To illustrate the ideas, we refer to Fig. 4.3 and consider only x1 components. We may choose any edge in the first and last (seventh) pattern, since these do not involve Γ. We must choose a vertical edge for the other five patterns. Patterns 2 and 6 require the lower edge, and Patterns 4 and 5 require the upper edge. We can choose either vertical edge in the third pattern. The definition of π : (H 1 (Ω))2 → Vh is now immediate: it is the interpolant of the modified finite element degrees of freedom of Vh (4.1)– (4.2), using the Scott-Zhang operator to define corner points, using edges as noted above near Γ to avoid discontinuities in the tangential velocity. Lemma 5.1.2. Let R ∈ Th and hR = diam(R). Suppose that R ⊂ Ω` , where ` is either s or d. Let R0 be the union of R and its nearest neighbor elements 31

that are also in Ω` . There exists some constant C independent of h, so that the following hold. ¯ ⊂ Ωd , and v ∈ (H r (R0 ))2 , then (a) If either R ⊂ Ωs or R r−j |πv − v|j,R ≤ C|v|r,R0 hR ,

1 ≤ r ≤ 2, j = 0, 1 .

(b) If R ⊂ Ωd and ∂R ∩ Γ 6= ∅, and v ∈ (H 1 (R0 ))2 , then kπv − vk0,R ≤ Ckvk1,R0 hR . (c) For v ∈ V such that vs ∈ (H 2 (Ωs ))2 and vd ∈ (H 1 (Ωd ))2 , kπv − vk1,s + kπv − vk0,d ≤ C{kvk2,s + kvk1,d }h . (d) PW ∇ · v = PW ∇ · πv. Proof. Result (c) follows from (a) and (b). Result (d) follows from the edge flux degrees of freedom (4.2) and a simple application of the divergence theorem: Z

R

∇ · v dx =

Z

∂R

v · ν ds =

Z

∂R

πv · ν ds =

Z

R

∇ · πv dx .

It remains only to show (a) and (b). ˜ h . Since Suppose element R ∈ Th is unmodified from that in the space V π ˜ and π agree on unmodified elements, we have (a) and (b) by Lemma 5.1.1(b). The restriction of R0 to Ωs or Ωd follows by treating the Scott-Zhang operator analysis from the appropriate side of each chosen edge, as discussed above and illustrated in Fig. 4.3. 32

˜ h. Suppose now element R ∈ Th is modified from that in the space V Let π ˜R be the unmodified Stokes operator defined above, except that at corner points where a basis function was removed from R when forming Vh , π ˜R should be defined using an appropriate edge of R, so that it is defined over Ω` . We illustrate by referring to Fig. 4.3 and x1 -components. If ` = d and a Pattern 2 modification arises, we use the top vertical edge to define π ˜ R for the purposes of this proof. If ` = d and a Pattern 4 or 5 modification arises, or if ` = s and a Pattern 5 modification arises, then we need to take the lower edge to define ˜ h. π ˜R for the purposes of this proof, so π ˜R maps onto V Note that v − πv = (v − π ˜R v) + (˜ πR v − πv) = (v − π ˜R v) + ((˜ πR v) − πR (˜ πR v)) , where πR is the same as π except that the Scott-Zhang operator is not used. For Darcy elements, by Lemma 5.1.1, we need only show the result (b) for the operator πR and for v = vh ∈ Vh (R) = Q1,2 (R) × Q2,1 (R). But πR is a linear projection on the finite dimensional space Vh (R), so it is bounded in the L2 (R)-norm. Moreover, a scaling analysis ([15, p. 117] or [16]) shows that it is bounded independently of h. Since πR preserves polynomials of degree 0, we have (b) by the Bramble-Hilbert Lemma [10, 16]. For the modified Stokes elements, we have that πR preserves polynomials of degree 1, so (a) follows. This completes the proof.

33

5.2

A Convergence Analysis We now present an a priori analysis of the approximation error. The

ˆ h be analysis on Ωd is relatively delicate, and follows the ideas in [5]. Let V ˆ h be the the lowest order Raviart-Thomas space, and let π ˆ : (H 1 (Ωd ))2 → V usual Raviart-Thomas projection operator [14, 31]. Among other properties, ˆ h be (L2 (Ωd ))2 projection. The ∇·π ˆ = PW ∇·. Let PVˆ : (L2 (Ωd ))2 → V following lemma is shown in [5]. ˜ h , then π Lemma 5.2.1. If v ∈ V ˆ v = PVˆ v. Take v ∈ Vh and substitute π ˆ v = PVˆ v for v in (3.15) to obtain µ(PVˆ (K −1 u), v)d − (PW p, ∇ · v)d − hPΛˆ pd , v · νi = (PVˆ f, v)d ,

(5.1)

where PΛˆ is projection onto piecewise constants over the interface grid of rectangle edges, since π ˆ v · ν is piecewise constant. Now combine this equation with (3.14) and (3.17), and subtract the finite element method (3.16)–(3.17) posed over the space Vh × Wh with w ∈ Wh to obtain 2µ(D(u − uh ), Dv)s + µhαK −1/2 (us − uh,s ) · τ, vs · τ i + µ(K −1 (u − uh ), v)d − (PW p − ph , ∇ · v) = (PVˆ f − f, v)d + µ(K −1 u − PVˆ (K −1 u), v)d + (p − PW p, ∇ · v)s − hpd − PΛˆ pd , v · νi , (∇ · (u − uh ), w) = 0 .

(5.2) (5.3)

34

Let us take v = πu − uh ∈ Vh and w = PW p − ph ∈ Wh . The sum of the equations leads to 2µ(D(u − uh ), D(u − uh ))s + µhαK −1/2 (us − uh,s ) · τ, (us − uh,s ) · τ i + µ(K −1 (u − uh ), (u − uh ))d = (PVˆ f − f, πu − uh )d + µ(K −1 u − PVˆ (K −1 u), πu − uh )d + (p − PW p, ∇ · (πu − uh ))s − hpd − PΛˆ pd , (πu − uh ) · νi + 2µ(D(u − uh ), D(u − πu))s + µhαK −1/2 (us − uh,s ) · τ, (us − πus ) · τ i + µ(K −1 (u − uh ), (u − πu))d .

(5.4)

Since |∇ · v| ≤ |Dv|, for some ² > 0 as small as we like and C > 0, it is straightforward to estimate kD(u − uh )k20,s + k(us − uh,s ) · τ k20,Γ + ku − uh k20,d © ≤ C kPVˆ f − f k20,d + kK −1 u − PVˆ (K −1 u)k20,d + kp − PW pk20,s + kpd − PΛˆ pd k20,Γ

+ kD(u − πu)k20,s + ku − πuk20,d + k(us − πus ) · τ k20,Γ + ²kπu − uh k21,s .

ª

(5.5)

We require a Korn inequality for V. Lemma 5.2.2. If ω is a Lipschitz domain and v ∈ (H 1 (ω))2 , then there is some constant C such that ª © kvk1,ω ≤ C kDvk0,ω + kv · τ k0,∂ω . 35

Proof. We use a relatively standard proof by contradiction technique for proving this variant of Korn’s inequality (see, e.g., [13, Proof of Thm. 9.2.16]). The proof is based on the direct sum decomposition (H 1 (ω))2 = H ⊕ R , where ½ ¶ ¾ Z Z µ ∂v2 ∂v1 1 2 H = v = (v1 , v2 ) ∈ (H (ω)) : v dx = dx = 0 − ∂x2 ∂x1 ω ω are the rotation free vectors and ¶¾ ½µ ¶ µ ¶ µ −x2 0 1 , , R = span x1 1 0 are the infinitesimal rigid motions. Following the standard proof (as in [13, Proofs of Thm. 9.2.16 and Cor. 9.2.22]), we are led to the following requirement: If v ∈ R and v · τ = 0, then v = 0. Since v = (c1 −bx2 , c2 +bx1 ), Stokes Theorem immediately implies that b = 0. But then v is constant, and the result is trivial. Thus, we can bound the left side of (5.5) from below by a multiple of ku − uh k21,s + ku − uh k20,d . Applying standard approximation results for the various projection operators, we are led to the error estimate © ª ku−uh k1,s +ku−uh k0,d ≤ Ch kf k1,d +kuk2,s +kuk1,d +kpk1,s +kpd k1,Γ . (5.6) We also prove an estimate for the pressure. Let v ∈ (H 1 (Ω))2 satisfy ∇ · v = p − ph and kvk1 ≤ Ckp − ph k0 ; such a v exists by [6]. Substitute 36

πv ∈ Vh as test function in (5.2), and note that PW ∇ · πv = PW p − ph . We are led to the estimate © kp − ph k20 ≤ C ku − uh k1,s + ku − uh k0,d + kPVˆ f − f k20,d

ª + kK −1 u − PVˆ (K −1 u)k20,d + kp − PW pk20 + kpd − PΛˆ pd k20,Γ ª © + ² kπvk21,s + kπvk20,d , (5.7)

where again ² > 0 is as small as we wish. Now π is bounded on (H 1 (Ωs ))2 , so kπvk1,s ≤ Ckvk1,s ≤ Ckp − ph k0 , and kπvk0,d ≤ kvk0,d + kv − πvk0,d ≤ kvk0,d + Chkvk1,d ≤ Ckp − ph k0 , so we can remove the last two terms on the right side of (5.7) and apply standard approximation results. Collecting this estimate, (5.6), and using (5.3), we have the following theorem. Theorem 5.2.3. There is some constant C such that ku − uh k1,s + ku − uh k0,d + kp − ph k0 © ª ≤ Ch kf k1,d + kuk2,s + kuk1,d + kpk1,s + kpk1,d + kpd k1,Γ . Moreover, PW ∇ · uh = PW q.

37

Chapter 6 Computational Results

We present some simple test cases involving smooth solutions to verify the convergence rates, and then we present examples involving the Pipe Creek sample.

6.1

A Remark on the Solution Procedure As noted earlier in Chapter 4, the resulting linear system is symmetric

and has a completely regular structure. However, the matrix is quite illconditioned; moreover, it has a saddle point structure. We briefly remark on our solution strategy, which is very effective for problems up to grid sizes of perhaps 128 × 128. We chose to implement a solver for the scheme using an inexact Uzawa technique [11, 12]. For a rectangular grid with nx ×ny elements, the matrix problem is of the  Axx ATxy BxT

form     Axy Bx ux fx     Ayy By uy = f y  , T By 0 p q

where ux represents the (nx + 1)(2ny + 1) nodal values of the x1 velocity components vix , uy represents the (2nx + 1)(ny + 1) nodal values of the x2 velocity components viy , and p represents the nx ny nodal values of the pressure. 38

Except for modification for boundary conditions, x x Axx,ij = 2µ(Dvix , Dvjx )s + µhαK −1/2 vs,i · τ, vs,j · τ i + µ(K −1 vix , vjx )d , y y Ayy,ij = 2µ(Dviy , Dvjy )s + µhαK −1/2 vs,i · τ, vs,j · τ i + µ(K −1 viy , vjy )d ,

Axy,ij = 2µ(Dvix , Dvjy )s , so both Axx and Ayy are positive definite. Since these submatrices arise from a two dimensional rectangular grid, they are banded with a reasonable band size, so direct factorization is feasible for problems not too large. The inexact Uzawa procedure starts with an initial guess for the solution, say p0 = 0, ux = 0, and uy = 0, and then for ` = 1, 2, ..., it defines iteratively `−1 − Bx p`−1 ) , u`x = A−1 xx (fx − Axy uy

(6.1)

T `−1 u`y = A−1 − By p`−1 ) , yy (fy − Axy ux

(6.2)

p` = p`−1 + βM (BxT ux + ByT uy − q) ,

(6.3)

where β > 0 is the Uzawa parameter and M is some preconditioner for the system. A relatively good preconditioner is needed. We took · µ ¶ µ ¶¸−1 ¡ T ¢ A−1 0 Bx T xx , M = Bx By 0 A−1 By yy

(6.4)

which is invertible (after modification for the compatibility condition that p is defined only up to a constant). This computation is quite expensive, since the matrix is full, but, again, direct factorization is feasible for problems not 39

too large. (For larger problems, some inexact inverse could be used.) This preconditioner is exact when Axy = 0 and β = 1. We found that the Jacobi preconditioner given by inverting only the diagonal of (6.4) performed very poorly, requiring sometimes hundreds of thousands of iterations to converge, and worsening greatly with the size of the problem. On the other hand, M solved problems on grids of size 8 × 8 up to 64 × 64 using only around 100 iterations (the worse case took under 500 iterations).

6.2

Some Simply Constructed Examples In our first set of examples, Ωs = (0, 1/2) × (0, 1), Ωd = (1/2, 1) × (0, 1),

and Γ is the line x = 1/2. It is difficult to construct solutions that satisfy the entire Darcy-Stokes system (3.4)–(3.12). If u and p are chosen somehow, we can easily satisfy (3.4)–(3.7) by defining f and q appropriately. Moreover, rather than requiring the solution to satisfy the outer boundary conditions (3.11)–(3.12), we can simply allow for a more general and nonhomogeneous set; again, the nonhomogeneous terms are defined from the solution. The difficulty is finding a solution satisfying the interface conditions (3.8)–(3.10). In this section, we simply use the same trick of generalizing the equations to include a nonhomogeneous term. That is, we replace (3.9)–(3.10) by 2ν · Dus · τ = −αK −1/2 us · τ + g1 2µν · Dus · ν = ps − pd + g2 40

on Γ ,

(6.5)

on Γ .

(6.6)

The construction is now clear: choose u satisfying (3.8), and then define f , q, g1 , g2 , and the outer boundary conditions from the solution. These test cases are summarized in Table 6.1. The variational form for this modified system has only a small change: (3.16) now includes the two terms hg2 , v · νi + µhg1 , vs · τ i on the right side. Case

p

1 2

us µ

0 x

e sin(x + y) 2

3

cos(x y)

4

−y 4 ex

µ

1 2

0 − y2

¶

cos(xy) ex+y

¶

cos(xy) 0

¶

¶ µ ¶ sin(x2 y) ex+y ¶ ¶ µ µ y 4 ex y 4 ex ey cos(2x) 4y 3 ex µ

sin(x2 y) cos(x2 y)

µ

ud µ ¶ 0 0

Table 6.1: The numerical Test Cases 1–4. ˜ h of Fortin [5, 17], we If we use the unmodified finite element space V see poor convergence results in Table 6.2. This is due to the fully continuous approximation of the velocity. However, our constructed solutions have a discontinuity in u · τ on Γ. The modified space Vh defined in Chapter 4 corrects this defect, as seen by the convergence rates in Tables 6.3 and 6.4. In Table 6.3, we have used uniform grids of size 8 × 8, 16 × 16, 32 × 32, and 64 × 64. In Table 6.4, we have randomly perturbed the points of the uniform grid by plus or minus one quarter of the uniform cell spacing, except that the center lines have been unperturbed so as to resolve Γ. We clearly see at least O(h) convergence for the pressure and velocity as measured in 41

the L2 -norm, and at least O(h) convergence for the gradient of the velocity on Ωs , as proved in Theorem 5.2.3. In fact, we see more from these computational results. It appears that there is some superconvergence of order perhaps O(h3/2 ) for the error PW p − ph in the L2 -norm, although it is not the usual O(h2 ) that mixed methods often produce. Moreover, on uniform grids only, we see perhaps O(h1/2 ) convergence in the L2 -norm of ∇(u − uh ) over all of Ω. Such convergence was observed for the unmodified elements when solving Darcy systems on uniform grids (see [5]). Finally, we see convergence in the L2 -norm of ∇ · (u − uh ) of order perhaps O(h) on uniform grids and O(h1/2 ) in general. Case kEp k0 kPW Ep k0 kEu k0 k∇Eu k0 k∇ · Eu k0 1 2 3 4

1.2081 0.9997 1.0082 0.9961

1.2081 0.9690 1.0780 1.0161

0.5084 0.4982 0.5003 0.5618

-0.4986 -0.5032 -0.5054 -0.4743

0.5298 0.5196 0.5174 0.6228

˜ h for the Table 6.2: Observed convergence rates using the unmodified space V errors Ep = p − ph and Eu = u − uh for test cases 1–4 on a uniform grid. Case kEp k0 kPW Ep k0 kEu k0 k∇Eu k0 k∇Eu k0,s k∇ · Eu k0 1 2 3 4

2.004 1.001 1.060 1.038

2.004 1.509 1.610 1.703

2.001 1.431 1.431 1.437

1.000 0.431 0.419 0.412

1.000 1.258 1.002 1.037

1.000 0.975 0.982 0.965

Table 6.3: Observed convergence rates using the modified space Vh for the errors Ep = p − ph and Eu = u − uh for Test Cases 1–4 on a uniform grid.

42

Case kEp k0 kPW Ep k0 kEu k0 k∇Eu k0 k∇Eu k0,s k∇ · Eu k0 1 2 3 4

1.983 0.987 1.038 1.006

1.983 1.458 1.596 1.664

1.963 1.137 1.157 1.173

0.988 0.114 0.114 0.159

0.988 1.239 0.999 1.024

0.988 0.616 0.682 0.781

Table 6.4: Observed convergence rates using the modified space Vh for the errors Ep = p − ph and Eu = u − uh for Test Cases 1–4 on a randomly perturbed grid.

6.3

Some Smooth Examples with Corners ¡ ¢ ¡ In our second set of examples, we take Γ = {1/2}×(0, 1/2] ∪ [1/2, 1)×

¢ {1/2} . That is, Γ has a corner and separates off the lower right quarter of the

square, (1/2, 1) × (0, 1/2), which will be Ωd in test case 5 and Ωs in test case 6. Note that Test Case 6 uses a modified element near the central corner (see Fig. 4.3, Pattern 2). Overall, these two cases require all modification patterns depicted in Fig. 4.3, except for the checkerboard Pattern 5. In this set of examples, we fully satisfy the system (3.4)–(3.10). We handle the equations (3.4)–(3.7) by defining f and q and the external boundary conditions (3.11)–(3.12) as above, once u and p are fixed. We first choose some us satisfying (3.9), and then take pd constant and find some ps that satisfies (3.10). Finally, ud is chosen to satisfy (3.8). Our test cases are summarized in Table 6.5. From Table 6.6, we see very good convergence. Actually, the L2 and H 1 -errors for u − uh are somewhat better than expected. We see clearly from these two test cases that there is no convergence difficulty with modifying the

43

ps

Case 5 6

e

−2xy

−

pd 1 −1/2 e 2

2e−y

0 0

us

ud

¶ xe−y + 12 e x(e−2xy + e−x ) ye−x + 21 e−y y(4y 2 e−x + e−y ) ¶ µ µ ¶ 2e−2xy (2x + 1)e−y −2e−y (y − 21 )3 e−x − 2e−1/2 ¶ −x

µ

µ

( 12 , 1) ×(0, 21 ) Ωd Ωs

Table 6.5: The numerical Test Cases 5–6. basis over a Darcy element. Case kEp k0 kPW Ep k0 kEu k0 k∇Eu k0 k∇Eu k0,s k∇ · Eu k0 5 6 5 6

1.000 1.001 1.000 1.000

2.066 1.895 1.988 1.771

2.005 1.993 2.006 1.983

1.007 1.007 1.011 1.003

1.033 1.005 1.004 0.999

1.004 1.000 1.011 1.000

Grid uniform uniform perturbed perturbed

Table 6.6: Observed convergence rates using the modified space Vh for the errors Ep = p − ph and Eu = u − uh for Test Cases 5–6.

6.4

Some Simple Layered Examples In our last set of examples, Ωs = (0, 1) × (0, 1/2), Ωd = (0, 1) × (1/2, 1),

and Γ is the line y = 1/2. We use the analytical solution of [4]. These test cases are summarized in Table 6.7. Case ps

pd

7

0

0

8

y 1−y

us

ud ¶ ¶ µ 1 (−2y 2 + y + 1) 4 0 0 µ ¶ µ ¶ 0 0 2 2

µ1

Table 6.7: The numerical Test Cases 7–8.

44

Test Case 7 has a solution (p, u) that is in the finite dimensional space Vh , so it is solved exactly up to rounding error. The solution to Test Case 8 is not in the space, and only ph = PW p is computed exactly. However, as shown in Table 6.8, the convergence is as good as or better than that expected from the earlier test cases. Case kEp k0 kPW Ep k0 kEu k0 k∇Eu k0 k∇Eu k0,s k∇ · Eu k0 8 8

1.000 0.988

— —

2.415 2.482

1.424 1.449

1.424 1.449

2.000 1.977

Grid uniform perturbed

Table 6.8: Observed convergence rates using the modified space Vh for the errors Ep = p − ph and Eu = u − uh for test case 8.

6.5

A Lower Dimension Modification In this section we further modify Vh so that extra tangential continuity

is removed from the Darcy side of the space. That is, remove the corner degrees of freedom (4.1) whenever the corner has Darcy elements surrounding it (see ¯ h . The effect Fig. 4.3—we modify only Pattern 1). Call the resulting space V ˆ h strictly inside Ωd , the full Fortin Stokes is to use Raviart-Thomas elements V ˜ h strictly inside Ωs , and our modified elements Vh as transition elements V elements near Γ. We may lose some global basis functions, and therefore also the regular sparsity structure of the matrix (although this regular structure can be recovered with some so called “slave nodes” that are set to zero, if desired). ¯ h is defined analogously The definition of the operator π ¯ : (H 1 (Ω))2 → V 45

as to that in §5.1, and a similar error estimate to Lemma 5.1.2 holds. Moreover, an a priori error analysis will yield a result analogous to Theorem 5.2.3. Case kEp k0 kPW Ep k0 kEu k0 k∇Eu k0,s k∇ · Eu k0 1 2 3 4 5 6 8

2.004 1.000 1.009 1.011 1.006 1.001 1.000

2.004 1.936 1.950 1.925 1.787 1.920 —–

2.001 0.994 1.000 0.998 0.993 0.969 2.415

1.000 1.262 1.002 1.054 1.037 1.005 1.424

1.000 0.801 0.971 0.914 0.739 0.560 2.000

¯ h for the Table 6.9: Observed convergence rates using the modified space V errors Ep = p − ph and Eu = u − uh for Test Cases 1–6 and 8 on a uniform grid. Case kEp k0 kPW Ep k0 kEu k0 k∇Eu k0,s k∇ · Eu k0 1 2 3 4 5 6 8

1.983 0.986 0.996 0.982 1.006 1.001 0.988

1.983 2.041 2.141 1.935 1.712 1.812 —–

1.963 0.971 0.986 0.975 0.990 0.955 2.482

0.988 1.243 0.998 1.042 1.008 1.000 1.449

0.988 0.802 0.962 0.903 0.740 0.567 1.977

¯ h for the Table 6.10: Observed convergence rates using the modified space V errors Ep = p − ph and Eu = u − uh for Test Cases 1–6 and 8 on a randomly perturbed grid. In Tables 6.9 and 6.10 we show the convergence of this modified scheme. These test cases are the same as in the previous section, but note that we omit the H 1 -errors on Ω, since now there are discontinuities in Ωd and no convergence can be expected. We also do not show convergence rates for Test Case 7, because it is essentially solved exactly up to rounding error. We see 46

the expected convergence rates of at least O(h) for the pressure and velocity as measured in the L2 -norm, and at least O(h) convergence for the gradient of the velocity on Ωs , as proved in the analogue to Theorem 5.2.3. It appears that there is some superconvergence of order perhaps O(h2 ) for the error PW p − ph in the L2 -norm. Finally, we see convergence in the L2 -norm of ∇ · (u − uh ) of order perhaps only order O(h1/2 ) in general. There does not appear to be any advantage to using a uniform grid. These rates are consistent with those from the previous scheme using Vh , except that previously the PW p − ph error only converged at the rate O(h3/2 ). The magnitude of the pressure errors in the L2 -norm were very comparable. However, the velocity and divergence errors are a bit better for the previous scheme. This would be expected, since the velocity polynomial ¯ h . The solution time of space of Vh is somewhat richer on Ωd than that of V the two schemes is also very comparable. Thus, although the modification of ¯ h in this section results in a smaller space and is the finite element space V simple to implement, it does not appear to be better than the previous space Vh .

6.6

Pipe Creek Sample Here we show examples of the method applied to the Pipe Creek Sample

described in Chapter 2. Figures 6.2 and 6.8 show the permeability of the center quarter of the 120th slice (Figure 6.1) and the 180th slice (Figure 6.7) of the CT scan of the sample. The rock was assigned a permeability of about 20 47

0

0.01

Rock 0.02

Mud

Y (m)

0.03

0.04

Vug

0.05

0.06

0.07

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

X (m)

Figure 6.1: CT scan of slice 120 of Pipe Creek sample.

Figure 6.2: Permeability of center quarter of slice 120.

millidarcies for these examples. We ran the code with the mud (about two Darcies) in the vugs as shown in Figures 6.4, 6.6, 6.10, 6.12 and with no mud in the vugs in Figures 6.3, 6.5, 6.9, 6.11. The code was run with a constant pressure difference of one psi from left to right with no-flow boundaries on top and bottom.

48

0.01

0.01

0.02

0.02

0.03

0.03

Y (m)

0

Y (m)

0

0.04

0.04

0.05

0.05

0.06

0.06

0.07

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.07

0

0.01

0.02

0.03

X (m)

0.04

0.05

0.06

0.07

X (m)

Figure 6.3: Clean pressure.

Figure 6.4: Muddy Pressure.

0

0

0.01

0.01

0.02

Y (m)

0.03

0.04

0.05

Speed 5.0E-02 5.0E-03 5.0E-04 5.0E-05 5.0E-06 5.0E-07 5.0E-08 5.0E-09 5.0E-10 5.0E-11 5.0E-12 5.0E-13 5.0E-14

0.02

0.03

Y (m)

Speed 5.0E-02 5.0E-03 5.0E-04 5.0E-05 5.0E-06 5.0E-07 5.0E-08 5.0E-09 5.0E-10 5.0E-11 5.0E-12 5.0E-13 5.0E-14

0.04

0.05

0.06

0.06

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0

X (m)

Figure 6.5: speed.

Clean velocity and

0.01

0.02

0.03

0.04

0.05

0.06

0.07

X (m)

Figure 6.6: Muddy speed and velocity.

49

0

0.01

Mud

0.02

0.03

Y (m)

Rock

0.04

Vug

0.05

0.06

0.07

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

X (m)

Figure 6.7: CT scan of slice 180 of Pipe Creek sample.

Figure 6.8: Permeability of center quarter of slice 180.

0.01

0.01

0.02

0.02

0.03

0.03

Y (m)

0

Y (m)

0

0.04

0.04

0.05

0.05

0.06

0.06

0.07

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.07

X (m)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

X (m)

Figure 6.9: Clean pressure.

Figure 6.10: Muddy Pressure.

50

0

0

0.01

0.01

0.02

Y (m)

0.03

0.04

0.03

0.04

0.05

0.05

0.06

0.06

0

0.01

0.02

0.03

0.04

0.05

0.06

Speed 5.0E-05 1.5E-05 5.0E-06 1.5E-06 5.0E-07 1.5E-07 5.0E-08 1.5E-08 5.0E-09 1.5E-09 5.0E-10

0.02

Y (m)

Speed 5.0E-05 1.5E-05 5.0E-06 1.5E-06 5.0E-07 1.5E-07 5.0E-08 1.5E-08 5.0E-09 1.5E-09 5.0E-10

0.07

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

X (m)

X (m)

Figure 6.11: Clean velocity and speed.

Figure 6.12: Muddy speed and velocity.

51

Chapter 7 Computational Results for the Homogenized Model

A macro-model of fluid flow in a vuggy medium has recently been derived from the micro-model through the mathematical theory of homogenization [4]. This model provides the effective permeability of a vuggy medium. In this chapter we present two computational studies to illustrate and verify this macro-model, see also [3]. In the first study, we illuminate the nature of the effective permeability itself by considering vuggy media with (i) layered vugs, (ii) meandering vug channels, (iii) constricted vug channels, and (iv) disconnected vugs. We find that vug connectivity is the most critical variable in predicting macroscopic properties. In the second study, we consider the macro-model. We compare fluid flow in a vuggy medium on the micro-scale to that on the macro-scale. We find support for the macro-model, and again find that vug interconnectivity is the critical variable to preserve in the upscaling process.

52

7.1

A Macro-Model and the Effective Permeability The details of deriving the macro-model are in [4]. For completeness, we

give a brief overview. One assumes that the geometric vug and pore structure of Ω are periodic of period Y , where Y is a reference cell, normally a rectangle in 2-D and a “brick” in 3-D. The reference cell is part vug Ys and part matrix Yd , with an interface γ between these regions. Thus, Ω is a tessellation. For 0 < ² ≤ 1, we create Ω² by tessellating the scaled cells ²Y . The homogenization problem is then to determine the behavior of fluid flow in Ω² as ² → 0, that is, as the fine scale becomes negligible. We must also scale both viscosity µ and permeability K by ²2 , since as ² → 0, flow paths (in our case vugs) become constricted, and a corresponding decrease in viscosity is required to maintain flow rates [36]. In the limit as ² vanishes, the fluid velocity u and pressure p converge ¯ and pressure p¯. What is important is (weakly) to a macroscopic velocity u ¯ and p¯ can be found by solving the macro-model that u ˜ −1 u ¯ + ∇¯ µK p=f

in Ω ,

(7.1)

¯=q ∇·u

in Ω .

(7.2)

¯ and p¯ satisfy a Darcy equation on all of Ω, with an effective permeThat is, u ˜ independent of the fluid viscosity. Moreover, the tensor K ˜ is ability matrix K symmetric and positive definite, and it can be computed from the solution of a Darcy-Stokes problem posed on Y involving K, α, and the vug geometry.

53

˜ one solves the following homogeTo find the effective permeability K, nized cell problem for each ej , the standard Cartesian unit vector in the jth direction, using periodic boundary conditions on ∂Y . It is a Darcy-Stokes system, and gives a periodic flux ωj (and a periodic pressure φj , which we may discard): − 2∇ · Dωj,s + ∇φj,s = ej

in Ys ,

(7.3)

∇ · ωj,s = 0

in Ys ,

(7.4)

K −1 ωj,d + ∇φj,d = ej

in Yd ,

(7.5)

∇ · ωj,d = 0

in Yd ,

(7.6)

ωj,s · νs = ωj,d · νs

on γ ,

(7.7)

on γ ,

(7.8)

on γ .

(7.9)

α 2νs · Dωj,s · τi = − √ ωj,s · τi K 2νs · Dωj,s · νs = φj,s − φj,d

˜ from the formula We then compute the matrix K ½Z ¾ Z 1 ˜ i,j = ωj,s,i dv + ωj,d,i dv , K V Yd Ys

(7.10)

where ωj,l,i is the ith component of the vector ωj,l and V is the volume (or area in 2-D) of Y .

7.2

Examples of Effective Permeabilities In this section we consider the solution to the homogenized cell problem

(7.3)–(7.9), and in particular the effective permeability that results from the formula (7.10). All our results are in 2-D. 54

The base case for our studies in this section is an 8 cm by 8 cm sample with Beavers-Joseph coefficient α = 1, rock matrix permeability K = 10 md, and vugs with aperture δ = 1 cm. We will consider varying these parameters and, especially, the vug configuration. 7.2.1

A Layered Medium We begin with a study of a horizontally layered medium, which has

a square reference cell Y = (0, `) × (0, `) with Ys = (0, `) × (0, δ) and Yd = (0, `) × (δ, `) (see Fig. 7.1), where ` = 8 cm and usually δ = 1 cm. In fact, this problem can be solved analytically [4]. The effective permeability that results is diagonal, and √ µ ¶ K 2 1 1 3 ˜ δ + δ + K(` − δ) , K11 = ` 12 2α ˜ 22 = ` K . K `−δ

(7.11) (7.12)

Thus this subsection can be viewed more as a test that the code approximates the solution accurately. We begin with a convergence study on the base case. We show the results of varying the computational grid spacing h in Table 7.1. As can be seen, we obtain good convergence for even modest values of h. We generally take h = 0.25 cm henceforth. Next we vary the Beavers-Joseph coefficient α. Results are tabulated in Table 7.2. We get excellent agreement with the analytical formula (7.11) for ˜ 11 , the x-permeability along the vug channel. Of course, the y-permeability K 55

Figure 7.1: The layered medium. The reference cell is on the left, the periodic medium is on the right. h (cm) 1.00000 0.50000 0.25000 0.12500 0.00625 0.03125 0.00000

˜ 11 (md) K 1.05550e+9 1.05549e+9 1.05549e+9 1.05548e+9 1.05548e+9 1.05548e+9 1.05548e+9

˜ 22 (md) K 11.4286 11.4327 11.4332 11.4334 11.4335 11.4335 11.4286

Table 7.1: Numerical convergence for the layered medium. The h = 0 results are the true analytical values from (7.11)–(7.12). is unaffected by the tangential slip due to α. Results of varying the rock matrix permeability K are given in Table 7.3, and again excellent agreement with the analytical formulas (7.11)–(7.12) results. Perhaps a more interesting experiment is to vary the vug aperture δ. Results are shown in Table 7.4. Large aperture vugs result in effective xpermeabilities of a million to a billion times greater than that of the rock. However, even a tiny aperture of about 0.3 mm results in an x-permeability 56

α 0.01 0.10 1.00 10.0 100.

computed ˜ 11 (md) K 1.06197e+9 1.05610e+9 1.05549e+9 1.05543e+9 1.05542e+9

analytical ˜ 11 (md) K 1.06171e+9 1.05605e+9 1.05548e+9 1.05543e+9 1.05542e+9

computed ˜ 22 (md) K 11.4332 11.4332 11.4332 11.4332 11.4332

Table 7.2: Effect of varying the Beavers-Joseph coefficient α for the layered medium. K (md) 1 10 100 1000

computed ˜ 11 (md) K 1.05544e+9 1.05549e+9 1.05563e+9 1.05610e+9

analytical ˜ 11 (md) K 1.05544e+9 1.05548e+9 1.05562e+9 1.05605e+9

computed ˜ 22 (md) K 1.1475 11.433 114.29 1142.9

analytical ˜ 22 (md) K 1.1429 11.429 114.29 1142.9

Table 7.3: Effect of varying the rock matrix permeability K for the layered medium. of some thousand times greater than K. Of course, the y-permeability is relatively unaffected by the presence of the vug, as the fluid is constrained to flow through the 10 md rock. 7.2.2

Flow through a Meandering Vug System Except perhaps fractures, natural vugs are unlikely to maintain their

orientation for long distances. We thus consider next the configuration depicted in Fig. 7.2, with a meandering vug channel of aperture δ = 1 cm. ˜ 11 = 7.05195e+8 md Here, the effective permeability is diagonal, with K

57

δ (cm) 2.00000 1.00000 0.50000 0.25000 0.12500 0.06250 0.03125

computed ˜ 11 (md) K 8.44364e+9 1.05549e+9 1.31946e+8 1.64955e+7 2.06252e+6 2.57967e+5 3.22905e+4

analytical ˜ 11 (md) K 8.44360e+9 1.05548e+9 1.31943e+8 1.64949e+7 2.06236e+6 2.57926e+5 3.22802e+4

computed ˜ 22 (md) K 13.3664 11.4327 10.6672 10.3227 10.1587 10.0787 10.0392

analytical ˜ 22 (md) K 13.3333 11.4286 10.6667 10.3226 10.1587 10.0787 10.0392

Table 7.4: Effect of varying the vug aperture δ for the layered medium.

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Figure 7.2: The meandering vug medium. The reference cell is on the left, the periodic medium is on the right. ˜ 22 = 17.8306 md. Compare this to the layered case, for which K ˜ 11 = and K ˜ 22 = 11.4286 md. From the layered case, the meandering 1.05548e+9 md and K case has a decrease in x-permeability and an increase in y-permeability, as one should expect. Thus vug channel tortuosity has an important effect on the overall flow properties of the effective medium.

58

7.2.3

Flow through a Constricted Vug System Natural vugs are unlikely to maintain their aperture, so we consider

next a vug system which includes constrictions; it is illustrated in Fig. 7.3. The vug has aperture δ = 1 cm, except the constriction, which is only 2 mm ˜ 11 = 3.64777+7 md and K ˜ 22 = in aperture (and 2 cm in length). We obtain K ˜ 11 = 11.2495 md. Comparing these to the system without a constriction (K ˜ 22 = 11.4286 md), we see that the y-permeability is 1.05548e+9 md and K essentially unchanged, but the x-permeability is reduced by a factor of about 30. In fact, the x-permeability for a layered medium with a 2 mm channel is 8.44588e+6 md, which is only about 4 times smaller. We conclude that small constrictions have a significant effect on the overall flow.

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Figure 7.3: The constricted vug medium. The reference cell is on the left, the periodic medium is on the right.

7.2.4

Flow through a Disconnected Vug System Finally, since some natural vuggy media are likely to have disconnected

vugs, we study the effect of vug disconnectivity on overall effective permeabil59

ity. We saw in the previous test that a medium with nearly disconnected vugs nevertheless results in a very large effective permeability. However, if the medium is disconnected, even by very small plugs between the vugs, we obtain permeabilities close to those for the matrix rock. We show this in a test on the configuration depicted in Fig. 7.4. In this configuration, the vugs are disconnected in the x-direction by a small plug of width that varies from 2 mm to 1 cm. Again, the vug aperture is 1 cm, so the plug is quite small. The computed effective permeabilities are given in Table 7.5. They are all within one order of magnitude of K. Thus, indeed, we have seen that vug connectivity or disconnectivity has a dramatic effect on effective flow properties.

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