二相流在破裂介質中的宏觀模式(II)

行政院國家科學委員會專題研究計畫 成果報告

二相流在破裂介質中的宏觀模式(2/2)

計畫類別： 個別型計畫

計畫編號： NSC94-2115-M-009-008-

執行期間： 94 年 08 月 01 日至 95 年 07 月 31 日

執行單位： 國立交通大學應用數學系(所)

計畫主持人： 葉立明

報告類型： 完整報告

報告附件： 出席國際會議研究心得報告及發表論文

處理方式： 本計畫可公開查詢

中 華 民 國 95 年 11 月 3 日

91/10/30 修訂

中文摘要

這是一個二年期的計劃。我們建議探討在破裂介質中的二相流的宏觀模式。

一個已知的事實是二相流在獨立的區塊中的流動雖然很緩慢，但它對流體在整體

上的表現卻佔了十分重要的角色，它具有所謂的調節功能。表現在外的一種形式

是在不同區塊會有不同的 time scale 的現象。此現象在宏觀模式下的單孔模式

(single-porosity model)中無法看出，但期望在宏觀模式下的雙孔模式

(dual-porosity model)可看出來。在之前的研究中，我們考慮了區塊為小尺寸及

中尺寸的情形，並證明了它們會收歛到某種的雙孔介質模式。且這些模式在區塊

與整體之間確有不同的 time scale 的現象。我們希望繼續先前的工作討論柱形區

塊及混和型區塊所構成的破裂介質的情形。目的就是想找出各種不同區塊的微觀

模式所對應的宏觀模式。

關鍵詞：單孔模式、雙孔模式。

英文摘要

This is a two-year project. We propose to investigate macroscopic model

for two-phase flows in fractured media. It is known that although flow moves

slow in matrix blocks, it has profound influence on global flow behaviors.

Usually, different time scales in different regions can be observed. This

phenomenon does not show in single-porosity models, but it is expected to

be captured in dual-porosity models. In previous study, we consider

small-sized block case and medium-sized block case, and prove they converge

to some dual-porosity models. They do show different time-scales in matrix

blocks and fracture system. We plan to continue this effort to study

column-block type fracture media and mixed-block fracture media. The purpose

is to find the relations between microscopic models and macroscopic models

for two-phase flows in fractured media.

Key words：single-porosity model、dual-porosity model.

背景及目的

在很多的應用問題上須要了解流體在破裂介質中的運動。譬如地下污染源

(污水或核廢料)的問題及多孔介質中多相流的問題。事實上，組成半導體晶片

的 transistor(如 MOSFET [37])本身也是一種破裂介質，因此電子流在

transistor 中的運動也對應到微觀下的流體在破裂介質中的運動。不過我們有

興趣的是多孔介質中二相流的問題。流體在破裂介質中的運動有一特殊的現

91/10/30 修訂

象，就是在不同的區塊中流體的變化會有不同的 time-scale 的特性

[30,31,33]。由於二相流在破裂介質中的運動和介質的 porosity、

permeability、geometry、及流體的性質等有關。因此描述破裂介質中二相流

變化的微觀模式十分複雜且很難做分析。即使想用計算的方式去找出微觀模式

的近似解也因為計算量太過龐大而不可行。目前對二相流在破裂介質中的問題

的研究都是研究它所對應的較簡化的宏觀模式。在一般不均勻的介質中(非破裂

介質)，流體的運動的宏觀模式可用 single-porosity model 描述。在已知的文

獻中，大部份的地下污染源的問題及多孔介質中多相流的問題都是屬於這類模

式。然而 single-porosity model 並不適用於破裂介質中。一種粗淺的說法是

在微觀下描述二相流在破裂介質中運動的一些係數，會有急劇振動的情形，此

急劇的變化在 single-porosity model 是無法看出。因此，就有工程的學者提

出 dual-porosity model 的概念。經過多次實驗測試的結果，dual-porosity

model 較 single-porosity model 更能準確描述流體在破裂介質中的變化。不過

相對付出的是前者比後者的方程式更複雜。目前並沒有太多探討這類問題的數

學結果。此外在 dual-porosity model 中正確的 interface condition 也是一

個問題。目前常用的 interface condition 是由物理直覺得到的並無理論基礎。

我們則希望借由數學工具建立起一套有系統的二相流在微觀與在宏觀下的模式

之間的對應關係及其理論基礎。

二相流在破裂介質中的微觀數學模式是由不連續係數的退化型拋物線與橢

圓方程式所組成[46,48]。此問題在工程上有很多的文獻討論，但他們著重在

modelling，數值計算，或資料統計的部份，而沒有數學理論。在數學界，不連

續係數的方程早在 60 年代已有人開始討論，不過他們只著重在輕微的不連續的

問題上[35]。若有急劇變化的情形則會忽略係數較小的部份不計，因此可以大

大的簡化問題的困難度。在我們的情形以上兩種假設都不可行。所以產生微觀

模式下二相流問題的解的存在性、穩定性等的証明的困難。這方面的參考資料

目前並不多。Poisson equation 及 heat equation 的情形可參考[32,36]。在

一般不均勻介質下的宏觀模式(即 single-porosity model)的部份，已知的結果

較多。解的存在性、唯一性、穩定性、數值計算方法、及物理性質可參考

[6,7,10,12,20,21,25,26,27,39,40]及其內的參考資料。解的 regularity 的部

份可參考[4,5,13,28,34]。至於破裂介質的宏觀模式大多是由物理性質及不嚴

謹的方式推導出一些 dual-porosity model [8,9,17,18,30,31,33]，但並無數

學的證明。少數一些 dual-porosity model 的數學結果(但屬於較簡單的例子)

可參考[11,15]。

在過去的研究中，我們討論了二相流在小尺寸與中尺寸的破裂介質中微觀

模式與宏觀模式的關聯[46,48]。我們希望在接下來的三年計劃中能沿續先前的

工作，討論在不同尺寸及混合尺寸下二相流的微觀模式及其對應的宏觀模式。

除了希望建立起一套有系統的二相流在微觀與在宏觀下的模式之間的對應關係

及其理論基礎外，也希望能將此結果應用到晶片中的 transistor 的電子流的問

91/10/30 修訂

題上。研究的方向是先考慮柱形區塊所構成的破裂介質的微觀模式的解的

well-posedness 問題，再考慮其對應的宏觀模式，接著再結合之前的結果(即小

尺寸及中尺寸的模式)討論混合尺寸下的問題。Triple-porosity model(即含有

三種不同 porosity 尺寸的模式)也是我們討論的範圍。底下我們列出想考慮的

一個 model problem。

We consider a porous medium

Ω

∈

ℜ

3

, which is a two-connected domain with a

periodic structure. Let

Y

:=

[

0

,

1

]

3

be a cell consisting of a matrix block domain

Y

_m

completely surrounded by a connected fracture domain

Y , and we denote by

f

Γ the

matrix-fracture interface in the cell

Y . Let

χ

( y

)

be the characteristic function of

m

Y extended Y -periodically to all of

ℜ . The medium Ω contains two

2

subdomains,

Ω and

εf

Ω , representing the system of fracture plans and matrix

εm

blocks respectively, and satisfying

Ω

εm

⊂

{

x

∈

Ω

|

χ

(

x

/

ε

)

=

1

},

Ω

εf

=

Ω

\

Ω

εm

.

Let

Γ

ε

:

=

∂

Ω

εf

I

∂

Ω

εm

I

Ω

be that part of the interface of

∂

Ω

εm

with

∂

Ω

εf

that is interior to

Ω .

For the fracture subdomain

Ω , we denote porosity by

εf

Φ , absolute

ε

permeability by

K , saturation of oil phase by

ε

S

ε

∈

[

0

,

1

]

, capillary pressure by

)

(

ε

γ

S

, the relative permeability by

Λ

_α

(

S

ε

)

, phase pressure by

P

_αε

, and a function

depending on gravity by

G

_αε

for

α

=

w,

o

. We use

φ

ε

,

κ

ε

,

s

ε

,

ν

(

s

ε

),

λ

_α

(

s

ε

),

p

_αε

,

g

_αε

for

α

=

w,

o

, in subdomain

Ω to represent same quantities as those denoted by

εm

upper case symbol in fracture subdomain. Let

ϖ

(>

0

)

be a constant. The

conservation of mass in each phase, with the Darcy’s law, can be written as, in

Ω ,

εf

0

>

t

,

−

Φ

ε

(

x

)

∂

_t

S

ε

−

∇

⋅

(

K

ε

(

x

)

Λ

_w

(

S

ε

)

∇

(

P

_wε

−

G

ε_w

))

=

0

,

Φ

ε

(

x

)

∂

_t

S

ε

−

∇

⋅

(

K

ε

(

x

)

Λ

_o

(

S

ε

)

∇

(

P

_oε

−

G

_oε

))

=

0

,

γ

(

S

ε

)

=

P

₀ε

−

P

_wε

,

in

Ω ,

εm

t

>

0

,

(

)

,

(

(

)

(

)

,

)

(

(

)

(

)

(

))

0

2

∇

⋅

∇

−

∇

⋅

∇

−

=

−

∂

−

_φ

ε ε

_ε

ϖ

_κ

ε

_λ

ε ε

_κ

ε

_λ

ε ε ε w w z w z w y x w y x t

s

x

s

p

x

s

p

g

x

91/10/30 修訂

(

)

,

(

(

)

(

)

,

)

(

(

)

(

)

(

))

0

2

∇

⋅

∇

−

∇

⋅

∇

−

=

−

∂

ε ϖ ε ε ε ε ε ε ε ε

_ε

_κ

_λ

_κ

_λ

φ

x

t

s

xy

x

o

s

xy

p

o z

x

o

s

z

p

o

g

o

,

ν

(

s

ε

)

=

p

₀ε

−

p

ε_w

,

Phase fluxes and pressures are required to be continuous on interface

Γ ,

ε

t

>

0

,

o

w

,

=

α

,

K

ε

x

_α

S

ε xy

P

_αε

ν

v

ε

ϖ

k

ε

x

λ

_α

s

ε x,y

p

_αε

ν

v

2 ,

(

)

(

)

)

(

)

(

Λ

∇

=

∇

,

P

_αε

=

p

_αε

,

where

ν

v

is the unit vector outer normal to

Γ . Boundary

ε

∂

Ω

_of

Ω includes

1

Γ

,

2

Γ

, which satisfying

Γ

₁

I

Γ

₂

=

φ

,

∂

Ω

=

Γ

₁

U

Γ

₂

. Boundary conditions are given by,

for

α

=

w

,

o

,

K

ε

(

x

)

Λ

_α

(

S

ε

)

∇

(

P

_αε

−

G

_αε

)

n

v

=

0

, on

Γ

₁

,

P

_αε

=

P

b,_α

, on

Γ

2

,

where

nv is the unit vector outer normal to

Γ

₁

. Initial conditions are

S

(

x

,

0

)

S

0

(

x

)

ε ε

₌

, in

Ω

εf

,

s

ε

(

x

,

0

)

=

s

₀ε

(

x

)

, in

Ω ,

ε_m

我們想問當

ε 很小時，以上的微觀模式所對應的宏觀模式為何?

研究方法、進行步驟及執行進度

第一步是討論微觀模式的解的 well-posedness 的問題。在這部份，須要仔

細研究不連續係數的退化型拋物線與橢圓方程式[16,29]。古典的拋物線與橢圓

方程式理論不適用於這問題。為了估計不連續截面的變化，也須要用到

psudodifferential operator [41]及 boundary integral method[14]。在函數

空間方面 Holder space 或 Sobolev space 並不是正確的空間，要嘗試 Besov

space 或其他函數空間才可[2,38,42]。第二步則是找出二相流的微觀模式與宏

觀模式的關係，這將要借助 homogenization 或 multiple-scale convergence

的技巧[1,3]。

91/10/30 修訂

結果與討論

這兩年我們完成了以下的工作:第一年找出了二相流在柱形區塊所構成的

破裂介質的微觀模式與宏觀模式的關係。此宏觀模式基本上是由退化型拋物線

與橢圓方程式所組成。它和二相流在小尺寸與中尺寸的破裂介質中的運動模式

最大的不同是重力在此方程式中所佔的比重。由於二相流是在柱形區塊所構成

的破裂介質中運動重力成了引導流體運動的重要因子。很自然的，方程式也變

得十分複雜。第二年我們找出了兩種可混合流體在中尺寸區塊所構成的破裂介

質的宏觀模式。此模式是由均勻拋物線與橢圓方程所組成。不過此拋物線方程

需用到橢圓方程式的解的導數，是個高度非線性的拋物線與橢圓方程組。底下

附上的兩篇論文就是我們這兩年的硏究結果。

計畫成果自評部份

這次的研究內容與原計畫相符合程、也達成預期目標情況、研究成果將發表

於國際學術期刊上。

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Arch. Rational Mech. Anal., 153(2000)

91--151.

[37] Peter Markowich.

The Stationary Semiconductor Device Equations. Springer-Verlag,

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[38] S. M. Nikol'skii.

Approximation of Functions of Several Variables and Imbedding

Theorems. Springer-Verlag, Berlin, 1975.

[39] Nochetto R. H. Error estimate for multidimensional sigular parabolic problems.

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Appl. Math., 4, 1987.

[40] Peaceman D. W.

Fundamentals of Numerical Reservoir Simulation. Elsevier, New York,

91/10/30 修訂

[41] Michael E. Taylor.

Psudodifferential Operators. Princeton University, 1981.

[42] Hans Triebel.

Interpolation Theory, Function Spaces, Differential Operators. North

Holland, 1978.

[43] L. M. Yeh. Convergence of A Dual--Porosity Model for Two--phase Flow in Fractured

Reservoirs.

Mathematical Methods in Applied Science, 23, 777--802, 2000.

[44] L. M. Yeh. Relations between phase mobilities and capillary pressures

of two--phase flows in fractured media.

Fluid Flow and Transport in Porous Media:

Mathematical and Numerical Treatment (Zhangxin Chen, Richard E. Ewing ed.),

Contemporary Mathematics, 295, 159--171, 2002.

[45] L. M. Yeh. On two-phase flows in fractured media.

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Manuscript

f

c NCTU

Tall Block Models for Two-Phase flows in Fractured Media

Li-Ming Yeh

Department of Applied Mathematics, National Chiao Tung University Hsinchu, 30050, Taiwan, R.O.C.

Email: [email protected]

Two-phase, incompressible, immiscible flow in fractured media with tall matrix blocks is concerned. Suppose  denotes horizontal size ratio of matrix blocks to whole medium, and suppose the horizontal widths of the fracture planes and matrix blocks are in same order. As  goes to 0, microscopic model for the two-phase flow problem converges to 1) a dual-porosity model if permeability ratio of matrix blocks to fracture planes is of order 2_{; 2) a single-porosity model for fracture flow if the ratio is smaller than order }2_;

3) another type of single-porosity model if the ratio is greater than order 2_.

Keywords: dual-porosity model, fractured media

1. Introduction

Homogenization for two-phase, incompressible, immiscible flow in fractured media with tall matrix blocks is concerned. Within a fractured medium there is an in-terconnected system of fracture planes dividing the porous rock into a collection of matrix blocks. The fracture planes, while very thin, form paths of high perme-ability. Most of the fluids reside in matrix blocks, where they move very slow. Let  be the horizontal size ratio of tall matrix blocks to the whole medium, and let the horizontal widths of the fracture planes and matrix blocks be in same order. In case permeability ratio of matrix blocks to fracture planes is of order 2_,

micro-scopic models for the two-phase flow problem converge to a dual-porosity model as  tends to 0. For the macroscopic model, a fractured medium is regarded as a porous medium consisting of two superimposed continua, a continuous fracture sys-tem and a discontinuous syssys-tem of matrix blocks. Matrix blocks play the role of a global source distributed over the entire medium. The immiscible two-phase flow is formulated by conservation of mass principles for each continum plus sources from tall matrix blocks. This problem was also considered by formal asymptotic expan-sion in [8]. If the ratio is smaller than order 2_{, the microscopic models approach}

a single-porosity model for fracture flow. If the ratio is greater than order 2_{, then}

microscopic models tend to another type of single-porosity model. Our intention is to prove the convergence of the microscopic models.

2 Tall Block Models

Rest of the paper is organized as follows: In next section§2, we state microscopic model for two-phase flow in fractured media. Notation and assumption will be given in_{§3. Then in §4, we present our main results. Some known results needed for our} main results will be recalled in §5. Proof of main result is in §6. In §6, we need to use the convergence of oil saturation in matrix blocks. The proof is lengthy and tedious, so we present it in last section§7.

2. Microscopic Model for Tall Matrix Blocks

Let Y _{≡ [0, 1]}2 _{be a cell consisting of a matrix block domain Y}

m completely

surrounded by a connected fracture domain Yf. Xm(y) is the characteristic function

of Ym, extended Y -periodically to all of<2. ˜Ω⊂ <2 contains two subdomains, ˜Ωf

and ˜Ω

m. ˜Ωm⊂ {˜x ∈ ˜Ω|Xm(˜x/) = 1}, ˜Ωf = ˜Ω\ ˜Ωm. Let ˜Γ ≡ ∂ ˜Ωf∩ ∂ ˜Ωm∩ ˜Ω.

Boundary of ˜Ω includes two parts ˜Γ1and ˜Γ2satisfying ˜Γ1∪˜Γ2= ∂ ˜Ω and ˜Γ◦1∩˜Γ◦2=∅.

Porous medium considered is a cylindrical aquifer Ω _{≡ ˜}Ω_{× [0, H] ⊂ <}3 _{and is}

assumed to be a two-connected domain with a periodic structure. It contains two subdomains, Ω

f ≡ ˜Ωf× [0, H] and Ωm ≡ ˜Ωm× [0, H], representing the system of

fracture planes and matrix blocks respectively. Let Γ ≡ ˜Γ× [0, H] be that part

of the interface of Ω

m with Ωf that is interior to Ω. Both Γ1 ≡ ˜Γ1× [0, H] and

Γ2≡ ˜Γ2× [0, H] are part of lateral boundary of Ω.

In fracture subdomain Ω

f, porosity is denoted by Φ, absolute permeability by

K_{, saturation of oil phase by S}_{, capillary pressure by Υ(S}_{), relative permeability}

by Λα(S), phase pressure by Pα, and a density-gravity term by Gα for α = w, o.

φ_{, k}_{, s}_{, υ(s}_{), λ}

α(s), pα, gαfor α = w, o, in subdomain Ωm represent same

quan-tities as those denoted by upper case symbol in fracture subdomain. Conservation of mass in each phase are written as, in Ω

f, t > 0, −Φ∂tS− ∇ · (KΛw(S)∇(Pw − Gw)) = 0, (2.1) Φ∂tS− ∇ · (KΛo(S)∇(Po− Go)) = 0, (2.2) Υ(S) = Po− Pw, (2.3) in Ω m, t > 0, −φ_∂ ts− ∇ · kI2$λw(s)∇(pw− Gw)
= 0, (2.4) φ∂ts− ∇ · kI2$λo(s)∇(po− Go)
= 0, (2.5) υ(s) = po− pw, (2.6) where_Id

 is a diagonal matrix defined byId≡

   d _{0 0} 0 d ₀ 0 0 1 

. Phase fluxes and pressures are required to be continuous on interface Γ, t > 0, α = w, o,

K_Λ

α(S)∇(Pα− Gα)· ~ν= kI2$λα(s)∇(pα− Gα)· ~ν, (2.7)

Tall Block Models 3

where ~ν _{is the unit vector normal to Γ}

. Boundary conditions are, for α = w, o,

KΛα(S)∇(Pα− Gα)· ~n = 0 on Γ1, (2.9) K_Λ α(S)∂x3(P  α− Gα)|x3=0,H = k _λ α(s)∂x3(p  α− Gα)|x3=0,H = 0, (2.10) P α= Pb,α on Γ2, (2.11)

where ~n is the unit vector outer normal to Γ1. Initial conditions are

S(0, x) = S0(x) in Ωf, (2.12)

s(0, x) = s0(x) in Ωm. (2.13)

3. Notation and Assumption For any x ∈ <3_{, x = (˜x, x}

3) where ˜x∈ <2. ˜Ω(2)≡ {˜x ∈ ˜Ω : dist(˜x, ∂ ˜Ω) > 2},

˜ Ω

m ≡ {˜x : ˜x ∈ (Ym+ j) ⊂ ˜Ω(2) for j∈ Z2}, ˜Ωf ≡ ˜Ω\ ˜Ωm, and ˜Ω ≡ {z :

z ∈ (Y + j), (Ym+ j) ⊂ ˜Ω(2) for j∈ Z2}. Ω ≡ ˜Ω× [0, H], Ωi ≡ ˜Ωi× [0, H],

YH

m ≡ Ym×[0, H], Q ≡ Ω×Y , Qm≡ Ω×Ym,Qi≡ Ω×Yi, i = f, m. Bt≡ (0, t)×B

forB = YH

m, Γ,Q, Qm, Ω, Ωi,Qi i = f, m.

<+0 ≡ <+∪ {0}. Time difference is defined to be ∂hψ(t) ≡

ψ(t+h)−ψ(t) h . For a

set_{B, XB}is a characteristic function of_{B. ψ(t, x, y) ∈ L}r_(ΩT_{; L}r

per(Y )), 1 < r <∞,

coincides with a function in Lr₍

QT_{) extended by Y -periodicity in y to the whole}

of _<2_{. For}

B = Yf, Ym, we define Lr(ΩT; Lperr (B)) ≡ {ψ ∈ Lr(ΩT; Lrper(Y )) :

ψ(t, x, y) = 0 if y ∈ Y \ B}. W0i,r(Ω) ≡ {ψ ∈ Wi,r(Ω) : ψ|Γ2 = 0} if i ∈ N and

r > 1,_{U ≡ W}01,2(Ω),U2 ≡ U × U, dual X ≡ dual space of X, sl (resp. 1− sr) is

residual matrix oil (resp. water) saturation. Lq,r_(ΩT₎

≡ Lr_{(0, T ; L}q_(Ω)).

If Υ : [0, 1) _{→ <}+0 (resp. υ : [sl, sr) → <+0) is onto and strictly increasing,

Υ−1 _{(resp. υ}−1_{) denotes the inverse function of Υ (resp. υ). Then we define}

J : [sl, sr)→ [0, 1) by J (z) ≡ Υ−1(υ(z)), and denote byJ−1 the inverse function

of_{J .} Pb,c≡ Pb,o− Pb,w, Sb≡ Υ−1(Pb,c), Λ≡ Λw+ Λo, λ≡ λw+ λo,      R(z) ≡R0z ΛwΛo Λ dΥ dS(ξ)dξ for z∈ [0, 1), A(z) ≡R0z q ΛwΛo Λ (Υ−1(ξ))dξ for z∈ [0, ∞), M(z) ≡Rszl λwλo λ dυ ds(ξ)dξ for z∈ [sl, sr). (3.1)

ϑ∈ (0, 1/8) is a number such that R0_{is increasing (resp. decreasing) in (0, ϑ) (resp.}

(1_{− ϑ, 1)).}

Next let us assume the following conditions: For α = w, o,

A1. Γ26= ∅, Ym⊂ <2is a bounded smooth domain, and Ω⊂ <3is open, bounded,

and connected with Lipschitz boundary, A2. K_{, G}

α(x3)∈ W1,∞(Ω), ∂tPb,α∈ L2(0, T ; H1(Ω)), Pb,α ∈ C(0, T ; C1,d1(Ω)),

S

4 Tall Block Models A3. K_{, k}_{, Λ, λ∈ [d} 3, d4], Sb, S0,J (s0)∈ (d5, 1− d5) and d5∈ (0, 1), A4. φ_{= φ(}x ), k= k( x

), where φ, k are smooth Y -periodic functions,

A5. Λw, λw (resp. Λo, λo) : [0, 1] → [0, 1] are continuous and decreasing (resp.

increasing), Λw(1− z) ∝ zd6, Λo(z)∝ zd7 for z∈ (0, ϑ), Λ_Λα(J (z)) =λ_λα(z),

A6. Υ : [0, 1)_{→ <}+₀ (υ : [sl, sr)→ <+0) is onto, increasing, and a locally Lipschitz

continuous function, and inf

z∈[0,1) dΥ dS(z) > 0, dΥ dS(J (z)) dυ ds(z) , Φ_{, φ} ∈ [d8, d9] for z ∈ [sl, sr], d_d9₈ ∼ 1,

A7. Λ3/2o (z)≤R_z2z(A(Υ(2z)) − A(Υ(ξ)))dξ for z ∈ (0, ϑ) and

Λ3/2w (1− z) ≤R₁1_−2z−z(A(Υ(ξ)) − A(Υ(1 − 2z)))dξ for z ∈ (0, ϑ),

A8. _|Λα(z1)− Λα(z2)| ≤ d10 p (_R(z1)− R(z2))(z1− z2) for any z1, z2∈ [0, 1], A9. max z∈[0,1]|Λ(z) − 1| + maxz∈[sl,sr]|λ(z) − 1| ≤ d 11 (d11 only depends on Ω, K, k), A10. ΛoΛw(z)≤ d12z|1 − z|pR0(z),R0(z)∝ zm|1 − z|m1 for z∈ (0, ϑ) ∪ (1 − ϑ, 1) and m, m1> 1,

where m, m1, di, i = 1,· · · , 12 are positive constants.

Remark 3.1 From A1, Ω

f is an open, bounded, and connected domain with

Lip-schitz boundary. In A2, the density-gravity terms G

w, Go are functions depending

on x3 variable. Initial and boundary saturations are away from two end points 0

and 1 (see A3). A5 implies that relative permeability Λw (resp. λw) in the

neigh-bor of end point 1 has similar properties as Λo (resp. λo) in the neighbor of end

point 0. Relative phase mobilities in fractures and matrix blocks behave similar. A6 requires that fracture capillary pressure increases as fast as capillary pressure of matrix blocks. Usually, derivative of capillary pressure Υ0_{(z) (resp. υ}0_{(z)) tends to}

infinity as z→ 0 or 1 (resp. sl or sr). A10 allows parabolic equations considered

are degenerate at end points 0 and 1, a characteristic of a porous medium equation. Indeed, it also impliesR0 _{∈ L}∞_{(0, 1). A7-8,10 are the restrictions on relative}

per-meability and capillary pressure in fractures. Indeed, if d6, d7 (see A5) are large

enough (depending on capillary pressure), A7-8,10 hold. One may also note that because of A5-10, Λo andR0 at the end point 0 have similar properties as Λw and

R0 _{at the end point 1.}

4. Main Result

In this section, we present the limit models of (2.1–2.13) as  → 0. Roughly speaking, the limit models are fracture flow equations plus interior sources from matrix blocks. The source terms depend on how fast the matrix permeability tends to 0 as → 0. For 0 < $ < 1 case, matrix permeability tends to 0 very slow and saturation variation in fracture system and in matrix blocks is almost simultaneous.

Tall Block Models 5

So the limit model is a single-porosity model with sources from matrix blocks. For $ = 1 case, saturation variation in fracture system and in matrix blocks is not simultaneous and the limit model is a dual-porosity model. In this case, domain acts as a porous medium consisting of two superimposed continua, a continuous fracture system Ω and a discontinuous system of matrix blocks_Qm. Primary flow

occurs in fracture system Ω, and each point x ∈ Ω is associated with a matrix block Ym. Flow in matrix blocks plays the role of a global source in the whole

fracture system. The model includes two systems of equations, one for flow in fracture system and the other for flow in matrix block system. The two systems are coupled through nonlinear sources. For 1 < $ case, matrix permeability tends to 0 so fast that matrix blocks play no roles in the limit model. The limit model is a single-porosity model containing only fracture flow equations without matrix sources.

4.1. For $ = 1 case

Let Ω _{⊂ <}3 _{be a fractured medium. Equations for fracture flow are, for x}

∈ Ω, t > 0,

−Φ∂tS− ∇ · (KΛw(S)∇(Pw− Gw)) = qw, (4.1)

Φ∂tS− ∇ · (KΛo(S)∇(Po− Go)) = qo, (4.2)

Υ(S) = Po− Pw. (4.3)

Φ is porosity, K is permeability field, S is oil saturation, Υ(S) is capillary pressure curve, Λα (α = w, o) is relative permeability curve of α-phase, Pα denotes phase

pressure, Gαis a function depending on density, gravity, and position, and qαis the

matrix-fracture source.

Above each point x ∈ Ω is suspended topologically a matrix block Ym ⊂ <2.

Equations for flow in a matrix block are, for x_{∈ Ω, y ∈ Y}m, t > 0,

−φ∂ts− ∂y,x3· (kλw(s)∂y,x3(pw− Gw)) = 0, (4.4)

φ∂ts− ∂y,x3· (kλo(s)∂y,x3(po− Go)) = 0, (4.5)

υ(s) = po− pw. (4.6)

Here functions s, pw, p0are defined in space domainQmand ∂y,x3= (∂y1, ∂y2, ∂x3).

Each lower case symbol denotes the quantity on Ymcorresponding to that denoted

by an upper case symbol in the fracture system equations. The matrix-fracture sources are given by, for x∈ Ω, t > 0,

qα= −1

|Ym|

Z

Ym

(σαφ∂ts− ∂x3(kλα(s)∂x3(pα− Gα))) dy, (4.7)

where σw =−1, σo = 1, and|Ym| is the volume of Ym. Boundary conditions are,

for t > 0, α = w, o,

6 Tall Block Models

KΛα(S)∂x3(Pα− Gα)|x3=0,H = kλα(s)∂x3(pα− Gα)|x3=0,H = 0, (4.9)

Pα= Pb,α for x∈ Γ2, (4.10)

where ~n is the unit vector outward normal to Γ1. On interface, pressures are

continuous, that is, for t > 0, x_{∈ Ω, y ∈ ∂Y}m, α = w, o,

pα(t, x, y) = Pα(t, x). (4.11)

Initial conditions are

S(0, x) = S0(x) for x∈ Ω, (4.12)

s(0, x, y) = s0(x) for x∈ Ω, y ∈ Ym. (4.13)

Theorem 4.1 Under A1_{−10, a subsequence of solutions of the microscopic models} (2.1–2.13) converges in two-scale sense to a solution of (4.1–4.13) (see next section for the definition of convergence in two-scale sense).

4.2. For 0 < $ < 1 case Equations are, for x_{∈ Ω, t > 0,}

−Φ∂tS− ∇ · (KΛw(S)∇(Pw− Gw)) = qw, (4.14)

Φ∂tS− ∇ · (KΛo(S)∇(Po− Go)) = qo, (4.15)

Υ(S) = Po− Pw= υ(s). (4.16)

Φ, K, S, Υ(S), υ(s), Λα, Pα, Gα, and qα (α = w, o) are the same quantities as

those in $ = 1. The matrix-fracture sources are given by, for x_{∈ Ω, t > 0,} qα= −1

|Ym|

Z

Ym

(σαφ∂ts− ∂x3(kλα(s)∂x3(Pα− Gα))) dy, (4.17)

where σw =−1, σo = 1, and|Ym| is the volume of Ym. Boundary conditions are,

for t > 0, α = w, o,

KΛα(S)∇(Pα− Gα)· ~n = 0 for x∈ Γ1, (4.18)

KΛα(S)∂x3(Pα− Gα)|x3=0,H = 0, (4.19)

Pα= Pb,α for x∈ Γ2, (4.20)

where ~n is the unit vector outward normal to Γ1. Initial condition is

S(0, x) = S0(x) for x∈ Ω. (4.21)

Theorem 4.2 Under A1_{−10, a subsequence of solutions of the microscopic models} (2.1–2.13) converges in two-scale sense to a solution of (4.14–4.21) (see next section for the definition of convergence in two-scale sense).

Tall Block Models 7

4.3. For $ > 1 case

Equations are, for x_{∈ Ω, t > 0,}

−Φ∂tS− ∇ · (KΛw(S)∇(Pw− Gw)) = 0, (4.22)

Φ∂tS− ∇ · (KΛo(S)∇(Po− Go)) = 0, (4.23)

Υ(S) = Po− Pw. (4.24)

Φ, K, S, Υ(S), υ(s), Λα, Pα, Gα, and qα (α = w, o) are the same quantities as

those in $ = 1. Boundary conditions are, for t > 0, α = w, o,

KΛα(S)∇(Pα− Gα)· ~n = 0 for x∈ Γ1, (4.25)

KΛα(S)∂x3(Pα− Gα)|x3=0,H = 0, (4.26)

Pα= Pb,α for x∈ Γ2, (4.27)

where ~n is the unit vector outward normal to Γ1. Initial condition is

S(0, x) = S0(x) for x∈ Ω. (4.28)

Theorem 4.3 Under A1_{−10, a subsequence of solutions of the microscopic models} (2.1–2.13) converges in two-scale sense to a solution of (4.22–4.28) (see next section for the definition of convergence in two-scale sense).

5. Some Known Results

Lemma 5.1 [1] Let 1≤ r < ∞ and A1 hold. There is a constant d13(Yf, r) and a

linear continuous extension operator Π: W1,r(Ωf)∩ L∞(Ωf)→ W1,r(Ω)∩ L∞(Ω)

such that if ϕ∈ W1,r_(Ω f)∩ L∞(Ωf) and d14≤ ϕ ≤ d15, then    Πϕ = ϕ in Ωf almost everywhere, kΠϕkW1,r_(Ω)≤ d₁₃kϕk_W1,r_(Ω f), d14≤ Πϕ≤ d15.

Definition 5.1 For a given  > 0 and 1≤ r < ∞, we define a dilation operator “ ” mapping a measurable function ϕ_{∈ L}r_(Ω,T

m ) to a measurable function ϕ∈ Lr(QTm) by, for (t, ˜x, x3, y)∈ QTm, ϕ(t, ˜x, x3, y)≡  ϕ(t, `<sub>(˜x) + y, x</sub> 3) if (`(˜x) + y, x3)∈ Ωm, 0 elsewhere, where `_(˜x)

≡ j if ˜x ∈ (Y + j), j ∈ Z2_{, denoting the lattice translation point of}

-cell domain containing ˜x.

Definition 5.2 A sequence of functions ϕ in Lr(ΩT), 1 < r <∞, is said to two-scale converge to ϕ in Lr_(ΩT_{; L}r

per(Y )) if, for any function ψ∈ C0∞(ΩT; Cper∞ (Y )),

we have lim →0 Z ΩT ϕ(t, x)ψ(t, x, ˜x/)dxdt = Z QT ϕ(t, x, y)ψ(t, x, y)dydxdt,

8 Tall Block Models

denoted by ϕ _{* ϕ}2

∈ Lr_(ΩT_{; L}r

per(Y )). Besides lim→0kϕkLr_(ΩT₎ =kϕk_Lr_(QT₎,

ϕ _{is said to two-scale converge to ϕ in L}r_(ΩT_{; L}r

per(Y )) strongly, and denoted by

ϕ 2_{→ ϕ ∈ L}r_(ΩT_{; L}r

per(Y )) strongly.

6. Proof of Main Result

A1-10 are assumed from now on. Let us derive a weak formulation of (2.1–2.6). Multiplying (2.1) and (2.4) by η as well as (2.2) and (2.5) by ζ, integrating over ΩT_{, and employing boundary conditions (2.7) and (2.9), we obtain}

− Z Ω,T_f Φ∂tSη + Z Ω,T_f KΛw(S)∇(Pw − Gw)∇η − Z Ω,T m φ_∂ tsη + Z Ω,T m k I2$  λw(s)∇(pw− Gw)∇η = 0, (6.1) Z Ω,T f Φ_∂ tSζ + Z Ω,T f K_Λ o(S)∇(Po− Go)∇ζ + Z Ω,Tm φ∂tsζ + Z Ω,Tm kI2$  λo(s)∇(po− Go)∇ζ = 0, (6.2)

for smooth functions η, ζ∈ L2_{(0, T ;U). Next we define global pressure [11] as}

   P ≡1 2  P o+ Pw + RΥ(S₎ 0 Λo Λ(Υ−1(ξ))− Λw Λ (Υ−1(ξ))
dξ, p≡ 1 2  po+ pw+ Rυ(s) 0 λo λ(υ−1(ξ))− λw λ (υ−1(ξ))
dξ, (6.3) Pbis defined as Pin (6.3)1except replacing Po, Pw, Υ(S) by Pb,o , Pb,w , Pb,c

respec-tively. Then_∇P₌Λw

Λ (S)∇Pw+ΛΛo(S)∇Po and∇p=λλw(s)∇pw+λλo(s)∇po

by (2.3) and (2.6). (6.2) can be rewritten as Z Ω,T_f Φ∂tSζ + Z Ω,T_f K Λo(S)∇(P− Go) +∇R(S)
∇ζ + Z Ω,Tm φ∂tsζ + Z Ω,Tm k_I2$ λo(s)∇(p− Go) +∇M(s)
∇ζ = 0. (6.4)

See_{§3 for R, M. Summing (6.1) and (6.2), we obtain, for η ∈ L}2_{(0, T ;}

U), Z Ω,T_f K Λ(S)∇(P − G o)− Λw(S)∇(Gw− Go)
∇η + Z Ω,Tm k_I2$ λ(s)∇(p− Go)− λw(s)∇(Gw− Go)
∇η = 0. (6.5) For ζ∈ L2_{(0, T ;U) ∩ H}1_(ΩT_{), ζ(T ) = 0,} Z Ω,T_f Φ∂tSζ + Φ(S− S0)∂tζ =− Z Ω,Tm φ∂tsζ + φ(s− s0)∂tζ. (6.6)

Tall Block Models 9

(6.1–6.6), (2.3), (2.6), (2.8), (2.11) form a weak formulation of (2.1–2.13).

Next we consider a regularized problem. Let v be a small number satisfying 0 < v < d5

4. Extend Λα (α = w, o) constantly and continuously to < and define

Λα,v, Λv, λα,v, λv as  Λα,v(z)≡ Λα 0.5(0.5z−v−v)
, Λv ≡ Λw,v+ Λo,v, λv(z)≡ Λv(J (z)), λα,v(z) = Λα,v(J (z)). (6.7)

By A2-3, there exist smooth functions S

0,v, Sb,v, s0,v such that S0,v , Sb,v,J (s0,v)∈ (d5/2, 1− d5/2), S0,v |Γ2 = Sb,v|Γ2(t = 0), (6.8)  S 0,v, Sb,v, s0,v→ S0, Sb, s0 in L2(0, T ; H1(Ω)), ∂tΥ(Sb,v)→ ∂t(Pb,o− Pb,w) in L1(ΩT), as v→ 0. (6.9)

The regularized problem is: Find{S vXΩ f + s  vXΩ m, P  vXΩ f+ p  vXΩ m} satisfying Φ_∂ tSvXΩ f + φ _∂ tsvXΩ m ∈ dual L 2_{(0, T ;} U), (6.10) v_{≤ S} vXΩ f+J (s  v)XΩ m ≤ 1 − v, (6.11) R(SvXΩ f +J (s  v)XΩ m)− R(S  b), PvXΩ f+ p  vXΩ m− Pb∈ L 2_{(0, T ;} U), (6.12) Z Ω,T_f Φ∂tSvζ + Z Ω,T_f K Λo,v(Sv)∇(Pv− Go) +∇R(Sv)
∇ζ + Z Ω,Tm φ∂tsvζ + Z Ω,Tm k_I2$ λo,v(sv)∇(pv− Go) +∇M(sv)
∇ζ = 0, (6.13) Z Ω,T_f K Λv(Sv)∇(Pv− Go)− Λw,v(Sv)∇(Gw− Go)
∇η + Z Ω,Tm kI2$  λv(sv)∇(pv− Go)− λw,v(sv)∇(Gw− Go)
∇η = 0, (6.14) S vXΩ f(0, x) + s  vXΩ m(0, x) = S  0,vXΩ f + s  0,vXΩ m, (6.15) for any ζ, η_{∈ L}2_{(0, T ;}

U). It is easy to see that (6.13) is a nondegenerate (depending on v) parabolic equation, and (6.13–6.14) imply, if S

w,v≡ 1 − Sv, 0 = Z Ω,T_f Φ∂tSw,v ζ + K Λw,v(1− Sw,v)∇(Pv− Gw)− ∇R(1 − Sw,v )
∇ζ + Z Ω,T m φ_∂ tsw,vζ + kI2$ λw,v(1− sw,v)∇(pv− Gw)− ∇M(1 − sw,v)
∇ζ.(6.16) By [4, 5, 6, 9, 12, 20, 22, 29], it is known

Lemma 6.1 Under (6.8–6.9), there exist functions S

v, Pv in Ωf and sv, pv in Ωm

satisfying (6.10–6.15) for each v,  as well as there exist functions S_{, P}_{, P} α in

Ω

f and s, p, pα in Ωm for α = w, o satisfying (6.1–6.6), (2.3), and (2.6–2.11).

ˇ S v ≡ SvXΩ f +J (s  v)XΩ m is in L

10 Tall Block Models and, as v→ 0, ( _ˇ S v→ ˇS≡ SXΩ f +J (s ₎ XΩ m pointwise, R( ˇS v), PvXΩ f+ p  vXΩ m → R( ˇS _{), P} XΩ f+ p  XΩ m in L 2_{(0, T ; H}1_(Ω)). Moreover, 0 < S_{< 1, s} l< s< sr, and X α=w,o  kpΛα(S)∇PαkL2_(Ω,T f )+kI $  p λα(s)∇pαkL2_(Ω,T m )  +_{k |∇P}_{| + |∇R(S})_{| + |∇A}_{| kL}2_(Ω,T f ) +k |I$  ∇p| + |I$∇M(s)| + |I$∇A| kL2_(Ω,T m )≤ c, where_A ≡  A(Υ(S_{)) if x} ∈ Ω f, A(υ(s_{)) if x} ∈ Ω m.

and c is a constant independent of .

Lemma 6.2 For any β, τ satisfying 2≤ β0≤ β − 2 ∈ N, d_β5₀ ≤ ϑ, and τ ≤ T , the

following inequality holds:

sup t≤τ  {x ∈ Ω : ˇS(t)_{≤ µ or 1 − µ ≤ ˇS}(t)_} ≤ c0|c0τ| β−β0 (β_{− β}0)(β−β0)fβ , (6.17) where µ_≡ d5 2β, lim

β→∞fβ= 1, and c0 is a constant independent of τ, β, , µ.

Proof: Let us defineLµ,Kµ, cKµ as

         Lµ(z)≡  1 if µ_{≤ z ≤ 2µ,} 0 elsewhere,

Kµ(z)≡R_A(Υ(2µ))z Lµ(Υ−1(A−1(ξ)))dξ for z∈ [0, A(∞)),

c Kµ(z)≡R

z

A(Υ(2µ))(LµΛ_Λo)◦ (Υ−1(A−1(ξ)))dξ for z∈ [0, A(∞)).

By 2µ ≤ d5

2 and A2-3,5, we take ζ = Kµ(A) ∈ L2(0, T ;U) in (6.4) and η =

c Kµ(A)∈ L2(0, T ;U) in (6.5) to obtain Z Ω,τ f Φ Kµ(A)∂tS+ Z Ω,τ f K_Λ o(S)Lµ(S)∇Υ(S)∇A + Z Ω,τm φKµ(A)∂ts+ Z Ω,τm kI2$  Λo(u)Lµ(u)∇υ(s)∇A ≤ c1  Z Ω,τ_f KΛo(S)Lµ(S)|∂x3A  | + Z Ω,τm kΛo(u)Lµ(u)|∂x3A  |  , (6.18)

where u_{≡ J (s}_{) and constant c}

1 is independent of , µ. Suppose

Z

Kµ(A) Φ∂tSXΩ,τ_f + φ∂tsXΩ,τm

Tall Block Models 11 (6.18–6.19) imply Z Ω,τ_f KΛo(S)Lµ(S)|∂x3A  | + Z Ω,τm kΛo(u)Lµ(u)|∂x3A  | ≤ c2  Z Ω,τ_f KΛ3/2o Lµ(S) 1 2 Z Ω,τ_f KΛo(S)Lµ(S)∂x3Υ(S _)∂ x3A  1 2 +c2  Z Ω,τm kΛ3/2o Lµ(u) 1 2 Z Ω,τm kΛo(u)Lµ(u)∂x3υ(s _)∂ x3A  1 2 , (6.20)

where constant c2is independent of , µ. A3 and (6.18–6.20) imply

Z Kµ(A) Φ∂tSXΩ,τ f + φ _∂ tsXΩ,τ m
≤ c3 Z Ωτ Λ3/2o Lµ( ˇS). (6.21) Let us define Z(S, s, µ)_≡ ( ΦRS 2µ Kµ(A(Υ(ξ)))dξ in Ω  f, φRs J−1_(2µ)Kµ(A(υ(ξ)))dξ in Ωm. (6.21) implies _Z Ωτ ∂tZ(S, s, µ)≤ c4 Z Ωτ Λ3/2o Lµ( ˇS). (6.22)

(6.22) and A6-7 yield that, if 0≤ t1≤ t2≤ T ,

Z t2 t1 Z Ω ∂tZ(S, s, µ)≤ c4 Z t2 t1 Z ΩZ(S _{, s}_{, 2µ), (6.23)}

where c4 is independent of t1, t2, µ, . Define

F_{(τ, µ)} ≡ _Λ 1 o(µ)3/2 sup t≤τ Z ΩZ(S _{, s}_{, µ).}

A5 and (6.23) imply that, for 0≤ t1≤ t2≤ T ,

F(t2, µ)− F(t1, µ)≤ c5(t2− t1)F(t2, 2µ),

where c5 is independent of t1, t2, µ, . By induction and A3, one obtains, for j ∈

N, jh_{≤ T ,} F_(jh,d5 2β)≤ (β − β0+ 1) j−1_|c 5h|β−β0F(jh, d5 2β0). (6.24) If j = β−β0

log(β−β0) and τ = jh in (6.24), then

F_(τ,d5 2β)≤ |c5τ|β−β0 (β_{− β}0)(β−β0)fβF _(τ, d5 2β0), (6.25)

12 Tall Block Models

where fβ→ 1 as β → ∞. Define B(t) ≡ {x ∈ Ω : ˇS(t, x)≤d₂β5}. (6.25) implies

sup t≤τ Z XB(t)≤ c6F(τ, d5 2β)≤ c6|c5τ|β−β0 (β_{− β}0)(β−β0)fβF _(τ, d5 2β0),

where constant c6 is independent of τ, β, , µ. So proof of first part of (6.17) is

completed. The other part can be proved in a similar way, so we skip it. Lemma 6.3 If r∈ (1, 2), kP αkLr_{(0,T ;W}1,r_(Ω f))+kI $  ∇pαkLr_(Ω,T m )≤ c, where α =

w, o and c is a constant independent of . Moreover, if $_{≤ 1, then kp}

αkLr_(Ω,T m )≤ c.

Proof: We define, for 2_{≤ β}0∈ N,

( B1+β0 ≡ {(t, x) ∈ Ω ,T f : d5 22+β0 ≤ S}, Bβ≡ {(t, x) ∈ Ω,Tf : 2dβ+15 ≤ S< d₂β5} if 2 + β0≤ β ∈ N.

A5, Lemmas 6.1-6.2, and H¨older inequality imply k∇P okrLr_(Ω,T f )≤ k p Λo(S)∇PokrL2_(Ω,T f )kΛ −1 o (S)k r/2 Lr/(2−r)(Ω,T f ) ≤ c1  Z Ω,T_f |Λ o(S)| −r 2−r ∞ X β=1+β0 XBβ 2−r 2 ≤ c2 (indep. of ). (6.26)

Similar argument will give k∇P

wkLr_(Ω,T f )+

P

α=w,okI$∇pαkLr_(Ω,T

m ) ≤ c. By

boundary condition A2, kP αkLr_(Ω,T f ) ≤ c, α = w, o. By Lemma 5.1, (2.8), and $ _{≤ 1, kp} α− ΠPαkLr_(Ω,T m ) ≤ k∂x1(p  α− ΠPα)kLr_(Ω,T m ) ≤ c. So kp  αkLr_(Ω,T m ) is bounded.

Lemma 6.4 For r_{∈ [1, ∞) and sufficiently small δ,} kδ2∂−δS ∂−δ_A_kLr_{((δ,T )×Ω} f)+kδ 2_∂−δ_s _∂−δ_A kLr_{((δ,T )×Ω} m)≤ cδ 1/r_{, (6.27)}

where c is independent of , δ. See§4 for notation ∂−δ_.

Proof: Note ζ(t, x) ≡ Rmax(t,δ)min(t+δ,T )δ ∂−δ A 

− A(Pb,c)
(τ, x)dτ ∈ L2(0, T ;U) by

A2-3 and Lemma 6.1. Take ζ above in (6.2) to get, by Fubini’s theorem, A2, and Lemma 6.1, Z T δ Z Ω f Φδ2∂−δS∂−δ_A(τ, x) + Z T δ Z Ω m φδ2∂−δs∂−δ_A(τ, x) = Z Ω,T_f Φ∂tS(t, x)ζ + Z Ω,Tm φ∂ts(t, x)ζ + Z T δ Z Ω f Φδ2∂−δS∂−δ_A(Pb,c) + Z T δ Z Ω m φδ2∂−δs∂−δ_A(Pb,c)≤ cδ,

where c is independent of , δ. So we prove (6.27) for r = 1 case. (6.27) for r > 1 case follows directly becauseA_{, ˇS}_{are bounded and (6.27) for r = 1 holds.}

Tall Block Models 13

Lemma 6.5 A subsequence of Π(A|Ω

f) converges toA

∗_{in L}2_(ΩT_{) and pointwise.}

Proof: This is due to A6,10, Lemmas 5.1, 6.1-6.4, and compactness principle. Lemma 6.6 s_{, p}_{, p}

α (α = w, o) satisfy, for almost all x∈ ˜Ω,

φ∂ts− ∂y,x3· kI 2$−2  ∂y,x3M(s) + λo(s)∂y,x3(p− Go)

= 0, (6.28) ∂y,x3·  kI2$−2 λ(s)∂y,x3p− X λα(s)∂y,x3Gα
 = 0, (6.29) −φ∂ts− ∂y,x3· kI 2$−2  λw(s)∂y,x3(p  w− Gw)
= 0, (6.30) φ∂ts− ∂y,x3· kI 2$−2  λo(s)∂y,x3(p  o− Go)
= 0, (6.31) in L2_{(0, T ; H}−1_(YH m)). Proof: Let ˆζ_{∈ L}2_{(0, T ; C}∞ 0 (YmH)). For x∈ Ω, y ∈ <2, we define ˇ ζ(t, x, y)_≡  ˆ ζ(t,y−`<sub></sub>(˜x), x3) for y∈ Ym+ `(˜x), 0 elsewhere. Then we plug ζ(t, x)_{≡ X}(Ym+j)(˜x)ˇζ(t, x, ˜x) for j ∈ Z

2 _{into (6.4). Since supp ζ}

⊂ (0, T )_{× (Y}m+ j)× [0, H], Z T 0 Z H 0 Z (Ym+j) φ∂tsζ + kI2$ λo(s)∇(p− Go) +∇M(s)
∇ζ = 0.

Since ˜x_{∈ (Y}m+ j), `(˜x) = j. Changing variable y = x−`˜

 (˜x)  gives Z T 0 Z YH m φ∂tsζ + kˆ I2$−2 ∂y,x3M(s) + λo(s)∂y,x3(p− Go)
∂y,x3ζ = 0,ˆ (6.32)

for almost all ˜x_{∈ (Y}m+ j), j ∈ Z2. Actually, by Definition 5.1, (6.32) holds for

˜

x∈ ˜Ω_{, i.e., (6.28). (6.29–6.31) can be proved in a similar way.}

Remark 6.2 By Lemmas 5.1, 6.5, if we define S

≡ Υ−1_(A−1_(Π (A|Ω f))) and S_≡  Υ−1(A−1_(A∗_{)) if A}∗_<A(∞), 1 if A∗_=A(∞), then 0≤ S, S≤ 1.

Lemma 6.7 There is a r_{∈ (1, 2) and a subsequence of {S}_{, s}_{, S}

0, s0, φ, k, Pα, p α, α = w, o} such that, as  → 0,                            XΩ fP  α 2 *_XYf(y)Pα(t, x) where Pα∈ L r_{(0, T ; W}1,r_{(Ω)), P} α= Pb,α in Γ2, XΩ f∇P  α 2

*_XYf(y)(∇Pα+ ∂yPα,1(t, x, y)) where Pα,1∈ L

r_(ΩT_{; L}r per(Yf)), XΩ fS  0 2 * S0∈ L2(Ω; L2per(Yf)), S

→ S strongly in L2_(ΩT_{) and pointwise,}

XΩ fS  2_{→ X} Yf(y)S(t, x) strongly, XΩ ms  0 2 → s0∈ L2(Ω; L2per(Ym)) strongly, p α* pα weakly in Lr(ΩT; W1,r(Ym)).

14 Tall Block Models

Proof: By Lemma 5.1 and Lemma 6.3, ΠPα is bounded in Lr(0, T ; W1,r(Ω)). So

a subsequence of ΠPα converges weakly to limit Pα ∈ Lr(0, T ; W1,r(Ω)). Since

ΠPα = Pb,αin Γ2, Pα= Pb,α in Γ2. Rest of proof are due to A2-4,6,10, Lemmas

6.1, 6.3, 6.5, and [3].

Lemma 6.8 s _{converges to s in L}2₍

QT

m) if 0 < $≤ 1.

Proof of this lemma is lengthy, and will be postponed untill the last five sections. Lemma 6.9 If $ = 1, then po− pw = υ(s), Po− Pw = Υ(S), and pα(t, x, y) =

Pα(t, x) for x∈ Ω, y ∈ ∂Ym, α = w, o. If $ < 1, then υ(s) = Po− Pw= Υ(S) and

pα(t, x, y) = Pα(t, x) for x∈ Ω, y ∈ Ym, α = w, o.

Proof: First we consider $ = 1 case. Note 0_{≤ S < 1, s}l ≤ s < sr by Egoroff’s

theorem [25] and Lemmas 6.1-6.2, 6.7-6.8. Since p

o− pw= υ(s), we get po− pw=

υ(s) by Lemmas 6.7-6.8. Similarly, one can derive Po−Pw= Υ(S). By Lemmas 5.1,

6.3 and (2.8), (ΠPα)|Ω m−p



α∈ Lr(ΩT; W 1,r

0 (Ym)) for 1 < r < 2. So, a subsequence

of (ΠPα)|Ω m− p



α converges weakly to XYm(y)Pα(t, x)− pα ∈ L

r_(ΩT_{; W}1,r 0 (Ym))

by Lemma 6.7. So, pα(t, x, y) = Pα(t, x) for y ∈ ∂Ym. Results for $ < 1 case can

be obtained by similar argument as above, so we skip it.

Now we consider the limit model of (2.1–2.13) as → 0. Plug into (6.1) and (6.6) a test function η = ˆζ(t, x) + ˆη(t, x,x˜_) where ˆζ _{∈ C}∞

0 (ΩT), ˆη ∈ C0∞(ΩT; Cper∞(Y )) to obtain 0 = Z Ω,T_f ΦS(∂tζ + ∂ˆ tη) + Kˆ Λw(S)∇(Pw − Gw)(∇ˆζ + ∂xη + ∂ˆ yη)ˆ + Z Ω,Tm φs(∂tζ + ∂ˆ tη) + kˆ I2$λw(s)∇(pw− Gw)(∇ˆζ + ∂xη + ∂ˆ yη)ˆ + Z Ω f ΦS0(ˆζ + ˆη)(0) + Z Ω m φs0(ˆζ + ˆη)(0). By A2 and Lemma 6.7, K_Λ w(S) converges to K∗Λw(S) in Lr(ΩT), r <∞ strongly.

Passing to two-scale limit, we get, by A2-4, Lemmas 6.3-6.9, Theorem 2.28 of [2], Theorem 1.8 of [3], and [8, 10], Z QT f Φ∗S∂tζ + Kˆ ∗Λw(S)(∇Pw+ ∂yPw,1− ∇Gw)(∇ˆζ + ∂yη)ˆ =₋ Z QT m φs∂tζ +ˆ Fw∗∂x3ζˆ− Z Qf Φ∗S0ζ(0)ˆ − Z Qm φs0ζ(0),ˆ where Fw∗ ≡ (_kλ w(s)∂x3(Pw− Gw) if 0 < $ < 1, kλw(s)∂x3(pw− Gw) if $ = 1, An L2 _{function if $ > 1.} (6.33) Apply Green’s theorem in t variable to get

− Z

QT f

Tall Block Models 15 = Z QT m φ∂ts ˆζ− Fw∗∂x3ζ +ˆ Z Qf Φ∗(S(0)_{− S}0)ˆζ(0) + Z Qm φ(s(0)_{− s}0)ˆζ(0). So we have, in ΩT_, (S(0)− S0) Z Yf Φ∗dy + Z Ym φ(s(0)− s0)dy = 0, (6.34)

and the choice of ˆη = 0 gives, in ΩT_,

Z Yf Φ∗dy∂tS +∇ Z Yf K∗Λw(S)(∇Pw+ ∂yPw,1− ∇Gw) =− Z Ym (φ∂ts + ∂x3F ∗ w) dy. (6.35)

The choice of ˆζ = 0 gives, by A2-3 and Lemma 6.7, 

∂2

yPw,1= 0 inQf,

(∂x˜Pw+ ∂yPw,1)· ~νy = 0 on ∂Ym, (6.36)

where ~νy is the unit vector outward normal to ∂Ym. Let ~ej be the unit vector in

jth direction. We denote by Ξ the tensor whose (i, j) component is ∂ϕj/∂yi, where

ϕj is a periodic solution in Y of the auxiliary problem



∆yϕj= 0 in Yf,

∂yϕj· ~νy=−~ej· ~νy on ∂Ym.

Pw,1 of (6.36) is given by the product Pw,1=Pjϕj(y)∂xjPw. So (6.35) becomes

Φ∂tS +∇ · (KΛw(S)∇(Pw− Gw)) = −1 |Ym| Z Ym (φ∂ts + ∂x3F ∗ w)dy, (6.37) where Φ_≡ 1 |Ym| R YfΦ

∗_{dy and K is a diagonal matrix satisfying}

K11= K22= K∗ |Ym| Z Yf (I + Ξ(y))dy, K33= |Yf|K ∗ |Ym| .

Proceeding as the proof of (6.37), we obtain, by (6.2), −Φ∂tS +∇ · (KΛo(S)∇(Po− Go)) = 1 |Ym| Z Ym (φ∂ts− ∂x3F ∗ o)dy, (6.38) where Fo∗≡ (_kλ o(s)∂x3(Po− Go) if 0 < $ < 1, kλo(s)∂x3(po− Go) if $ = 1, An L2 _{function if $ > 1.} (6.39) Matrix sources for 0 < $ < 1 case is clear from (6.33), (6.39), and Lemma 6.9. Next we consider the matrix source terms for $_{≥ 1 cases.}

16 Tall Block Models

6.1. For $ = 1 case

By (6.30) of Lemma 6.6, we have, for any η_{∈ L}2_(ΩT_{; H}1 0(Ym)), Z Q,T m φ∂tsη + Z Q,T m kλo(s)∂y,x3(po− Go)∂y,x3η = 0.

As _{→ 0, by Lemmas 6.7-6.8, one obtains} Z QT m φ∂ts η + Z QT m kλo(s)∂y,x3(po− Go)∂y,x3η = 0. (6.40)

In a similar way, we obtain, by (6.31), Z QT m φ∂ts η− Z QT m kλw(s)∂y,x3(pw− Gw)∂y,x3η = 0. (6.41)

By (6.37–6.41) and Lemmas 6.7-6.9, it is easy to show Theorem 4.1. 6.2. For $ > 1 case

By (6.30) of Lemma 6.6, we have, for any η_{∈ L}2_(ΩT_{; H}1 0(Ym)), Z Q,T m φ∂tsη + Z Q,T m k_I2$−2λo(s)∂y,x3(p  o− Go)∂y,x3η = 0.

As → 0, by Lemmas 6.7-6.8, one obtains Z QT m φ∂ts η + Z QT m Fo∗∂x3η = 0.

So we get φ∂ts− ∂x3Fo∗ = 0. In a similar way, we obtain φ∂ts + ∂x3Fw∗ = 0.

Therefore we prove Theorem 4.3.

Rest of this work is to prove Lemma 6.8. 7. Convergence of s Remark 7.3 Define        G ≡ _υ−1( A−1_(Π (A|Ω f))) if ΠA _< A(∞), sr if ΠA=A(∞), G ≡  υ−1_(A−1_(A∗_{)) if A}∗_<A(∞), sr if A∗=A(∞).

See Lemma 6.5 for_A∗_{. By Lemma 6.7, A1,3, Theorem 2.28 of [2], and [3, 8, 10],}

it is easy to see that   

kM(G₎

kL2_{(0,T ;H}1_(Ω))are bounded independently of ,

M(G_| Ω m)→ M(G) strongly in L 2₍ QT m), M(G_| Ω m)− M(s _{)∈ L}2_(ΩT_{; H}1 0(Ym)). (7.1)

Tall Block Models 17

Assume that si_{, p}i_{, i = 1, 2 are two solutions of (6.28–6.29), and ζ, η are smooth}

functions satisfying

ζ(T ) = 0, ζ_|∂Ym×[0,H]= η|∂Ym×[0,H]= ∂x3ζ|x3∈{0,H}= ∂x3η|x3∈{0,H}= 0. (7.2)

Let x_{∈ Ω}1_{∩ Ω}2_{. By subtracting one solution from the other and integration by}

parts, we obtain Z YH,T m (s1− s2) φ∂ tζ +F1∂y,x3(kI 2$−2

 ∂y,x3ζ)− F2∂y,x3ζ− F3∂y,x3η

+ Z YmH,T (p1− p2) ∂ y,x3kI 2$−2  (λ(s1)∂y,x3η + λo(s 1)∂ y,x3ζ)
=_F4+F5, (7.3) where F1≡ µ + _M(s1)−M(s2) s1−s2 if s1 6= s2, 0 otherwise, (7.4) F2≡ ( k(λo(s1)−λo(s2))I2$−2∂y,x3(p2−G2o ) s1−s2 if s1 6= s2, 0 otherwise, (7.5) F3≡ ( P α k(λα(s1)−λα(s2))I2$−2∂y,x3(p2−G2α) s1_−s2 if s1 6= s2, 0 otherwise, (7.6) F4≡ µ Z YmH,T (s1− s2)∂ y,x3(kI 2$−2  ∂y,x3ζ), (7.7) F5≡ Z YmH,T 2$−2∂y  M(G1| Ω1m)− M(G 2| Ω2m)  k∂yζ  + Z YmH,T 2$−2∂y  (Π1P 1| Ω1m − Π2P 2| Ω2m) kλ(s 1)∂ yη + kλo(s1)∂yζ
 − X α∈{w,o} Z YmH,T kλα(s1)∂x3(G 1 α − Gα2)∂x3η − Z YmH,T kλo(s1)∂x3(G 1 o − Go2)∂x3ζ− Z YH m (s1 0 − s02)φζ(0). (7.8) Define eU1 ≡ {ζ : ζ ∈ H1(YmH,T)∩ L∞(0, T ; H1(YmH)), ζ|∂Ym×[0,H] = ∂x3ζ|x3=0,H =

ζ(0) = 0_{}. We consider the following auxiliary problem for fixed µ:}

Lemma 7.1 Let_F2,F3∈ L∞(YmH,T) and 0 < d18<F1< d19<∞. For (f1, f2)∈

L2_(YH,T

m )× L2(YmH,T), there is a unique (ζ, η)∈ eU1× L2(0, T ; H1(YmH)) such that

−φ∂tζ +F1∂y,x3(kI

2$−2

 ∂y,x3ζ)− F2∂y,x3ζ− F3∂y,x3η = f1, (7.9)

∂y,x3 kI 2$−2  (λ∂y,x3η + λo∂y,x3ζ)
= f2. (7.10) Moreover, sup τ≤TkI $−1  ∂y,x3ζ(τ )kL2(YmH)+k|I $−1  ∂y,x3η| + d 1/2 18 |∂y,x3(kI 2$−2  ∂y,x3ζ)|kL2_(YH,T m ) ≤ cd19,k(|F2| + |F3|)/F11/2kL∞(YmH,T)  k |f1|/F11/2+|f2| k_L2_(YH,T m ). (7.11)

18 Tall Block Models

Proof: This is proved by following the argument of Lemma 5.1 [29]. Finally we give the proof of Lemma 6.8.

Proof: For x∈ Ω1_∩Ω2_{, we take f}

1=M(s1)−M(s2) in (7.9) and f₂= p1−p2

in (7.10) to obtain solution (ζµ_{, η}µ_{) for each µ by (7.4–7.6), Remark 7.3, and Lemma}

7.1. After substitution t→ T − t for the solution (ζµ_{, η}µ_{), we plug it into (7.3) to}

obtain Z YmH,T (s1− s2)(M(s1)− M(s2)) + Z YmH,T |p1− p2|2=F 4+F5. (7.12)

By Lemmas 6.1, 7.1 and [13, 15, 26, 27], we see 1)F4 is bounded by cõ, where

c is a constant independent of µ, 1, 2; and 2) For fixed µ, F5 converges to 0 as

1, 2 tend to 0. So it is not difficult to show that M(s2) is a Cauchy sequence in

L2_(QT

m), which implies s2 is a Cauchy sequence in L2(QTm) as well.

Acknowledgement

This research is supported by the grant number NSC 93-2115-M-009-09 from the research program of National Science Council of Taiwan.

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