油气藏数值模拟
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•基本流动方程—多相流 •两相非混相流体
油相基本流动方程：
•o
k kro uo
po
oD
t
o
So
q~o
水相基本流动方程：
•w
kkrw uw
pw
wD
t
wSw
q~w
辅助方程
So Sw 1 Pc(Sw) Po Pw f (Sw)
数值模型的建立：微分方程的离散化 时间与空间离散
导数的差商逼近
production
2. Propane K-value adjusted to match bitumen-
propane saturation data. K pure C3 = 1.00 ; K
adjusted = 1.44
3. Heavy oil data chosen as average data for
b21
a21
c21
e11 e21
p1 f11
p2
f
21
b31 a31 c31 b41 a41 c41
e31 e41
p3 p4
f f
31 41
d12
a12 c12
e12
p5 f12
d22
b22 a22 c22
e22
p6
f
22
d32
b32 a32 c32
(i-1 , j)
(i, j)
y
(i+ 1 , j)
(i, j-1 )
x
F a c e j- 1 /2
计算传导率（ Transmissibility）
Consists of two parts Geometric part (flow area) Saturation and pressure dependent part (mobility)
j
二维自然排列时五对角矩阵结构示意图
Benefits of Simulation
Field management strategies Support geological & engineering analysis Add reserves Identify remedial measures & opportunities
Recommends work over for wells Each well goes from 4 BOPD to 400 BOPD Field now matches simulator forecast
Cut Away of Reservoir - See Inside
Oil is Red
( u ) x x
(
u)
|x1x 2
u(x
x ) x
u(
x)
|x1 2
x
u(x) u(x x) x
x x
1 (x x)
2
油油藏藏
有效 网格
网格系统
(i,j)
b
。
。。 a
(i,j)
无效网格
•有限差分方程的建立 空间离散
考虑二维椭圆型微分方程：
x
(x
p ) x
y
(y
p y
)
百度文库
q(x,
y,
L
FLOW IN
FLOW OUT
x
CONTROL VOLUME
单相流体基本渗流微分方程
x
Vx
y
Vy
z
Vz
t
q~
•V q~
t
三维空间单相渗流的达西定律
Vx
K u
p x
D x
Vy
K u
p y
D y
Vy
K u
p y
D y
V k p D
•V
q~
t
• k pD q~
t
多相多组分渗流基本微分方程
70% of oil in the reservoir left behind Reservoir Simulation
“see” inside reservoirs Low risk find trapped oil/gas learn where to drill decide best exploitation Helps produce more oil/gas High return and yield
Design of processes Transfer of of laboratory process to the field Design of field pilots Design & optimization of new field operations
Correct Mistakes
Waterflood implemented - little response Engineer believes wells are damaged, models field, gets waterflood forecast
Support Geological & Engineering Analysis
数值模拟的基本过程
建立数学模型
建立一套描述油藏流体渗流的偏微分方程组 。完整的数学模型包括定解条件（初始条件 和边界条件）。
建立数值模型
偏微分 方程组
非线性有限 差分方程组
线性代数 方程组
离散化
线性化
建立计算机模型 将各种数学模型的计算方法编制成计
算机程序，用计算机计算各种结果。
渗流数学模型的建立
Pi
Pi1
Q VAQ Ti1/ 2 (Pi Pi1) where
Ti1/ 2
ZiYi
Zi1Yi1 2
2KiKi1
xiKi1 xi1Ki
dij pij1 bij pi1 j aij pij cij pi1 j eij pij1 fij
矩阵AP = f
wi
a11 c11
Use harmonic average for permeability in series
i
i+1
KiPi Pi1/ 2 Ki1Pi1/ 2 Pi1
xi
xi+1
1/ 2xi
1/ 2xi1
计算传导率（ Transmissibility）
Solve for Pi+1/2
V
xi
2Ki Ki1 Ki1 xi1 Ki
五点差分格式
i,j+1
i-1,j
i,j
i+1,j
i,j-1
Apply material balance equation for oil and water for each gridblock
F a c e i-1 /2
(i, j+ 1 )
F a c e j+ 1 /2 F a c e i+ 1 /2
Only way that simulation will match performance of primary & waterflood is to use well damage factor in the simulator
Support Geological & Engineering Analysis
Water is Blue
Production decisions are more effective with simulation
History Match for Well, 2-14 (Water Cut & GOR)
Initial Sw
Year 12
Initial Sw
Waterflood Year 12
5. Straight lines rel perms
6. Temperature effects included
7. Diffusion of C3 into Oil D = .12 cm2/h
Additional Concepts
It is less expensive to add hydrocarbons from existing reservoirs than to find new ones. Simulation can increase recovery by at least 5% over the short term and even more over the longer term. Simulation provides a tool to evaluate field management strategies.
油藏流体及其与岩石作用的复杂性
岩石非均质性及孔隙结构（孔道大小、孔隙之间关系） 油、气和水的组分各不相同，各组分间存在相间传质， 流体性质随温度、压力的变化很大 油层中的流体与岩石相互作用产生物理化学现象如扩散 、吸附等 各种提高采收率方法的使用，如热力采油、化学驱、混 相驱以及各种增产措施如酸化等
produced oil:
- Molecular mass = 568 g/gmol
- Viscosity
= 25,000 cp
- Density
= .9798 g/cm3
- Porosity
= .35
- Permeability = 5000 md
4. Heavy oil not volatile
p)
0
y
x
re s e rv o ir's b o u n d a ry
5
4
3
y
j 2
1
1234567
x
i
P ro d u c tio n w e ll In je c t io n w e ll
Aerial View of a Reservoir with Superimposed Grid
•空间离散
油藏研究方法
直接观察法
如钻观察井、井下测试、井下电视、岩心实验 、开辟生产实验区等)
模拟法
模拟法
模拟法
物模
数模
数学模 拟
数学模型求解方法
解析法
数值方法
数值模拟
油气藏数值模拟
—科学开发油气田的关键技术
利用计算机模型模拟仿真油气复杂开发过程，动态重现 开发历史，预测未来开发动态，可在计算机上“多次” 模拟开发过程，进行油田开发方案优选、产量和地层压 力动态预报、寻找开发中后期剩余油分布和采收率预测 、经济效益预测以及对整个油田开发的重大问题进行决 策的一门有效的工具。
Benefits of Simulation
Increase production Improve operational efficiency Prevent mistakes Correct mistakes Negotiations & asset trades
Prevent Mistakes
一阶导数的 差商逼近
考察函数u(x), 其自变量的一阶导数可定义为 下面的各种极限：
u
u(x x) u(x)
lim
x x0
x
u u(x) u(x x)
lim
x x0
x
u lim u(x x) u(x x)
x x0
2x
前差商 后差商 中心差商
二阶偏导数的差商逼近
混合二阶 差商逼近
e32 p7 f32
d42
b42 a42
e42
p8
f
42
d13
a13 c13
p9 f13
d23
b23 a23 c23
p10
f 23
d33
b33 a33 c33 p11 f33
d43
b43
a43
p12
f 43
dij bij aij ci
eij
实际上就是通过渗流微分方程方程，借用大型计算机， 计算数学的求解，结合油藏地质、油藏描述、油藏工程 、试井等学科再现油田开发的过程，由此来解决油田实 际问题。
Why Simulation ?
Evaluating Reservoir Management Strategies
Simulation: Why Oil Companies Buy
New well drilled without simulation produces only water Simulation determines what can be done - success or abandon Saves field, saves operator
Support Geological & Engineering Analysis
VAPEX – ATHABASCA OIL – LAB
ASSUMPTIONS:
1. The two step process: asphaltene
precipitation and heavy oil production, is
modeled as one step process: heavy oil
Large Indonesian field 4km x 8km Eight sands, faulted, several hundred feet thick 60 years of history Severe loss of production
Support Geological & Engineering Analysis
Example Cases
Light Oil - Ecuador
Waterflood operation Identify poorly swept areas Value of additional oil $30 - 40 MM US Cost of study $250 M US Profitability ratio 140:1
油相基本流动方程：
•o
k kro uo
po
oD
t
o
So
q~o
水相基本流动方程：
•w
kkrw uw
pw
wD
t
wSw
q~w
辅助方程
So Sw 1 Pc(Sw) Po Pw f (Sw)
数值模型的建立：微分方程的离散化 时间与空间离散
导数的差商逼近
production
2. Propane K-value adjusted to match bitumen-
propane saturation data. K pure C3 = 1.00 ; K
adjusted = 1.44
3. Heavy oil data chosen as average data for
b21
a21
c21
e11 e21
p1 f11
p2
f
21
b31 a31 c31 b41 a41 c41
e31 e41
p3 p4
f f
31 41
d12
a12 c12
e12
p5 f12
d22
b22 a22 c22
e22
p6
f
22
d32
b32 a32 c32
(i-1 , j)
(i, j)
y
(i+ 1 , j)
(i, j-1 )
x
F a c e j- 1 /2
计算传导率（ Transmissibility）
Consists of two parts Geometric part (flow area) Saturation and pressure dependent part (mobility)
j
二维自然排列时五对角矩阵结构示意图
Benefits of Simulation
Field management strategies Support geological & engineering analysis Add reserves Identify remedial measures & opportunities
Recommends work over for wells Each well goes from 4 BOPD to 400 BOPD Field now matches simulator forecast
Cut Away of Reservoir - See Inside
Oil is Red
( u ) x x
(
u)
|x1x 2
u(x
x ) x
u(
x)
|x1 2
x
u(x) u(x x) x
x x
1 (x x)
2
油油藏藏
有效 网格
网格系统
(i,j)
b
。
。。 a
(i,j)
无效网格
•有限差分方程的建立 空间离散
考虑二维椭圆型微分方程：
x
(x
p ) x
y
(y
p y
)
百度文库
q(x,
y,
L
FLOW IN
FLOW OUT
x
CONTROL VOLUME
单相流体基本渗流微分方程
x
Vx
y
Vy
z
Vz
t
q~
•V q~
t
三维空间单相渗流的达西定律
Vx
K u
p x
D x
Vy
K u
p y
D y
Vy
K u
p y
D y
V k p D
•V
q~
t
• k pD q~
t
多相多组分渗流基本微分方程
70% of oil in the reservoir left behind Reservoir Simulation
“see” inside reservoirs Low risk find trapped oil/gas learn where to drill decide best exploitation Helps produce more oil/gas High return and yield
Design of processes Transfer of of laboratory process to the field Design of field pilots Design & optimization of new field operations
Correct Mistakes
Waterflood implemented - little response Engineer believes wells are damaged, models field, gets waterflood forecast
Support Geological & Engineering Analysis
数值模拟的基本过程
建立数学模型
建立一套描述油藏流体渗流的偏微分方程组 。完整的数学模型包括定解条件（初始条件 和边界条件）。
建立数值模型
偏微分 方程组
非线性有限 差分方程组
线性代数 方程组
离散化
线性化
建立计算机模型 将各种数学模型的计算方法编制成计
算机程序，用计算机计算各种结果。
渗流数学模型的建立
Pi
Pi1
Q VAQ Ti1/ 2 (Pi Pi1) where
Ti1/ 2
ZiYi
Zi1Yi1 2
2KiKi1
xiKi1 xi1Ki
dij pij1 bij pi1 j aij pij cij pi1 j eij pij1 fij
矩阵AP = f
wi
a11 c11
Use harmonic average for permeability in series
i
i+1
KiPi Pi1/ 2 Ki1Pi1/ 2 Pi1
xi
xi+1
1/ 2xi
1/ 2xi1
计算传导率（ Transmissibility）
Solve for Pi+1/2
V
xi
2Ki Ki1 Ki1 xi1 Ki
五点差分格式
i,j+1
i-1,j
i,j
i+1,j
i,j-1
Apply material balance equation for oil and water for each gridblock
F a c e i-1 /2
(i, j+ 1 )
F a c e j+ 1 /2 F a c e i+ 1 /2
Only way that simulation will match performance of primary & waterflood is to use well damage factor in the simulator
Support Geological & Engineering Analysis
Water is Blue
Production decisions are more effective with simulation
History Match for Well, 2-14 (Water Cut & GOR)
Initial Sw
Year 12
Initial Sw
Waterflood Year 12
5. Straight lines rel perms
6. Temperature effects included
7. Diffusion of C3 into Oil D = .12 cm2/h
Additional Concepts
It is less expensive to add hydrocarbons from existing reservoirs than to find new ones. Simulation can increase recovery by at least 5% over the short term and even more over the longer term. Simulation provides a tool to evaluate field management strategies.
油藏流体及其与岩石作用的复杂性
岩石非均质性及孔隙结构（孔道大小、孔隙之间关系） 油、气和水的组分各不相同，各组分间存在相间传质， 流体性质随温度、压力的变化很大 油层中的流体与岩石相互作用产生物理化学现象如扩散 、吸附等 各种提高采收率方法的使用，如热力采油、化学驱、混 相驱以及各种增产措施如酸化等
produced oil:
- Molecular mass = 568 g/gmol
- Viscosity
= 25,000 cp
- Density
= .9798 g/cm3
- Porosity
= .35
- Permeability = 5000 md
4. Heavy oil not volatile
p)
0
y
x
re s e rv o ir's b o u n d a ry
5
4
3
y
j 2
1
1234567
x
i
P ro d u c tio n w e ll In je c t io n w e ll
Aerial View of a Reservoir with Superimposed Grid
•空间离散
油藏研究方法
直接观察法
如钻观察井、井下测试、井下电视、岩心实验 、开辟生产实验区等)
模拟法
模拟法
模拟法
物模
数模
数学模 拟
数学模型求解方法
解析法
数值方法
数值模拟
油气藏数值模拟
—科学开发油气田的关键技术
利用计算机模型模拟仿真油气复杂开发过程，动态重现 开发历史，预测未来开发动态，可在计算机上“多次” 模拟开发过程，进行油田开发方案优选、产量和地层压 力动态预报、寻找开发中后期剩余油分布和采收率预测 、经济效益预测以及对整个油田开发的重大问题进行决 策的一门有效的工具。
Benefits of Simulation
Increase production Improve operational efficiency Prevent mistakes Correct mistakes Negotiations & asset trades
Prevent Mistakes
一阶导数的 差商逼近
考察函数u(x), 其自变量的一阶导数可定义为 下面的各种极限：
u
u(x x) u(x)
lim
x x0
x
u u(x) u(x x)
lim
x x0
x
u lim u(x x) u(x x)
x x0
2x
前差商 后差商 中心差商
二阶偏导数的差商逼近
混合二阶 差商逼近
e32 p7 f32
d42
b42 a42
e42
p8
f
42
d13
a13 c13
p9 f13
d23
b23 a23 c23
p10
f 23
d33
b33 a33 c33 p11 f33
d43
b43
a43
p12
f 43
dij bij aij ci
eij
实际上就是通过渗流微分方程方程，借用大型计算机， 计算数学的求解，结合油藏地质、油藏描述、油藏工程 、试井等学科再现油田开发的过程，由此来解决油田实 际问题。
Why Simulation ?
Evaluating Reservoir Management Strategies
Simulation: Why Oil Companies Buy
New well drilled without simulation produces only water Simulation determines what can be done - success or abandon Saves field, saves operator
Support Geological & Engineering Analysis
VAPEX – ATHABASCA OIL – LAB
ASSUMPTIONS:
1. The two step process: asphaltene
precipitation and heavy oil production, is
modeled as one step process: heavy oil
Large Indonesian field 4km x 8km Eight sands, faulted, several hundred feet thick 60 years of history Severe loss of production
Support Geological & Engineering Analysis
Example Cases
Light Oil - Ecuador
Waterflood operation Identify poorly swept areas Value of additional oil $30 - 40 MM US Cost of study $250 M US Profitability ratio 140:1