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III
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VII
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VIII
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IX
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X
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XI
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XII
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XIII
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XIV
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XV
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XVI
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XVII
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𝐿(𝒑, 𝒎) = 𝐿(𝒑(𝒎), 𝒎) = 0

𝐿(𝒑, 𝒎)

34
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𝑁𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑
1
𝐽(𝑝, 𝑚) = ∑ 𝜔𝑖 (𝑑 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 (𝑝) − 𝑑 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 )2
2
𝑖=1

𝐽(𝑝, 𝑚)

𝜔𝑖

𝑑 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 (𝑝)

𝑑 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑

35
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min 𝐽(𝑝, 𝑚) => ∇𝐽(𝑝, 𝑚) = 0

𝑚

36
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38
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40
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𝑑𝐽(𝑚) 𝐽(𝑚 + ∆𝑚) − 𝐽(𝑚 − ∆𝑚)

=
𝑑𝑚 2∆𝑚

𝑑𝐽 𝜕𝐽 𝑑𝑝 𝜕𝐽
= ⋅ +
𝑑𝑚 𝜕𝑝 𝑑𝑚 𝜕𝑚

𝑑𝐽 𝜕𝐽 𝑑𝑝 𝜕𝐽
= ⋅ + =0
𝑑𝑚 𝜕𝑝 𝑑𝑚 𝜕𝑚

𝑑𝑝
𝑑𝑚

41
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𝜕𝐿 𝜕𝐿 𝑑𝑝
+ ⋅ =0
𝜕𝑚 𝜕𝑝 𝑑𝑚

𝜕𝐿 𝑑𝑝 𝜕𝐿
⋅ =−
𝜕𝑝 𝑑𝑚 𝜕𝑚

𝑑𝑝
𝑢=
𝑑𝑚

𝜕𝐿
𝐴=
𝜕𝑝

𝜕𝐿
𝑓=−
𝜕𝑚

42
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𝜕𝐽
𝑔𝑇 =
𝜕𝑝

𝐴⋅𝑢 =𝑓
𝑇
𝑔 ∙𝑢

𝐴⋅𝑢 =𝑓

43
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𝑔𝑇 ∙ 𝑢

𝐴𝑇 ⋅ 𝜆 = 𝑔
𝜆𝑇 ⋅ 𝑓 𝑔𝑇 ⋅ 𝑢 =
(𝐴𝑇 ⋅ 𝜆)𝑇 ⋅ 𝑢 = 𝜆𝑇 ⋅ (𝐴 ⋅ 𝑢) = 𝜆𝑇 ⋅ 𝑓

𝐴𝑇 ⋅ 𝜆 = 𝑔

𝜆𝑇 ⋅ 𝑓

𝑔𝑇 ⋅ 𝑢 = (𝐴𝑇 ⋅ 𝜆)𝑇 ⋅ 𝑢 = 𝜆𝑇 ⋅ (𝐴 ⋅ 𝑢) = 𝜆𝑇 ⋅ 𝑓

𝜕𝐽 𝑑𝑝
𝑔𝑇 ∙ 𝑢 = ⋅
𝜕𝑝 𝑑𝑚
𝜕𝐿
𝜆𝑇 ⋅ 𝑓 = 𝜆𝑇 ⋅ (− )
𝜕𝑚

44
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𝜕𝐿 𝑇 𝜕𝐽 𝑇
( ) ⋅λ=( )
𝜕𝑝 𝜕𝑝

∇𝐽(𝒑, 𝒎)

∇𝐽(𝒑, 𝒎)
𝒙 = (𝒑, 𝒎)

𝒙𝑘+1 = 𝒙𝑘 − 𝛼𝑘 ⋅ ∇J(𝒙𝑘 )

𝛼𝑘

45
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𝛼𝑘

𝛼𝑘 = arg min 𝐽(𝒙𝑘 − 𝛼 ⋅ ∇𝐽(𝒙𝑘 ))

𝛼≥0

∇𝒗 𝐽(𝑥) 𝒗
‖𝒗‖ = 1

𝐽(𝒙 + 𝜀 ⋅ 𝒗) = 𝐽(𝒙) + ∇𝐽𝑇 (𝒙) ⋅ (𝒙 + 𝜀 ⋅ 𝒗 − 𝒙) + 𝑜(𝜀)

= 𝐽(𝒙) + 𝜀 ⋅ ∇𝐽𝑇 (𝒙) ⋅ 𝒗 + 𝑜(𝜀)

𝒗
∇𝐽(𝒙) 𝒗

𝐽(𝒙 + 𝜀 ⋅ 𝒗) − 𝐽(𝒙)
∇𝒗 𝐽(𝑥) = lim = ∇𝐽𝑇 (𝒙) ⋅ 𝒗
𝜀→0 𝜀

46
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max ∇𝒗 𝐽(𝑥) = max ∇𝐽𝑇 (𝒙) ⋅ 𝒗 = ‖∇J(𝒙)‖ ⋅ ‖𝒗‖ ⋅ max cos(∇J(𝒙), 𝒗)

𝒗 𝒗 𝒗
= ‖∇J(𝒙)‖

max cos(∇𝐽(𝒙), 𝒗) = 1
𝒗
𝒗 𝒗 ∥ ∇𝐽(𝒙)

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Sensitivity Explorer (SenEx)

Workflow

Input (BASE CASE)

Simulation

Mismatch (OF) estimation

Acceptable Final Model

match? YES

NO

Sensitivity coefficients calculation

Model parameters modification

Export of the next case

52
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𝑂𝐹 = 𝐽 (𝑝, 𝑚) = ∑ 𝜔𝑖 ⋅ 𝑀𝑖
𝑖

𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 2 𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 2
𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑
𝛼𝑜 (𝑂𝑖,𝑡 − 𝑂𝑖,𝑡 ) 𝛼𝑤 (𝑊𝑖,𝑡 − 𝑊𝑖,𝑡 )
𝑀𝑖 = ∑ [ +
𝛽𝐿,𝑖 𝛽𝐿,𝑖
𝑡
𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 2
𝛼𝑔 (𝐺𝑖,𝑡 − 𝐺𝑖,𝑡 )
+
𝛽𝑔,𝑖
𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑑 2
𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑
𝛼𝑝 (𝑝𝑖,𝑡 − 𝑝𝑖,𝑡 )
+ ]
𝛽𝑝,𝑖

53
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𝑂𝐹

𝛼𝑖 ; 𝜔𝑖

𝑀𝑖

𝛽𝑖

𝑜, 𝑤, 𝑔

𝜕𝐽 (𝑝, 𝑚)
𝜕𝑚

54
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𝐽(𝑝, 𝑚)

𝒙𝑘+1 = 𝒙𝑘 − 𝛼𝑘 ⋅ ∇J(𝒙𝑘 )

𝛼𝑘

∇J(𝒙𝑘 )

55
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𝑥 𝑘+1
𝑥0

56
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𝝓 𝝓

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𝑉𝑝 𝑉𝑏 − 𝑉𝑠
𝜙= =
𝑉𝑏 𝑉𝑏

𝑉𝑝

𝑉𝑏

𝑉𝑠

75
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∆𝑃

𝑘 𝑘 ∆𝑝
𝑞 = −𝐴 ∇𝑝 = −𝐴
𝜇 𝜇 𝐿

77
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𝑞 𝑘
𝑣= = − ∇𝑝
𝐴 𝜇

𝑞 𝑘
𝑣𝑟𝑒𝑎𝑙 = 𝑣/𝜙 = =− ∇𝑝
𝐴𝜙 𝜇𝜙

∇𝑝

𝑣 𝑣𝑟𝑒𝑎𝑙

78
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𝑐𝑚3 [𝑐𝑚2 ][𝐷] [𝑎𝑡𝑚]

[ ]=
𝑠 [𝑐𝑝] [𝑐𝑚]

𝑚3 [𝑚2 ][𝑚2 ] [𝑃𝑎]

[ ]=
𝑠 [𝑃𝑎. 𝑠] [𝑚]

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𝑘 ⋅ 𝑘𝑟𝛼 ∆𝑝𝛼
𝑞𝛼 = ⋅𝐴⋅
𝜇𝛼 ∆𝐿

82
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𝑞𝛼

𝑘𝛼 

𝜇𝛼 

∆𝑝𝛼 

∆𝐿

83
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𝑘𝛼 (𝑆𝛼 )
𝑘𝑟𝛼 =
𝑘

84
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𝑘𝑟𝛼 

𝑘𝛼 (𝑆𝛼 ) 

85
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𝑃𝑐 = 𝑝𝑛𝑤 − 𝑝𝑤

88
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𝑃𝑐

𝑝𝑛𝑤

𝑝𝑤

4𝜎12 cos 𝜃 2𝜎12 cos 𝜃

𝑃𝑐 = =
𝑑 𝑟

𝑃𝑐

𝜎12

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𝑛𝑤
𝑛𝑤 𝑆𝑤 − 𝑆𝑤𝑐
𝑘𝑟𝑤 (𝑆𝑤 ) = 𝑘𝑟𝑤𝑛 ∙ (𝑆𝑤𝑒𝑓𝑓 ) = 𝑘𝑟𝑤𝑛 ∙ ( )
1 − 𝑆𝑜𝑟 − 𝑆𝑤𝑐

94
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1 − 𝑆𝑤 − 𝑆𝑜𝑟 𝑛
𝑘𝑟𝑜 (𝑆𝑤 ) = 𝑘𝑟𝑜𝑛 ∙ (1 − 𝑆𝑤𝑒𝑓𝑓 )𝑛𝑛𝑤 = 𝑘𝑟𝑜𝑛 ∙ ( ) 𝑛𝑤
1 − 𝑆𝑜𝑟 − 𝑆𝑤𝑐

𝑆𝑤

𝑆𝑤𝑒𝑓𝑓

𝑘𝑟𝑤 (𝑆𝑤 )

𝑘𝑟𝑜 (𝑆𝑤 )

𝑘𝑟𝑤𝑛

𝑘𝑟𝑜𝑛

𝑆𝑤𝑐

𝑆𝑜𝑟

𝑛𝑤

𝑛𝑜

95
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96
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𝑃𝑐 𝑘
𝐽(𝑆𝑤 ) = √
𝜎12 cos 𝛩 𝜙

𝐽(𝑆𝑤 )

𝑃𝑐

𝜎12

98
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𝑘
𝜙

99
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𝜙𝑒𝑓𝑓
𝜙𝑛 = ( )
1 − 𝜙𝑒𝑓𝑓

𝑅𝑄𝐼
𝐹𝑍𝐼 =
𝜙𝑛

𝑆𝑤 − 𝑆𝑤𝑐 𝑐𝑤
𝐴𝑤 ∙ (1 − ) + 𝑆𝑤𝑑 ∙ 𝑟𝑖 + 𝑏𝑖
𝑆𝑤𝑑 − 𝑆𝑤𝑐
𝑃𝑐 = 𝑆𝑤 ∙ 𝑟𝑖 + 𝑏𝑖

1 − 𝑆𝑤 − 𝑆𝑜𝑟 𝑐𝑜
𝐴𝑜 ∙ (1 − ) + 𝑆𝑜𝑑 ∙ 𝑟𝑖 + 𝑏𝑖
{ 1 − 𝑆𝑜𝑑 − 𝑆𝑜𝑟

100
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𝑆𝑤𝑐 ≤ 𝑆𝑤 < 𝑆𝑤𝑑

𝑆𝑤𝑑 ≤ 𝑆𝑤 ≤ 𝑆𝑜𝑑

𝑆𝑜𝑑 < 𝑆𝑤 ≤ 1 − 𝑆𝑜𝑟

𝐴𝑤 𝐴𝑜

𝑟𝑖

𝑆𝑤 : 𝑆𝑤𝑐 < 𝑆𝑤𝑑 < 𝑆𝑜𝑑 < 1 − 𝑆𝑜𝑟

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ϕ
ϕ

113
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Rock Type Adjusting History

Matching Workflow

Input
(BASE CASE)

Simulation

Mismatch (OF) estimation

Correction

Acceptable Final Model

match? YES

Validation

Adjustment of
Rock Type NO

Sensitivity coefficients calculation

Model parameters modification

Export of the next case

114
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ϕ
ϕ

ϕ
ϕ

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Rock Type Adjusting History

Matching Workflow

Input
(BASE CASE) Identification of
New SATNUM array
Rock Type regions

Simulation

Mismatch (OF) estimation

Correction

Acceptable Final Model

match? YES

Validation

Adjustment of
Rock Type NO

Sensitivity coefficients calculation

Model parameters modification

Export of the next case

121
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𝑋 2 + 𝑌 2 = 𝑅2

𝑋 2 𝑌2
+ =1
𝑅2 𝑅2

(0,0)

𝑅 = √𝑋 2 + 𝑌 2

122
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𝑙
𝑙 =𝑑⋅𝑅 (0,0) 𝑑=
𝑅

𝑋 2 + 𝑌2 = 𝑙2

𝑋 2 + 𝑌 2 = (𝑑 ⋅ 𝑅)2

𝑋 2 𝑌2
+ = 𝑑2
𝑅2 𝑅2

𝑙
𝑑= 𝑙=𝑑⋅
𝑅
𝑅 (0,0) 𝑑
𝑑 = 1,
𝑅 𝑑<1 𝑑>1
𝑝
𝑅𝑝 (0,0)
𝑅𝑝
𝑑𝑝 =
𝑅

(0,0)

123
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𝑋 2 𝑌2
+ =1
𝑎2 𝑏 2

𝑎 𝑏 𝑎≥
𝑏

𝑙 =𝑑⋅𝑎 𝑚 =𝑑⋅𝑏

𝑋 2 𝑌2
+ =1
𝑙 2 𝑚2

𝑋2 𝑌2
+ =1
(𝑑 ⋅ 𝑎)2 (𝑑 ⋅ 𝑏)2

𝑋 2 𝑌2
+ = 𝑑2
𝑎2 𝑏 2

𝑙 𝑚
𝑑= =
𝑎 𝑏
𝑑
𝑑 = 1,

124
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𝑑<1
𝑑>1
𝑝 𝑎𝑝
𝑏𝑝 (0,0)
𝑎𝑝 𝑏𝑝
𝑑𝑝 = =
𝑎 𝑏

(𝑥𝐶 , 𝑦𝐶 )
𝜃 𝑥
𝑎

2
((𝑥 − 𝑥𝐶 ) ⋅ cos 𝜃 + (𝑦 − 𝑦𝐶 ) ⋅ sin 𝜃)
𝑎2
(−(𝑥 − 𝑥𝐶 ) ⋅ sin 𝜃 + (𝑦 − 𝑦𝐶 ) ⋅ cos 𝜃)2
+ =1
𝑏2

(𝑥, 𝑦) (𝑥𝐶 , 𝑦𝐶 ) 𝜃

𝑥 𝑥𝐶 cos 𝜃 −sin 𝜃 𝑋
(𝑦) = (𝑦 ) + ( )⋅( )
𝐶 sin 𝜃 cos 𝜃 𝑌

𝑥 = 𝑥𝐶 + 𝑋 ⋅ cos 𝜃 − 𝑌 ⋅ sin 𝜃

𝑦 = 𝑦𝐶 + 𝑋 ⋅ sin 𝜃 + 𝑌 ⋅ cos 𝜃

125
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𝑋 = (𝑥 − 𝑥𝐶 ) ⋅ cos 𝜃 + (𝑦 − 𝑦𝐶 ) ⋅ sin 𝜃

𝑌 = −(𝑦 − 𝑥𝐶 ) ⋅ sin 𝜃 + (𝑦 − 𝑦𝐶 ) ⋅ cos 𝜃

2
((𝑥 − 𝑥𝐶 ) ⋅ cos 𝜃 + (𝑦 − 𝑦𝐶 ) ⋅ sin 𝜃)
𝑎2
(−(𝑥 − 𝑥𝐶 ) ⋅ sin 𝜃 + (𝑦 − 𝑦𝐶 ) ⋅ cos 𝜃)2
+ = 𝑑2
𝑏2

(𝑥𝐶 , 𝑦𝐶 ) 𝑎
𝑏 𝑎≥𝑏 𝜃
𝑥
𝑑
𝑝 𝑎𝑝
𝑏𝑝 (𝑥𝐶 , 𝑦𝐶 )
𝜃
𝑎𝑝 𝑏𝑝
𝑑𝑝 = =
𝑎 𝑏

126
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𝑥
𝒙𝑇 ⋅ 𝒙 = ( 𝑥 𝑦) ⋅ ( ) = 𝑥 2 + 𝑦 2 = 𝑅 2
𝑦

𝑥 2 𝑦2
+ = 𝑑2
𝑅2 𝑅2

2 𝑥
𝒙𝑇 ⋅ 𝐴 ⋅ 𝒙 = ( 𝑥 𝑦 ) ⋅ ( 1⁄𝑅 0
) ⋅ (𝑦 ) = 𝑑 2
0 1⁄𝑅 2

𝑥 2 𝑦2
+ = 𝑑2
𝑎2 𝑏 2

2 𝑥
𝒙𝑇 ⋅ 𝐴 ⋅ 𝒙 = ( 𝑥 𝑦) ⋅ (1⁄𝑎 0
2 ) ⋅ (𝑦 ) = 𝑑 2
0 1⁄ 𝑏

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(𝑥𝐶 , 𝑦𝐶 ) 𝜃

2
((𝑥 − 𝑥𝐶 ) ⋅ cos 𝜃 + (𝑦 − 𝑦𝐶 ) ⋅ sin 𝜃)
𝑎2
(−(𝑥 − 𝑥𝐶 ) ⋅ sin 𝜃 + (𝑦 − 𝑦𝐶 ) ⋅ cos 𝜃)2
+ = 𝑑2
𝑏2

𝒙𝑇 ⋅ 𝐴 ⋅ 𝒙 = (𝑥 − 𝑥𝐶 𝑦 − 𝑦𝐶 )
cos 2 𝜃 sin2 𝜃 1 1
+ ( 2 − 2 ) ⋅ cos 𝜃 ⋅ sin 𝜃
⋅ 𝑎2 𝑏2 𝑎 𝑏
1 1 sin2 𝜃 cos 2 𝜃
( − ) ⋅ cos 𝜃 ⋅ sin 𝜃 +
( 𝑎2 𝑏 2 𝑎2 𝑏2 )
𝑥 − 𝑥𝐶
⋅ (𝑦 − 𝑦 ) = 𝑑 2
𝐶

𝒙 𝑇 ⋅ 𝐴 ⋅ 𝒙 = 𝒙𝑇 ⋅ 𝑅 ⋅ 𝐷 ⋅ 𝑅 𝑇 ⋅ 𝒙
2
= (𝑥 − 𝑥𝐶 𝑦 − 𝑦𝐶 ) ⋅ (cos 𝜃 − sin 𝜃) ⋅ (1⁄𝑎 0
)
sin 𝜃 cos 𝜃 0 1⁄𝑏 2
cos 𝜃 sin 𝜃 𝑥 − 𝑥𝐶 2
⋅( ) ⋅ (𝑦 − 𝑦 ) = 𝑑
−sin 𝜃 cos 𝜃 𝐶

cos 𝜃 − sin 𝜃
𝑅=( ) 𝐷=
sin 𝜃 cos 𝜃
2
1⁄𝑎 0
( )
0 1⁄ 𝑏 2

𝐴 𝑅−1 𝑅−𝑇

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𝑅 −1 = 𝑅𝑇 , 𝑅−𝑇 = (𝑅𝑇 )−1 =

(𝑅𝑇 )𝑇 = 𝑅

𝐴 = 𝑅 ⋅ 𝐷 ⋅ 𝑅𝑇 ⟹ 𝑅 −1 ⋅ 𝐴 ⋅ 𝑅−𝑇 = (𝑅 −1 ⋅ 𝑅) ⋅ 𝐷 ⋅ (𝑅𝑇 ⋅ 𝑅 −𝑇 ) = 𝐷

𝐷 = 𝑅𝑇 ⋅ 𝐴 ⋅ 𝑅

ϕ
𝝁𝑅𝑇 ϕ

−1 (𝒙
𝑑𝑀,𝑅𝑇 (𝒙, 𝝁𝑅𝑇 ) = √(𝒙 − 𝝁𝑅𝑇 )𝑇 𝚺RT − 𝝁𝑅𝑇 )

−𝟏
𝚺𝑹𝑻 ϕ

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(𝑥 ′ , 𝑦 ′ )

(𝜎𝑥2′ , 𝜎𝑦2′ )

1/𝜎𝑥2′ 0
̃𝑹𝑻
𝚺 −𝟏
=( 2)
0 1/𝜎𝑦′

2 2
1/𝜎𝑥2′ 0 𝑥′ 𝑥′ 𝑦′
(𝑥 ′ 𝑦′) ⋅ ( ) ⋅ ( ) = ( ) + ( 2
) = 𝑑𝑀,𝑅𝑇
0 1/𝜎𝑦2′ 𝑦′ 𝜎𝑥 ′ 𝜎𝑦 ′

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𝜎
𝑥̅ ~𝑁 (𝜇, )
√𝑛

𝑛
2
1
𝑠 = ⋅ ∑(𝑥𝑖 − 𝑥̅ )2
𝑛−1
𝑖=1

𝜎 𝑠
𝜇 = 𝑥̅ ± 𝑧𝛼/2 ≈ 𝑥̅ ± 𝑡𝛼/2
√𝑛 √𝑛

𝑧𝛼/2 𝑡𝛼/2

𝑧𝛼/2 ≈ 1.96

𝑛 ≥ 30

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𝜒𝑘2

2
𝑛 ≥ 30 𝑇𝑘,𝑛−1

𝑛 < 30

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𝑃 ((𝒙 − 𝝁𝑅𝑇 )𝑇 ⋅ 𝚺RT ⋅ − 𝝁𝑅𝑇 ) ≤ 𝐶𝐷𝐹χ−1

−1 (𝒙
2 (1 − 𝛼 )) = 1 − 𝛼 = 0.95
2

−1
𝑃 ((𝒙 − 𝝁𝑅𝑇 )𝑇 ⋅ 𝚺RT ⋅ (𝒙 − 𝝁𝑅𝑇 ) ≤ 𝐶𝐷𝐹𝑇−1
2 (1 − 𝛼 )) = 1 − 𝛼 = 0.95
2,n−1

ϕ 𝑛>
30 𝒩(0,1)
𝑛 ≤ 30
𝑡𝑛−1

𝛼
𝑃 (𝜇𝑅𝑇 − 𝑡𝑛−1,𝛼 ⋅ 𝜎𝑅𝑇 ≤ 𝑥 ≤ 𝜇𝑅𝑇 + 𝑡𝑛−1,𝛼 ⋅ 𝜎𝑅𝑇 ) = 1 − 2 ⋅ = 0.95
2 2 2

𝛼
𝑃 (𝜇𝑅𝑇 − 𝑧𝛼 ⋅ 𝜎𝑅𝑇 ≤ 𝑥 ≤ 𝜇𝑅𝑇 + 𝑧𝛼 ⋅ 𝜎𝑅𝑇 ) = 1 − 2 ⋅ = 0.95
2 2 2

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2
((𝑥 − 𝑥𝑖 ) ⋅ cos 𝜃𝑖 + (𝑦 − 𝑦𝑖 ) ⋅ sin 𝜃𝑖 )
𝑎𝑖2
(−(𝑥 − 𝑥𝑖 ) ⋅ sin 𝜃𝑖 + (𝑦 − 𝑦𝑖 ) ⋅ cos 𝜃𝑖 )2
+ = 𝑑𝑖2
𝑏𝑖2

(𝑥𝑖 , 𝑦𝑖 ) 𝑖 𝑎𝑖
𝑏𝑖 𝑎𝑖 ≥ 𝑏𝑖 𝜃𝑖
𝑥

𝑑12 = 𝑑22

𝐴 ⋅ 𝑥 2 + 2 ⋅ 𝐵 ⋅ 𝑥 ⋅ 𝑦 + 𝐶 ⋅ 𝑦2 + 2 ⋅ 𝐷 ⋅ 𝑥 + 2 ⋅ 𝐸 ⋅ 𝑦 + 𝐹 = 0

𝑑12 = 𝑑22

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𝑑12 = 𝑑22

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ф~𝒩(𝜇ф , 𝜎𝜙2 )

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log 𝑘~𝒩(𝜇𝑘 , 𝜎𝑘2 )

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RELATIVE PERMEABILITY
krwRT1 kroRT1
0.8
0.7
0.6
kr [fraction]

0.5
0.4
0.3
0.2
0.1
0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Sw [fraction]

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CAPILLARY PRESSURE
PcowRT1 PcowRT2
0.4

0.3

0.3
Pcow [bar]

0.2

0.2

0.1

0.1

0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Sw [fraction]

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kr Pc

ϕ k kr

Pc

kr
ϕ Pc
k Pc
k kr

kr Pc

kr

ϕ Pc

kr Pc

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(22=|{k,𝜙}| − 1) × (22=|{𝑘𝑟 ,𝑝𝑐 }| − 1) = 9

k, 𝜙

𝑘𝑟 , 𝑝𝑐

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𝑅2
𝑓 𝑦

𝑆𝑆𝑟𝑒𝑠
𝑅2 = 1 −
𝑆𝑆𝑡𝑜𝑡

𝑆𝑆𝑟𝑒𝑠
𝐹𝑉𝑈 = 1 − 𝑅2 =
𝑆𝑆𝑡𝑜𝑡

𝐹𝑉𝑈[%] = 100 ⋅ 𝐹𝑉𝑈

𝑆𝑆𝑟𝑒𝑠 = ∑(𝑦𝑖 − 𝑓𝑖 )2
𝑖

𝑆𝑆𝑡𝑜𝑡 = ∑(𝑦𝑖 − 𝑦̅)2

𝑖

𝑆𝑆𝑟𝑒𝑠 𝑆𝑆𝑡𝑜𝑡

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∑𝑖 (𝜙𝑖𝑡𝑟𝑢𝑒 − 𝜙𝑖 )2
𝐹𝑉𝑈𝜙 [%] = 100 ⋅
∑𝑖(𝜙𝑖𝑡𝑟𝑢𝑒 − 𝜙̅𝑡𝑟𝑢𝑒 )2

∑𝑖(𝑘𝑖𝑡𝑟𝑢𝑒 − 𝑘𝑖 )2
𝐹𝑉𝑈𝑘 [%] = 100 ⋅ 2
∑𝑖(𝑘𝑖𝑡𝑟𝑢𝑒 − 𝑘̅ 𝑡𝑟𝑢𝑒 )

∑𝑖 (log10 𝑘 𝑡𝑟𝑢𝑒
𝑖 − log10 𝑘 𝑖 )2
𝐹𝑉𝑈log10 𝑘 [%] = 100 ⋅ 2
∑𝑖(log10 𝑘 𝑡𝑟𝑢𝑒
𝑖 − ̅̅̅̅̅̅̅̅̅
log10 𝑘 𝑡𝑟𝑢𝑒 )

𝜙 𝑖 , 𝑘𝑖
𝜙𝑖𝑡𝑟𝑢𝑒 , 𝑘𝑖𝑡𝑟𝑢𝑒

𝐹𝑉𝑈 𝜙, 𝑘

∑𝑖 (𝜙𝑖𝑡𝑟𝑢𝑒 − 𝜙𝑖 )2 + ∑𝑖 (𝑘𝑖𝑡𝑟𝑢𝑒 − 𝑘𝑖 )2
𝐹𝑉𝑈𝜙,𝑘 [%] = 100 ⋅ 2
∑𝑖(𝜙𝑖𝑡𝑟𝑢𝑒 − 𝜙̅𝑡𝑟𝑢𝑒 )2 + ∑𝑖(𝑘𝑖𝑡𝑟𝑢𝑒 − 𝑘̅ 𝑡𝑟𝑢𝑒 )

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𝐹𝑉𝑈𝜙,log10 𝑘 [%] = 100

∑𝑖(𝜙𝑖𝑡𝑟𝑢𝑒 − 𝜙𝑖 )2 + ∑𝑖(log10 𝑘 𝑡𝑟𝑢𝑒
𝑖 − log10 𝑘 𝑖 )2
⋅ 2
∑𝑖 (𝜙𝑖𝑡𝑟𝑢𝑒 − 𝜙̅𝑡𝑟𝑢𝑒 )2 + ∑𝑖(log10 𝑘 𝑡𝑟𝑢𝑒
𝑖 − ̅̅̅̅̅̅̅̅̅
log10 𝑘 𝑡𝑟𝑢𝑒 )

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𝑛𝐹𝐼𝑁𝐴𝐿
𝑉𝑅𝑇 [%] = ⋅ 100
𝑛𝐵𝐴𝑆𝐸

𝑛𝐹𝐼𝑁𝐴𝐿

𝑛𝐵𝐴𝑆𝐸

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𝑛𝑣𝑎𝑙𝑖𝑑
𝑉𝑀𝐷 [%] = ⋅ 100
𝑛𝑡𝑜𝑡𝑎𝑙

𝑛𝑣𝑎𝑙𝑖𝑑

𝑛𝑡𝑜𝑡𝑎𝑙

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𝑘𝑒𝑓𝑓 = 𝑘 ⋅ 𝑘𝑟

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