1 2 Equations EM
August 8, 2017 | Author: Hassan | Category: N/A
Short Description
1 2 Equations EM...
Description
Reservoir Simulation - Equations
Data review
• Why run a flow simulation ? • Mathematical & Numerical considerations • ECLIPSE Reminder
Introduction
Etienne MOREAU
History matching
Space & Time Discretisation Reservoir description Fluid description Initialisation Aquifer & Well representation Flow description
Production Forecast
• • • • • •
Outline
2
© 2010 - IFP Training
EP - Reservoir Simulation - Equations - E.M.
Mathematical Considerations General Overview
Example 2: Transport Equation
− Pore & Fluid compressibility − Reservoir permeability & porosity
• Hypothesis: One phase flow, no gravity, low compressibility • Main unknown : Pressure vs space & time • Main parameters
Example 1: Diffusivity Equation
Mathematical & Numerical considerations
3
© 2010 - IFP Training
• Hypothesis: Two phase incompressible flow • Main unknown : Saturation vs space & time • Main parameters: − Filtration Velocity − Reservoir Porosity − Fractional flow (fluids’ relative mobility, fluids’ density, Relative permeability & capillary pressure)
4
© 2010 - IFP Training
EP - Reservoir Simulation - Equations - E.M.
Diffusivity Equation: Main Hypothesis & Basic Laws Hypothesis
X
c = Fluid compressibility ρ = Fluid Density P = Fluid Pressure
X=L
• Flow property: One phase flow, no gravity effect • Fluid behaviour: Slightly compressible fluid
X=0
Basic Equations (Fluid behaviour)
1 dρ c= = Cte ρ dP
5
© 2010 - IFP Training
Q(x) = Flow Rate along the flow line k = Reservoir Permeability A = Section opened to flow µ = Fluid Viscosity P = Fluid Pressure along the flow line x = Distance along the flow line
t = Time
ρ = Fluid Density x = Distance along the flow line
φ = Reservoir Porosity
Basic Equations (Material Balance)
k A dP Q(x) = − × µ dx
Basic Equations (Flow Equation)
Diffusivity Equation: Main Hypothesis & Basic Laws
EP - Reservoir Simulation - Equations - E.M.
ρ {Q(x ) − Q(x + dx )} dt = d(ρ A φ dx )
EP - Reservoir Simulation - Equations - E.M.
© 2010 - IFP Training
6
Flow Term k A ∂ 2P dx dt µ ∂x 2
Diffusivity Equation: Material Balance Equation
ρ {Q(x ) − Q(x + dx )} dt = ρ
Accumulation Term
dt
) dP A dx dt
dρ dφ + φ A dx dt d ( ρ φ A dx ) = ρ dt dt dP dρ dρ dP dP dφ dφ dP = × = φ cp = × = ρ cf ; dt dt dP dt dt dt dP dt d ( ρ φ A dx ) = ρ φ ( c p + c f
Diffusivity Equation k ∂ 2P ∂P = φ(c p + c f ) µ ∂x 2 ∂t
7
© 2010 - IFP Training
X
ρ o = Oil Density
X=L
ρ o = Cte
ρ w = Water Density
Basic Equations (Fluid behaviour)
X=0
• Flow Geometry: Two phase flow, Constant total rate • Fluid behaviour: Incompressible fluids
Hypothesis
Transport Equation: Main Hypothesis & Basic Laws
EP - Reservoir Simulation - Equations - E.M.
ρ w = Cte
8
© 2010 - IFP Training
EP - Reservoir Simulation - Equations - E.M.
f o (x ) = Oil Fractional flow
f w (x ) = Water Fractional flow
Q w (x ) = Water Flow
Q = Total Flow Rate = Cte
Transport Equation: Main Hypothesis & Basic Laws Basic Equations (Flow Equation)
Q (x ) f (x ) = w w Q f o (x ) = 1 − f w (x )
φ = Reservoir Porosity ρ w , Sw = Water Density & Saturation
A = Section opened to total flow
Q w (x) = Water Flow Rate
Basic Equations (Water Material Balance) ρ w {Q w (x ) − Q w (x + dx )} dt = ρ w A d(Sw φ dx ) x = Distance along the flow line t = Time
© 2010 - IFP Training
9
Transport Equation: Material Balance Equation
Transport Equation
w
w
∂Sw dx dt ∂t
∂f w ∂f w ∂Sw = × ∂x ∂Sw ∂x
ρ w d(A φ Sw dx ) = ρ w A φ
Accumulation Term
w
Flow Term ∂f ρ {Q (x ) − Q (x + dx )} dt = − ρ Q w dx dt ∂x w
EP - Reservoir Simulation - Equations - E.M.
∂Sw Q ∂f w ∂Sw × + φ =0 ∂t A ∂Sw ∂x
10
© 2010 - IFP Training
EP - Reservoir Simulation - Equations - E.M.
Mathematical Considerations Diffusivity Equation
Mathematical Expression (1D flow)
• 1D horizontal flow, Slightly compressible fluid
Hypothesis
Diffusivity Equation: Mathematical Properties (1/3)
11
© 2010 - IFP Training
∂ 2 P φµc ∂P − =0 ∂x 2 k ∂t φ, k = Reservoir Porosity & Permeability
;
k K= = Hydraulic Diffusivity φµc
----------
µ = Fluid Viscosity , c = total Compressibility (pores + fluid)
∂ 2 P ∂P K − =0 ∂x 2 ∂t
12
© 2010 - IFP Training
EP - Reservoir Simulation - Equations - E.M.
Diffusivity Equation: Mathematical Properties (2/3) 2D Flow (rectangular coordinates) k ∂ 2 P ∂ 2 P ∂P + =0 − φ µ c ∂x 2 ∂y 2 ∂t
2D Radial flow (radial circular coordinates)
k ∂ 2 P 1 ∂P ∂P + − = 0 φ µ c ∂r 2 r ∂r ∂t
13
© 2010 - IFP Training
Steady State Solution ∂P =0 ∂t
∂P = Cte ∂t
Transient Solution ∂P ≠ Cte ∂t
∂ 2P = Cte ∂x 2
∂ 2P =0 ∂x 2
K
∂ 2 P ∂P = ∂x 2 ∂t
K
⇒
K
In any case Solutions of the diffusivity equation depend on
⇒
Semi Steady State Solution
⇒
Diffusivity Equation: Mathematical Properties (3/3)
EP - Reservoir Simulation - Equations - E.M.
• Initial conditions • Boundary conditions
14
© 2010 - IFP Training
EP - Reservoir Simulation - Equations - E.M.
P (x, t ) = a + b x
Diffusivity Equation: Steady State Solution (1/3)
⇒
; ;
∂P (0, t ) = b ∂x ∂P (1, t ) = b ∂x
Pressure versus space & time ∂P ∂ 2P =K =0 ∂t ∂x 2
Boundary Conditions P (0, t ) = a P (1, t ) = a + b
EP - Reservoir Simulation - Equations - E.M.
© 2010 - IFP Training
15
Diffusivity Equation: Steady State Solution (2/3)
EP - Reservoir Simulation - Equations - E.M.
© 2010 - IFP Training
16
Diffusivity Equation: Steady State Solution (3/3)
17
© 2010 - IFP Training
Boundary Conditions
;
;
P (1, t ) = a + b +
∂P c ( 1, t ) = b + ∂x K c +ct 2K
∂P ∂ 2P ∂P c =K 2 =c ⇒ = b + x + f (t ) ∂t ∂x ∂x K c 2 P (x, t ) = a + b x + x +ct 2K
Pressure versus space & time
Diffusivity Equation: Semi-Steady State Flow (1/3)
EP - Reservoir Simulation - Equations - E.M.
∂P (0, t ) = b ∂x P (0, t ) = a + c t
EP - Reservoir Simulation - Equations - E.M.
© 2010 - IFP Training
18
Diffusivity Equation: Steady State Solution (2/3)
EP - Reservoir Simulation - Equations - E.M.
© 2010 - IFP Training
19
Diffusivity Equation: Steady State Solution (3/3)
EP - Reservoir Simulation - Equations - E.M.
© 2010 - IFP Training
20
Two examples are considered
Diffusivity Equation: Transient Solution (1/5)
Example 1
Example 2
Boundary conditions
∂P ( 0, t ) = Cte ; P(L, t ) = Pi ∂x
• Initial Pressure Constant • Inflow & Outlet Pressure constant with time Initial Condition : P(x,0) = Pi 0 < x < L
• Initial Pressure Constant • Inlet and outlet Pressure constant with time
Initial Condition : P(x,0) = Pi 0 < x < L Boundary conditions P(0, t ) = Pi + 1 ; P(L, t ) = Pi
21
© 2010 - IFP Training
EP - Reservoir Simulation - Equations - E.M.
Diffusivity Equation: Transient Solution (2/5)
EP - Reservoir Simulation - Equations - E.M.
© 2010 - IFP Training
22
Diffusivity Equation: Transient Solution (3/5)
EP - Reservoir Simulation - Equations - E.M.
© 2010 - IFP Training
23
Diffusivity Equation: Transient Solution (4/5)
EP - Reservoir Simulation - Equations - E.M.
© 2010 - IFP Training
24
Diffusivity Equation: Transient Solution (5/5)
EP - Reservoir Simulation - Equations - E.M.
25
© 2010 - IFP Training
Mathematical Considerations Transport Equation
© 2010 - IFP Training
26
Transport Equation: Mathematical properties (1/2) Flow Equations
w
∂z ∂P U = − k M × o −ρ g o o o ∂x ∂x ∂P ∂Pc o, w ∂z U = − k M × o − −ρ g w ∂x ∂x ∂x w
Fractional Flow
∂Pc M k ∂z w, o w fw = + Mw + (ρw − ρo ) g Mw + Mo U ∂x ∂x
27
© 2010 - IFP Training
Iso Saturation Equation S (x, t ) = Cte w
Iso Saturation velocity
⇒
∂Sw U df w = ∂x φ dSw
∂Sw ∂S dx + w dt = 0 ∂x ∂t
∂f ∂S ∂S U w × w = φ× w ∂Sw ∂x ∂t
Transport Equation (1D)
Transport Equation: Mathematical properties (2/2)
EP - Reservoir Simulation - Equations - E.M.
dx ∂Sw =− ∂t dt
28
© 2010 - IFP Training
EP - Reservoir Simulation - Equations - E.M.
Sw 1
0 0
EP - Reservoir Simulation - Equations - E.M.
Transport Equation: Solution Examples
x
With Pc
Without Pc
Saturation front
29
© 2010 - IFP Training
Mathematical Considerations General Equations
© 2010 - IFP Training
30
General Equations: Black Oil Model
31
© 2010 - IFP Training
32
General Equations: Black Oil Model
S o = 1 − S w − Sg
Pg = Po + Pc g,o (Sg )
Pw = Po − Pc w,o (Sw )
© 2010 - IFP Training
Water Pressure
Gas Pressure
Oil Saturation
EP - Reservoir Simulation - Equations - E.M.
EP - Reservoir Simulation - Equations - E.M.
General Equations: Black Oil Model
ab o
= ∆m oa
Material Balance Equation (Oil)
b
∑Q
ab w
= ∆m aw
Material Balance Equation (Water)
∑Q b
+ Q g,abd ) = ∆m ga + ∆m g,a d
Material Balance Equation (Gas) ab g
© 2010 - IFP Training
33
© 2010 - IFP Training
34
General Equations: Compositional Model
∑ (Q b
EP - Reservoir Simulation - Equations - E.M.
EP - Reservoir Simulation - Equations - E.M.
Oil Saturation
Gas Pressure
Water Pressure
n
i =1
p
35
General Equations: Compositional Model
S o = 1 − S w − Sg
Pg = Po + Pc g,o (Sg )
Pw = Po − Pc w,o (Sw )
x =
o
(S
+ Sg ) z p
So + Sg K p
= ∆m aw
;
yp = K p x p
p = 1 to N
© 2010 - IFP Training
36
General Equations: Compositional Model
=1 ; p
a a + ∑ Q ab p,g = ∆m p,o + ∆m p,g
ab w
© 2010 - IFP Training
n
p
ab p,o
b
b
∑Q
Material Balance Equation (Water)
b
∑Q
Material Balance Equation (HC component)
i =1
∑x = ∑y
Gas Oil Equilibrium
EP - Reservoir Simulation - Equations - E.M.
EP - Reservoir Simulation - Equations - E.M.
Numerical Considerations Diffusivity Equation
Space Discretisation
(1D
horizontal
∂ 2 P φµc ∂ P − =0 ∂x 2 k ∂t
Diffusivity Equation compressible fluid)
flow,
Slightly
Diffusivity Equation: Space & Time Discretisation
37
© 2010 - IFP Training
EP - Reservoir Simulation - Equations - E.M.
© 2010 - IFP Training
38
horizontal
flow,
(
)
Slightly
Diffusivity Equation: Space & Time Discretisation (1D
)
∂ 2 P φµc ∂ P − =0 ∂x 2 k ∂t
Diffusivity Equation compressible fluid)
Space Discretisation
(
∂P ∂P x 1 x i+ 1 − ∂ 2P ∂ ∂P 2 ∂x i− 2 ( xi)= (x i ) ≈ ∂ x ∆x ∂x 2 ∂x ∂x ∂ 2P 1 P − Pi Pi − Pi − 1 Pi + 1 − 2P i + Pi − 1 xi)≈ − = ( i +1 ∂x 2 ∆x ∆x ∆x ∆x 2
39
© 2010 - IFP Training
Explicit Scheme ∂ 2 P n ∂P n i K = i (t n ) ∂x 2 ∂t
n i
;
)
n +1 n ∂Pin (t ) ≈ Pi − Pi n ∂t ∆t
∆t ∆x ≤ 1 2K
∆t =P +K Pin+1 − 2Pin + Pin−1 ∆x 2
Diffusivity equation becomes :
P
n +1 i
Stability condition : Reminder
(
Diffusivity Equation: Space & Time Discretisation
EP - Reservoir Simulation - Equations - E.M.
• No linear algebraic system to be solved • Reduced numerical dispersion
40
© 2010 - IFP Training
EP - Reservoir Simulation - Equations - E.M.
;
n +1 n ∂Pin +1 (t ) ≈ Pi − Pi n ∂t ∆t
Diffusivity Equation: Space & Time Discretisation Implicit Scheme ∂ 2 Pin +1 ∂Pin +1 (t n ) K = ∂x 2 ∂t
Diffusivity equation becomes :
Always Stable
Reminder
∆t ∆t n +1 ∆t n +1 n −K P n +1 + 1 + 2K Pi − K 2 Pi +1 = Pi i 1 − ∆x 2 ∆x 2 ∆x
• Linear algebraic system to be solved • Risk of numerical dispersion
41
© 2010 - IFP Training
*
∆t −K 2 ∆x ∆t 1 + 2K 2 ∆x
*
*
∆t −K 2 ∆x
*
*
*
∆t 1 + 2K 2 ∆x
*
*
∆t −K 2 ∆x *
∆t −K 2 ∆x
* ∆t ∆x 2
*
−K
* ∆t 1 + 2K 2 ∆x ∆t ∆x 2 −K
Pn 2 n −1 ∆t D P2 − K G (t n ) ∆x n P3 P3n −1 * * * * × Pn = Pin −1 i * * * * ∆t −K P n −1 ∆x 2 n N 2 − ∆t PN −2 1 + 2K 2 ∆t n −1 ∆x n PN −1 + K (∆x )2 PD (t n ) PN −1
Implicit Schema: Linear system to be solved
Diffusivity Equation: Space & Time Discretisation
EP - Reservoir Simulation - Equations - E.M.
∆t 1 + K ∆x 2 ∆t −K 2 ∆x
EP - Reservoir Simulation - Equations - E.M.
© 2010 - IFP Training
42
Numerical Considerations Transport Equation
Space Discretisation
(1D
horizontal
∂f ∂S ∂S U w × w =φ w ∂Sw ∂x ∂t
Diffusivity Equation compressible fluid)
flow,
Slightly
Transport Equation: Space & Time Discretisation
43
© 2010 - IFP Training
EP - Reservoir Simulation - Equations - E.M.
© 2010 - IFP Training
44
Transport Equation: Space & Time Discretisation Transport Equation (1D flow, Two incompressible fluids)
∂f ∂S ∂S U w × w = φ× w ∂Sw ∂x ∂t Space Discretisation
S (x ) − Sw (x i −1 ) ∂S w ( x )= w i i ∆x ∂x
45
© 2010 - IFP Training
= φ×
Explicit Scheme
∂f w,n i ∂x
∂S ∂t
=S
n w,i
;
∂S
∆t
Snw,+i1 − Snw,i w,i ( t n)≈
)
∂t
(
U ∆t + × f w,n i − f w,n i-1 φ ∆x
Transport equation becomes
U
w,i ( t n)
Transport Equation: Space & Time Discretisation
EP - Reservoir Simulation - Equations - E.M.
S
n +1 w,i
EP - Reservoir Simulation - Equations - E.M.
© 2010 - IFP Training
46
∂f w,n +i1 ∂x
∂S ∂t
w,i ( t n)
;
∂S ∆t
Snw,+i1 − Snw,i
w,i ( t n)≈
∂t
Transport Equation: Space & Time Discretisation
= φ×
Implicit Scheme
U
(
)
U ∆t − × f w,n +i 1 − f w,n +i 1-1 = Snw,i φ ∆x
Transport equation becomes
S
n +1 w,i
EP - Reservoir Simulation - Equations - E.M.
47
© 2010 - IFP Training
Numerical Considerations General Equations
© 2010 - IFP Training
48
3 main flow directions
Bf
ab f
General Equations: Space Discretisation
I , J , K-1
I+1, J , K
I , J+1 , K
1 cell can communicate with 6 neighbours
I , J-1, K
I-1, J , K
I , J , K+1
General Equations: Space Discretisation
49
© 2010 - IFP Training
6
b =1
= ∑Q
6
(
; Bf = Fluid Volume Factor
b =1
= ∑T
ab
(
)
Kr ab f Pfb − Pfa + ρ f g ∆z ab µf
)
Flow related to phase “f”, cell “a” & it’s 6 neighbours
ρf = Fluid Stock Density
Tab = Transmissivity between cells « a » and « b » Krf = Relative permeability µf = Viscosity
Pf = Fluid Pressure ρf = Fluid Density g= gravity acceleration ∆zab = za – zb Depth difference between cells « a » and « b »
ab f
ρ Krab Q = f,s Tab f Pfb − Pfa + ρf g ∆zab µf
Flow related to phase “f” between two adjacent cells
EP - Reservoir Simulation - Equations - E.M.
Q
a f
50
© 2010 - IFP Training
EP - Reservoir Simulation - Equations - E.M.
ab
General Equations: Time Discretisation
a
n
6
∆t = ∆m a = m a (t n +1 ) − m a (t n )
Material Balance Equation (Oil, water or gas) 6
b =1
∑Q Explicit Schema a
n +1
6
ab
b =1
ab
n
m (t ) = m (t ) + ∑ Q (t ) ∆t
Implicit schema a
n +1
n +1
a
n
© 2010 - IFP Training
51
General Equations: Time Discretisation
b =1
m (t ) − ∑ Q (t ) ∆t = m (t )
6
b =1
∑Q
∆t = ∆m ap = m ap (t n +1 ) − m ap (t n )
Usual calculations are
• Beginning of the time step ( explicit schema) • End of the time step ( implicit schema)
Parameters in the Flow term can be evaluated at
ab p
Material Balance Equation (Component p)
EP - Reservoir Simulation - Equations - E.M.
• Implicit for pressure only • Implicit for pressure and saturation • Implicit for pressure, saturation and other parameters (density, viscosity, volume factor, ....)
52
© 2010 - IFP Training
EP - Reservoir Simulation - Equations - E.M.
)
)
Black Oil Model: Material Balance (Oil) Oil flow between two adjacent cells
(
ρ Krab Q = os Tab o Pob − Poa + ρo g ∆zab Bo µo ab o
Po = Oil pressure ρo = Oil density g= gravity acceleration ∆zab = za - zb, depth difference oil between cells « a » and « b » Tab = Transmissivity between cells « a » and « b »
(
6 Kr ab ∑ T ab o Pob − Poa + ρo g ∆z ab µo
© 2010 - IFP Training
53
Black Oil Model: Material Balance (Oil)
b =1
Kro = relative permeability µo = viscosity Bo = formation volume factor ρos = stock tank oil density
Flow Term 6 ρ ∑ Qoab = o s Bo b =1
Accumulation Term
Va = volume Φa = porosity ρoa = oil density Soa = oil saturation
of cell « a »
moa = Va Φa ρoa Soa
Oil Mass contained in cell “a”
EP - Reservoir Simulation - Equations - E.M.
]
a ∆ρa ∆Φ ∆ma = Va Sa ρa + Φa o ∆Pa + Φa ρa ∆Sa o o o o o ∆P ∆P o
[
∆mao = ρoa Va Φa Soa (Cp + Cw ) ∆Poa + ∆Soa
54
© 2010 - IFP Training
EP - Reservoir Simulation - Equations - E.M.
Flow Term ab o
6 ρ ∑ Q = os Bo b =1
Accumulation Term
)
Black Oil Model: Material Balance (Oil)
(
= ∆m oa
]
6 Kr ab ∑ T ab o Pob − Poa + ρ o g ∆z ab µo b =1
[
ab o
∆m ao = V a ρ oa Φ a Soa (C p + C o ) ∆Poa + ∆Soa
55
© 2010 - IFP Training
(
Pw = Po − Pc w,o
(
= ∆m aw
)
]
6 Kr ab ∑ T ab w Pob − Pc bw,o − Pwa + Pcaw,o + ρ w g ∆z ab µw
)
Black Oil Model: Material Balance (Water)
b =1
∑Q
6
Material Balance Equation
Flow Term 6 ρ ∑ Qabw = w s Bw
[
Material Balance Equation 6
b =1
∑Q
ab w
∆m aw = V a ρ aw Φ a Saw (C p + C o ) ∆Poa − ∆Pc aw,o + ∆Saw
Accumulation Term
b =1
b =1
EP - Reservoir Simulation - Equations - E.M.
EP - Reservoir Simulation - Equations - E.M.
© 2010 - IFP Training
56
Free gas
ρ g,s
Bg
6
∑T g,s
ab
6
Kr
ab g
µg
Solution gas
+ Pcg,b o − Pga + Pcg,a o + ρg g ∆zab +
)
Black Oil Model: Material Balance (Gas)
b o
(P (
))
)
ab ab Kro Pob − Poa + ρo g ∆zab = ∆mga µo
( (
∆mga = Va ∆ Φa ρga Sga + ρgs R s Soa
b=1
Rs∑ T
b=1
ρ
(
(
)
)
(
)
∆ρa ∆R g s ∆Pga = Va ρga Sga + ρgs R s Soa ∆Φa + Va Φa Sga + Soa ρgs ∆P ∆P + Va Φaρga ∆Sga + Va Φaρgs Rs ∆Soa
57
© 2010 - IFP Training
EP - Reservoir Simulation - Equations - E.M.
View more...
Comments
