Method of developing a petroleum reservoir from a flow model calibrated through pressure and saturation map scaling
A method of developing a petroleum reservoir from a flow model constrained by production data and seismic data is disclosed. It is used to define a development scheme and to start or continue the reservoir development. Production data and seismic data associated with a fine grid are acquired during the development of the reservoir. Production data, as well as pressure and saturation values, are simulated in each cell of the flow model. A pressure value and a saturation value are assigned to each cell of the fine grid. A pressure equation is therefore solved locally on the fine grid and the saturations are calculated by writing the continuity of the phase flows between neighboring lithofacies, the conservation of the pore volume and the conservation of the phase volume. The maps obtained are then used to simulate seismic data.
1. A method for optimizing development of an underground medium containing fluids, from a flow model including a first grid forming a set of cells discretizing the medium, wherein production data and seismic data associated with a second grid are measured during the development of the medium with each cell of first grid containing a set of cells of the second grid, comprising:
using a flow simulator implemented in software executed by a computer to simulate production data and pressure and saturation values in each cell of the first grid;
assigning a pressure value to each cell of the second grid;
assigning a saturation value to each cell in the second grid while carrying out for each cell of the first grid identifying zones of a set of connected cells of the second grid with each cell of identical lithofacies being a same rock type and determining, for each identified zone, a saturation value which is assigned to the cells of the zone by solving a first system of equations expressing a continuity of fluid flows between neighboring identified zones and a second system of equations expressing a conservation of a pore volume in each identified zone and a conservation of a volume of each fluid in the cells of the first grid;
simulating seismic data in the second grid using pressure and saturation values associated with each cell of the second grid; and
modifying the flow model to minimize a difference between the measured production data and the simulated production data and to minimize a difference between the measured seismic data and the simulated seismic data.
2. A method as claimed in claim 1 , wherein the flow model is modified by modifying petrophysical property values associated with each cell of the second grid, which is used with the modified values to define an optimum development scheme for the underground medium and the underground medium is then developed according to the development scheme.
3. A method as claimed in claim 1 , wherein the first system of equations involves only relative permeabilities of the fluids in each identified zone.
4. A method as claimed in claim 2 , wherein the first system of equations involves only relative permeabilities of the fluids in each identified zone.
5. A method as claimed in claim 3 , wherein the first system of equations is obtained by writing the following equation for each connected zone n:
kr
α
1
,
Rn
(
S
α
1
,
Rn
)
kr
α
1
,
Rn
(
S
α
1
,
Rn
)
+
kr
α
2
,
Rn
(
S
α
2
,
Rn
)
=
kr
α
1
,
Rn
+
1
(
S
α
1
,
Rn
+
1
)
kr
α
1
,
Rn
+
1
(
S
α
1
,
Rn
+
1
)
+
kr
α
2
,
Rn
+
1
(
S
α
2
,
Rn
+
1
)
with:
kr α,Rn being relative permeability of fluid phase α; and
S α,Rn being saturation of fluid phase α in an identified zone R n .
6. A method as claimed in claim 3 , wherein the first system of equations is obtained by writing the following equation for each connected zone n:
kr
α
1
,
Rn
(
S
α
1
,
Rn
)
kr
α
1
,
Rn
(
S
α
1
,
Rn
)
+
kr
α
2
,
Rn
(
S
α
2
,
Rn
)
=
kr
α
1
,
Rn
+
1
(
S
α
1
,
Rn
+
1
)
kr
α
1
,
Rn
+
1
(
S
α
1
,
Rn
+
1
)
+
kr
α
2
,
Rn
+
1
(
S
α
2
,
Rn
+
1
)
with:
kr α,Rn being relative permeability of fluid phase α; and
S α,Rn being saturation of fluid phase α in an identified zone R n .
7. A method as claimed in claim 1 , wherein the second system of equations is written as follows:
{
(
S
α
1
,
R
1
∑
i
⋐
I
,
i
⋐
R
1
Vol
i
·
ϕ
f
,
i
)
+
…
+
(
S
α
1
,
Rn
∑
i
⋐
I
,
i
⋐
Rn
Vol
i
·
ϕ
f
,
i
)
=
S
α
1
,
I
·
Vol
I
·
ϕ
c
,
I
…
(
S
α
p
,
R
1
∑
i
⋐
I
,
i
⋐
R
1
Vol
i
·
ϕ
f
,
i
)
+
…
+
(
S
α
p
,
Rn
∑
i
⋐
I
,
i
⋐
Rn
Vol
i
·
ϕ
f
,
i
)
=
S
α
p
,
I
·
Vol
I
·
ϕ
c
,
I
∑
p
S
α
p
,
R
1
=
1
…
∑
p
S
α
p
,
Rn
=
1
with:
α1, . . . αp being fluids 1 to p
R1, . . . Rn being identified zones 1 to n
i being cells of the second grid included in a cell l of the first grid
Vol i being volume of a cell i of the second grid
Vol l being volume of a cell l of the first grid
φ f,i being porosity of a cell i of the second grid
φ c,l being porosity of a cell l of the first grid
S α,R being saturation of fluid phase α in an identified zone R
S α,l being saturation of fluid phase α in cell l, calculated by said flow simulation.
8. A method as claimed in claim 2 , wherein the second system of equations is written as follows:
{
(
S
α
1
,
R
1
∑
i
⋐
I
,
i
⋐
R
1
Vol
i
·
ϕ
f
,
i
)
+
…
+
(
S
α
1
,
Rn
∑
i
⋐
I
,
i
⋐
Rn
Vol
i
·
ϕ
f
,
i
)
=
S
α
1
,
I
·
Vol
I
·
ϕ
c
,
I
…
(
S
α
p
,
R
1
∑
i
⋐
I
,
i
⋐
R
1
Vol
i
·
ϕ
f
,
i
)
+
…
+
(
S
α
p
,
Rn
∑
i
⋐
I
,
i
⋐
Rn
Vol
i
·
ϕ
f
,
i
)
=
S
α
p
,
I
·
Vol
I
·
ϕ
c
,
I
∑
p
S
α
p
,
R
1
=
1
…
∑
p
S
α
p
,
Rn
=
1
with:
α1, . . . αp being fluids 1 to p
R1, . . . Rn being identified zones 1 to n
i being cells of the second grid included in a cell l of the first grid
Vol i being volume of a cell i of the second grid
Vol l being volume of a cell l of the first grid
φ f,i being porosity of a cell i of the second grid
φ c,l being porosity of a cell l of the first grid
S α,R being saturation of fluid phase α in an identified zone R
S α,l being saturation of fluid phase α in cell l, calculated by said flow simulation.
9. A method as claimed in claim 3 , wherein the second system of equations is written as follows:
{
(
S
α1
,
R
1
∑
i
⋐
I
,
i
⋐
R
1
Vol
i
·
ϕ
f
,
i
)
+
…
+
(
S
α
1
,
Rn
∑
i
⋐
I
,
i
⋐
Rn
Vol
i
·
ϕ
f
,
i
)
=
S
α
1
,
I
·
Vol
I
·
ϕ
c
,
I
…
(
S
α
p
,
R
1
∑
i
⋐
I
,
i
⋐
R
1
Vol
i
·
ϕ
f
,
i
)
+
…
+
(
S
α
p
,
Rn
∑
i
⋐
I
,
i
⋐
Rn
Vol
i
·
ϕ
f
,
i
)
=
S
α
p
,
I
·
Vol
I
·
ϕ
c
,
I
∑
p
S
α
p
,
R
1
=
1
…
∑
p
S
α
p
,
Rn
=
1
with:
α1, . . . αp being fluids 1 to p
R1, . . . Rn being identified zones 1 to n
i being cells of the second grid included in a cell l of the first grid
Vol i being volume of a cell i of the second grid
Vol l being volume of a cell l of the first grid
φ f,i being porosity of a cell i of the second grid
φ c,l being porosity of a cell l of the first grid
S α,R being saturation of fluid phase α in an identified zone R
S α,l being saturation of fluid phase α in cell l, calculated by said flow simulation.
10. A method as claimed in claim 4 , wherein the second system of equations is written as follows:
{
(
S
α1
,
R
1
∑
i
⋐
I
,
i
⋐
R
1
Vol
i
·
ϕ
f
,
i
)
+
…
+
(
S
α
1
,
Rn
∑
i
⋐
I
,
i
⋐
Rn
Vol
i
·
ϕ
f
,
i
)
=
S
α
1
,
I
·
Vol
I
·
ϕ
c
,
I
…
(
S
α
p
,
R
1
∑
i
⋐
I
,
i
⋐
R
1
Vol
i
·
ϕ
f
,
i
)
+
…
+
(
S
α
p
,
Rn
∑
i
⋐
I
,
i
⋐
Rn
Vol
i
·
ϕ
f
,
i
)
=
S
α
p
,
I
·
Vol
I
·
ϕ
c
,
I
∑
p
S
α
p
,
R
1
=
1
…
∑
p
S
α
p
,
Rn
=
1
with:
α1, . . . αp being fluids 1 to p
R1, . . . Rn being identified zones 1 to n
i being cells of the second grid included in a cell l of the first grid
Vol i being volume of a cell i of the second grid
Vol l being volume of a cell l of the first grid
φ f,i being porosity of a cell i of the second grid
φ c,l being porosity of a cell l of the first grid
S α,R being saturation of fluid phase α in an identified zone R
S α,l being saturation of fluid phase α in cell l, calculated by said flow simulation.
11. A method as claimed in claim 5 , wherein the second system of equations is written as follows:
{
(
S
α1
,
R
1
∑
i
⋐
I
,
i
⋐
R
1
Vol
i
·
ϕ
f
,
i
)
+
…
+
(
S
α
1
,
Rn
∑
i
⋐
I
,
i
⋐
Rn
Vol
i
·
ϕ
f
,
i
)
=
S
α
1
,
I
·
Vol
I
·
ϕ
c
,
I
…
(
S
α
p
,
R
1
∑
i
⋐
I
,
i
⋐
R
1
Vol
i
·
ϕ
f
,
i
)
+
…
+
(
S
α
p
,
Rn
∑
i
⋐
I
,
i
⋐
Rn
Vol
i
·
ϕ
f
,
i
)
=
S
α
p
,
I
·
Vol
I
·
ϕ
c
,
I
∑
p
S
α
p
,
R
1
=
1
…
∑
p
S
α
p
,
Rn
=
1
with:
α1, . . . αp being fluids 1 to p
R1, . . . Rn being identified zones 1 to n
i being cells of the second grid included in a cell l of the first grid
Vol i being volume of a cell i of the second grid
Vol l being volume of a cell l of the first grid
φ f,i being porosity of a cell i of the second grid
φ c,l being porosity of a cell l of the first grid
S α,R being saturation of fluid phase α in an identified zone R
S α,l being saturation of fluid phase α in cell l, calculated by said flow simulation.
12. A method as claimed in claim 6 , wherein the second system of equations is written as follows:
{
(
S
α1
,
R
1
∑
i
⋐
I
,
i
⋐
R
1
Vol
i
·
ϕ
f
,
i
)
+
…
+
(
S
α
1
,
Rn
∑
i
⋐
I
,
i
⋐
Rn
Vol
i
·
ϕ
f
,
i
)
=
S
α
1
,
I
·
Vol
I
·
ϕ
c
,
I
…
(
S
α
p
,
R
1
∑
i
⋐
I
,
i
⋐
R
1
Vol
i
·
ϕ
f
,
i
)
+
…
+
(
S
α
p
,
Rn
∑
i
⋐
I
,
i
⋐
Rn
Vol
i
·
ϕ
f
,
i
)
=
S
α
p
,
I
·
Vol
I
·
ϕ
c
,
I
∑
p
S
α
p
,
R
1
=
1
…
∑
p
S
α
p
,
Rn
=
1
with:
α1, . . . αp being fluids 1 to p
R1, . . . Rn being identified zones 1 to n
i being cells of the second grid included in a cell l of the first grid
Vol i being volume of a cell i of the second grid
Vol l being volume of a cell l of the first grid
φ f,i being porosity of a cell i of the second grid
φ c,l being porosity of a cell l of the first grid
S α,R being saturation of fluid phase α in an identified zone R
S α,l being saturation of fluid phase α in cell l, calculated by said flow simulation.
13. A method as claimed in claim 1 , wherein a pressure value is assigned to each cell of the second grid by partitioning the first grid into a set of connected subdomains and solving, for each subdomain, an equation of conservation of total volume of fluids in the cells of the second grid included in the subdomain.
14. A method as claimed in claim 2 , wherein a pressure value is assigned to each cell of the second grid by partitioning the first grid into a set of connected subdomains and solving, for each subdomain, an equation of conservation of total volume of fluids in the cells of the second grid included in the subdomain.
15. A method as claimed in claim 3 , wherein a pressure value is assigned to each cell of the second grid by partitioning the first grid into a set of connected subdomains and solving, for each subdomain, an equation of conservation of total volume of fluids in the cells of the second grid included in the subdomain.
16. A method as claimed in claim 4 , wherein a pressure value is assigned to each cell of the second grid by partitioning the first grid into a set of connected subdomains and solving, for each subdomain, an equation of conservation of total volume of fluids in the cells of the second grid included in the subdomain.
17. A method as claimed in claim 5 , wherein a pressure value is assigned to each cell of the second grid by partitioning the first grid into a set of connected subdomains and solving, for each subdomain, an equation of conservation of total volume of fluids in the cells of the second grid included in the subdomain.
18. A method as claimed in claim 6 , wherein a pressure value is assigned to each cell of the second grid by partitioning the first grid into a set of connected subdomains and solving, for each subdomain, an equation of conservation of total volume of fluids in the cells of the second grid included in the subdomain.
19. A method as claimed in claim 7 , wherein a pressure value is assigned to each cell of the second grid by partitioning the first grid into a set of connected subdomains and solving, for each subdomain, an equation of conservation of total volume of fluids in the cells of the second grid included in the subdomain.
20. A method as claimed in claim 8 , wherein a pressure value is assigned to each cell of the second grid by partitioning the first grid into a set of connected subdomains and solving, for each subdomain, an equation of conservation of total volume of fluids in the cells of the second grid included in the subdomain.
21. A method as claimed in claim 9 , wherein a pressure value is assigned to each cell of the second grid by partitioning the first grid into a set of connected subdomains and solving, for each subdomain, an equation of conservation of total volume of fluids in the cells of the second grid included in the subdomain.
22. A method as claimed in claim 10 , wherein a pressure value is assigned to each cell of the second grid by partitioning the first grid into a set of connected subdomains and solving, for each subdomain, an equation of conservation of total volume of fluids in the cells of the second grid included in the subdomain.
23. A method as claimed in claim 11 , wherein a pressure value is assigned to each cell of the second grid by partitioning the first grid into a set of connected subdomains and solving, for each subdomain, an equation of conservation of total volume of fluids in the cells of the second grid included in the subdomain.
24. A method as claimed in claim 12 , wherein a pressure value is assigned to each cell of the second grid by partitioning the first grid into a set of connected subdomains and solving, for each subdomain, an equation of conservation of total volume of fluids in the cells of the second grid included in the subdomain.
25. A method as claimed in claim 13 , wherein the conservation equation is written as follows:
div( K f ( n w +n o +n g ) f,c ∇P )=0 in Ω loc
P=P B loc in B loc
with:
K f being permeability defined on the second grid
η α,f,c being mobility of a fluid α
P B loc being pressure on a boundary B loc of subdomain Ω loc .
26. A method as claimed in claim 1 , wherein:
if porosity of a cell of the second grid is lower than a first fixed threshold, no pressure value is assigned to the cell and flows with neighboring cells are considered to be zero; and
if the permeability of a cell of the second grid is lower than a second fixed threshold, the cell is assigned a pressure value equal to a pressure before development of the medium and flows with neighboring cells are considered to be zero.
27. A method as claimed in claim 1 , wherein:
if the porosity of an identified zone is lower than a first fixed threshold, no saturation value is assigned to the zone; and
if the permeability of an identified zone is lower than a second fixed threshold, the zone is assigned a saturation value equal to a saturation before development of the medium.