Introduction to Reservoir Simulation As Practiced in Industry

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Introduction to Reservoir Simulation as Practiced in Industry

Knut–Andreas Lie

SINTEF Digital, Mathematics and Cybernetics / Department of Mathematical Sciences, NTNU, Norway

Computational Issues in Oil Field Applications Tutorials March 21–24, 2017, IPAM, UCLA

1 / 52 Petroleum reservoirs

Naturally occurring ﬂammable liquid/gases found in geological formations

I Originating from organic sediments that have been compressed and ’cooked’ to form hydrocarbons that migrated upward in sedimentary rocks until limited by a trapping structure

I Found in shallow reservoirs on land and deep under the seabed

I Only 30% of the reserves are ’conventional’; remaining 70% include shale oil and gas, heavy oil, extra heavy oil, and oil sands.

Uses of (reﬁned) petroleum:

I Fuel (gas, liquid, solid)

I Alkenes manufactured into plastics and compounds

I Lubricants, wax, paraﬃn wax

I Pesticides and fertilizers for Johan Sverdrup, new Norwegian ’elephant’ discovery, 2011. agriculture Expected to be producing for the next 30+ years

2 / 52 Production processes

Gas

Oil Caprock

Aquifer w/brine

Primary production – puncturing the ’balloon’ When the ﬁrst well is drilled and opened for production, trapped hydrocarbon starts ﬂowing toward the well because of over-pressure

3 / 52 Production processes

Gas injection Gas

Oil Water injection

Secondary production – maintaining reservoir flow As pressure drops, less hydrocarbon is flowing. To maintain pressure and push more profitable hydrocarbons out, one starts injecting water or gas into the reservoir, possibly in an alternating fashion from the same well.

3 / 52 Production processes

Gas injection Gas

Oil Water injection

Enhanced oil recovery

Even more crude oil can be extracted by gas injection (CO2, natural gas, or nitrogen), chemical injection (foam, polymer, surfactants), microbial injection, or thermal recovery (cyclic steam, steam ﬂooding, in-situ combustion), etc.

3 / 52 Why reservoir simulation?

To estimate reserves and support economic and operational decisions

To this end, reservoir engineers need to:

I understand reservoir and ﬂuid behavior

I quantify uncertainty

I test hypotheses and compare scenarios

I assimilate data

I optimize recovery processes

4 / 52 1 a geological model – volumetric grid with cell/face properties describing the porous

rock formation 1

0.8

0.6

∂t(φbwSw) + ∇ · (bw~uw) = bwqw 2 0.4 a flow model – describes how fluids flow ∂t[φ(bwSo + bgrvSg)] + ∇ · (bo~uo + bgrv~ug) = boqo + bgrvqg in a porous medium (conservation laws + 0.2 ∂t[φ(bgSg + borsSo)] ∇ · appropriate closure relations) + (bg~ug + bors~uo) = bgqg0 + borsqo 0 0.2 0.4 0.6 0.8 1

3 a well model – describes ﬂow in and out of the reservoir, in the wellbore, ﬂow control devices, surface facilities

Reservoir models

Somewhat simpliﬁed, consist of three parts:

5 / 52 1

0.8

0.6

3 a well model – describes ﬂow in and out of the reservoir, in the wellbore, ﬂow control devices, surface facilities

Reservoir models

Somewhat simpliﬁed, consist of three parts:

1 a geological model – volumetric grid with cell/face properties describing the porous rock formation

5 / 52 3 a well model – describes ﬂow in and out of the reservoir, in the wellbore, ﬂow control devices, surface facilities

Reservoir models

Somewhat simpliﬁed, consist of three parts:

1 a geological model – volumetric grid with cell/face properties describing the porous

rock formation 1

0.8

0.6

5 / 52 Reservoir models

Somewhat simpliﬁed, consist of three parts:

1 a geological model – volumetric grid with cell/face properties describing the porous

rock formation 1

0.8

0.6

3 a well model – describes ﬂow in and out of the reservoir, in the wellbore, ﬂow control devices, surface facilities

5 / 52 Geologic model: sedimentary rocks

Mineral particles broken oﬀ by weathering and erosion Transported by wind or water to a place where they settle and accumulate into a sediment, building up in lakes, rivers, sand deltas, lagoons, choral reefs, etc

6 / 52 Geologic model: sedimentary rocks

Erosion Deposition

Flood plain

Mud Sand

Gravel

Layered structure with diﬀerent mixtures of rock types with varying grain size, mineral type, and clay content

Thin beds that stretch hundreds or thousands of meters, typically horizontally or at a small angle. Gradually buried deeper and consolidated

6 / 52 Geologic model: sedimentary rocks

Normal dip-slip fault Reverse dip-slip fault Strike-slip fault

Geological activity will later fold, stretch, and fracture the consolidated rock

6 / 52 Geologic model: sedimentary rocks

Structural trap: anticline Stratigraphic traps

Sandstone encased Gas in mudstone Unconformity Oil

Pinch out Impermeable rock

Permeable rock with brine

Fault trap Salt dome

Fault

Impermeable salt

6 / 52 Geologic model: sedimentary rocks

Outcrops of sedimentary rocks from Svalbard, Norway. Length scale: ∼100 m

6 / 52 Geologic model: sedimentary rocks

Layered geological structures typically occur on both large and small scales

6 / 52 Porous media ﬂow – a multiscale problem

The scales that impact ﬂuid ﬂow in subsurface rocks range from

I the micrometer scale of pores and pore channels

I via dm-m scale of well bores and laminae sediments

I to sedimentary structures that stretch across entire reservoirs Porous rocks are heterogeneous at all length scales (no scale separation)

−→

7 / 52 Porous media ﬂow – a multiscale problem

−→

7 / 52 Flow model: representative elementary volume

Porosity: V φ = v Vv + Vr

The assumption of a representative elementary volume (REV) is essential in macro- scale modeling of porous media. Here illustrated for porosity.

8 / 52 I Darcy’s law: ~u = −K(∇p − ρg∇z) empirical law for describing processes on an unresolved scale.

Similar to Fourier’s law (heat) [1822], Ohm’s law (electric current) [1827], Fick’s law (concentration) [1855], except that we now have two driving forces

Governing equations for ﬂuid ﬂow

In its simplest form – two main principles I Conservation of mass ∂ Z I Z m dx + F~ · ~nds = r dx ∂t V ∂V V m=mass, F~ =ﬂow rate, r=ﬂuid sources

9 / 52 Governing equations for ﬂuid ﬂow

In its simplest form – two main principles I Conservation of mass ∂ Z I Z m dx + F~ · ~nds = r dx ∂t V ∂V V m=mass, F~ =ﬂow rate, r=ﬂuid sources

I Darcy’s law: ~u = −K(∇p − ρg∇z) empirical law for describing processes on an unresolved scale.

Similar to Fourier’s law (heat) [1822], Ohm’s law (electric current) [1827], Fick’s law (concentration) [1855], except that we now have two driving forces

9 / 52 Darcy’s law and permeability

In reservoir engineering: K ~u = − ∇p − ρg∇z µ Intrinsic permeability K measures ability to transmit ﬂuids Anisotropic and diagonal by nature, full tensor due to averaging. Reported in units Darcy: 1 d = 9.869233 · 10−13 m2

Fluid velocity: Darcy’s law is formulated for volumetric flux, i.e., volume of fluid per total area per time. The fluid velocity is volume per area occupied by fluid per time, i.e., ~u . ~v = φ

Theoretical basis (M. K. Hubbert, 1956): Darcy’s law derived from the Navier–Stokes equations by averaging, neglecting intertial and viscous eﬀects

10 / 52 Assume constant density ρ, unit ﬂuid viscosity µ, and neglect gravity g −→ ﬂow equation on mixed form

∇ · ~u = q, ~u = −K∇p

or as a Poisson equation with variable coeﬃcients

−∇K∇p = q

Single-phase, incompressible ﬂow

Model equations for single-phase ﬂow:

∂(φρ) K + ∇ · ρ~u = q, ~u = − ∇p − ρg∇z ∂t µ

11 / 52 Single-phase, incompressible ﬂow

Model equations for single-phase ﬂow:

∂(φρ) K + ∇ · ρ~u = q, ~u = − ∇p − ρg∇z ∂t µ Assume constant density ρ, unit ﬂuid viscosity µ, and neglect gravity g −→ ﬂow equation on mixed form

∇ · ~u = q, ~u = −K∇p

or as a Poisson equation with variable coeﬃcients

−∇K∇p = q

11 / 52 Insert into conservation equation

∂(φρ) K = ∇ · ρ ∇p ∂t µ ∂p c ρ ρ (c + c )φρ = f ∇p · K∇p + ∇ · (K∇p) r f ∂t µ µ

If cf is suﬃciently small, so that cf ∇p · K∇p ∇ · (K∇p), we get

∂p 1 = ∇ · K∇p,c = c + c ∂t µφc r f

Single-phase, slightly compressible ﬂow

Introduce compressibilities for rock and ﬂuid

dφ dρ = c φ, = c ρ dp r dp f

12 / 52 If cf is suﬃciently small, so that cf ∇p · K∇p ∇ · (K∇p), we get

∂p 1 = ∇ · K∇p,c = c + c ∂t µφc r f

Single-phase, slightly compressible ﬂow

Introduce compressibilities for rock and ﬂuid

dφ dρ = c φ, = c ρ dp r dp f Insert into conservation equation

∂(φρ) K = ∇ · ρ ∇p ∂t µ ∂p c ρ ρ (c + c )φρ = f ∇p · K∇p + ∇ · (K∇p) r f ∂t µ µ

12 / 52 Single-phase, slightly compressible ﬂow

Introduce compressibilities for rock and ﬂuid

dφ dρ = c φ, = c ρ dp r dp f Insert into conservation equation

∂(φρ) K = ∇ · ρ ∇p ∂t µ ∂p c ρ ρ (c + c )φρ = f ∇p · K∇p + ∇ · (K∇p) r f ∂t µ µ

If cf is suﬃciently small, so that cf ∇p · K∇p ∇ · (K∇p), we get

∂p 1 = ∇ · K∇p,c = c + c ∂t µφc r f

12 / 52 Mass conservation per grid cell: Z I Z ∇ · ~udx = ~u · ~nds = q dx Ωi ∂Ωi Ωi X ui,k = qi k Pressure is cell-wise constant, ﬂux is continuous across cell interfaces

Numerical discretization

Assumption: a grid G consisting of a collection of polyhedral cells {Ωi}

Ωi

~ni,k

Γi,k

13 / 52 Numerical discretization

Assumption: a grid G consisting of a collection of polyhedral cells {Ωi}

Mass conservation per grid cell: Z I Z ∇ · ~udx = ~u · ~nds = q dx Ωi ∂Ωi Ωi Ωi X ui,k = qi ~ni,k k Γi,k Pressure is cell-wise constant, ﬂux is continuous across cell interfaces

13 / 52 Numerical discretization

Mass conservation per grid cell: Z I Z ∇ · ~udx = ~u · ~nds = q dx Ωi ∂Ωi Ωi Ωi X ui,k = qi ~ni,k k Γi,k Pressure is cell-wise constant, ﬂux is continuous across cell interfaces

Assume K is constant within each cell Z

ui,k = − K∇p · ~nik ds Ai,k Γ ik Kk Ki ~c (pi − πi,k)~cik i,k ~ni,k ≈ Ak K 2 · ~nik |~cik| = Ti,k pi − πi,k

13 / 52 P Mass conservation qi = k ui,k gives a linear system (P j Tij , k = i, Ap = q, where Aij = −Tik, k =6 i.

Numerical discretization

Assume K is constant within each cell Z

ui,k = − K∇p · ~nik ds Ai,k Γ ik Kk Ki ~c (pi − πi,k)~cik i,k ~ni,k ≈ Ak K 2 · ~nik |~cik| = Ti,k pi − πi,k

Next, we use continuity of ﬂux and pressure to eliminate the interface pressures ui,k = Tik pi − pk

13 / 52 Numerical discretization

Assume K is constant within each cell Z

ui,k = − K∇p · ~nik ds Ai,k Γ ik Kk Ki ~c (pi − πi,k)~cik i,k ~ni,k ≈ Ak K 2 · ~nik |~cik| = Ti,k pi − πi,k

Next, we use continuity of ﬂux and pressure to eliminate the interface pressures ui,k = Tik pi − pk P Mass conservation qi = k ui,k gives a linear system (P j Tij , k = i, Ap = q, where Aij = −Tik, k =6 i.

13 / 52 Grids: volumetric representation of the reservoir

The structure of the reservoir (geological surfaces, faults, etc) + well paths

The stratigraphy of the reservoir (sedimentary structures)

Petrophysical parameters (permeability, porosity, net-to-gross, . . . )

14 / 52 y

x z

Grids: mimicking geological processes

Deposition

Erosion

Petrophysics

Deformation

15 / 52 Grids: mimicking geological processes

Deposition

x z

Erosion

Petrophysics

Deformation

15 / 52 Grids: mimicking geological processes

Deposition

x z

Erosion

Petrophysics

Deformation

15 / 52 Grids: mimicking geological processes

Deposition

x z

Erosion

Petrophysics

Deformation

15 / 52 Petrophysical parameters

4 3500 x 10

Horizontal 3 Tarbert Vertical 3000 Ness 2.5

2500

2000

1.5

1500

1 1000

0.5 500

0 0 −4 −3 −2 −1 0 1 2 3 4 −8 −6 −4 −2 0 2 4 16 / 52 Research challenge: numerical robustness

Complex, unstructured grids with many obscure challenges Grid dictated by geology, not chosen freely to maximize accuracy of numerical discretization Topology is generally unstructured, non-neighboring connections Cells deviate strongly from box shape, high aspect ratios, many faces/neighbors, small faces, . . . Potential inconsistencies: bilinear vs tetrahedral surfaces

Petrophysics: Many orders of magnitude variations Strong discontinuities No clear scale separation (long and short correlations)

17 / 52 Research challenge: eﬃcient solvers

Large coeﬃcient variations, complex sparsity patterns, etc. Call for eﬃcient iterative solvers and preconditioning methods −→ good test problems for multigrid methods

18 / 52 pi πi,k pk K ~c ~n Ωi i,k i,k Ωk Γi,k

R uik = − Kxx∂xp + Kxy ∂y p + Kxz ∂z p ds Γik

Here, ∂y p and ∂z p cannot be estimated from pi and pk −→ transverse flux Kxy py and Kxz pz neglected −→ inconsistent scheme Many methods developed to amend this (mortar) mixed finite elements multipoint flux approximation (MFPA) mimetic finite difference vertex approximate gradient (VAG) nonlinear TPFA

Research challenge: consistent discretizations

Problem: standard ﬁnite-volume methods are not consistent unless the grid is K orthogonal

19 / 52 Many methods developed to amend this (mortar) mixed finite elements multipoint flux approximation (MFPA) mimetic finite difference vertex approximate gradient (VAG) nonlinear TPFA

Research challenge: consistent discretizations

Problem: standard ﬁnite-volume methods are not consistent unless the grid is K orthogonal

pi πi,k pk K ~c ~n Ωi i,k i,k Ωk Γi,k

R uik = − Kxx∂xp + Kxy ∂y p + Kxz ∂z p ds Γik

Here, ∂y p and ∂z p cannot be estimated from pi and pk −→ transverse ﬂux Kxy py and Kxz pz neglected −→ inconsistent scheme

19 / 52 Research challenge: consistent discretizations

Problem: standard ﬁnite-volume methods are not consistent unless the grid is K orthogonal

pi πi,k pk K ~c ~n Ωi i,k i,k Ωk Γi,k

R uik = − Kxx∂xp + Kxy ∂y p + Kxz ∂z p ds Γik

19 / 52 Research challenge: consistent discretizations

Problem: standard ﬁnite-volume methods are not consistent unless the grid is K orthogonal

pi πi,k pk K ~c ~n Ωi i,k i,k Ωk Γi,k

R uik = − Kxx∂xp + Kxy ∂y p + Kxz ∂z p ds Γik

19 / 52 Example: comparison of consistent methods

Example: 3D Voronoi grid adapting to branching well. Anisotropic and spatially varying permeability

Method dof nnz ratio cond TPFA 9026 126002 13.96 9.64e+02 NTPFA 11920 280703 23.55 2.92e+07 MFD 60321 1538305 25.50 9.37e+15 VEM1 52350 3404977 65.04 4.91e+11 VEM2 227459 38770593 170.45 —

20 / 52 Solution: use a linear inﬂow-performance relation q = J pR − pbh

Here, pbh is ﬂowing pressure in wellbore and pR average pressure in cell

Wells: ﬂow in and out of the reservoir

5–40 in

2rw

20–200 m

Inﬂow and outﬂow take place on a subgrid scale, with large variations in pressure over short distances.

21 / 52 Wells: ﬂow in and out of the reservoir

5–40 in

2rw

20–200 m

Inflow and outflow take place on a subgrid scale, with large variations in pressure over short distances. Solution: use a linear inflow-performance relation q = J pR − pbh

Here, pbh is ﬂowing pressure in wellbore and pR average pressure in cell

21 / 52 Integrating around a small cylinder surrounding the well, I q = ~u · ~nds = −2πhC

Wells: analytic subscale model

p a

r −4 −2 0 2 4 a p bhp

−100 −80 −60 −40 −20 0 20 40 60 80 100

Pseudo-steady, radial ﬂow. Mass conservation in cylinder coordinates

1 ∂(ru) = 0 −→ u = C/r. r ∂r

22 / 52 Wells: analytic subscale model

p a

r −4 −2 0 2 4 a p bhp

−100 −80 −60 −40 −20 0 20 40 60 80 100

Pseudo-steady, radial ﬂow. Mass conservation in cylinder coordinates

1 ∂(ru) = 0 −→ u = C/r. r ∂r Integrating around a small cylinder surrounding the well, I q = ~u · ~nds = −2πhC

22 / 52 (volumetric average pressure p = pa at ra = 0.472rd)

2πKh = pa − pbh µ ln(rd/rw) − 0.75

Solution

2πKh q = pd − pbh µ ln(rd/rw)

Wells: analytic subscale model

p a

r −4 −2 0 2 4 a p bhp

−100 −80 −60 −40 −20 0 20 40 60 80 100

Insert into Darcy’s law and integrate from wellbore radius rw to drainage radius rd at which p = pd is constant:

q K dp Z pd dp Z rd dr u = − = − −→ 2πKh = 2πrh µ dr qµ r pbh rw

22 / 52 (volumetric average pressure p = pa at ra = 0.472rd)

2πKh = pa − pbh µ ln(rd/rw) − 0.75

Wells: analytic subscale model

p a

r −4 −2 0 2 4 a p bhp

−100 −80 −60 −40 −20 0 20 40 60 80 100

Insert into Darcy’s law and integrate from wellbore radius rw to drainage radius rd at which p = pd is constant:

q K dp Z pd dp Z rd dr u = − = − −→ 2πKh = 2πrh µ dr qµ r pbh rw Solution

2πKh q = pd − pbh µ ln(rd/rw)

22 / 52 Wells: analytic subscale model

p a

r −4 −2 0 2 4 a p bhp

−100 −80 −60 −40 −20 0 20 40 60 80 100

Insert into Darcy’s law and integrate from wellbore radius rw to drainage radius rd at which p = pd is constant:

q K dp Z pd dp Z rd dr u = − = − −→ 2πKh = 2πrh µ dr qµ r pbh rw

Solution (volumetric average pressure p = pa at ra = 0.472rd)

2πKh 2πKh q = pd − pbh = pa − pbh µ ln(rd/rw) µ ln(rd/rw) − 0.75

22 / 52 Analytic model valid in neighboring blocks qµ qµ qµ p = p + ln∆x/r − = p + lne−π/2∆x/r bh 2πKh w 4Kh bh 2πKh w Peaceman’s formula:

π 2πKh − p p q = p − pbh , re = e 2 ∆x∆y ≈ 0.20788 ∆x∆y µ ln re/rw

Wells: analytic subscale model

Producer pW p pE Injector

Quarter ﬁve-spot ∆y pS

Repeated ﬁve-spot −→ symmetric solution. Discretize Poisson’s equation:

Kh qµ − 4p − pW − pN − pW − pS = q −→ p = pE − µ 4Kh

23 / 52 qµ = p + lne−π/2∆x/r bh 2πKh w Peaceman’s formula:

π 2πKh − p p q = p − pbh , re = e 2 ∆x∆y ≈ 0.20788 ∆x∆y µ ln re/rw

Wells: analytic subscale model

Producer pW p pE Injector

Quarter ﬁve-spot ∆y pS

Repeated ﬁve-spot −→ symmetric solution. Discretize Poisson’s equation:

Kh qµ − 4p − pW − pN − pW − pS = q −→ p = pE − µ 4Kh Analytic model valid in neighboring blocks qµ qµ p = p + ln∆x/r − bh 2πKh w 4Kh

23 / 52 Peaceman’s formula:

π 2πKh − p p q = p − pbh , re = e 2 ∆x∆y ≈ 0.20788 ∆x∆y µ ln re/rw

Wells: analytic subscale model

Producer pW p pE Injector

Quarter ﬁve-spot ∆y pS

Repeated ﬁve-spot −→ symmetric solution. Discretize Poisson’s equation:

Kh qµ − 4p − pW − pN − pW − pS = q −→ p = pE − µ 4Kh Analytic model valid in neighboring blocks qµ qµ qµ p = p + ln∆x/r − = p + lne−π/2∆x/r bh 2πKh w 4Kh bh 2πKh w

23 / 52 Wells: analytic subscale model

Producer pW p pE Injector

Quarter ﬁve-spot ∆y pS

Repeated ﬁve-spot −→ symmetric solution. Discretize Poisson’s equation:

Kh qµ − 4p − pW − pN − pW − pS = q −→ p = pE − µ 4Kh Analytic model valid in neighboring blocks qµ qµ qµ p = p + ln∆x/r − = p + lne−π/2∆x/r bh 2πKh w 4Kh bh 2πKh w Peaceman’s formula:

π 2πKh − p p q = p − pbh , re = e 2 ∆x∆y ≈ 0.20788 ∆x∆y µ ln re/rw

23 / 52 Wells: many complications

There are several known extensions to Peaceman’s well model: p Diagonal permeability tensor K → KxKy

Rectangular grid cells (more complex formula for re) Horizontal wells, off-centered wells, multiple wells, . . . Near-well effects (permeability increase/reduction) Other grid types and discretization schemes Despite obvious limiting assumptions, Peaceman’s model is used rather uncritically in industry. Need for more accurate/robust/versatile models. . . In general: need to describe flow within wellbore and annulus, downhole equipment, surface facilities, control strategies (choking, reinjection) involving complex logic, . . .

24 / 52 What can you do with single-phase ﬂow?

Run basic diagnostics of your model to establish basic timelines, volumetric connections, measures of dynamic heterogeneity, etc

Forward time of ﬂight Residence time

Can be computed by tracing streamline or by ﬁnite-volume methods solving steady transport equations ~u · ∇h = f(x)

25 / 52 What can you do with single-phase ﬂow?

Run basic diagnostics of your model to establish basic timelines, volumetric connections, measures of dynamic heterogeneity, etc

Inﬂuence regions Well-pair regions

Can be computed by tracing streamline or by ﬁnite-volume methods solving steady transport equations ~u · ∇h = f(x)

25 / 52 Flow diagnostics

< 1%

20% 22%

< 1% I1, P1 I2, P1 I1, P2 3% I2, P2 I1, P3 I2, P3 23% I1, P4 I2, P4 18%

13%

26 / 52 Flow diagnostics

Allocation by connection Allocation by connection Well allocation factors Well allocation factors P6 2 P4 2 4 4 6 6 8 8 P5 P3 10 10 P2 12 12 Connection # Connection # 14 P3 14 16 16 18 18 P4 20 20 0 500 1000 1500 0 500 1000 Well: I6 Accumulated flux [m3/day] Well: I4 Accumulated flux [m3/day]

26 / 52 Flow diagnostics

qi F

normalize

Vi Φ

26 / 52 Flow diagnostics

F 1 − F Ev

Φ td td F-Φ diagram Fractional recovery Sweep eﬃciency

26 / 52 Model reduction: ﬂow-based upscaling

−∇ · (K∇p) = f, in Ω

Subdivide grid into coarse blocks. For each block B, we seek a tensor K∗ such that Z Z K∇p dx = K∗ ∇p dx, B B Typical approach: ﬂow-based upscaling That is, we use Darcy’s law on the coarse scale

u¯ = −K∗∇p

to relate the net ﬂow rate u¯ through Many alternatives, few are suﬃciently accurate and robust B to the average pressure gradient ∇p inside B. See talks by Y. Efendiev and H. Tchelepi

27 / 52 Multiphase ﬂow

Hydrocarbon typically consists of diﬀerent

chemical species like methane, ethane, Water propane, etc. Common modelling practice to group ﬂuid components into phases, i.e., a mixture of components having similar Oil ﬂow properties. Grain Most common phases: aqueous, liquid, and vapor

28 / 52 Fundamental physics: wettability

Immiscible phases separated by a inﬁnitely thin surface having associated surface tension Water wet Oil wet

Oil σow

Water θ θ σ σ σ os ws os Solid

Contact angle θ: determined by balance of adhesive and cohesive forces

Young’s equation (energy balance): σow cos θ = σos − σws

Water generally shows greater aﬃnity than oil to stick to the rock surface −→ reservoirs are predominantly water-wet systems

29 / 52 Fundamental physics: capillary pressure

Diﬀerent equilibrium pressure in two phases separated by curved interface:

2πr σ cos θ 2σ cos θ πr2 gh(ρ − ρ ) p = p − p = = = l a = ∆ρgh c n w πr2 r πr2 | {z } | {z } upward force downward force

θ θ

30 / 52 Fundamental physics: drainage (primary migration)

Saturation: fraction of pore volume filled by a given fluid phase Drainage: non-wetting fluid displacing wetting fluid, controlled by widest non-invaded pore throat

pcnw

primary drainage

Swr pe

31 / 52 Fundamental physics: drainage (primary migration)

Saturation: fraction of pore volume filled by a given fluid phase Drainage: non-wetting fluid displacing wetting fluid, controlled by widest non-invaded pore throat

pcnw

primary drainage

Swr pe

31 / 52 Fundamental physics: drainage (primary migration)

Saturation: fraction of pore volume filled by a given fluid phase Drainage: non-wetting fluid displacing wetting fluid, controlled by widest non-invaded pore throat

pcnw

primary drainage

Swr pe

31 / 52 Fundamental physics: drainage (primary migration)

Saturation: fraction of pore volume filled by a given fluid phase Drainage: non-wetting fluid displacing wetting fluid, controlled by widest non-invaded pore throat

pcnw

primary drainage

Swr pe

31 / 52 Fundamental physics: imbibition (hydrocarbon recovery)

Imbibition: wetting fluid displaces non-wetting fluid, controlled by the size of the narrowest non-invaded pore. Will not follow the same capillary curve −→ hysteresis (cause: trapped oil droplets, different wetting angle for advancing and receding interfaces)

pcnw

Snr

primary drainage primary imbibition

Swr pe

EOR: inject substances to alter wetting properties to mobilize immobile oil, Sor → 1 32 / 52 Extensions of model equations to multiphase ﬂow

Three-phase Darcy velocities (Muskat, 1936):

Kα(Sα) ~uα = − ∇pα − ραg∇z µα Assuming each phase consists of only one component, the mass-balance equations for each phase read (Muskat, 1945):

∂(φρ S ) α α + ∇ · ρ ~u = q ∂t α α α

Macro-scale capillarity concept (Leverett, 1941):

r φ pc(Sw) = J σ cos θ K

33 / 52 1

krw kro 0.8 This gives Darcy’s law on the form 0.6 Kkrα ~uα = − ∇pα − ραg∇z 0.4 µα 0.2 = −Kλα ∇pα − ραg∇z

0 0 0.2 0.4 0.6 0.8 1

Relative permeability

Eﬀective permeability experienced by one 1 phase is reduced by the presence of other krw kro 0.8 phases. Relative permeabilities 0.6

krα = krα(Sα1 ,...,Sαm ), 0.4 are nonlinear functions that attempt to 0.2 account for this eﬀect. Notice that 0 X 0 0.2 0.4 0.6 0.8 krα < 1 α

34 / 52 Relative permeability

Eﬀective permeability experienced by one 1 phase is reduced by the presence of other krw kro 0.8 phases. Relative permeabilities 0.6

krα = krα(Sα1 ,...,Sαm ), 0.4 are nonlinear functions that attempt to 0.2 account for this eﬀect. Notice that 0 X 0 0.2 0.4 0.6 0.8 krα < 1 α 1

krw kro 0.8 This gives Darcy’s law on the form 0.6 Kkrα ~uα = − ∇pα − ραg∇z 0.4 µα 0.2 = −Kλα ∇pα − ραg∇z

0 0 0.2 0.4 0.6 0.8 1

34 / 52 Commercial reservoir simulators: insert functional relationships pc = Pc(Sw) and ρα and φ as function of pα, and discretize with backward Euler in time and the two-point scheme in space In academia: common practice to rewrite the equations to better reveal their mathematical nature

General ﬂow equations for two-phase ﬂow

Gathering the equations, we have

∂(φρ S ) α α + ∇ · ρ ~u = q , α = {w, n} ∂t α α α Kkrα ~uα = − ∇pα − ραg∇z µα

pc = pn − pw,Sw + Sn = 1

35 / 52 General ﬂow equations for two-phase ﬂow

Gathering the equations, we have

∂(φρ S ) α α + ∇ · ρ ~u = q , α = {w, n} ∂t α α α Kkrα ~uα = − ∇pα − ραg∇z µα

pc = pn − pw,Sw + Sn = 1

Commercial reservoir simulators: insert functional relationships pc = Pc(Sw) and ρα and φ as function of pα, and discretize with backward Euler in time and the two-point scheme in space In academia: common practice to rewrite the equations to better reveal their mathematical nature

35 / 52 Fractional ﬂow formulation

Choose Sw and pn as primary unknowns, consider incompressible ﬂow (i.e., ρ is constant and can be divided out)

∂S φ α + ∇ · ~u = q . ∂t α α

36 / 52 Fractional ﬂow formulation

Choose Sw and pn as primary unknowns, consider incompressible ﬂow (i.e., ρ is constant and can be divided out)

∂S φ α + ∇ · ~u = q . ∂t α α Sum mass-conservation equations:

∂ φ (S + S ) + ∇ · (~u + ~u ) = q + q ∂t n w n w n w

36 / 52 Fractional ﬂow formulation

Choose Sw and pn as primary unknowns, consider incompressible ﬂow (i.e., ρ is constant and can be divided out)

∂S φ α + ∇ · ~u = q . ∂t α α Sum mass-conservation equations:

∂ φ (Sn + Sw) + ∇ · (~un + ~uw) = qn + qw −→ ∇ · ~u = q ∂t | {z } | {z } | {z } ≡1 =~u =q

36 / 52 Fractional ﬂow formulation

Choose Sw and pn as primary unknowns, consider incompressible ﬂow (i.e., ρ is constant and can be divided out)

∂S φ α + ∇ · ~u = q . ∂t α α Sum mass-conservation equations:

∂ φ (Sn + Sw) + ∇ · (~un + ~uw) = qn + qw −→ ∇ · ~u = q ∂t | {z } | {z } | {z } ≡1 =~u =q

Sum Darcy equations

~u = ~un + ~uw = −(λn + λw)∇pn + λw∇pc + (λnρn + λwρw)g∇z

36 / 52 Fractional ﬂow formulation

Choose Sw and pn as primary unknowns, consider incompressible ﬂow (i.e., ρ is constant and can be divided out)

∂S φ α + ∇ · ~u = q . ∂t α α Sum mass-conservation equations:

∂ φ (Sn + Sw) + ∇ · (~un + ~uw) = qn + qw −→ ∇ · ~u = q ∂t | {z } | {z } | {z } ≡1 =~u =q

Sum Darcy equations

~u = ~un + ~uw = −(λn + λw)∇pn + λw∇pc + (λnρn + λwρw)g∇z | {z } =λ

36 / 52 Fractional ﬂow formulation

Choose Sw and pn as primary unknowns, consider incompressible ﬂow (i.e., ρ is constant and can be divided out)

∂S φ α + ∇ · ~u = q . ∂t α α Sum mass-conservation equations:

∂ φ (Sn + Sw) + ∇ · (~un + ~uw) = qn + qw −→ ∇ · ~u = q ∂t | {z } | {z } | {z } ≡1 =~u =q

Sum Darcy equations

~u = ~un + ~uw = −(λn + λw)∇pn + λw∇pc + (λnρn + λwρw)g∇z | {z } =λ

Inserted into ∇ · ~u = q gives pressure equation −∇ · (λK∇pn) = q − ∇ λw∇pc + (λnρn + λwρw)g∇z

36 / 52 Fractional ﬂow formulation

Choose Sw and pn as primary unknowns, consider incompressible ﬂow (i.e., ρ is constant and can be divided out)

∂S φ α + ∇ · ~u = q . ∂t α α Sum mass-conservation equations:

∂ φ (Sn + Sw) + ∇ · (~un + ~uw) = qn + qw −→ ∇ · ~u = q ∂t | {z } | {z } | {z } ≡1 =~u =q

Sum Darcy equations

~u = ~un + ~uw = −(λn + λw)∇pn + λw∇pc + (λnρn + λwρw)g∇z | {z } =λ

Inserted into ∇ · ~u = q gives pressure equation −∇ · (λK∇pn) = q − ∇ λw∇pc + (λnρn + λwρw)g∇z | {z } | {z } Poisson only function of Sw

36 / 52 Solve for ~uw and insert into conservation equation ∂S φ w + ∇ · f ~u + λ ∆ρg∇z = q − ∇ · f λ P 0 ∇S ∂t w n w w n c w

Setting Pc ≡ 0 and g ≡ 0 for simplicity

−∇Kλ(S)∇p) = q, ~u = −Kλ(S)∇p,

φ∂tS + ∇· (~uf(S)) = 0

System of one elliptic pressure equation and one hyperbolic saturation equation. Typically: solved sequentially with specialized methods

Fractional ﬂow formulation

Multiply phase velocity by mobility of other phase and subtract λn~uw − λw~un = λwλnK ∇pc + (ρw − ρn)g∇z]

37 / 52 Setting Pc ≡ 0 and g ≡ 0 for simplicity

−∇Kλ(S)∇p) = q, ~u = −Kλ(S)∇p,

φ∂tS + ∇· (~uf(S)) = 0

System of one elliptic pressure equation and one hyperbolic saturation equation. Typically: solved sequentially with specialized methods

Fractional ﬂow formulation

Multiply phase velocity by mobility of other phase and subtract λn~uw − λw~un = λwλnK ∇pc + (ρw − ρn)g∇z]

Solve for ~uw and insert into conservation equation ∂S φ w + ∇ · f ~u + λ ∆ρg∇z = q − ∇ · f λ P 0 ∇S ∂t w n w w n c w

37 / 52 Fractional ﬂow formulation

Multiply phase velocity by mobility of other phase and subtract λn~uw − λw~un = λwλnK ∇pc + (ρw − ρn)g∇z]

Solve for ~uw and insert into conservation equation ∂S φ w + ∇ · f ~u + λ ∆ρg∇z = q − ∇ · f λ P 0 ∇S ∂t w n w w n c w

Setting Pc ≡ 0 and g ≡ 0 for simplicity

−∇Kλ(S)∇p) = q, ~u = −Kλ(S)∇p,

φ∂tS + ∇· (~uf(S)) = 0

System of one elliptic pressure equation and one hyperbolic saturation equation. Typically: solved sequentially with specialized methods

37 / 52 Buckley–Leverett solution for 1D displacement

M = 1 M = 5 M = .2

S2 S + f(S) = q, f(S) = ,M = µ /µ t x S2 + M(1 − S)2 w n Here, M = .2 gives poor local displacement eﬃciency, M = 5 gives very good

38 / 52 Simulation examples: quarter-ﬁve spot

t=0.20 PVI t=0.40 PVI t=0.60 PVI t=0.80 PVI Sw 1

0.8

0.6

0.4

0.2

5 x 10 1 Sw in completion 2.5 Water cut initial oil in place 0.8

0.6

2 0.4 initial oil rate

0.2

1.5 0

0 0.2 0.4 0.6 0.8 1

250

1 200

150 0.5

100 water breakthrough

50 Oil rate [m3/day]

0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2

39 / 52 Simulation examples: quarter-ﬁve spot

4 years 8 years 12 years 16 years 20 years

0.9

0.8 ratio 1:10

0.7

0.6

0.5 ratio 1:1

0.4

0.3

0.2 ratio 10:1 0.1

39 / 52 Simulation examples: quarter-ﬁve spot

cumulative oil production wcut: Water fraction at reservoir conditions 18000 1 initial oil in place P (Ratio 1:10) 0.8 16000 P (Ratio 1:1)

0.6 P (Ratio 10:1) 14000 0.4 12000 0.2

0 10000 1000 2000 3000 4000 5000 6000 7000 Time (days)

8000 3 −5 Oil surface rate [m /s] x 10

6000 2.5

P (Ratio 1:10) 2 4000 P (Ratio 1:1) 1.5 P (Ratio 10:1) 2000 1 P (Ratio 1:10) P (Ratio 1:1) 0.5 0 P (Ratio 10:1) 1000 2000 3000 4000 5000 6000 7000 0 Time (days) 1000 2000 3000 4000 5000 6000 7000 Time (days)

39 / 52 150 days 670 days

810 days 1020 days

1500 days 4500 days

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 40 / 52 High permeability on top Low permeability on top

120 days 120 days

360 days 360 days

1500 days 1500 days

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

40 / 52 Equations: conservation for phases or components? Choose components to avoid source terms for mass transfer

∂ X X X φ c` ρ S + ∇ · c` ρ ~u + J~` = c` ρ q , ∂t α α α α α α α α α α α α α Here, J is diﬀusion, e.g., Fickian

~` ` ` Jα = −ραSαDα∇cα,

More in talk by K. Jessen

41 / 52 Conservation equations: ∂t φboSo + ∇ · bo~uo = boqo ∂t φbwSw + ∇ · bw~uw = bwqw ∂t φ bgSg + borsoSo + ∇ · bg~ug + borso~uo = bgqg + borsoqo

s s Dissolved gas in oil: rso = Vg /Vo . Similarly: oil vaporized in gas rsg

Black-oil equations

Hydrocarbon components lumped together to a light ’gas’ and a heavier ’oil’ pseudocomponent at surface conditions

W O G A X reservoir L X X surface V X X

Simple PVT: formation-volume factors, s s Bα = Vα/Vα = ρα/ρα or bα = 1/Bα

42 / 52 Black-oil equations

Hydrocarbon components lumped together to a light ’gas’ and a heavier ’oil’ pseudocomponent at surface conditions

W O G A X reservoir L X X surface V X X

Simple PVT: formation-volume factors, s s Bα = Vα/Vα = ρα/ρα or bα = 1/Bα

Conservation equations: ∂t φboSo + ∇ · bo~uo = boqo ∂t φbwSw + ∇ · bw~uw = bwqw ∂t φ bgSg + borsoSo + ∇ · bg~ug + borso~uo = bgqg + borsoqo

s s Dissolved gas in oil: rso = Vg /Vo . Similarly: oil vaporized in gas rsg

42 / 52 Example: ﬂuid model from SPE9

Relative permeability for oil

Oil relative permeability curves 1 Oil−Water system Oil−Gas system 0.9

0.8

0.7

0.6 r

k 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 1 Oil saturation

J. E. Killough (1995). Ninth SPE comparative solution project: A reexamination of black-oil simulation. SPE Reservoir Simulation Symposium 43 / 52 Example: ﬂuid model from SPE9

Water formation volume factor 1.007

1.006

1.005

Formation-volume factors (inverse densities): 1.004

1.003 Oil formation volume factor 1.002

1.001 1.12 1

0.999 0 100 200 300 400 500 1.1 Gas formation volume factor 0.018

0.016

1.08 0.014 saturated 0.012

under-saturated 0.01 1.06

o 0.008 B 0.006 1.04 0.004 0.002

0 0 100 200 300 400 500 1.02 Rock compressibility 1.005

1.004

1 1.003

1.002

0.98 1.001 50 100 150 200 250 300 350 400 450 500 Pressure [bar] 1

0.999

0.998

J. E. Killough (1995). Ninth SPE comparative solution project: A reexamination of black-oil0.997 simulation. SPE Reservoir Simulation Symposium 0 100 200 300 400 500

43 / 52 Example: ﬂuid model from SPE9

Viscosities:

−3 Water viscosity x 10 −3 1.5 x 10 Oil viscosity 1.2

1.15

0.5

1.1 0 0 100 200 300 400 500 −5 Gas viscosity o x 10

µ 2.2

2.1 1.05 2

1.9

1.8

1 1.7

1.6

1.5

1.4 0.95 1.3 0 100 200 300 400 500 50 100 150 200 250 300 350 400 450 500 Pressure

43 / 52

J. E. Killough (1995). Ninth SPE comparative solution project: A reexamination of black-oil simulation. SPE Reservoir Simulation Symposium Black-oil: discretization and linearization

Discretization: backward Euler in time, two-point ﬂux-approximation with upstream mobility in space. ∂E Newton’s method for nonlinear equation: δxi = E(xi) ∂xi

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ s s s ∂po ∂sw ∂sg ∂sg ∂qw ∂qo ∂qg ∂pbh

s Ew Eqw

s Eqo

s Eqg Eg

Ectrl

44 / 52 For larger models, pressure subsystem should be solved with algebraic multigrid Time-step control chop if too large changes in variables chop if convergence failure more advanced logic to maintain targeted iteration count

Elaborate logic for well control and surface facilities

Black-oil: solution strategies

Solution procedure s s s 1. Eliminate well variables qo, qw, qg, and pbh 2. Set ﬁrst block-row equal to sum of block-rows, leave out rows that may harm diagonal dominance in block (1,1) 3. Set up two-stage preconditioner: −1 – M1 : solves pressure subsystem −1 – M2 : ILU0 decomposition of the full system 4. Solve full system with GMRES using −1 −1 preconditioner M2 M1 5. Recover remaining variables

45 / 52 Black-oil: solution strategies

Solution procedure For larger models, pressure subsystem s s s should be solved with algebraic 1. Eliminate well variables q , q , q , o w g multigrid and pbh 2. Set ﬁrst block-row equal to sum of Time-step control block-rows, leave out rows that may chop if too large changes in harm diagonal dominance in block variables (1,1) chop if convergence failure 3. Set up two-stage preconditioner: more advanced logic to maintain −1 – M1 : solves pressure subsystem targeted iteration count −1 – M2 : ILU0 decomposition of the full system Elaborate logic for well control and 4. Solve full system with GMRES using surface facilities −1 −1 preconditioner M2 M1 5. Recover remaining variables

45 / 52 Example: SPE 9 benchmark

Grid with 9000 cells 1 water injector, rate controlled, switches to bhp 25 producers, oil-rate controlled, most switch to bhp Appearance of free gas due to pressure drop Production rates lowered to 1/15 between days 300 and 360

7 PROD13 x 10 PROD13 2.5 2.5 2

/s) 2 3 1.5 1.5 1 Pressure (Pa) Gas rate (m 1 0.5

0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Time (years) Time (years) x 107 PROD18 PROD18 2.5 2.5 MRST ECLIPSE 2

2 /s) 3 1.5 1.5 1

Pressure (Pa) MRST Gas rate (m 0.5 1 ECLIPSE

0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Time (years) Time (years)

46 / 52 Example: the Voador ﬁeld

prod 7 South wing of the reservoir (Petrobras) prod 3

Gradients obtained through adjoint simulations prod 6 Validate: open-source / commercial simulator: prod 1 prod 2 prod 4, 5 injector – 20 years of historic data – virtually identical results – main challenge: needed to reverse-engineer description of wells. . .

producer 1 injector producer 1 injector

0 2000 4000 6000 8000 0 2000 4000 6000 8000 0 2000 4000 6000 8000 0 2000 4000 6000 8000 producer 2 producer 3 producer 2 producer 3

0 2000 4000 6000 8000 0 2000 4000 6000 8000 0 2000 4000 6000 8000 0 2000 4000 6000 8000 producer 4 producer 5 producer 4 producer 5

0 2000 4000 6000 8000 0 2000 4000 6000 8000 0 2000 4000 6000 8000 0 2000 4000 6000 8000 producer 6 producer 7 producer 6 producer 7

0 2000 4000 6000 8000 0 2000 4000 6000 8000 0 2000 4000 6000 8000 0 2000 4000 6000 8000 bottom-hole pressure water rate 47 / 52 Multisegment wells

More accurate modelling: In nodes: – Network models – pressure p m – Represent annulus – mass fractions xw , m m – Flow inside wellbore xo , xg – (Autonomous) inﬂow control devices In segments: m – Artiﬁcial lift, etc – mass rates v

48 / 52 Multisegment wells

Discrete mass conservation in nodes:

V 0 m m xcρ − xc ρ + div vc − qc = 0 ∆t | {z } |{z} | {z } source term (well accumulation in- and outﬂux to neighboring nodes control, connections)

Discrete pressure drop equations in segments:

grad(p) − g avg(ρ) grad(z) − hvm, uw(ρ), uw(µ) = 0 | {z } | {z } gravity term heuristic pressure drop term 48 / 52 Example: eﬀect of modeling annulus

×10 4

3 Uniform, no annulus Uniform, annulus Thief zones, no annulus 2 Thief zones, annulus Gas production [Mscf/day] SPE10, no annulus SPE10, annulus 1 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Time [days]

49 / 52 Enhanced Oil Recovery: polymer ﬂooding

Simple model: introduce extra immiscible component and mixture law

Polymer transported in water:

krw(S) ~uw = − K ∇pw − ρwg∇z µw,eﬀ(c) Rk(c)

krw(S) ~up = − K ∇pw − ρwg∇z µp,eﬀ(c)Rk(c)

Conservation of polymer component:

a ∂t φ(1 − Sipv)cbwS + ρr C (1 − φ) + ∇ · cbw~up = qp

50 / 52 Enhanced Oil Recovery: polymer ﬂooding

Simple model: introduce extra immiscible component and mixture law

Polymer transported in water:

krw(S) ~uw = − K ∇pw − ρwg∇z µw,eﬀ(c) Rk(c)

Todd–Longstaﬀ mixing:

krw(S) 1−c/c ~u = − K ∇p − ρ g∇z 1 = m p w w µ ω 1−ω µp,eﬀ(c)Rk(c) w,eﬀ µm(c) µw + c/cm ω 1−ω viscosity enhancement µm(c) µp Conservation of polymer component:

a ∂t φ(1 − Sipv)cbwS + ρr C (1 − φ) + ∇ · cbw~up = qp

50 / 52 Enhanced Oil Recovery: polymer ﬂooding

Simple model: introduce extra immiscible component and mixture law

Polymer transported in water:

krw(S) ~uw = − K ∇pw − ρwg∇z µw,eﬀ(c) Rk(c)

krw(S) ~up = − K ∇pw − ρwg∇z µp,eﬀ(c)Rk(c)

Conservation of polymer component:

Todd–Longstaﬀa mixing: ∂t φ(1 − Sipv)cbwS + ρr C (1 − φ) + ∇ · cbw~up = qp

1 = 1−c/cm µ ω 1−ω w,eﬀ µm(c) µw + c/cm ω 1−ω viscosity enhancement permeability reduction µm(c) µp

50 / 52 Enhanced Oil Recovery: polymer ﬂooding

Simple model: introduce extra immiscible component and mixture law

Polymer transported in water:

krw(S) ~uw = − K ∇pw − ρwg∇z µw,eﬀ(c) Rk(c)

krw(S) ~up = − K ∇pw − ρwg∇z µp,eﬀ(c)Rk(c)

Conservation of polymer component:

Todd–Longstaﬀa mixing: ∂t φ(1 − Sipv)cbwS + ρr C (1 − φ) + ∇ · cbw~up = qp

1 = 1−c/cm µ ω 1−ω w,eﬀ µm(c) µw + c/cm ω 1−ω viscosity enhancement permeability reduction µm(c) µp inaccessible pore space

50 / 52 Enhanced Oil Recovery: polymer ﬂooding

Simple model: introduce extra immiscible component and mixture law

Polymer transported in water:

krw(S) ~uw = − K ∇pw − ρwg∇z µw,eﬀ(c) Rk(c)

krw(S) ~up = − K ∇pw − ρwg∇z µp,eﬀ(c)Rk(c)

Conservation of polymer component:

Todd–Longstaﬀa mixing: ∂t φ(1 − Sipv)cbwS + ρr C (1 − φ) + ∇ · cbw~up = qp

1 = 1−c/cm µ ω 1−ω w,eﬀ µm(c) µw + c/cm ω 1−ω viscosity enhancement permeability reduction µm(c) µp inaccessible pore space adsorption

50 / 52 Summary

Geological models: complex unstructured grids having many obscure challenges Flow models: system of highly nonlinear parabolic PDEs with elliptic and hyperbolic sub-character Well models: subscale models, complex logic, strong impact on ﬂow Validation and availability in software

Challenges: Main point of grid: describe stratigraphy and structural architecture, i.e., not chosen freely to maximize accuracy of numerical discretization Industry standard: corner-point / stratigraphic grids Grid topology is generally unstructured, with non- neighboring connections Geometry: deviates (strongly) from box shape, high aspect ratios, many faces/neighbors, small faces, . . . Potential inconsistencies since faces are bilinear or tetrahedral surfaces

51 / 52 Summary

1 k 0.9 rw k Geological models: complex unstructured grids ro 0.8 k rog having many obscure challenges k row 0.7 k rg Flow models: system of highly nonlinear parabolic 0.6 PDEs with elliptic and hyperbolic sub-character 0.5

0.4 Well models: subscale models, complex logic,

0.3 strong impact on ﬂow 0.2 Validation and availability in software 0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Challenges: Delicate balances: viscous forces, gravity, capillary, . . . Strong coupling between ’elliptic’ and ’hyperbolic’ variables (small scale: capillary, large scale: gravity) Large variation in time constants and coupling Orders-of-magnitude variations in permeability Parameters with discontinuous derivatives Path-dependence: hysteretic parameters Sensitive to subtle changes in interpolation of tabulated physical data Monotonicity and mass conservation 51 / 52 Summary

producer 1 injector Geological models: complex unstructured grids having many obscure challenges 0 2000 4000 6000 8000 0 2000 4000 6000 8000 producer 2 producer 3 Flow models: system of highly nonlinear parabolic PDEs with elliptic and hyperbolic sub-character

0 2000 4000 6000 8000 0 2000 4000 6000 8000 producer 4 producer 5 Well models: subscale models, complex logic, strong impact on ﬂow

0 2000 4000 6000 8000 0 2000 4000 6000 8000 producer 6 producer 7 Validation and availability in software

0 2000 4000 6000 8000 0 2000 4000 6000 8000 Challenges: prod 7 Near singular radial ﬂow in near-well zone (much larger

prod 3 ﬂow than inside reservoir) Induce nonlocal connections prod 6

prod 1 prod 2 Completely diﬀerent multiphase ﬂow inside wellbore injector prod 4, 5 Coupling to surface facilities Abrupt changes in driving forces Control strategies with intricate logic which is highly sensitive to state values . . .

51 / 52 Summary

0 2000 4000 6000 8000 0 2000 4000 6000 8000 producer 4 producer 5 Well models: subscale models, complex logic, strong impact on ﬂow

0 2000 4000 6000 8000 0 2000 4000 6000 8000 producer 6 producer 7 Validation and availability in software

0 2000 4000 6000 8000 0 2000 4000 6000 8000 Challenges: prod 7 New methods tend to be immature and too simpliﬁed

prod 3 Researchers: incompressible ﬂow and explicit methods. Industry: implicit methods for compressible ﬂow prod 6

prod 1 prod 2 Industry relies on a few software providers and has prod 4, 5 injector strong faith in software with (undocumented) safe- guards and algorithmic choices Oil companies seldom give away data Realistic models involve a large number of intricate de- tails (Eclipse has 2–3000 keywords. . . )

51 / 52 http://www.sintef.no/MRST 52 / 52 Originally: developed to support research on multiscale methods and mimetic discretizations first public release as open source, April 2009 Today: general toolbox for rapid prototyping and verification of new computational methods wide range of applications two releases per year each release has from 900 (R2013a) to 2100 (R2015a) unique downloads Users: academic institutions, oil and service companies large user base in USA, Norway, China, Brazil, United Kingdom, Iran, Germany, Netherlands, France, Canada, . . . Publications: used in 24 PhD theses and 59 master theses used in more than 100 scientific papers by people outside of SINTEF

http://www.sintef.no/MRST 52 / 52