SPE-119200-PA (Thermal Streamline Simulation for Hot Waterflooding)

372 June 2010 SPE Reservoir Evaluation & Engineering

Thermal Streamline Simulation for Hot Waterflooding

Z. Zhu, SPE, and M.G. Gerritsen, SPE, Stanford University; and M.R. Thiele, SPE, Streamsim/Stanford U.

SummaryWe explore the extension of streamline simulation to thermal-recovery processes. For each global timestep, we first compute the pressure field on an Eulerian grid. We then solve for the advective parts of the mass balance and energy equations along the individual streamlines. At the end of each global timestep, we account for the nonadvective terms of the transport equations on the Eulerian grid along with gravity, using the operator splitting method. We included temperature-dependent viscosity and account for thermal expansion of the fluids. We tested our streamline simulator on 2D heterogeneous quarter five-spot problems and compared the results with those computed by a commercial thermal simulator both for accuracy and computational efficiency. We present sensitivity studies for effects of fluid compressibility, gravity, and thermal-conductivity. For the cases investigated, our thermal streamline simulator is capable of producing accurate results at a computa-tional cost that is significantly lower than that of existing Eulerian simulators.

IntroductionThermal enhanced recovery techniques account for more than 60% of the US enhanced oil recovery production. Steamflooding and hot waterflooding are the most widely used, but other processes such as in-situ combustion and steam-assisted gravity drainage are applied also.

Planning and management of these processes generally make extensive use of thermal reservoir simulations. To our knowledge, all commercial and academic thermal simulators are finite volume based codes that use either a fully implicit (FIM) timestepping method or an adaptive implicit method (Aziz and Settari 1979). The computational cost of thermal processes is very high because of the strongly nonlinear flow, and it becomes even higher when considering heterogeneous domains. As a result, running optimi-zation and/or sensitivity studies on grids with desirable numerical resolution becomes prohibitively expensive. There is an urgent need for fast approximate solvers for thermal problems. The essence of this work is to explore the extension of streamline simu-lation to thermal enhanced recovery processes. Our aim is to design a fast and effective simulator with sufficient accuracy for use in common reservoir simulation studies, such as well placement, optimization, and history matching.

Streamline-based flow simulation (SL) has been especially successful in the simulation of isothermal oil/water systems for large geologically complex and strongly heterogeneous systems that are challenging for more traditional modeling techniques (Batycky et al. 1997; Martin and Wegner 1979). The success of streamline simulation is based on the physical observation that in heterogeneous reservoirs the time scale at which fluids flow along streamlines is often much smaller than the time scale at which the streamline locations change significantly. For isothermal problems, this allows decoupling of the costly 3D transport problem into a set of 1D advection problems along streamlines. Streamline simula-tion uses a dual-grid approach: The equations governing pressure are solved on an Eulerian or fixed 3D grid, which we will refer to

as the pressure grid. The equations governing transport are solved along the individual streamlines of the streamline grid. Crossflow effects, including gravity and diffusivity, can be included through operator splitting techniques (Bratvedt et al. 1996; Berre et al. 2002; Berenblyum et al. 2003).

The computational efficiency of streamline methods for many reservoir settings makes them especially suitable for history match-ing and optimization problems, and commercial streamline simu-lators are applied for such purposes (Thiele and Batycky 2006). Streamline simulation has recently been extended to handle (near-) miscible gas injection processes (Crane et al. 2000; Gerritsen et al. 2005; Thiele et al. 1997), flow in fractured reservoirs (Donato and Blunt 2004), and compressible flow (Beraldo et al. 2008; Osako 2006). Streamline simulation can also be parallelized very effec-tively (Batycky et al. 2009; Gerritsen et al. 2009).

Previous work by Pasarai and Arihara (2005) proposed the streamline simulation for hot waterflooding simulations, where the mass and energy equation are solved along the streamlines using an implicit scheme. Here, we suggest an extension: At each global timestep, we first solve for pressure on an Eulerian grid. We then solve the advective parts of the mass balance and energy equations together along streamlines using either an implicit or an explicit solver, including compressibility effects. At the end of a global timestep, we account for the nonadvective parts of the mass balance and energy equations along with gravity by using operator splitting. In this work, we focus on hot waterflooding and include tempera-ture dependent viscosity and thermal compressibility effects. Steam injection requires additional techniques to handle the strong volume changes, and that is a subject of ongoing research.

We first introduce the governing equations. Then, we present our thermal streamline formulation, and we discuss how to account for volume changes, heat conduction, and gravity. Finally we present our results. We test our hot waterflood streamline simulator on a quarter five-spot hot waterflooding problem and compare its accu-racy and efficiency with a commercial simulator (STARS 2004). We explore the sensitivity on viscosity as a function of temperature, heat conduction, gravity, and number of pressure updates is performed also. We show that the streamline method is able to produce comparable results to commercial code but at a signifi-cantly reduced computational cost.

Governing EquationsGeneral Compositional Thermal Formulation. The governing equations for thermal, compositional porous media fl ow include nc component conservation equations, one energy conservation equation, and one volume conservation equation. The mass balance equation for the ith component is given by

+ =

= =

t

y S y qi j j jj

n

i j jj

n

i

p p

, ,

1 1u j

, . . . . . . . . . . . . . . . . . . . (1)

where np is the number of total phases, is the porosity of the porous medium, yi,j is the mole fraction of component i in phase j, j is the phase molar density, Sj the volumetric phase saturation, uj is the velocity of phase j, and qi is the mass source or sink term.

The energy conservation equation is

( ) +

+ + =

t

U S U hr j jj

n

j j j

p

11

u j kk T qcj

n

h

p

( ) ==

1

,

. . . . . . . . . . . . . . . . . . . . . . . . (2)

Copyright 2010 Society of Petroleum Engineers

This paper (SPE 119200) was accepted for presentation at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, USA, 24 February 2009, and revised for publication. Original manuscript received for review 3 November 2008. Revised manuscript received for review 21 June 2009. Paper peer approved 13 July 2009.

June 2010 SPE Reservoir Evaluation & Engineering 373

where Ur is the internal energy of rock, Uj is the internal energy of phase j, hj is the enthalpy of phase j, and kc is the heat conductivity. qh is the heat source or sink term.

Finally, the volume conservation expresses that the fluid must fill the pore space; that is,

V Vp jj

np

=

=

1

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

where Vj is the volume of phase j.The balance on the total fluid volume leads to the pressure

equation

cPt

VN

yt ti

i j jj

np +

= , u j1jj

n

t

tj j

j

n

t

t

p

pVU

h VU

q

=

=

+

=

1

1 u j

hh

t

ij

n

iVN

qp

+

=

1

. . . . . . . . . . . . . . (4)

Here, ct is the total compressibility of the fluid and qi and qh are the source or sink terms for composition and energy. The partial derivatives

VN

t

i

and

VU

t

t

express the dependency of total volume

on composition and total internal energy, respectively.For simplicity, we first consider the case without capillary

effects. The multiphase extension of Darcys law gives

u j = ( )k P g Drjj

j

k , . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

with P the pressure, k the permeability tensor, krj the relative per-meability of phase j, j the phase viscosity of phase j, j the mass density, and D the depth of the reservoir.

Governing Equations for Two-Phase Hot Waterfl oods. We note that in this work, we focus on hot waterfl ooding and will assume that the water and oil present in the reservoir are immiscible. For two phases in an immiscible water/oil system, the mass and energy conservation simplify to

+ =

+ =

tS q

tS q

t

w w w w

o o o o

u

u

w

o

,

,

11( ) +

+ + =

U S U h kr j j jj w o

j j c,

u j ( ) ==

T qj w o

h,

,

. . . . . . . . . . . . . . . . . . . . . . . . (6)and the pressure equation becomes

cPt

q qt

o

o

w

ww

w

o

o

+ ( ) + ( ) = +

1 1

u uo w

, . . . . . . . . . . (7)

where c SP

SPt

w

w

w o

o

o=

+

is the total compressibility.

If compressibility effects and thermal expansion are sufficiently small to be neglected, the pressure equation is given by the steady-state form:

( ) =kt P 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)where

t

rw

w

ro

o

kT

kT

= ( ) + ( ) is the total fluid mobility, k is the permeability tensor, and the fluid viscosities w(T) and o(T) are functions of temperature. The mass transport can be simplified further as

+ =St

ju j 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

Both the compressible and the incompressible formulations are implemented in our work. We start with the simplified incompress-ible hot waterflooding simulation and then include the compress-ibility effects.

The major thermal effects for thermal enhanced recovery proc-ess include reduced oil viscosity, volume expansion, wettability changes, and oil/water interfacial tension changes (Prats 1982). The viscosity reduction is generally the primary mechanism for thermal enhanced recovery, followed by fluid thermal expansion. We consider both effects in our work. The viscosity of water and oil can be entered either through tables or from the correlation (Prats 1982)

j j jA B T= ( )exp , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)where Aj and Bj are empirical parameters for the temperature dependent viscosities of water and oil.

We obtain water and oil liquid densities from the commonly used relationship

j jsc

j sc j scT P c P P a T T, exp( ) = ( ) ( ) , . . . . . . . . . . . (11)where cj and aj are the compressibility and thermal expansion coefficients, respectively.

Thermal Streamline FormulationWe first give the general framework of our thermal streamline simulator for hot waterflooding, which is also illustrated in Fig. 1. Further details of the individual stages are discussed in subsequent sections.

1. Given boundary conditions (well conditions) and initial or current solutions, the pressure equation (Eq. 7) is solved on the 3D (Eulerian) pressure grid. With the pressure known, the velocity is computed explicitly using Darcys law (Eq. 5). We use a finite-volume method to discretize the 3D pressure equation.

2. Given the total phase velocity field

u ut j==

j

np

1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)

Streamlines are traced using Pollocks analytical tracing method (Pollock 1988). During tracing of a streamline, the time of flight (TOF) equation (Eq. 14) of the entry and exit points of each grid cell crossed by the streamline is recorded. The initial 1D TOF grid along each streamline is highly irregular, which makes it difficult to solve the equations numerically. As a result, the grid is post-proc-essed to a regular TOF grid to make it more amenable to solve the advective portions of the transport equations along the streamlines. During the mapping from the original highly irregular grid to the regular grid, the properties are averaged, weighted by the TOF, to form the new properties on the regular grid.

3. Solution variables (composition, temperature and pressure) are mapped from the pressure grid onto the streamlines. For this step we use a piecewise linear representation with slope limiting is implemented for mapping from the pressure grid to the streamline (Mallison et al. 2004).

4. The advective parts of the mass balance equations and energy equation (Eq. 15) are then solved along individual streamlines using an appropriate numerical method and for a predefined global timestep over which the streamlines are assumed to be fixed.

5. At the end of the global timestep, the newly computed solu-tion variables are mapped back from the streamlines to the pressure grid. The gravity and heat conduction effects are accounted for on the Eulerian grid using an operator splitting approach. This finishes the global timestep. The process is now restarted from Step 1.

Streamline simulation uses two timesteps: a global timestep between pressure updates over which streamlines are assumed fixed, and local timesteps used when solving for the 1D mass and energy transport along the streamlines. The global timestep size can be estimated using fluid mobility changes, crossflow effects (gravity, capillary crossflow, and viscous crossflow) and diffusion effects (heat conduction). However, the proper number of streamlines


remains an unsolved problem and we have conducted extensive sensitivity studies to ensure we have properly captured the primary nonlinear effects for the problem we study. One problem arises because of mapping errors that occur from the mapping of properties to/from the Eulerian grid from/to the streamlines. In general, there is a compromise between pressure accuracy, which requires more pressure updates, and mapping error accumulations, which benefit from fewer pressure updates.

1D Advective Equations Along Streamlines. For ease of repre-sentation, we ignore the source terms in the equations. By rear-ranging the governing mass and energy equations and splitting the gravity fl ux from the pressure driven fl ux, we obtain the governing equations along each streamline, given by

+

+ ( ) +

=

( ) +

tC

sF F G

z

tU

i i ii

t

u u

u

t t

t

g 0,

+ ( ) +

=

==

s

Fh Fh Gzi i i ii

n

i

n

ucc

ut11

0g , . . . . . (13)

where Ci is the mass concentration, Ut is the total internal energy, Fi is the overall fractional flow function, Gi and GU are the gravity term for mass and energy transport. Spatial derivatives in these equations are taken with respect to the streamline arc length s. Following the traditional streamline formulation, we can instead use the TOF , given by

= ut0s

d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14)

Assuming that the rock is incompressible, the streamline equations can then be written as

+

+

( )+

=

( ) +

tC F F G

z

tU

i i ii

t

ut g 0,

Fh Fh G

zi ii

n

i ii

n

Uc c

= =

+ ( ) + =1 1 0ut g

.

. . . . . . . . . . . . . . . . . . . . . . . (15)

In the compressible case, the nonzero divergence of the veloc-ity is taken into account as source or sink terms in the mass and

energy transport equations, given, respectively, by Fi ( )ut

and

Fhi ii

nc

=

( )1

ut

.

We use both a first order explicit and a first order implicit method to solve these equations [Eq. 15 combined with the stand-ard first order single point upwinding (SPU) scheme.

Mappings. Solution variables are mapped from the streamlines to the pressure grid at the end of each global timestep, and from the pressure grid to the new streamlines at the start of the next global timestep. Mapping also takes place from the original irregular TOF grid to the post-processed regular grid. The composition and energy are averaged, weighted by TOF, onto the regular grids. Care must be taken when designing mappings because they can lead to smoothing of the solutions, as well as mass balance errors. Map-ping errors are not as great a concern for temperture (or energy) as for saturations (or compositions): The temperature fi eld is gener-ally smooth because of the heat conduction (diffusion transport), whereas saturation and composition may contain sharp gradients (nonlinear hyperbolic transport). To alleviate the numerical dif-fusion that is often introduced by the mapping from the pressure

1. Solve for pressure

5. Map solution back, accountfor nonadvective processes 2. Find velocity, trace streamlines

3. Map solution to streamline4. Update solution (advective part)

Fig. 1The steps in streamline simulation.


grid to the streamlines, we apply a piecewise linear interpolation of the variables based on total variation diminishing (TVD) slope limiting, as proposed by Mallison et al. (2004). When mapping back from streamlines to the pressure grid, we employ the standard running sum approach (Batycky 1997): For any pressure grid cell, we compute the solution variables as a weighted average over the N streamlines crossing the cell; that is,

C Ci ii

N

cell ==

1

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (16)

where Ci is a solution variable. The weights i are chosen to reflect the relative volumetric contribution of streamline i to the cell (Batycky 1997). We choose i = qii, with qi the volumetric flux of the streamline and i the TOF of the streamline that intersects that gridblock (Batycky 1997). In the incompressible case, qi is usually calculated as a portion of the total volumetric flux from the injection well. For compressible flow, the flux qi is calculated according to

q q ei odi

=

( ) v

0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (17)

which accounts for the compressibility of the total phase velocity field (Osako 2006).

Accounting for Volume Changes. The streamline approach fol-lowed here can be interpreted as a streamtube approach, in which each streamline has an associated pore volume. Streamlines are equivalent to streamtubes if one considers the relationship between a streamtube and the central streamline within a streamtube. As mass and energy are moved along the streamlines, the fl uid shrinks or expands because of the pressure and temperature dependence of the phase densities. We model the volume discrepancy between fl uid volume and pore volume using a dimensionless velocity approach similar to the one suggested by Mallison (2004) and Dindoruk (1992). We set

( ) +

( ) + ( ) =

=+

tC u F F

u u

i d i i

d k d

1 0

12

ut ,

, ,kk tV V

+

( )12

1 fluid cell , . . . . . . . . . . . . . . . . (18)

where Vfluid is the fluid volume, Vcell is the cell pore volume, and is a small relaxation parameter that represents the influence of fluid volume changes on the pressure field (magnitude of the flux).

For hot waterflooding, this dimensionless velocity approach is effective because the volume discrepancy is relatively small

because of the fluid density only being weak functions of absolute pressure and temperature.

Accounting for Heat Conduction and Gravity. At the end of a global timestep, we solve the diffusive part of the energy equation on the Eulerian pressure grid, while holding the other solution vari-ables, including pressure and saturations/compositions, constant in each gridblock. Gravity effects are also accounted for at this time, using operator splitting (Bratvedt et al. 1996).

After these corrections, the fluid volume will generally not fit the cell pore volume with a small discrepancy. We estimate the magnitude of these volume discrepancies and include them as source or sink terms in the 3D pressure equation in the subsequent global timestep. The discretized form of the pressure equation is now

cP P

t

V Vt

Ptn n

fn

pn

n+

+

=

( )

+ ( )1 1Tn . . . . . . . . . . . . . . (19)Here Vf

n represents the fluid volume and Vp

n is the pore volume

at time t n.The operator splitting applied to the energy and mass bal-

ance equations introduces an additional volume error. This error is generally acceptable provided the processes considered are advection dominated and changes in primary variables are small because of the operator splitting step. Our sensitivity test in the Results section confirmthe applicability of operator splitting for hot waterflooding.

ResultsWe test our thermal streamline simulator for hot water injection in a 2D quarter five-spot configuration. The heterogeneous domain is 500500 m2, which we discretize into 5050 cells The perme-ability is taken from the SPE 10 Comparative Solution Project (Christie et al. 2001) and is shown in Fig. 2. We use quadratic relative permeability curves krw = Sw

2 and kro = So

2 for water and oil

phases, respectively. The rock and reservoir parameters are listed in Table 1. We compare our simulation results with a commercial thermal simulator.

Incompressible Hot Waterfl ooding. Since the compressibility of water and oil liquid is usually small, we fi rst assume the liquids to be incompressible and we explore the temperature dependence of viscosity, using the relationship given by Eq. 10. We start the simulation with an initial oil saturation equal to 0.9, and an initial temperature of the oil and water set to be 20C. The temperature of the injected hot water is 80C. As in-situ oil is heated by the injected water, the oil viscosity is reduced dramatically, reduc-ing the mobility ratio M = o/w from M = 10 at 20C to M = 1 at 80C. Pressure controlled injection and production wells are implemented in all our simulations. The other fl uid properties for water and oil are listed in Table 2.

The base pressure grid in the streamline simulator is chosen as 5050. For comparison, we show finite volume simulator results for grids varying from 5050 to 400400. Comparisons of the water saturation, temperature and pressure fields are shown in Figs. 3 through Fig. 5, respectively. Here, we used 90 streamlimes and 20 global timesteps for an equivalent of 0.29 pore volumes injected (PVI).

The results show that our SLS results are with accuracy comparable to that of the finite volume simulator. In fact, the

Fig. 2The permeability field.

TABLE 2VISCOSITY RELATIONSHIPS

Viscosity coefficient, j=Ajexp(Bj/T)

Aj (cp) Bj (K)

Water 0.5 0.0 Oil 0.6541 10 5 3969

TABLE 1RESERVOIR PROPERTIES

Reservoir dimension (m2) 500 500 Thickness (m) 1 0.4 Rock thermal capacity [kJ/(m3K)]

2300

Rock thermal conductivity [kJ/(mdK)]

302


Fig. 3The water-saturation field.

Fig. 4The temperature field.

Fig. 5The pressure field.

TABLE 3FLUID PARAMETERS FOR INCOMPRESSIBLE HOT WATERFLOODING

Injection-well pressure, Pinj (kPa) 24 000 Production-well pressure, Pprod (kPa) 14 000 Injection-well water temperature, Tinj (C) 80 Initial oil temperature, Tinit (C) 20 Water thermal conductivity, [kJ/(mdK)] 51.8 Oil thermal conductivity [kJ/(mdK)] 51.8 Water density (kg/m3) 1004.26 Oil density (kg/m3) 981.76 Water heat capacity (kJ/kg) 4.19 Oil heat capacity (kJ/kg) 2.02


5050 SL solution appears closer to the 400400 reference than to the 5050 finite volume simulation result, showing reduced numerical diffusion and the ability to resolve channeling through the domain.

We considered both FIM and explicit SPU timestepping meth-ods along 1D streamlines. The results are shown in Fig. 6. The FIM 1D solver leads to slightly increased diffusion, but otherwise the results are comparable.

The sensitivity of the solution to the number of global timesteps in our case is demonstrated in Table 4. The table shows that the accuracy of the method depends on the number of global timesteps chosen. Too many pressure updates lead to increased mapping errors. On the other hand, too few pressure updates decrease the accuracy of the transport calculations because we are not captur-ing the temporal velocity field sufficiently well. How to determine the optimal number of pressure updates a priori for a specific reservoir process is an open research question and this is subject to ongoing research.

We also tested the sensitivity of the streamline simulator accuracy to the viscosity relationship j = Ajexp(Bj/T). The water viscosity is still assumed constant with respect to temperature. Of course, as the viscosity increases, the reservoir needs a larger driving force and hence a higher pressure difference between injec-tion and production wells. Both 5050 thermal streamline results and 5050 FD simulator results are compared with the reference 400400 FD result. The relative error is calculated according to

Fig. 6The comparison between 1D FIM transport solver and 1D SPU explicit transport solver for thermal SL simulation (5050).

TABLE 4ERRORS OF DIFFERENT VISCOSITY PROPERTY COMPARED WITH REFERENCE RESULT (400 400 STARS RESULT) USING L2 NORM

M=10 (20C) to M=1 (80C) PVI= 0.32

Number of global timesteps 2 5 10 20 40

2 5 10 20 40

2 5 10 20 40

Streamline error in the saturation 18.1% 12.8% 10.1% 9.1% 11.4%

Streamline error in the temperature 12.1% 8.0% 6.5% 7.1% 7.2% Error in the saturation of finite-difference (FD) simulator (50 50) with reference result 10.0% Error in the temperature of FD simulator (50 50) with reference result 5.9%

M=100 (20C) to M=1 (80C) PVI= 0.18

Number of global timesteps

Streamline error in the saturation 25.0% 22.6% 19.6% 17.7% 18.4% Streamline error in the temperature 22.1% 16.0% 12.5% 10.7% 10.5% Error in the saturation of FD simulator (50 50) with reference result 17.6%

Error in the temperature of FD simulator (50 50) with reference result 11.0%

M=1,000 (20C) to M=10 (80C) PVI= 0.10

Number of global timesteps

Streamline error in the saturation 45.4% 35.6% 29.7% 26.2% 30.5% Streamline error in the temperature 35.7% 28.2% 22.5% 19.2% 17.5% Error in the saturation of FD simulator (50 50) with reference result 18.8% Error in the temperature of FD simulator (50 50) with reference result 13.6%

error = XX

. L2 norm is used here for error calculations. The

surface cumulative production is also plotted.Our results show that increasing exponents in the temperature

dependent viscosity relationship j = Ajexp(Bj/T) lead to larger errors in the streamline simulations. Keeping the total velocity fixed during a global timestep in the case of such highly viscous oil leads to more errors. Also since we are using the initial veloc-ity field un for the transport between time level t n and t n+1, we are underestimating the water production. We can improve the accuracy by increasing the number of pressure updates.

Compressible Hot Waterfl ooding. For hot waterfl ooding prob-lems with low compressibility, we use the dimensionless-velocity approach discussed in the previous. The initial oil saturation in this case is set at 1.0. All other variables are the same as in the preced-ing test case. The temperature dependent viscosity relationship is the same as the M=10 (80C) to M=1 (20C) test case. The fl uid density calculations parameters are listed in Table 5. The refer-ence pressure is Psc = 100 kPa, and the reference temperature is Tsc = 273.15 K.

The results are shown in Figs. 10 through 12. The compress-ible thermal streamline simulator is able to account for the effect of volume changes and again yields results with similar accuracy when compared to the 400400 FD simulator results.


Time (day)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Water Injection (Streamline)Water Injection (STARS 5050)Water Injection (STARS 400400)

Wat

er In

ject

ion

(1

103 m

3 )

104104

40

35

30

25

20

15

10

5

0

Time (day)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Water Production (STARS 5050)Oil Production (STARS 5050)Water Production (STARS 400400)Oil Production (STARS 400400)Water Production (Streamline)Oil Production (Streamline)

Pro

duct

ion

(1

103 m

3 )35

30

25

20

15

10

5

0

Fig. 7The surface cumulative production and injection for M=10 test case.

Time (day)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


Wat

er In

ject

ion

(1

103 m

3 )

104104

30

25

20

15

10

5

0

Time (day)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


Pro

duct

ion

(1

103 m

3 )

20

18

16

14

12

10

8

6

4

2

0

Fig. 8The surface cumulative production and injection for M=100 test case.


Time (day)0 0.5 1 1.5 2 2.5 3 3.5

Wat

er In

ject

ion

(1

103 m

3 )

104


1040 0.5 1 1.5 2 2.5 3 3.5

Pro

duct

ion

(1

103 m

3 )

35

30

25

20

15

10

5

0

25

20

15

10

5

0

Fig. 9The surface cumulative production and injection for M=1,000 test case.


We have also explored the sensitivities of larger thermal com-pressibility for hot waterflooding. Usingthe previous case, we have increased the water and oil compressibility to cw=co= 5106/kPa, aw = ao = 5103/K. The production and injection results of SL and the reference solutions are reported in Table 6.

Once again, the thermal streamline simulation yields a solu-tion comparable to the reference. We can also observe the thermal compressibility effects in the oil production from our results. Compared with the low compressibility case, the high compress-ibility system needs less amount of water injected to produce the equivalent amount of oil at the producer.

Gravity Effect. Gravitational effects are included through operator splitting (Bratvedt et al. 1996). We test the approach on a tilted version (30 tilt) of our test reservoir for incompressible fl uids. The results are shown in Figs. 13 and 14.

The result show that by changing the depth of the reservoir, we have accelerated and decelerated the transport of water and oil in the reservoir. However, in this case the water and oil density dif-ference is relatively small, leading to a weak gravity segregation effect between the phases.

Heat Conduction Effect. The streamline method is best suited for simulation of an advection dominated fl ow process. Nonadvective

TABLE 5COEFFICIENTS OF DENSITY CALCULATIONS

Coefficients of density calculations

(kg/m3) cj (1/kPa) aj (1/K)

Water 998 1.0 10 7 1.0 10 4

Oil 972 1.0 10 7 1.0 10 4

effects (gravity, heat diffusion, capillarity) are accounted for using operator splitting. As the heat Peclet number, Pe, decreases, we expect the performance of the streamline simulator to worsen. The heat Peclet number is the ratio of heat advection to conduc-

tion. Here, we set Pe =

L u

Cv

, where L is the distance between

the injector and producer, the average heat conductivity, Cv the average total heat capacity, and u the average velocity along the diagonal direction. We have tested the original incompressible test problem for different Pe (Pe=1.5, Pe=15, Pe=150) by arbitrarily changing the heat conductivity of the fl uid and rock. Again, both the 5050 streamline and 5050 FD simulation results are com-pared with the refi ned 400400 FD simulation results. The results are listed in Table 7.

For Pe=15 and Pe=150, the SL errors are comparable with the FD simulation for both saturation and temperature fields because the flow is advection dominated. For the smaller Pe (Pe=1.5), the SLS error in the temperature is larger than the FD simulator temperature error (8.4% for SL compared to 5.3% for FD). We believe this is because of solving the heat conduction and heat convection on dif-ferent fractional timesteps. The saturation results show a weaker relationship with Pe. This is because mass transport is purely advec-tive flow; thus, it is less sensitive to the operator splitting.

Extension to 3D Problems and Steam-Injection Simulations. Extension to 3D reservoir problems and to steam follows the same principles and procedures as the 2D cases described here. A 3D pressure solve will be performed initially, followed by streamline tracing in the 3D space from injector to producer.

Then, the mass and energy transport will be calculated along these 1D streamlines. Eventually, the solution will be mapped back to the Eulerian grid. The gravity effects and heat conduction will be accounted for on the 3D Eulerian grid using operator splitting.

Fig. 10The saturation field of compressible streamline simulation.

Temperature (C)

Fig. 11The temperature field of compressible streamline simulation.


For steam-injection simulations, the flow physics willbe significantly more complicated. The greatest challenge is that the volume changes are now much more severe, as steam condenses into the liquid phase or water vaporizes into the gas phase. We hereby propose a streamline simulation framework that is differ-ent from the classical streamline approach. Initially, a 3D pressure solve is performed to define the flow paths or streamlines in the 3D space. Then, it will be necessary to solve the 1D pressure equation along with the transport equations on the 1D streamline grid, to account for the phase changes. At the end of the global timestep, the mass and energy will again be mapped back to the Eulerian grid. A new global timestep will be launched and the process repeated. This approach is the subject of ongoing research.Computational Cost Comparison. The computational cost asso-ciated with streamline simulation is generally considerably lower than for Eulerian simulation methods. For a fully implicit Eulerian simulation, the computational costs can be estimated as

T c N c N k nFIM = ( ) + ( ) solver jacobi time3 3

32

sstep, . . . . . . . . . . . . (20)

where N is the number of total unknowns, k the average number of Newton loops, and ntimestep the total number of timesteps. In

104Time (day)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


Pro

duct

ion

(1

103 m

3 )35

30

25

20

15

10

5

0

Fig. 12The surface cumulative production comparison of compressible streamline simulation.

TABLE 6EFFECTS OF THERMAL COMPRESSIBILITY ON THE PRODUCTION (SURFACE CONDITION)

Small compressibility: cw=co=1 10 7/kPa, aw=ao=1 10 4/K

Oil produced (104m3) Water produced (104m3) Water injected (104m3)

SLS (50 50) 3.218 0.183 3.373 STARS (50 50) 3.121 0.193 3.299 STARS (200 200) 3.312 0.359 3.669

Large compressibility: cw=co=5 10 6/kPa, aw=ao=5 10 3/K

Oil produced (104m3) Water produced (104m3) Water injected (104m3) SLS (50 50) 3.126 0.143 2.872 STARS (50 50) 3.068 0.166 2.852 STARS (200 200) 3.266 0.304 3.143

Fig. 13The saturation field for results with gravity.

Fig. 14The saturation field for results with gravity.


contrast, the simulation time for streamline simulation can be estimated as

T n c n M tSL = +( )timestep streamline other , . . . . . . . . . . . . . . . . . . (21)where nstreamline is the number of total streamlines, M the average cost of transport solve for each streamline, and tother is the time spent on the pressure solve, streamline mappings, and tracings. In compositional and/or thermal processes, the most significant contribution to the computational cost comes from the 1D trans-port calculations. Therefore, the streamline method gives rise to a near linear scaling of run time with the number of active cells. It makes streamline simulations extremely attractive for solving large, complex, and heterogeneous reservoir simulation problems that are dominated by advection.

ConclusionWe have presented a thermal streamline framework for two-phase hot waterflooding simulation with the thermal effects of tempera-ture dependent viscosity and thermal compressibility. Our method is based on an operator splitting approach in which the advective parts of the governing equations are solved along the streamlines while the heat conduction and gravity solved onthe original Eul-erian grid.

The streamline results are compared with a commercial finite-volume simulator. We have shown that the method is capable of producing comparable results for quarter five-spot test problems at a lower computational cost. Various sensitivity tests were performed to study the accuracy and robustness of the method. Our results provide guidelines for implementing the streamline simulation for thermal recovery and as a fast proxy with reasonable accuracy.

ReferencesAziz, K. and Settari, A. 1979. Petroleum Reservoir Simulation. Essex, UK:

Elsevier Applied Science Publishers.Batycky, R.P. 1997. A Three-Dimensional Two-Phase Field Scale Stream-

line Simulator. PhD dissertation. Stanford University, Stanford, Cali-fornia, USA (January 1997).

Batycky, R.P., Blunt, M.J., and Thiele, M.R. 1997. A 3D Field-Scale Streamline-Based Reservoir Simulator. SPE Res Eng 12 (4): 246254. SPE-36726-PA. doi: 10.2118/36726-PA.

Batycky, R.P., Frster, M., Thiele, M.R., and Stben, K. 2009. Parallel-ization of a Commercial Streamline Simulator and Performance on Practical Models. Paper SPE 118684 presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, USA, 24 February. doi: 10.2118/118684-MS.

Beraldo, V.T., Blunt, M.J., and Schiozer, D.J. 2008. Compressible Stream-line-Based Simulation With Changes in Oil Composition. SPE Res Eval & Eng 12 (6): 963973. SPE-115983-PA. doi: 10.2118/115983-PA.

Berenblyum, R.A., Shapiro, A.A., Jessen, K., Stenby, E.H., and Orr, F.M. Jr. 2003. Black Oil Streamline Simulator With Capillary Effects. Paper SPE 85037 presented at the SPE Annual Technical Conference and Exhibition, Denver, 58 October. doi: 10.2118/84037-MS.

Berre, I., Dahle, H.K., Karlsen, K.H., and Nordhaug, H.F. 2002. A Stream-line Front Tracking Method for Two- and Three-Phase Flow Including Capillary Forces. Contemporary Mathematics 295: 4961.

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Crane, M., Bratvedt, F., Bratvedt, K., and Olufsen, R. 2000. A Fully Compo-sitional Streamline Simulator. Paper presented at the SPE Annual Tech-nical Conference and Exhibition, Dallas, 14 October. doi: 10.2118/63156-MS.

Dindoruk, B. 1992. Analytical theory of multiphase, multicomponent displacement in porous media. PhD dissertation, Stanford University, Stanford, California, USA.

Donato, G.D. and Blunt, M.J. 2004. Streamline-Based Dual-porosity Simu-lation of Reactive Transport and Flow in Fractured Reservoirs. Water Resour. Res. 40: W04203. doi: 10.1029/2003WR002772.

Gerritsen, M.G., Jessen, K., Mallison, B.T., and Lambers, J. 2005. A Fully Adaptive Streamline Framework for the Challenging Simulation of Gas Injection Processes. Paper SPE 97270 presented at the SPE Annual Technical Conference and Exhibition, Dallas, 912 October. doi: 10.2118/97270-MS.

TABLE 7HEAT PE AND ITS INFLUENCE ON THE SIMULATION RESULT

Pe=1.5 Streamline error in the saturation 9.4%

FD-simulator error in the saturation (50 50) with reference result (400 400)

8.4%

Streamline error in the temperature 8.4%

FD-simulator error in the temperature (50 50) with reference result (400 400)

5.3%

Pe=15 Streamline error in the saturation 9.4%

FD-simulator error in the saturation (50 50) with reference result (400 400)

8.7%



7.8%

Pe=150 Streamline error in the saturation 9.3% FD-simulator error in the saturation (50 50) with reference result

(400 400) 10.8%



11.1%


Gerritsen, M.G., Lf, H., and Thiele, M.R. 2009. Parallel implementations of streamline simulators. Computational Geosciences 13 (1): 135149. doi: 10.1007/s10596-008-9113-y.

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Zhouyuan Zhu is a PhD candidate in energy resources engi-neering at Stanford U. E-mail: [email protected]. His inter-ests include reservoir simulation and heavy oilin particular streamline method and in-situ combustion simulation. He holds a BS degree from Tsinghua U., China and an MS degree from Stanford U. Margot Gerritsen is an associate professor in the department of energy resources engineering at Stanford U. E-mail: [email protected]. She holds BS and MSc degrees from Delft U. of Technology and a PhD degree from Stanford U. in scientific computing and computational math-ematics. Gerritsen specializes in the development of numeri-cal algorithms for fluid flow applications, and in particular in enhanced oil recovery processes. Marco Thiele is cofounder of Streamsim Technologies and a consulting professor at Stanford U. E-mail: [email protected]. He holds BS and MSc degrees from the U. of Texas at Austin and a PhD degree from Stanford U., all in petroleum engineering. Thiele is a recipient of the 1996 SPE Cedrick K. Ferguson Medal, serves on the SPE Books Committee and is an Associate Editor for SPE Res Eval & Eng.

Documents

SPE-119200-PA (Thermal Streamline Simulation for Hot Waterflooding)