Streamline-based History Matching of Arrival

Times and Bottom-Hole Pressure Data for

Multicomponent Compositional Systems

Shusei Tanaka

November, 2015

Motivation

Streamlines0.0

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WCT/GOR

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Time [Days]

BHP

Observation

Initial

2/21

Update in geological model

What can we tell prior to injected water breakthrough?

• Pressure data needs to be incorporated within streamline framework

How to generalize streamline-based history matching to multicomponent compositional system?

• Need to develop a technique to integrate pressure and component information

Objectives

Propose a novel approach to compute pressure and injected gas arrival time sensitivity using streamlines• Analytic sensitivity of bottom hole pressure (BHP), GOR and injection

component

• Application to compositional, gas injection problem

Simultaneous inversion of BHP, GOR and primary injection component• Integrate gas production data with injector and producer BHP

• Application to the Brugge benchmark case

3/21

Streamline-Based Inverse Modeling

1. Run simulation with prior models 2. Trace streamlines and map underline grid property

4. Update reservoir parameters

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t

Time [Days]

𝛿𝑡

Observation

𝜕𝑡

𝜕𝑘𝑖

3. Calculate parameter sensitivities along streamlines 4/21

𝛿𝑡: Travel time shift

SL-Based Travel Time Sensitivity(Vasco, Yoon and Datta-Gupta, 1999)

TOF(t ): Travel time of a neutral tracer along streamline

ik

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Time [Days]Travel time shift

injectorProducer

𝜏 𝑥, 𝑦, 𝑧 = න𝐼𝑛𝑙𝑒𝑡

(𝑥,𝑦,𝑧)𝜙

𝑢𝑑𝜉

𝛿𝒕

𝜕𝜏

𝜕𝑘𝑖=𝜕∆𝜏𝑖𝜕𝑘𝑖

= −න𝜉𝑖

𝜙

𝜆𝑘2 𝛻𝑝𝑑𝜉 = −

∆𝜏𝑖𝑘𝑖

𝜕𝑡

𝜕𝑘𝑖= −

𝜕𝑆

𝜕𝜏

𝜕𝜏

𝜕𝑘𝑖∙𝜕𝑆

𝜕𝑡

−1

=1

𝑓′(𝑆)

𝜕𝜏

𝜕𝑘𝑖

• Time of flight sensitivity:

• Water-cut travel time sensitivity:

5/21

SL-Based Bottom Hole Pressure Sensitivity

Use local pressure drop along streamline and underline property

Rate-Rate constraint

Rate-BHP constraint

• Bottom hole pressure sensitivity along streamline

ikinjector

Producer

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Time [Days]Production BHP

i

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ppppp

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p

......21

𝜕𝑝𝑏ℎ𝑝

𝜕𝑘𝑖= ±

𝜕∆𝑝𝑖𝜕𝑘𝑖

≈ ±∆𝑝𝑖𝑘𝑖

𝜕𝑝𝑏ℎ𝑝

𝜕𝑘𝑖≈ ±

𝜏𝑖𝜏

𝜕∆𝑝𝑖𝜕𝑘𝑖

≈ ±𝜏𝑖𝜏

∆𝑝𝑖𝑘𝑖

6/21

Sensitivity Comparison: 3-phase Gas Injection

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Analytical (Stremaline)

Adjoint Method

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Analytical (Stremaline)

Adjoint Method

Inj: Gas Rate

Prd: Rate

Producer BHP sensitivity to k

Injector BHP sensitivity to k

7/21

AdjointStreamline

Sensitivity Comparison: 2D Areal Example

Injector BHP sensitivity by k

P4 BHP sensitivity of by k

Permeability field(Wells by rate constraint)

Adjoint Proposed

8/21

Injector

P1

P2 P3

P4

Extension to Multicomponent System:

Gas and Component Sensitivity Equation

𝜕𝑡

𝜕𝑘𝑖= −

𝜕𝜏𝜕𝑘𝑖

𝜕𝜕𝜏

𝑆𝑔𝑏𝑔 + 𝑅𝑆𝑆𝑜𝑏𝑜

𝜕𝜕𝜏

𝐹𝑔𝑏𝑔 + 𝑅𝑆𝐹𝑜𝑏𝑜 +𝑐𝜙

𝐹𝑔𝑏𝑔 + 𝑅𝑆𝐹𝑜𝑏𝑜

• Travel time sensitivity of gas (GOR)

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R [

-]

Time [Days]

Observed

Initial

Updated

9/21

𝜕𝑓𝑘𝜕𝑘𝑖

=𝜕𝑓𝑘𝜕𝑧𝑘

𝜕𝑧𝑘𝜕𝜏

𝜕𝜏

𝜕𝑘𝑖

• Amplitude sensitivity of the production component (primary injection component):

GOR is informative, but applicable only if pressure data is matched

Limited data points but able to match breakthrough of injected gas component at producer

GOR

Mole fraction

Travel time toMaximize overlap

Amplitude match of each data point

Sensitivity Comparison:

2D Areal CO2 Flood, Gas Travel Time

NumericalPermeability field SL-based AnalyticalP4

10/21

Injector

P1

P2 P3

Sensitivity Comparison:

2D Areal CO2 Flood, CO2 Amplitude

Numerical SL-based Analytical

11/21

Permeability fieldP4

Injector

P1

P2 P3

Permeability Update by Iterative Linearized

Inversion (LSQR)

• Define a penalized misfit:

• Iterative inversion to update parameters :

Advantages:• Post process to compute sensitivities analytically along streamlines • High resolution updates in permeability to match well data

Injection component GOR - Smoothness

- Consistency with prior

static model

Scaled by stdev

min 𝛿𝐝𝑖𝑐 − 𝐒𝑖𝑐𝛿𝐤 + 𝛿𝐝𝐺𝑂𝑅 − 𝐒𝐺𝑂𝑅𝛿𝐤 + 𝛿𝐝𝑏ℎ𝑝 − 𝐒𝑏ℎ𝑝𝛿𝐤 + 𝛽1 𝐈𝛿𝐤 + 𝛽2 𝐋𝛿𝐤

𝐒𝑖𝑐𝐒𝐺𝑂𝑅𝐒𝑏ℎ𝑝𝛽1𝐈𝛽2𝐋

∆𝐤 =

𝛿𝐝𝑖𝑐𝛿𝐝𝐺𝑂𝑅𝛿𝐝𝑏ℎ𝑝00

Pressure

12/21

History Matching: Multicomponent Gas

Injection

• 7 component 5-spot CO2 injection• Matching injection BHP, production GOR and CO2 mole fraction

Reference model Initial model

Inj

P1

P2P3

P4

13/21

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Time [Days]

Observed P1 Observed P2 Observed P3 Observed P4 Initial P1 Initial P2 Initial P3 Initial P4

Initial Data Misfit

Injector BHP(Amplitude, average)

Production GOR(Travel Time)

Producer CO2 composition

(Amplitude, every point)

14/21

observed

Calculated

Reduction in Objective Functions

Data mismatch wrt. iteration(Normalized RMSE)

15/21

Data Misfit After History Match

Injector BHP Production GOR Producer CO2composition

16/21

Updated Permeability Field

Reference model Initial model Updated

17/21

Injector

P1

P2 P3

P4

DecreasedIncrased

History Matching of Brugge Benchmark Field

• Use simulation result of realization 77 as observed data• Use realization 1 as initial model• Match GOR, BHP and end time production CO2 (at 10 yrs)

Reference model Initial model

Producer

CO2 Injector

18/21

Change in Permeability

19/21

Reference k Initial k

Change of k, GOR Change of k, using all data

High perm at middle layer

Reduction in Data Misfit per Well

20/21

Pressure RMS error(30 wells)

GOR RMS error(10 producers)

Individual well

Mean

Production CO2 mole fraction at 10 yrs

(20 producers)

Conclusion

21/21

We have developed a novel Streamline-based method to integrate pressure data into prior geologic models

• Can be applied to field data prior to breakthrough with water/gas injection multicomponent system

The method offers the same advantages as prior streamline work:• Analytic calculation of sensitivities comparable with Adjoint-

based calculation• Requires single flow simulation per iteration • Applicable with conventional Finite Difference simulations