Dual Mesh Method in UpscalingDual Mesh Method in Upscaling

Pascal Audigane and Martin Blunt Imperial College

London

SPE Reservoir Simulation Symposium, Houston, 3-5 February 2003

Fine MeshLarge memory and time requirements

Coarse Mesh

Upgridding+

Upscaling

As the techniques for reservoir modeling are improved, very refined meshes to represent the reservoir can be generated. However, time and memory constraints often prevent flow simulation on this grid, forcing the use of an upscaled, coarser grid.

Watercut

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frac

tiona

l flo

wUsing a coarser grid introduces errors in predictions of recovery

An example of the difference obtained for waterflooding simulation performed on a fine mesh (30 x 30 cells) and a coarse mesh (10 x 10 cells). Geometric averaging of absolute permeability.

fine

coarse

The problem with upscaling

The aim of the Dual Mesh Method

To reduce this error so that the simulation performed with the coarse mesh will provide a good estimate in little time

How?

By reintroducing the fine mesh information

In this study, only waterflooding simulation is considered.

Previous work1. Verdière and Guérillot (1996): dual mesh method with

geometric averaging for upscaled properties. 2D, no wells, no gravity.

2. Guedes and Schiozer (1999): 2D with gravity.3. Gautier and Blunt (1999): nested gridding in the context of

streamlines. 3D with gravity and wells.

This work: -3D with gravity and wells. -Different upscaling methods. -First step towards a general multi-scale upscaling technique.

Uses a standard IMPES approach to reservoir simulation, but….

Solving the Pressure field with an implicit scheme is the most time consuming step of the simulation, hence :

1- Pressure field is solved on the coarse mesh

Saturation is updated with an explicit scheme and can be easily handle on grids containing large number of grid-blocks:

2- Saturation field is updated on the fine mesh

The method is implemented in 3 steps…

The Dual Mesh MethodThe Dual Mesh Method::

STEP 1STEP 1

The pressure field is found on the coarse mesh with all the properties upscaled using any type of upscaling technique.

• Fine Mesh • Coarse Mesh

Upscaling

PressureVelocity

Darcy’s law

1- Coarse pressure value as a Dirichlet condition on the centre of the coarse cell

2- Coarse flux weighted by interblock transmissivities on the fine cell faces

• Coarse Mesh

A refinement technique is used to compute the total velocity on the fine mesh using coarse scale information

STEP 2STEP 2

For each coarse grid block, the following pressure solve is done using:

Performing STEP 2 for all coarse grid blocks, the velocity field can be recomputed on the entire fine grid, and the saturation field can be up-dated on the fine grid. Then the simulation can be performed for the next time step with the new saturation field (STEP 1).

STEP 3STEP 3

Darcy flow field used to update saturation explicitly at the fine scale:

•For the fine cells in contact with coarse cell face, the weighted flux is applied.

•Inside each coarse grid block, the flux across each fine cell face is deduced from the pressure gradient.

• Mass balance is ok

1. Pressure solver method (PSM):

For each coarse grid block each effective transmissivity component is deduced from the flux obtained with a pressure drop applied on two opposite faces of the cell and no flux for the other faces

2. Geometrical average (GA):perform on the product between absolute permeability and total mobility

Pi (pressure drop)

Teffi = Qcal / Pi

Dir i

One Coarse Grid block:

No flux

Two different upscaling techniques:

Heterogeneous Medium, 2D Horizontal, 2 PhasesHeterogeneous Medium, 2D Horizontal, 2 Phases

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The watercut estimates are better using the dual mesh method in comparison with the simulation performed on the coarse mesh only

Permeability : 80 % 100 mD (Black)20 % 1 mD (White)

Porosity: 0.2Fine mesh: 30 x 30 Coarse mesh: 10 x 10 Reservoir size: 300 m x 300 m x 1m

Water injection: 5 m^3 / day

Fluid production: 76 bars

finecoarse

Watercut comparisons

Dual Mesh Method

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Saturation Field Comparison after 300 daysSaturation Field Comparison after 300 daysFine GridFine Grid

PSM casePSM case

Coarse GridCoarse Grid

GA caseGA case

Lack of precision for the simulation performed only on the coarse mesh

Improvement of the saturation field simulation especially for the PSM case.

With the dual mesh method

Injection Production

2D with Gravity2D with GravityFine: 60 x 60

Coarse: 10 x 10

Gravity number :

Ng = gravity / viscous = 0.3

Permeability: 80 % 100 mD (White)

20% 0.1 mD (Red)

Porosity: 0.2

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The dual mesh method using the pressure solve method (psm) is virtually identical to the fine mesh simulation

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Watercut comparisons

Dual Mesh Method

Three different scenarios have been set up based on the SPE Comparative project on upscaling (Christie & Blunt, 2001).

The original grid contains more than one million grid blocks but the size of the problem has been reduced to a fine mesh: 20x55x85 = 93,500 cells and a coarse mesh: 5x11x5= 275 cells.

upper lower

Porosity scale: blue <= 0.1, white = 0.15 and red >= 0.2

3D Results3D Results

Permeability: dual facies, 80% of high permeable zone of 100mD and 20% at 1mD. 1 injector and 1 producer well.

The same as 1 but with 1 central injector well and 4 producers at each corner.

Permeability and Porosity created for the SPE Comparative Project 2001: 1 central injector and 4 producers at each corner

Case 1

Case 2

Case 3

Case 1

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The Dual Mesh Method is very good at matching the fine mesh simulation especially with the use of the pressure solve method for upscaling (psm)

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Dual Mesh Method

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Case 2

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Case 3

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Cumulative oil Produced Comparison (case 3)Cumulative oil Produced Comparison (case 3)

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. Oil

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5 x 11 x 6 SPE 5 x 11 x 5 coarse 5 x 11 x 5 ga 5 x 11 x 5 psm30 x 55 x 17 SPE20 x 55 x 85 fine60 x 220 x 85 SPE

Time of Calculation

2D 2Dv 3D SPEFine mesh 30x30 60x60 30x30x30 20x55x85Coarse mesh 10x10 10x10 10x10x10 5x11x17

DMM 0.9 2.1 2.7 3.9NESTED 0.6 1.3 1.3 3.1COARSE 3.6 28 94 432

Speed Factor: time(fine) / time(method)

( > 1 ?), Gautier et al, 1999: 3 to 4 (in 3D with gravity)Verdiere and Guerillot, 1996: from 4 to 7 (in 2D, no gravity)

• Fine • Coarse

Upscaling

• Coarse

Next? A fully decoupled approachNext? A fully decoupled approach

Step 1Step 1- The same as before: upscaling from the fine to the coarse mesh

Step 2Step 2- The Pressure field AND THE SATURATION FIELD are updated on the coarse mesh.

Step 3Step 3- The saturation field is recomputed on the fine mesh inside each coarse grid block separately rather than

on the whole fine grid.

Why?Why?Less time-step restriction. Only perform fine grid simulation on selected cells.Save CPU time and memory

• Coarse

The whole simulation is done on the coarse mesh: pressure and sat update

ConclusionsConclusions

• Dual Mesh Method has been implemented in 3D with gravity effects and well boundary conditions.

• This method is able to reduce considerably the error between a conventional simulation performed on the chosen coarse mesh and the one performed on the initial fine mesh.

• The accuracy of the dual mesh method depends on the complexity of the reservoir heterogeneity

• Different up-scaling techniques can be applied.

• The time of calculations is reduced compared with the simulation performed on the fine mesh.

References- Christie, M A. and Blunt, M J., (2001), “Tenth SPE Comparative Solution

Poject: A comparison of Upscaling Techniques”, SPE 66599, presented at the SPE Reservoir Simulation Symposium, Houston, Texas, 11-14 February 2001

- Gautier, Y. and Blunt, M J.,(1999), “ Nested Gridding and Streamline-Based Simulation for Fast Performance Prediction”, Computational Gesosciences 3, 295-320

- Guedes, S S. and Schiozer, D J., (1999), “An Implicit Treatement of Upscaling in Numerical Reservoir Simulation”, SPE 51937, presented at the SPE Reservoir Simulation Symposium, Houston, Texas, 14-17 February 1999

- Verdière, S. and Guérillot, D., (1996), “Dual Mesh Method in Heterogeneous Porous Media”, 5th European Conference on the Mathematics of Oil Recovery, Leoben (September).