A general pore-to-reservoir transport simulator

Matthew E. Rhodes and Martin J. BluntPetroleum Engineering and Rock Mechanics Group

Department of Earth Science and Engineering

Outline

Motivation Transport Algorithm Upscaling

Strategy Validation Multiscale modelling Field Scale Simulation

Conclusions and Future Work

Motivation – Modelling Non-Reactive Contaminant Transport

Petroleum Engineering

Hydrology

Mass Balance

Characterise and solve numerically

Generic behaviour

Coarse Scale – Gaussian-like

Fine Scale – Anomalous Transport

Statistical Theory

FractalsADE – Constant

Parameters – Gaussian Behaviour

CTRW – Single parameter-

Anomalous Transport

Motivation – Field Scale Transport

We are interested in modelling single phase contaminant transport in porous

media. This can be of two forms: Gaussian-like plume

spreading Found in statistically

homogeneous media (rarely observed but often assumed)

Anomalous Transport Typical field scale profile Invariant concentration peak Early breakthrough Long tail arrival distributions

Motivation

We want to simulate single phase transport across all scales in reservoir systems without assuming an average PDE

We must account for the appropriate reservoir physics at each scale of interest

We therefore require an algorithm that allows us to upscale without presupposing the effective PDE for transport

Outline

Motivation Transport Algorithm Upscaling

Strategy Validation Multiscale modelling/ Field Scale Simulation

Conclusions and Future Work

Transport Algorithm - The Porous Medium

The first step in our algorithm involves the generation of a representative grid

We suggest converting the porous medium into a topologically equivalent network model of nodes connected by one-dimensional links

This is no different from current reservoir engineering approaches in which Cell Centres Nodes Cell Transmissibility Links

But then how do we model the fluid flow?

Transport Algorithm – Fluid Motion To model transport we couple CTRW formalism with Monte

Carlo simplicity In the CTRW framework, transport is viewed as a series of

discrete transitions from node to nearest node: This has the disadvantage of particles only being located

physically at the nodes But if this approximation can be tolerated, an increase in

computational efficiency can be derived using our method We can therefore move particles from point to point in a

time t But,

how do we calculate this t and determine the neighbour to which a particle would jump to?

Transport Algorithm - CTRW To address these issues, we use

the CTRW approach and define a probability, (t).dt, that a “particle” will arrive at a nearest neighbour in a time t+dt

In CTRW formalism, this (t) is usually assumed to be spatially constant

But we need to explicitly account for the system’s heterogeneity so how would we do this???

We assume that 1D ADE represents these jumps.

As such we can write the following for the each branch in the system:

2

2i i i

i i

C C CV D

t x x

x=0 is the central node Subscript k denotes a bond with a

node Lk units from the junction and a local velocity that is in the direction of the flux flowing through it

Subscript j denotes a bond with a node -Lj units from the junction and a local velocity that is in the opposite direction to the prevailing flux

Transport Algorithm - (t) cont’d

But we are interested in the arrival probability at the exit node!

This is equivalent to the flux arriving at each node which is given by:

But then how do we calculate t? We know that distribution of

transit times along each link is given by (t)

2

,

sinh

i

i

i ii x L

Pe

i ii

i i

i k j

Cs D

x

D B s es

L

We can sample this distribution by defining a cumulative distribution of arrival times in the Laplace domain:

invert this numerically and then follow a Monte Carlo approach similar to that of Sorbie and Clifford (1991)

We then use the final value theorem to obtain analytical expressions for the exit node

( ) ii

sF s

s

Transport Algorithm We first choose a uniform random number, z, between 0 to 1 We then calculate the probability of jumping to each node

and convert this to a cumulative probability, Pin defined by

the recurrence relation given below:

If z falls in the range Pi-1n ≤ z < Pi

n the particle will jump to node i, otherwise increase i

We can then normalise z and Fi(s) with respect to the actual branch probability, Pi using:

`

1

0

1

0 and 1

n ni i i

nN

P P P i N

P P

1 ; n

inin i

i i

F sz Pz F s

P P

Transport Algorithm

Finally we obtain the time that is equivalent to zn by numerically inverting Fn(s) using the Stehfest (1970) algorithm and employing the bisection method to solve the equation:

Cumulative Probability (Fn(t)) vs. Time/s

0.000.100.200.300.400.500.600.700.800.901.00

0 0.5 1 1.5 2 2.5 3 3.5

Time/s

Fn(t

)

0ni nF t z

Transit Time t

Outline

Motivation Transport Algorithm Upscaling

Strategy Validation Multiscale modelling Field Scale Simulation

Conclusions and Future Work

Upscaling - Strategy We empirically determine at each simulation scale of interest a

(t) that acts as a proxy for explicit reservoir heterogeneity We start at the pore scale where the transport physics are known At each length scale the system is represented as a lattice of

nodes connected by links (throats) This function we then use as an input to model a subsequent

reservoir scale

p (t)

g (t)

Upscaling – Strategy (p(t))

We start by determining the (t) at the pore scale where the transport physics is well understood Stokes Flow Molecular Diffusion

Bijeljic et. al (2004) developed a pore scale model of dispersion that explicitly tracked particles through a series of advective and diffusive displacements in a network model of pores and throats

They were able to obtain an excellent match to experimental results in the literature

10<Pe<400 -

Upscaling – Strategy (p(t)) Bijeljic and Blunt (2006) determined that the ensemble transit time

distribution averaged over every pore-to-pore transition and all possible statistically equivalent networks is:

This function was found to be a best fit for over 6 orders of magnitude in time and Pe number

As there is no long range heterogeneity we can model transport with an ensemble average network which is homogeneous and this single function

This is different to the ade(t) which does not account for the distribution of times that arise from velocity variation due to heterogeneity (must be explicitly defined to obtain the correct macroscopic behaviour)

Upscaling – Porescale Simulation

p(t) + 3D Homogeneous Lattice

ADE + 3D Heterogeneous Lattice

Results of Bijeljic et. al (2004)ADE + Topologically disordered Berea Network

Experimental Results

We launch 10000 particles and track their motion within the networks

Upscaling – Core – to – Field Scale Simulation (g(t))

It is not practical to model the field scale using a pore scale lattice

If we want to simulate transport on an explicit reservoir description where heterogeneity is defined we must determine a function to represent transport on every possible representation of sub-grid block scale heterogeneity applicable to our reservoir

We know that this transport now becomes dominated by advection with grid block scale heterogeneity controlling the spread of arrival times

So to expedite the process of determining this new (t) we use a multiscale modelling approach

Upscaling – Multiscale Modelling (g(t))

This approach requires that we extract two adjacent cubic grid blocks from our field scale model

We can calculate the flow field at the grid faces by imposing the known boundary conditions (wells) at the reservoir scale

We populate each block with a homogeneous lattice (50×50×50) and use the boundary fluxes as Neumann conditions to determine the flow rate within each bond

We then simulated transport by launching 10,000 particles which we tracked through the network using p(t) to determine the distribution of times to reach each face

We find that irrespective of the launching conditions, exit face or grid block pair: an exponential distribution is obtained.

1cm

Upscaling – Effect of Heterogeneity

We ran several numerical experiments and obtained a linear dependence of to the macroscopic Pe = Q/LDm

We also studied the effect of making the network more heterogeneous

We found for 1<<2 that the relationship remains linear but decreases in gradient

For = 0.5 we found a power law relationship with exponent 0.8

We then took these results and applied it to the field scale

Upscaling – Field Scale Simulation

Using the ADE and our new g(t) (=1.8) we model transport on the SPE10 reservoir description

The model contains long range correlation with permeability variations of four orders of magnitude

We used a Cartesian grid with 1,122,000 blocks (60×220×85)

We simulated transport using two different boundary conditions: 1 Injector (800 m3/day) 1

Producer (27,000 Kpa) Face Injection (x-z)

Our results compared extremely well to that of Di Donato et al. (2003) who also found using streamline simulation m=1.2

BC1 + ADE

BC1 + g(t)

BC2 + ADE

BC2 + g(t)

Upscaling –Effect of pore scale heterogeneity on the field scale

We investigated the effect of increasing pore scale heterogeneity on the field scale value of m

This had the effect of delaying the breakthrough time by about an order of magnitude

But had little effect on the late time distribution as the large scale heterogeneity controlled the transport

Outline

Motivation Transport Algorithm Upscaling

Strategy Validation Multiscale modelling Field Scale Simulation

Conclusions and Future Work

Conclusions

We presented a general transport algorithm that marries the elegance of CTRW formalism with the simplicity of Monte Carlo simulations

We applied our upscaling algorithm to use the known physics at the pore scale to model transport at the reservoir scale

Our algorithm is relatively fast with simulation times of order 10 minutes for SPE10 (1 million cells)

Future Work Demonstrate numerically that varying the

throat size distribution of the Berea network will change the pore scale

Perform a cm-m scale upscaling step for input to the field scale simulation Geostatistically generate a model to plausibly

represent the typical sub-grid block reservoir heterogeneity

Repeat the upscaling algorithm with the output of the pore scale simulations (g(t)) as an input for this new stage

Extend to multiphase?

THE END!!!!!

Thanks for your attention……

Any questions???