1. A general pore-to-reservoir transport simulator Matthew E. Rhodes and Martin J. Blunt Petroleum Engineering and Rock Mechanics Group Department of Earth Science and Engineering

2. Outline Motivation Transport Algorithm Upscaling  Strategy  Validation  Multiscale modelling  Field Scale Simulation Conclusions and Future Work

3. Motivation – Modelling Non-Reactive Contaminant Transport Petroleum Engineering Hydrology Mass Balance Characterise and solve numerically Coarse Scale – Gaussian-like Fine Scale – Anomalous Transport Generic behaviour Statistical Theory Fractals ADE – Constant Parameters – Gaussian Behaviour CTRW – Single parameter- Anomalous Transport

4. 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

5. 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

6. Outline Motivation Transport Algorithm Upscaling  Strategy  Validation  Multiscale modelling/  Field Scale Simulation Conclusions and Future Work

7. 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?

8. 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?

9. 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: x=0 is the central node Subscript k denotes a bond with a node L k 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 -L j units from the junction and a local velocity that is in the opposite direction to the prevailing flux

10. 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) 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

11. 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, P i n defined by the recurrence relation given below: If z falls in the range P i-1 n ≤ z < P i n the particle will jump to node i, otherwise increase i We can then normalise z and F i (s) with respect to the actual branch probability, P i using: `

12. Transport Algorithm Finally we obtain the time that is equivalent to z n by numerically inverting F n (s) using the Stehfest (1970) algorithm and employing the bisection method to solve the equation: Transit Time t

13. Outline Motivation Transport Algorithm Upscaling  Strategy  Validation  Multiscale modelling  Field Scale Simulation Conclusions and Future Work

14. 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)

15. 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 -

16. 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)

17. 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

18. 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

19. 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

20. Upscaling – Effect of Heterogeneity We ran several numerical experiments and obtained a linear dependence of  to the macroscopic Pe = Q/LD m 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

21. 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 m 3 /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)

22. 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

23. Outline Motivation Transport Algorithm Upscaling  Strategy  Validation  Multiscale modelling  Field Scale Simulation Conclusions and Future Work

24. 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)

25. 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?

26. THE END!!!!! Thanks for your attention…… Any questions???