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 Application – Generic Porous Media 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 reproduce this anomalous behaviour without assuming an average macroscopic PDE We must therefore account for relevant transport physics and reservoir heterogeneity at each scale of interest But to do this we require a framework that gives us the flexibility to represent these features effectively and efficiently

6. Outline Motivation Transport Algorithm Application – Generic Porous Media 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 of one dimensional links and nodes 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 being physically located only 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 To complete the system, we impose the following conditions: At time t = 0, a particle will just arrive at the central node. This can be represented as a delta function initial condition of the form: Sorbing boundary conditions at the extremities: Concentration continuity at the nodes: Flux conservation at the nodes:

11. Transport Algorithm -  (t) cont’d We then transform the “coupled” system of equations and conditions into Laplace space and solve for concentration:

12. Transport Algorithm -  (t) cont’d As we know there is concentration continuity at the nodes we can write the coefficients B k and B j in terms of the coefficient for branch 1 which we denote as B: We can obtain B(s) by invoking flux continuity at the node and then write the concentration distribution in each branch of our system as a function of the Pe in each other branch:

13. 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) But which branch must we sample?

14. Transport Algorithm – Transition Probability In the advective and diffusive limits this answer is trivial – flux weighting for advection or a random choice for diffusion But if there is equal competition of advection and diffusion what do we do? Fortunately F(t) has another use - As we injected a unit concentration into the junction, the ultimate fraction reaching each node would be equivalent to the probability, P i of a particle exiting said node. In other words: In Laplace space we can determine this probability as a function of branch Peclet (Pe = VL/D) number, using the Final Value theorem:

15. Transport Algorithm – Transition Probability – The Limits The resulting probabilities reduce trivially to the appropriate limits: As Pe number becomes very large, P i equals: While in the limit of small Pe, P i becomes: Flux weighted Random Transition

16. 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: `

17. 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

18. Transport Algorithm – But does it work? We attempted to reproduce our analytical solution and the resulting mixing rules by tracking 10 4 particles on a two branch grid In a time,  t a particle will move through a displacement r given by the sum of an advective (in the direction of V) and diffusive length (in a random direction): If a particle arrives at the junction during an advective step we route them by flux weighting but if it enters in the diffusive step we randomly choose an outlet node We then monitored the time for a particle to reach one of the exit nodes and the number of particles that leave the simulation through said node 0.01≤V 2 <10 D j = 1 m 2 s -1 L j = 1 m 0.01≤V 1 <10 D k = 1 m 2 s -1 L k = 1 m Direction of Velocity not flux

19. Transport Algorithm – and if there are more branches? We then tested our solution on a three branch grid We generated a structure with a central node and three nearest neighbours connected by three bonds (as shown below) We then repeated the procedure discussed on two suites of simulations. We kept the velocity in branch 2 constant and allowed continuity to determine the velocity in the rest:  Velocity in branch 1 = 0 ms -1  Velocity in branch 1 = 1 ms -1 0.01≤V 2 <10 ms -1 D 2 = 1 m 2 s -1 L 2 = 1 m 0.01+V 1 ≤V 3 <10 +V 1 ms -1 D 3 = 1 m 2 s -1 L 3 = 1 m V 1 = 0/1 ms -1 D 1 = 1 m 2 s -1 L 1 = 1 m Error in Bond 2 when V 1 =0 ms -1 Error in Bond 2 when V 1 = 1 ms -1

20. Transport Algorithm – But… What do we do if transport cannot be modelled by the ADE? We define/determine a  (t) which does  One such example is an empirical function produced from the pore scale simulations of Bijeljic et al. (2004): We can then repeat the methodology suggested earlier substituting this new function for the one determined analytically

21. Outline Motivation Transport Algorithm Application – Generic Porous Media Conclusions and Future Work

22. Application – Transport on Fractal Porous Media We applied our algorithm to a two and three dimensional bond percolation system with occupation probability, p = p c (0.5 and 0.2488 respectively) and L = 1000 and 100 We populated the grid then removed the spanning cluster from the system using the Hoshen and Kopelman (1976) algorithm We then numerically solved for the pressure field assuming:  unit conductance within the bonds and zero in the “empty links”,  a pressure of 1 bar at the bottom boundary and 0 at the top,  no flow through the other four and  the finite difference form of Darcy’s Law: Using this pressure we calculated the local velocity for each bond. We assumed a dispersion coefficient = 1m 2 s -1 and bond length = 1m We then initiated our algorithm by randomly assigning ten thousand particles to the bottom boundary of the system We moved these particles from node to node using both the analytical  (t) and the empirical  core (t) with  =1 The time to transit the system was reported and subsequently shown

23. Application – 2D Fractal Media and Breakthrough Curves We could identify little difference in the transit time distributions from the two simulations - the time shift being due to the choice of  We identified three distinct late time regimes shown by the dual peaks in our results and the change in gradient at late times. We see the first being due to transport along the portion of the backbone leading towards the exit node while the second and third is believed to be a result of a diversion into dead ends connected to the backbone as shown Analytical  t  Empirical  t 

24. Application – 3D Fractal Media and Breakthrough Curves Analytical  t  Empirical  t  Again we see little difference in the two simulations and three distinct time regimes shown by the dual peaks in our results and the change in gradient. It is again believed that these regions represent transport governed by advection, a mixture of advection and diffusion and finally diffusion.

25. Application – 3D Fractal Media and Breakthrough Curves Analytical  t  Empirical  t  In these simulations only two distinctly anomalous regimes were identified by the change in the slope of the breakthrough curves. The lack of a second peak is due to the large number of particles that were fed to the backbone due to the structure and our launching conditions. This prevented an adequate sampling of the connected dead ends and as such no third regime.

26. Conclusions We presented a general transport algorithm that marries the elegance of CTRW formalism with the simplicity of Monte Carlo simulations to on average follow the governing PDE or arrival distribution We applied our algorithm to fractal porous media and reproduced anomalous transport We note the algorithm is relatively fast with a simple cubic percolating lattice of size 100 3 running in:  Less than 15 mins for analytical function  Less than 1 minute for empirical function

27. Outline Motivation Transport Algorithm Application – Generic Porous Media Conclusions and Future Work

28. Future Work Extend current code to non-rectilinear grids as used in state-of-the art pore network models (Valvatne and Blunt 2004) Obtain a generic  (t,Pe) for transport in fractal media using the analytical solutions presented.

29. Future Work – Pore-to-Reservoir simulation Using first principle simulations Bijeljic et al. (2004) obtained a core scale distribution of transit times  core (t) We can upscale this function to the field scale by simulating transport in network models of intermediate size using the analytical mixing rules derived earlier We initiate our upscaling methodology by implementing this  core (t) in a transport simulation of a pore scale network We could then fit the resulting transport distribution to an analytical function  1 (t) whose parameters can be related to physical properties of the medium. This provides a method to account for local heterogeneity within the system

30. Future Work – Pore-to-Reservoir simulation We can then use this function in a grid-block scale simulation of the porous media From which we would obtain a new distribution,  2 (t) that can be employed for reservoir simulations This provides a simulation methodology that on average accounts for the reservoir physics and heterogeneity without making any assumptions on the governing PDE 1 m 100’s m  core (t)  1 (t)  2 (t)

31. THE END!!!!! Thanks for your attention…… Any Questions???