WaterSubject

Time Domain Random Walk (TDRW) methods have become very appealing for simulating mass transport in media with huge contrasts in their hydrodynamic properties. For instance, fractured rocks can be depicted as Discrete Fracture Networks (DFN) made up of connected 1D bonds, e.g. Properties in each bond are constant but may vary from one bond to the other. In the same way, complex flow (involving, e.g., non-Newtonian fluids) or Darcian flow in heterogeneous porous media can be mimicked by percolation networks or handled as a set of non-intertwined streamlines, each streamline being separated into successive segments of uniform hydrodynamic properties. In this context, the classical Eulerian approaches are hampered by constraints when discretizing the transport equations to avoid instability, numerical diffusion and non-physical oscillations (also called over/undershooting). These constraints have to deal with u?t/?x the Courant number,u?x/D the Peclet number and their ratio D?t/?x², with u [LT^?1] the fluid velocity, D [L²T^?1] the dispersion coefficient, and ?t, ?x the time step and space step of discretization. Usually, ?t is required to be small enough to avoid instability of explicit schemes or numerical diffusion and other over/undershoots with implicit schemes. On the other hand, Lagrangian methods moving particles in space or in time are not free from conceptual problems. For instance, they are hampered by non-linear problems, or must develop tricky algorithms when it is dealt with non-homogeneous boundary conditions. Note however that DFN approaches (and associated Lagrangian algorithms) have not been developed for that kind of things, but rather for problems mostly dominated by the huge contrasts in fluid velocities over very short distances. In this context, classical Random Walk (RW) methods moving particles in space for a given time step imply that this time step is small enough for the particles to sample the fluid velocity field accurately. If the latter spans a wide range of values, calculations may become very computer-time consuming, which jeopardizes the intuitive and versatile character of RW methods.

The TDRW method proposes to move particles in a single step between the inlet and the outlet of a 1D homogeneous segment. This is achieved by calculating the travel time probability density function (pdf) between the inlet and the outlet, and then, the outlet is merely the convolution in time between the inlet signal and the segment pdf. For a complete network of 1D segments, particles carry their successive arrival times at all the segment terminations and intersections experienced during the transport simulation. Note that the gain in terms of computation costs can be very important compared with classical RW, even for contrasted velocity fields needing a huge number of 1D segments for an accurate representation. Since particles are moved in time, the cost of a transition is even whatever the size of the segment and whatever the location of the particle in a high or a low velocity area. With classical RW methods in space, as stated above, the small time step required for a good sampling of high velocity areas makes the particles almost immobile in low velocity zones. It could be also questioned on the apparent incapacity of TDRW algorithms to accommodate transverse dispersion, since transport is 1D along segments. In fractured rocks, classical transverse dispersion is often negligible compared with dispersion due to variations in the orientation of the velocities. For instance, transport is modeled by 1D paths because flow is often channeled and mass-sharing at channels and/or fracture intersections generates a spatial spreading which is much more important than classical transverse dispersion in a channel. In heterogeneous porous media handled by means of non-intertwined stream-tube models, transverse dispersion can be simulated as done in petroleum engineering for the motion of oil saturation. A particle can pass from a tube to its direct neighbor according to empirical transition probabilities. Since these transitions between adjacent tubes are considered as instantaneous, they do not hinder TDRW algorithms which can be used as for networks of 1D segments.

In the following and for the sake of simplicity, it is merely dealt with equations and algorithms in a single homogeneous segment (however, the inversion of a simple network is proposed for illustration purpose at the end of Section). Extension to networks can be done easily by tracking particles (their arrival times) along successive segments and by adding rules to mass-sharing at segment intersections. Several TDRW algorithms have been developed and/or enhanced in the last twenty years and among them, the very recent work by Painter et al. proposes a unified conceptual framework able to combine advection–dispersion in mobile water and various types of retention mechanisms. As far as we know, their method is accurate since it is based on the convolution of exact pdfs of travel and retention times. But its inversion is probably hard to express provided one is interested in a precise procedure relying on analytical sensitivities to model parameters. This is why the present work has turned toward the TDRW method initially proposed by Delay and Bodin. Its calculation principles will be recalled hereafter in order to clearly understand how an accurate inversion method can be derived. Note that this TDRW method has been improved by adding two retention mechanisms: (1) diffusion into an infinite porous matrix, and (2) – sorption onto the solid with linear first-order kinetics.

Basically, the addition of retention mechanisms can be justified many ways. First, retention may actually occur at the local scale, for instance due to the trapping of the solute into dead ends of a fractured rock, the diffusion into the porous walls of a single fracture, the sticking onto these walls due to electrostatic forces, etc. Next, very heterogeneous media may be envisioned as partly homogenized at the macroscopic scale but discriminating between rapid pathways moving the solute by advection–dispersion and slow pathways acting as a feed-back onto rapid ones by adding retention times. Slow pathways are depicted by “artificial” retention mechanisms even if the latter do not really occur at the local scale. Actually, this philosophy inherits notions developed for dual-medium approaches to contrasted hydrodynamic behaviors. Following this philosophy may yield a separation of complex media into a simplified draining network and attached retention mechanisms. It is then obvious that this synthetic representation will require the inversion of the transport – retention macroscopic parameters for mimicking the behavior of a natural medium.

It has been stated above that TDRW methods are well suited to handle transport with contrasted properties. However, Lagrangian approaches are often considered irrelevant as regard inversion since the randomness of particle motion prohibits an accurate evaluation of the descent direction when seeking optimal solutions. The major aim of this work is to show that accurate inversion is feasible with a TDRW approach. To achieve this goal, analytical sensitivities to parameters have to be derived. Then, Section focuses on the inversion capabilities of the TDRW method handling advection–dispersion + a retention mechanism, i.e. matrix diffusion or sorption with first-order kinetics. It is shown with synthetic examples that parameter estimation is accurate provided it is guided by a prior guess on transport mechanisms. For instance, it is quite impossible to seek accurately kinetic rates in a sorption problem if these coefficients yield an almost instantaneous retention (local equilibrium assumption, LEA) compared with mean advection times. Thus, Section also discusses the accuracy of the sought parameters and their possible ranges according to different types of transport scenarios. In view of the very few attempts of transport inversion reported in the ongoing literature, it is obvious that the knowledge of the problem is much less advanced than for flow. Therefore, the complete inversion of a transport scenario over a complex flow field is hardly conceivable for the moment whatever the conceptual model used (continuous approach or DFN). However and for illustration purpose, the inversion of a quite simple example on a small DFN is reported at the end of Section.