We conducted four core-flood experiments on samples of a micritic, reef limestone from Abu Dhabi under conditions of constant flow rate. The pore fluid was water in equilibrium with CO2, which, because of its lowered pH, is chemically reactive with the limestone. Flow rates were between 0.03 and 0.1 mL/min. The difference between up and downstream pore pressures dropped to final values ≪1 MPa over periods of 3–18 h. Scanning electron microscope and microtomography imaging of the starting material showed that the limestone is mostly calcite and lacks connected macroporosity and that the prevailing pores are few microns large. During each experiment, a wormhole formed by localized dissolution, an observation consistent with the decreases in pressure head between the up and downstream reservoirs. Moreover, we numerically modeled the changes in permeability during the experiments. We devised a network approach that separated the pore space into competing subnetworks of pipes. Thus, the problem was framed as a competition of flow of the reactive fluid among the adversary subnetworks. The precondition for localization within certain time is that the leading subnetwork rapidly becomes more transmissible than its competitors. This novel model successfully simulated features of the shape of the wormhole as it grew from few to about 100 µm, matched the pressure history patterns, and yielded the correct order of magnitude of the breakthrough time. Finally, we systematically studied the impact of changing the statistical parameters of the subnetworks. Larger mean radius and spatial correlation of the leading subnetwork led to faster localization.
Recent interest in CO2 sequestration and water-alternate-CO2 flooding has spurred a series of studies on the reactive flow of carbonic acid in carbonate rock, including experiments [e.g., Luquot and Gouze, 2009; Noiriel et al., 2009; Grgic, 2011; Gouze and Luquot, 2011; Carroll et al., 2013; Menke et al., 2014], numerical simulations [e.g., Kang et al., 2002, 2003, 2010; Kalia and Balakotaiah, 2009; Molins et al., 2012; Hao et al., 2013; Maheshwari and Balakotaiah, 2013; Huber et al., 2014; Steefel et al., 2013; Ovaysi and Piri, 2013; Chen et al., 2014; Upadhyay et al., 2015], and field observations [e.g., White, 2013]. When high-pressure CO2 dissolves in water, the acidified fluid reacts with carbonate minerals such as calcite and dolomite. Thus, the flowing fluids change the porosity and permeability of the rock. On the other hand, the evolving flow field affects the penetration and the shape of the dissolution front. Depending on the rock characteristics [Kalia and Balakotaiah, 2009; Carroll et al., 2013] and the rates of injection and reaction [Fredd and Fogler, 1999; Luquot and Gouze, 2009], different dissolution patterns can form, ranging from uniform (or homogeneous) to localized (or heterogeneous) dissolution. The strong coupling between flow field and chemical reaction gives rise to various forms of dissolution instabilities [Scymczak and Ladd, 2006, 2011, 2014]. At the macroscale, these instabilities eventually grow into wormholes, i.e., tree-shaped, localized dissolution channels.
Among the different techniques commonly used for numerical simulation of localized dissolution are the continuum [e.g., Kalia and Balakotaiah, 2009; Hao et al., 2013; Maheshwari and Balakotaiah, 2013] and the pore scale approach [e.g., Kang et al., 2002, 2003, 2010; Molins et al., 2012; Huber et al., 2014; Steefel et al., 2013; Ovaysi and Piri, 2013; Chen et al., 2014; Pereira Nunes et al., 2016]. In the former the rock is modeled as being composed of blocks with continuum properties. Thus, the sample is discretized into a 3-D grid with each grid block assigned a porosity value either by referencing microtomography images or by random sampling. A functional dependence of permeability on porosity must be stipulated. Often, it is necessary to assign a high sensitivity of permeability to porosity in order to achieve localization [e.g., Lai et al., 2014], even though such extreme sensitivity is not necessarily supported by experimental evidence [e.g., Luquot and Gouze, 2009]. The second approach directly models the pore-scale physics using numerical methods such as lattice-Boltzmann [Kang et al., 2002, 2003, 2010; Huber et al., 2014], finite volume [Molins et al., 2012; Steefel et al., 2013], or Lagrangian moving particle [Ovaysi and Piri, 2013]. Here no porosity-permeability relationship needs to be assumed. Another pore-scale approach, which is particularly simple, idealizes the porosity as a network of conduits, whose shapes and sizes are assigned either from an assumed statistical distribution or microstructure images [Hoefner and Fogler, 1988; Bernab'e, 1996; Budek and Szymczak, 2012; Raoof et al., 2013; Menke et al., 2014, 2015].
In this study, injection experiments were conducted on samples of a limestone from Abu Dhabi using aqueous fluid saturated with CO2 under high pressure. We held the injection rate constant and measured the evolution of the pressure difference between the upstream and downstream reservoirs. We characterized the rock microstructure using 2-D scanning electron microscope (SEM) imaging and also performed 3-D microtomography scans of one of the samples before and after injection. Wormholes formed in all the experiments. In addition, we wanted to numerically simulate the processes experimentally observed, but the material and experimental conditions used here had several features detrimental to the continuum approach. In particular, we feared that the low-dissolving capacity of the carbonic acid would inhibit localization of dissolution in continuum simulations unless we assumed an unrealistically high sensitivity of permeability to porosity. We therefore opted to use a network simulation, particularly because it is the easiest of the pore-scale approaches to implement. Even so, limitations in computational resources required an additional simplification. Thus, we split the porosity into two separate networks, which we call leading and secondary. The leading network is constructed using higher mean pore size than the secondary and is therefore more susceptible to changes caused by reactive flow. The secondary network is formed of identical subnetworks so that we exchange a very large problem for two much smaller ones, respectively, associated with the leading network and a single secondary subnetwork. In this model, the leading and secondary networks compete for the flow of reactive fluid and wormhole formation results when the leading network wins the competition (a similar process was analyzed by Scymczak and Ladd ). Using this competing network model, we successfully generated the dissolution patterns and pressure histories observed in the experiments. The method is efficient in that our limited computational resources are focused on the portion of the pore space that experiences the fastest changes.
The paper is organized as follows. In section 2, we present sample characterization, experimental setup and the results of the experiments. In section 3, we analyze the reactive flow of carbonic acid through a single calcite pipe and test the simplest competing network model, in which the leading and secondary subnetworks are reduced to parallel pipes. We then consider more realistic 3-D competing subnetworks and investigate the effect of the different statistical parameters characterizing the networks. In the last two sections, the main results and potential extensions are discussed, and finally, our main conclusions are summarized.
2.1 Sample Characterizations
Samples for the core-flood experiments came from a micritic limestone found in a drill core from Abu Dhabi, UAE. Based on visual inspection and energy-dispersive X-ray spectroscopy, we estimated that the rock is 95% calcite, the remainder being secondary minerals such as quartz, dolomite, and salt. Backscattered electron images of the starting material (Figure 1) show shell fragments in a micritic matrix, suggesting that the rock originated in a marginal reef environment.
The fluid transport properties of rocks are determined by parameters including porosity, pore-size distribution, pore shape, and connectivity. Assuming that the rock is pure calcite, density measurements show that the porosity ϕ is about 13%. Grains vary in size and can be as large as a few hundreds of microns in diameter or less than a fraction of a micron. There is no simple relation to determine the average dimensions of an ensemble of irregularly shaped pores from the intersection of pores with a single plane. However, we can estimate an apparent radius, r*, using
where S is the area of an individual pore in the image. The number distribution of r* (Figure 2) is strongly skewed toward low values. The majority of the pores have an apparent pore radius near 1 µm. Bigger pores are relatively rare. We also used X-ray microtomography to characterize the pore geometry before and after a core-flood experiment for one sample (BAABSP-25, Figure 3). The data were acquired at the X-ray facility in Center for Nanoscale Systems at Harvard University. The resolution of the microtomography image is about 17 µm, implying that the connections between the pores are not resolved. This, together with the pore radius count (Figure 2), suggests that the dominant radius of the connections in the pore network is on the order of 1 µm. Indeed, the permeability, k, of the samples is on the order of a tenth of a millidarcy (≈10−15 m2), much less, for example, than the 40 mdarcy oolitic limestone considered in Luquot and Gouze . A characteristic pore radius, rc, of the starting rock can be estimated using the Kozeny-Carman model ( , assuming a tortuosity τh2 of 2); rc ranges between 0.1 and 1 µm.
2.2 Methods of Core-Flood Experiments
The apparatus used is schematically represented in Figure 4. Cylindrical limestone samples were ground to a diameter of 1.4 cm and length of 2.6 cm. Before insertion into the confining vessel, the samples were double jacketed with a sleeve of polyolefin inside a second sleeve of Teflon. The two jackets were used to prevent CO2 from leaking into the confining pressure system. In the upstream reservoir, distilled water and CO2 were mixed at 5.5 MPa through several cycles of shaking and standing to ensure that the fluid had reached equilibrium. The pressure in the downstream fluid collector was buffered by the CO2 gas tank. During an experiment, we first injected the CO2 saturated fluid through the upstream vent. The valve between the sample and the downstream fluid collector remained closed until the downstream pressure reached that of the gas tank. The injection pump functioned in constant volumetric flux rate mode with the downstream pressure kept constant at 5.5 MPa. The confining pressure was fixed at 20 MPa. Under these boundary conditions, the permeability at a given time can be calculated using the measured pressures at both ends of the sample. For the injection rates we used, the upstream pore pressure was at most 12 MPa. The upstream mixer has a maximal size of 15 mL. To inject more fluid, the valves at both ends of the sample were closed; a second charge of water was saturated with CO2, and then the injection was resumed.
2.3 Results of Experiments
Four experiments were conducted with constant flow rates of 0.11, 0.084, 0.046, and 0.036 mL/min corresponding to average Damkholer numbers ranging from about 30 to 80. For comparison, values of 1 to 5 were used in Luquot and Gouze . In all tests, the pressure head between up and downstream reservoirs decreased with time and the permeability increased (Figure 5). Occasional discontinuities in the differential pressure curves can be seen when the flow was halted so that the upstream reservoir could be replenished. These intervals are removed from the pressure history for clarity. In experiment BAABSP-22, an anomaly occurred after 12 h of injection when flow was resumed after such a halt. Its cause is unknown, but it is possible that some bubbles formed in the pore space owing to the reduction in fluid pressure. In all experiments, the rate of permeability increase was regular and stable at first but eventually accelerated quasi-exponentially. Additional permeability measurements were carried out at the end of tests BAABSP-20 and BAABSP-22 using increased flux rates of 0.2–0.5 mL/min (indicated by star symbols in Figure 5b). The microtomography image of sample BAABSP-25 taken after the experiment revealed that several localized dissolution channels connected to the upstream face had formed, only one of which perforated the entire sample and reached the downstream face (Figure 3b). In addition, newly created holes, visible to the naked eye, appeared at both ends of the samples after all four experiments. This morphology corresponds to the “inlet instability” analyzed in Scymczak and Ladd  and is consistent with the conditions of occurrence they derived (Szymczak and Ladd's porosity contrast parameter should be large in our 95% calcite rock). The localized dissolution channels conduct fluid much faster than the rock matrix, and the jumps in permeability most likely happened when the wormholes perforated the downstream end the samples. The injection rate f and the breakthrough time τ displayed a power law correlation, f ∝ τ−1.3, but the exponent may be uncertain owing to differences in initial permeability and the difficulty to determine the breakthrough time precisely (exponents on the order of −1.1 were observed in previous studies, A.J.C. Ladd, personal communication, 2016).
3 Numerical Modeling
The pore geometry of each sample cannot be known accurately at the relevant scales, and even if it were known, it cannot be included in its totality in a numerical model owing to computational limitations. Thus, we focused the subsequent modeling on reproducing the general flow behavior, including the main features of the pressure evolution (i.e., initially steady decrease followed by a sharp drop at the breakthrough time), the morphology of the wormholes, and the correct order of magnitude of the breakthrough time. We did not try to reproduce the pressure histories observed exactly. As stated earlier, we opted for the network simulation approach. Our premise is that the wormhole must stem from the heterogeneity of the pore space since uniform dissolution should be produced in a perfectly homogeneous pore space. Hence, we divide the sample pore space into two categories, a leading network susceptible of changing quickly and a set of identical secondary subnetworks that only change slowly. The susceptibility to reactive flow of the leading and secondary networks is adjusted using different pore radius distributions, the leading network having a higher mean pore radius. Hence, we substantially reduce the size of the numerical problem and also to explicitly include the transfer of flow from the secondary to the leading subnetworks that enables the incipient wormhole to grow [Scymczak and Ladd, 2006]. In the following we will first describe the reactive flow equations in an initially cylindrical pipe. We will then treat the simple case of a single leading pipe in parallel with a large number of identical secondary pipes (Figure 6a). The insight gained will enable us to carry out more complex and more realistic simulations using simple cubic 3-D networks (Figure 6b).
3.1 Reactive Flow in a Single Pipe
We consider an initially cylindrical pipe of length L. During the course of reactive flow, the pipe cross section changes, and the radius r is therefore defined as a function of the axial distance x and time t. The flow field in a pipe with a varying cross section cannot, in general, be expressed analytically. For tractability, we use the Reynolds approximation, i.e., Poiseuille's law is assumed to be applicable pointwise for deriving the analytic solution and piecewise in the numerical simulation. With this approximation we can write
where f is the volumetric flux, r the radius of the pipe, which may vary with position and time, η the viscosity of the fluid, and dp/dx the pressure gradient. The integrated form of equation (2) links the overall pressure drop Δp to the flux as follows:
where the proportionality factor T measures the pipe conductivity as its shape changes with time. The pipe radius changes with dissolution according to the following:
where M is the molar mass of calcite, ρ the density of calcite, C the concentration of calcium ions and R(C) the reaction rate in mol L−1 s−1 units. The reaction rate is modulated by mass transfer and limited by the degree of saturation:
where kS is the dissolution rate constant, kC the local mass transfer coefficient, and CS the solubility of calcite in the CO2 buffered solution. The mass transfer coefficient can be calculated as [Balakotaiah and West, 2002; Gupta and Balakotaiah, 2001; Maheshwari and Balakotaiah, 2013]:
where D is the molecular diffusivity of the ions, Re the Reynolds number, and Sc the Schmidt number ( and with u the average fluid velocity and ν the kinematic viscosity of the fluid). The asymptotic Sherwood number Sh∞ is a constant. The concentration of calcium ion along the pipe is governed by the convection-reaction equation:
Diffusion is not included here as the Péclet number is typically much bigger than 1, especially for the leading pipe. Note that variations of C in the radial direction need not be considered because of the small values of r compare to L, even after dissolution takes place. The values of physical constants in this study can be found in Table 1.
Table 1. Values of Physical Constants Used in This Study
To test the method, we considered the simplest possible system, i.e., a bundle of parallel pipes all connected to the up and downstream reservoirs but otherwise separated (Figure a). All pipes have the same length L. The radius of the leading pipe is rL, while each of the n secondary pipes has the same radius rS. Initially, rL and rS are independent of x and start with constant values RL and RS, respectively. As in the experiment, the downstream pressure and the total flux (F) are fixed. Owing to the in parallel arrangement, the total flux is the sum of the fluxes in the individual pipes. During reactive flow, the flux fL in the leading pipe is determined by the following:
where TL and TS are the proportionality factors (equation (4)) for the leading and secondary pipes, respectively.
When a pressure difference of 2 MPa (the initial value used in all simulations hereafter) is applied on a pipe with a diameter of 0.1–1000 µm and length L, the local mass transfer coefficient kC is more than 100 times bigger than the dissolution rate constant kS. Under these circumstances, the effect of local advection is negligible, and the reaction rate (equation (6)) can be simplified to the following:
This simplification becomes less valid as the pipe deviates from its initially uniform shape, and while it is not used in the later numerical simulations, nevertheless, it helps to understand the onset of localization. We define the following quantifiers:
where Δp0 is the pressure drop at the beginning of the simulation, L is the length of the pipes, and the subscript X can be either L or S, denoting the leading pipe and the secondary pipes, respectively. The reference radius R0, flux f0, speed u0, and time t0 all take the initial values of the leading pipe. Da is the Damkholer number that relates the convection time scale to the reaction time scale, and Nc is the acid capacity number that measures the volume of solid dissolvable in unit volume of acid. Equations. (8), (5), (9), and (10) can thus be written in nondimensional form as follows:
The initial conditions are
where the ratio β = RS/RL has values between 0 and 1 and measures how competitive the secondary pipes are compared to the leading pipe. When β approaches 0, the secondary pipes are so small that they experience negligible dissolution and remain unchanged; if β = 1, the secondary pipes have radii identical to the leading pipe, and the competition between the secondary and leading pipes is maximal. The boundary condition is as follows:
Accordingly, the evolution of the parallel pipe system is determined by three dimensionless numbers: the Damkholer number (Da), acid capacity number (Nc), and the initial radius contrast (β). Since we fix Δp0 = 2 MPa in all simulations, Da is inversely proportional to RL3, and the fraction of total flux initially carried by the leading pipe f0/F increases as RL4 (the number n of secondary pipes is a dependent parameter controlled by RL and β).
The reactive flow equations for the leading and secondary pipes can only be solved numerically. We used the finite volume method [Leveque, 2002], i.e., a pipe is discretized into segments (here 100) and each segment is supplemented with the volume integral forms of the differential equations to be solved. The finite volume method has the advantage of automatically enforcing mass conservation.
It is very instructive to examine the results obtained in the β ≈ 0 limiting case. We ran a simulation using Δp0 = 2 MPa, RL = 3.1 µm, F = 0.1 mL/min (corresponding to Da ≈ 11). We found that the pressure difference between upstream and downstream barely changed for several hours before suddenly dropping to nearly zero (Figure 7a). Simultaneously, the fraction of the total flux carried by the leading pipe jumped from less than 1% to nearly 100% (Figure 7a). During reactive flow, the leading pipe, owing to its growth, stole more and more flux from the secondary pipes. The extra flux acted as a positive feedback, enhancing further growth and leading to runaway enlargement of the leading pipe. At the end of the runaway interval, there was minimal pressure difference remaining, and the leading pipe accommodated almost all the flux. In order to understand this runaway process more precisely, we plot snapshots of the concentration and radius profiles of the leading pipe at different times (Figures 7b and 7C, respectively). It appears that runaway growth begins when the fluid at the downstream end of the leading pipe becomes undersaturated (at about t = 4.3 h; Figure 7b). Before this critical time, the leading pipe develops a trumpet shape with its widest cross section at the upstream end (Figure 7c). After the critical time, the leading pipe growth becomes more and more uniformly distributed along the pipe length (Figure 7c).
We also examined the effect of the radius ratio (β) and of the initial leading pipe radius (RL). Examples of pressure difference history are shown in Figure 8. It is clear that localization cannot occur in a perfectly homogeneous system (i.e., β = 1). On the other hand, even a small contrast in leading and secondary radii is capable of triggering the formation of a wormhole (e.g., β ≈ 0.9). Stronger competition from the secondary pipes (i.e., greater β) requires a larger leading pipe radius to produce a wormhole in a given amount of time. For example, for a breakthrough time of about 3 h (between the red and green lines in Figures 10a and 10b), an increase of β from 0 to 0.9 requires increasing RL from about 3.1–3.2 µm to nearly 3.6 µm. We also note that in order to produce the correct order of magnitude of breakthrough time (≈4 h), we needed values of RL slightly too high compared to the dominant pore radii inferred from permeability measurements and SEM micrographs (≈1 µm).
3.3 Competing 3-D Networks Model
In the parallel pipe model (Figure 6a), the pipes are all connected to the up and downstream reservoirs but disjoint otherwise. This is not realistic since the actual pore space in rocks is much more interconnected. However, the vast number of pipes needed in a literal account of the actual pore space of a macroscopic rock sample (≈1 cm3) cannot be handled computationally. A computationally less demanding model similar to the parallel pipe model is to frame the problem as a competition for reactive fluid among a leading and a number of identical secondary 3-D heterogeneous networks. As before, we assume in parallel arrangement of the leading and secondary networks, i.e., they are connected to the up and downstream reservoirs but disjoint otherwise (Figure 6b). The network structure used here is the combination of three mutually perpendicular arrays of pipes connected at nodes and forming a primitive cubic system. Hence, a network with M × M × N nodes has (M − 1) × M × N pipes along the x direction, M × (M − 1) × N pipes along the y direction, and M × M × (N − 1) pipes along the z direction (here assumed to be the nominal flow direction). Since the length of individual pipes is comparable to the length of the pipe segments considered in the parallel pipe model, we assumed that the individual pipes remained cylindrical with a uniform circular cross section throughout the simulation, although their radii will increase owing to dissolution. During the simulation the nodes are characterized by the current value of the fluid pressure and the pipes by their current radii and calcium concentrations. The flux in a pipe is assumed to obey Poiseuille's law. For example, the flux in the pipe along the x direction, connecting node (i, j, k) and node (i + 1, j, k) is as follows:
where f is the volumetric flux, l the pipe length, P the node pressure, rX the current radius of the pipe, and the subscript X now indicates the direction of the pipe. The pressure field of the whole network is calculated by solving a linear system of equations that balance mass at the nodes:
where K denotes the Poiseuille proportionality constants and P and Q are the pressure and source terms, respectively. We solved the pressure linear system using the PETSc library [Balay et al., 2014] and the BoomerAMG algebraic multigrid preconditioner [Yang, 2002]. The concentration in each pipe is governed by the reaction-convection equation and solved by the operator splitting technique [Valocchi and Malmstead, 1992]:
where reaction happens in the time interval [tn, tn+1/2), convection happens in [tn+1/2, tn+1), , |u| is the average flow velocity, Δt the time step size and is the volume average concentration of the influxes at the upstream node of the current pipe. Along with the concentration, the radius of the current pipe is updated according to the following:
For the sake of simplicity, the pipe radii were initialized according to independent uniform distributions, i.e.,
where μ denotes the mean and σ the standard deviation of the uniform distribution U(min, max). As before, the subscript X can be either L (leading) or S (secondary). We used the same boundary conditions at the sides as in the experiments, i.e., there was no flow across the boundaries in the transverse x and y directions.
In terms of computational time, the pressure equation (equation (18)) is much more expensive than the reaction-convection equation. To gain computational efficiency, the pressure field was updated every 40 time steps, while the concentration field was updated every time step. The implicit scheme (equation (20)) guarantees stability but not accuracy. Theoretically, a homogeneous 3-D network must behave identically to a homogeneous parallel pipe system. We were thus able to benchmark results obtained in 32 × 32 × 64 3-D network by comparison with those of a corresponding parallel pipe system. We verified that the pressure histories obtained for the 3-D networks converged toward those of the parallel pipe system as the time step was decreased. We chose the time step dt = 1/3600 such that 8 h of reactive flow can be simulated in a day of computation while limiting the discrepancy relative to the benchmark to 20%.
3.4 Results of the Simulations
Figure 9 displays an example of the evolution of the leading network with time when the secondary networks have very low conductivities and do not change with time (β = μS/μL ≈ 0, μL = 1 µm and σL = 0.46 µm). We observe that at early times, enhanced dissolution occurred in many sites on the upstream face but that most of these contending dissolution paths failed, eventually leaving only one throughgoing path, i.e., a wormhole, at the end of the simulation. The simulated dissolution pattern shown in Figure 9 is strikingly similar to the one experimentally observed (Figure 3b). As in the parallel pipe model, the onset of localization decreased monotonically with increasing mean leading pipe radius μL (Figures 10a and 10b). However, the effect of the standard deviation σL is more complex (Figures 10c and 10d). In particular, we note that networks generated with same random seed (hence, having an identical ordering of the pipe radii) and different values of σL did not produce a monotonic variation of the onset of localization (for example, the highlighted curve in Figure 10d has a maximum for σL = 0.29 µm). Examination of the corresponding dissolution patterns suggests that the local maxima discussed above were linked to switching among nearly equivalent potential localization paths. For example, Figure 11 shows the localized dissolution patterns corresponding to the simulations highlighted in Figure 10d. We see that the localized path is in the back left corner for σL lower than 0.23 µm, switches to front left for σL = 0.29 µm, and returns to the previous location for larger values of σL. These swings suggest that the calculations become more sensitive to finite size fluctuations as the contrast between the smallest and the largest pipes increases with σL. The thin black lines in Figure 10d help to visualize the ensemble-averaged behavior: the mean breakthrough time decreases monotonically for σL increasing up to about 0.17 µm and shows much smaller variations for larger values.
Another end-member case is when the leading network and the secondary network are identical (β = 1). In this case, there is no flux transfer from the secondary to the leading network. Unlike in the parallel pipe system (Figure 8), wormhole formation can still occur in this case owing to the heterogeneity of the networks (flux is internally transferred from the low conductivity paths to the high conductivity one that eventually grows into a wormhole). However, localization occurs more progressively than in the β ≈ 0 cases discussed earlier (Figure 12a). In intermediate cases a significant amount of progressive growth may occur before the eventual runaway localization, as is illustrated by the pressure histories of simulations with β = 0.96 shown in Figure 12b. The pressure difference decreases regularly at first and then drops suddenly due to wormhole formation, quite similar to the behavior observed in the experiments. In general, breakthrough times of the correct order of magnitude were obtained with values of μL more realistic than for the parallel pipe model (≈1 µm).
4.1 Flow Paths in Uniformly Distributed 3-D Networks
Any simple path that travels from the bottom to the top of the network without forming loops is a possible path for fluid flow. In an M × M × N network, such a simple path is composed of n (n > = N) pipes with radii (r1, r2, …, rn). In terms of fluid flow, each simple path is equivalent to a uniform pipe with a length N l and an effective radius equal to the following:
Thus, reffective can be used to rank the hydraulic conductivity of all possible simple paths in a lattice. Because of statistical variations, some paths have larger effective radii, are able to transmit a larger volume of reactive fluid, and grow faster by dissolution. Given the boundary conditions (vertical externally applied pressure gradient and nominally vertical flow), a strictly vertical path composed of N pipes represents a statistically typical path in a 3-D lattice. The probability distribution of the effective radius of a vertical path, rvertical, can be estimated by Monte Carlo simulations. The dashed lines in Figure 13 are such probability distributions for networks generated with three uniform distributions with the same mean (1 µm) and differing standard deviations (0.23, 0.12, and 0.06 µm, corresponding to U(0.6, 1.4), U(0.8, 1.2), and U(0.9, 1.1)). The mean values of rvertical are less than 1 µm and furthermore decrease as the variance increases, as should be expected because equation (23) is similar to a harmonic average. If we use Dijkstra's algorithm [Cormen et al., 2009] and expand the search to subvertical paths (containing a few horizontal pipes), it becomes possible to identify the most transmissive path in each network realization. The effective radius of the most transmissive path, roptimal, is different for each realization. The probability distribution of roptimal can also be found by Monte Carlo simulations (solid lines in Figure 13). The results are that the mean values of roptimal are larger than 1 µm and increase as the variance increases. Hence, increases in the variance of the radius distribution (in other words, increases in pore heterogeneity) lead to larger contrast between roptimal and rvertical. By analogy to the parallel pipe system, we would then expect that localization of fluid flow, and therefore localization of dissolution, is favored by a larger standard deviation of the radius distribution. Indeed, the simulated breakthrough times decreased as σL increased from near 0 to 0.17 (Figure 10d). However, as mentioned in section 4, this relationship is perturbed for higher values of σL by finite size statistical fluctuations.
Yen's algorithm [Yen, 1971] can be adapted to find the K most conductive paths in a 3-D lattice where any two of these paths differ by at least one pipe. The inset in Figure 13 shows the 50 most conductive paths in one network realization generated with μL = 1 µm and σL = 0.23 µm. They all coincide with the optimal path over a large portion of their length, and their effective radii lie in a very narrow range (within 0.001 µm of the optimal value). This result suggests that an optimal path in the 3-D lattice is much more competitive than a leading pipe with an identical effective radius in the parallel pipe model because the optimal path in 3-D networks can collaborate with neighboring high-conductivity paths and grow together. In order for localization to happen in the parallel pipe model in less than 8 h, we found that the leading pipe had to have an initial radius of at least 3 µm (Figure 8). In contrast, localization in the 3-D networks required a leading network with μL of only 1 µm. The difference probably reflects the collaborative effect of neighboring high-conductivity paths.
4.2 Wormhole Morphology
The morphology of the wormholes in the experiments and the simulations appeared qualitatively similar. As mentioned in section 2.3, there is a selection among incipient localized paths, eventually won by a single wormhole, consistent with the instability analysis of Scymczak and Ladd . Furthermore, both experimental and simulated wormholes displayed short branches growing from the main path, which became more frequent as the distance to the downstream end decreased. This enhanced that branching can be understood by analyzing wormhole formation in shorter samples and noting that Damkholer number decreases linearly with sample length. For example, we consider two parallel pipe systems of different lengths. If the concentration profile changes slowly with time, i.e., , the early time concentration profiles can be expressed as follows:
We verified that these steady state solutions indeed match the patterns observed in the numerical simulations (Figure 7b). We now consider two equally long pipes with slightly different radii. When the pipes are sufficiently long, the two concentration profiles at the onset of runaway growth are very different and the narrow pipe is quickly out matched. When the pipes are shorter, the concentration profiles difference is less marked, and the two pipes are able to grow simultaneously for a longer time. Thus, as the wormhole and undersaturated fluid move closer to the downstream boundary, more branches become competitive and grow.
We also note that in both the experiments and the simulations, only one throughgoing wormhole was formed. The selection of a single dominant wormhole probably owes to the boundary conditions prescribed, i.e., uniform fluid pressure along the planar ends of the sample, and to the finite dimensions of the sample and networks. The uniform fluid pressures at the sample ends allow fluid to be distributed among the different paths purely based on their hydraulic conductivity. In acid treatment of reservoir rock, the boundary conditions are likely very different (e.g., point source and radial flow), and the downstream end can be very far away from the injection source. To construct a model that would be successful for such scenarios, the effects of scaling and boundary conditions must be more closely examined.
4.3 Effect of Spatial Correlations
The 3-D networks studied thus far were initialized by independently distributed pipe radii. However, spatial correlations of pore characteristics are commonly observed in rocks [e.g., Blunt, 2001] and are the focus of this section. A network of M × M × N nodes with the radii drawn according to an arbitrary, spatially correlated distribution can be generated as follows. Suppose that Lcorr is the desired correlation length, i.e., there is negligible correlation between the radii of two pipes separated by more than Lcorr in either x/y/z direction. First, (M + Lcorr −1) × (M + Lcorr −1) × (N + Lcorr −1) numbers fijk are generated according to the standard normal distribution with mean 0 and variance 1. These numbers are then convoluted with a Lcorr × Lcorr × Lcorr mask wpqr to generate M × M × N new values gijk for each node:
Now each pipe is assigned the sum of the gijk values at its neighboring nodes. For example, the values for the pipes in the x direction are as follows:
Because hijk is a linear combination of independent normal random variables, it is also normally distributed. The values for two pipes within distance Lcorr are correlated since they share parts of their combinations. Each hijk can then be transformed by its cumulative distribution function into another random variable sijk obeying a standard uniform distribution, and, albeit distorted, the spatial correlation of the sijk field is largely preserved. Finally, pipe radii drawn according to spatially correlated, arbitrary distributions can be obtained from sijk. The values of the convolution mask determine the spatial correlation pattern and the variance of hijk. We generated 3-D networks, whose radii followed a spatially correlated, uniform distribution using a convolution mask of all ones, and the resulting correlation functions are shown in Figure 14a.
The probability distributions of rvertical and roptimal for correlated networks depend on Lcorr. The mean of rvertical is unchanged, but the variance increases as the correlation length increases, while both the mean and variance of roptimal increase with Lcorr. Since localization of flow is very sensitive to the mean effective radius (e.g., Figure 10c), wormhole formation is expected to be easier in correlated networks. Our simulations indeed showed that in correlated networks, a mean radius of less than 1 µm was sufficient to match the experimental breakthrough time. However, we could not quantify the effect of the correlation length precisely owing to relatively large finite size fluctuations, especially when Lcorr was a sizable fraction of the network length. Wormholes formed in correlated networks seemed to consist of several smaller wormholes bundled together and tended to be straighter than in uncorrelated networks (Figure 15).
We conducted core-flood experiments in a micritic reef limestone from Abu Dhabi using water saturated with CO2 at 5.5 MPa. The carbonic acid thus prepared was injected at constant rates between 0.036 and 0.11 mL/min. Microtomography images of the limestone before the experiment did not show connected macropores but revealed abundant microscale pores, which likely form the connected backbone of the pore space. Both the time evolution of the pressure head and a microtomography image made after the experiment demonstrated that localized channels (wormholes) formed during the experiments. We also tried to simulate wormhole formation numerically. To reduce the computational difficulties, we used a novel approach that divided the pore network in two parts, a leading network where the wormhole forms and a number of identical secondary networks that change more slowly. The leading and secondary networks are, in effect, competing for reactive flow, but the leading network wins the competition. Insight into the runaway dissolution instability was gained by analyzing the dissolution process in a system of parallel leading and secondary pipes. Runaway growth begins when the fluid reaching the leading pipe downstream end deviates from complete calcium saturation. The breakthrough of undersaturated fluid provides a kinetic switch [Groves and Howard, 1994] for increased flow. We found that the leading pipe did not need to be significantly wider than the secondary pipes for the runaway dissolution instability to occur as predicted by theoretical instability analyses [e.g., Scymczak and Ladd, 2011, 2014]. More realistic simulations were performed using 3-D heterogeneous networks. The results indicate that increasing the initial mean radius of the leading network or including finite spatial correlation facilitates wormhole formation. However, the effect of the variance of the radius distribution was more complex, owing to statistical fluctuations and the competition among highly conductive paths with nearly equal effective radii. Overall, the simulations produced pressure histories and dissolution patterns quite similar to the experimental observations, and the simulated breakthrough times agreed in order of magnitude with the ones observed.
The 3-D network model can be made more robust and realistic in several ways. First, we used simple cubic networks with a coordination number of 6. In this geometry, the nominal flow direction is perpendicular to two thirds of the pipes. Using other lattice geometries (e.g., face-centered cubic and body-centered cubic) and different coordination numbers would increase the generality of the model. Second, the networks studied, so far, were composed of cylindrical pipes whose radii were drawn according to simple uniform distributions. The pore space of rocks is much more complicated and, if discretized into individual conduits, has nontrivial pore shape and size distributions. It would be interesting to test the sensitivity of the model to more realistic distributions, preferably derived from high-resolution microtomography images. For example, Menke et al.  segmented the pore space into spherical nodal pores and cylindrical throats. Finally, when two adjacent pipes grow at the same time, the barrier between them can dissolve totally, and the pipes would merge into a single conduit [see Budek and Szymczak, 2012]. These events were not modeled here.
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division under award number DE-FG01-09ER14760. The microtomography scans were performed at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Infrastructure Network (NNIN), which is supported by the National Science Foundation under NSF award number ECS-0335765. CNS is part of Harvard University. The experimental data and simulation code are available at http://badshot.mit.edu/public/data.