Semi-analytical finite element method for simulating chemical dissolution-front instability problems in fluid-saturated porous media

PurposeThe objective of this paper is to develop a semi-analytical finite element method for solving chemical dissolution-front instability problems in fluid-saturated porous media.Design/methodology/approachThe porosity, horizontal and vertical components of the pore-fluid velocity and solute concentration are selected as four fundamental unknown variables for describing chemical dissolution-front instability problems in fluid-saturated porous media. To avoid the use of numerical integration, analytical solutions for the property matrices of a rectangular element are precisely derived in a purely mathematical manner. This means that the proposed finite element method is a kind of semi-analytical method. The column pivot element solver is used to solve the resulting finite element equations of the chemical dissolution-front instability problem.FindingsThe direct use of horizontal and vertical components of the pore-fluid velocity as fundamental unknown variables can improve the accuracy of the related numerical solution. The column pivot element solver is useful for solving the finite element equations of a chemical dissolution-front instability problem. The proposed semi-analytical finite element method can produce highly accurate numerical solutions for simulating chemical dissolution-front instability problems in fluid-saturated porous media.Originality/valueAnalytical solutions for the property matrices of a rectangular element are precisely derived for solving chemical dissolution-front instability problems in fluid-saturated porous media. The proposed semi-analytical finite element method provides a useful way for understanding the underlying dynamic mechanisms of the washing land method involved in the contaminated land remediation.

Dynamics of wormhole formation in fractured limestones

Reactive transport in porous or fractured media often results in an evolution of highly conductive flow channels, often referred to as “wormholes”. The most spectacular wormholes are caves in fractured limestone terrains. Here, a model of their early evolution is presented. The modeling domain is a two-dimensional square net consisting of one-dimensional fractures intersecting each other in a rectangular grid. Fractures have given width b and length l, and to each fracture a constant aperture width, a (homogeneous net), or an aperture width taken from a lognormal statistical distribution (heterogeneous net) is assigned. The boundary conditions are constant head h at the input driving the water downstream to the output at h=0. Linear dissolution kinetics, controlled by surface kinetics and diffusion, are active. First we discuss the simple case of a homogeneous net. Two steps in its evolution are observed. In the first, all fractures are widened evenly and a homogeneous even dissolution front progresses slowly into the aquifer. The second step is triggered by an instability when, due to small perturbations, some of the foremost fractures gain length compared to the neighboring ones. Then, these fractures discharge flow using the parallel resistances of the net. This way they attract more fresh aggressive water and their propagation is enhanced. Several wormholes (caves) are penetrating into the aquifer but only one reaches the output, whereas the others stop growing due to the redistribution of hydraulic heads caused by the leading wormhole. The mechanisms governing the evolution of a single wormhole are explored by increasing the aperture width of one selected input fracture by Δa≪a. In this case, only one single wormhole is created and inspection of the flow rates along it reveal the mechanism of flow enhancement in detail. If one uses a heterogeneous net, the first step of evolution is suppressed because of the large perturbations, and wormholes start to grow immediately. We have also modeled the case of several competing wormholes in a homogeneous net by inserting appropriate seeds. We find that there is a critical distance between the wormholes. Within this distance only one wormhole survives, whereas there is no interaction between them when they are separated by more than the critical distance. We also answer the following question: why do wormholes in a two-dimensional model exhibit breakthrough times at least 1 order of magnitude smaller than a one-dimensional model representing the aquifer by one single plane-parallel fracture of the same dimensions? Finally, we present several scenarios with non-homogeneous distribution of initial aperture widths. In these, a uniform dissolution front does not develop and wormholes start to grow immediately, which is more likely expected in nature.

Simulation of Dissolution Front Propagation During Acid Injection Into Porous Media: A CFD Approach

Acid is often injected into porous media to dissolve rock material and enhance flow capability of the rock. Most simulation studies on the propagation of the dissolution front are based on constant injection rate (CIR). Therefore, the objective of this work is to develop numerical model to study acid dissolution front under constant injection pressure (CIP) and also incorporate the effect of fluid temperature on acid–rock interaction. Commercial computational Fluid Dynamics (CFD) software (ANSYS Fluent) is used to solve three-dimensional acid-rock interaction model in cylindrical coordinates. In this work, correlated porosity and permeability distributions are generated. Effect of heat transfer between the injected fluid and the formation on fluid properties and surface reaction rate are accounted for in the model. The study confirmed that all types of acid dissolution patterns exist during constant injection pressure. CIP technique requires lower acid volume to achieve breakthrough in the conical and branched dissolution regimes, than that is required for CIR technique. In dominant wormhole pattern, both techniques require nearly the same acid volume to breakthrough. Thermal interaction between the injected fluid and the formation leads to change of surface reaction rate and physical properties of the fluid, such as viscosity, density, and diffusivity. Injection of cold fluid into heated formation leads to a higher wormhole density as found from experimental studies due to retardation of surface reaction rate. The model developed in this work accurately captures different dissolution patterns. The model shows that the acid volume required for wormhole breakthrough depends on the inlet conditions (CIR or CIP) and the thermal interaction between the injected fluid and formation. This modeling study attempts to answer the critical questions pertaining to the effect of temperature and injection conditions on acid-rock interaction.