Flow Modeling of Well Test Analysis for Porous–Vuggy Carbonate Reservoirs

Porous–vuggy carbonate reservoirs consist of both matrix and vug systems. This paper represents the first study of flow issues within a porous–vuggy carbonate reservoir that does not introduce a fracture system. The physical properties of matrix and vug systems are quite different in that vugs are dispersed throughout a reservoir. Assuming spherical vugs, symmetrically distributed pressure, centrifugal flow of fluids and considering media that is directly connected with wellbore as the matrix system, we established and solved a model of well testing and rate decline analysis for porous–vuggy carbonate reservoirs, which consists of a dual porosity flow behavior. Standard log–log type curves are drawn up by numerical simulation and the characteristics of type curves are analyzed thoroughly. Numerical simulations showed that concave type curves are dominated by interporosity flow factor, external boundary conditions, and are the typical response of porous–vuggy carbonate reservoirs. Field data interpretation from Tahe oilfield of China were successfully made and some useful reservoir parameters (e.g., permeability and interporosity flow factor) are obtained from well test interpretation.     

An Analytical Model of Porosity–Permeability for Porous and Fractured Media

The classic Kozeny–Carman equation (KC) uses parameters that are empirically based or not readily measureable for predicting the permeability of unfractured porous media. Numerous published KC modifications share this disadvantage, which potentially limits the range of conditions under which the equations are applicable. It is not straightforward to formulate non-empirical general approaches due to the challenges of representing complex pore and fracture networks. Fractal-based expressions are increasingly popular in this regard, but have not yet been applied accurately and without empirical constants to estimating rock permeability. This study introduces a general non-empirical analytical KC-type expression for predicting matrix and fracture permeability during single-phase flow. It uses fractal methods to characterize geometric factors such as pore connectivity, non-uniform grain or crystal size distribution, pore arrangement, and fracture distribution in relation to pore distribution. Advances include (i) modification of the fractal approach used by Yu and coworkers for industrial applications to formulate KC-type expressions that are consistent with pore size observations on rocks. (ii) Consideration of cross-flow between pores that adhere to a fractal size distribution. (iii) Extension of the classic KC equation to fractured media absent empirical constants, a particular contribution of the study. Predictions based on the novel expression correspond well to measured matrix and fracture permeability data from natural sandstone and carbonate rocks, although the currently available dataset for fractures is sparse. The correspondence between model calculation results and matrix data is better than for existing models.     

Behavior of Nonaqueous Phase Liquids in Fractured Porous Media under Two‐Phase Flow Conditions

After dense nonaqueous phase liquids (DNAPLs) travel downward through the subsurface, they typically come to rest on fractured bedrock or tight clay layers, which become additional pathways for DNAPL migration. DNAPLs trapped in fractures are continuous sources of groundwater contamination. To decide whether they can be left in place to dissolve or volatilize, or must be removed with active treatment, the movement of DNAPLs in fractured media must be understood at a fundamental level. This work presents numerical simulations of the movements of DNAPLs in naturally fractured media under twophase flow conditions. The flow is modeled using a multiphase network flow model, used to develop predictive capabilities for DNAPL flow in fractures. Capillary pressure–saturation–relative permeability curves are developed for twophase flow in fractures. Comparisons are made between the behavior in crystalline, almost impermeable rocks (e.g. granite) and more permeable rocks like sandstone, to understand the effects of the rock matrix on the displacement of the DNAPLs in the fracture. For capillarydominated flow, displacements occur as a sequence of jumps, as the invading phase overcomes the capillary pressure at downgradient apertures. Preferential channels for the displacement of nonaqueous phase are formed due to high fracture aperture in some regions.     

Modeling Kinetic Interphase Mass Transfer for Two-Phase Flow in Porous Media Including Fluid–Fluid Interfacial Area

Interphase mass transfer in porous media takes place across fluid–fluid interfaces. At the field scale, this is almost always a kinetic process and its rate is highly dependent on the amount of fluid–fluid interfacial area. Having no means to determine the interfacial area, modelers usually either neglect kinetics of mass transfer and assume local equilibrium between phases or they estimate interfacial area using lumped parameter approaches (in DNAPL pool dissolution) or a dual domain approach (for air sparging). However, none of these approaches include a physical determination of interfacial area or accounts for its role for interphase mass transfer. In this work, we propose a new formulation of two-phase flow with interphase mass transfer, which is based on thermodynamic principles. This approach comprises a mass balance for each component in each phase and a mass balance for specific interfacial area. The system is closed by a relationship among capillary pressure, interfacial area, and saturation. We compare our approach to an equilibrium model by showing simulation results for an air–water system. We show that the new approach is capable of modeling kinetic interphase mass exchange for a two-phase system and that mass transfer correlates with the specific interfacial area. By non-dimensionalization of the equations and variation of Peclet and Damköhler number, we make statements about when kinetic interphase mass transfer has to be taken into account by using the new physically based kinetic approach and when the equilibrium model is a reasonable simplification.     

Explicit Rate–Time Solutions for Modeling Matrix–Fracture Flow of Single Phase Gas in Dual-Porosity Media

One of the widely used methods for modeling matrix–fracture fluid exchange in naturally fractured reservoirs is dual porosity approach. In this type of modeling, matrix blocks are regarded as sources/sinks in the fracture network medium. The rate of fluid transfer from matrix blocks into fracture medium may be modeled using shape factor concept (Warren and Root, SPEJ 3:245–255, 1963); or the rate–time solution is directly derived for the specific matrix geometry (de Swaan, SPEJ 16:117–122, 1976). Numerous works have been conducted to study matrix–fracture fluid exchange for slightly compressible fluids (e.g. oil). However, little attention has been taken to systems containing gas (compressible fluid). The objective of this work is to develop explicit rate–time solutions for matrix–fracture fluid transfer in systems containing single phase gas. For this purpose, the governing equation describing flow of gas from matrix block into fracture system is linearized using pseudopressure and pseudotime functions. Then, the governing equation is solved under specific boundary conditions to obtain an implicit relation between rate and time. Since rate calculations using such an implicit relation need iterations, which may be computationally inconvenient, an explicit rate–time relation is developed with the aid of material balance equation and several specific assumptions. Also, expressions are derived for average pseudopressure in matrix block. Furthermore, simplified solutions (originated from the complex general solutions) are introduced applicable in infinite and finite acting flow periods in matrix. Based on the derived solutions, expressions are developed for shape factor. An important observation is that the shape factor for gas systems is the same as that of oil bearing matrix blocks. Subsequently, a multiplier is introduced which relates rate to matrix pressure instead of matrix pseudopressure. Finally, the introduced equations are verified using a numerical simulator.     

Convergence Results for Forchheimer’s Equations for Fluid Flow in Porous Media

The Forchheimer equations for non-slow flow in a saturated porous medium are studied. We prove the convergence results for both the first and the second Forchheimer coefficients.     

Transport of Polymer Particles in Oil–Water Flow in Porous Media: Enhancing Oil Recovery

Transport in Porous Media - 3D printing with powders offers the most analogous method to the natural way in which clastic reservoir rocks are formed, resulting in pore network textures and...     

Impact of the Buoyancy–Viscous Force Balance on Two-Phase Flow in Layered Porous Media

Motivated by geological carbon storage and hydrocarbon recovery, the effect of buoyancy and viscous forces on the displacement of one fluid by a second immiscible fluid, along parallel and dipping layers of contrasting permeability, is characterized using five independent dimensionless numbers and a dimensionless storage or recovery efficiency. Application of simple dimensionless models shows that increased longitudinal buoyancy effects increase storage efficiency by reducing the distance between the leading edges of the injected phase in each layer and decreasing the residual displaced phase saturation behind the leading edge of the displacing phase. Increased transverse buoyancy crossflow increases storage efficiency if it competes with permeability layering effects, but reduces storage efficiency otherwise. When both longitudinal and transverse buoyancy effects are varied simultaneously, a purely geometrical dip angle group defines whether changes in storage efficiency are dominated by changes in the longitudinal or transverse buoyancy effects. In the limit of buoyancy-segregated flow, we report an equivalent, unidimensional flow model which allows rapid prediction of storage efficiency. The model presented accounts for both dip and layering, thereby generalizing earlier work which accounted for each of these but not both together. We suggest that the predicted storage efficiency can be used to compare and rank geostatistical realizations, and complements earlier heterogeneity measures which are applicable in the viscous limit.     

Near-Critical Gas–Liquid Flow in Porous Media: Monovariant Model, Analytical Solutions and Convective Mass Exchange Effects

We examine a class of hydrocarbon reservoirs whose thermodynamic state remains close to the critical point during the all period of reservoir exploitation. Such a situation is typical for the so-called gas–condensate systems, in which the liquid phase is formed from gas when pressure decreases. Due to proximity to critical point, the mixture contains many components which are neutral with respect to the phase state. This determines a low thermodynamic degree of freedom of the system. As the results, the mathematical flow model allows a significant reduction in the number of conservation equations, whatever the number of chemical components. In the vicinity of a well, the system may be reduced to one transport equation for saturation. This nonlinear model yields exact analytical solutions when the flow is self-similar. In more general case of flow, we develop partially linearized solutions which are shown to be sufficiently exact. The spectrum of examined cases covers the flow in a medium with a sharp heterogeneity and a sharp variation in the flow rate. A significant relative gas flow past liquid gives rise to a convective mass exchange phenomenon which appears highly different from that observed in static. In the case of a medium discontinuity, the convective mass exchange gives rise to a phenomenon of condensate saturation billow formation. A sharp variation in the flow rate leads to a hysteretic behavior of the saturation field.     

Uniqueness of Specific Interfacial Area–Capillary Pressure–Saturation Relationship Under Non-Equilibrium Conditions in Two-Phase Porous Media Flow

The capillary pressure?Csaturation (P ^c?CS ^w) relationship is one of the central constitutive relationships used in two-phase flow simulations. There are two major concerns regarding this relation. These concerns are partially studied in a hypothetical porous medium using a dynamic pore-network model called DYPOSIT, which has been employed and extended for this study: (a) P ^c?CS ^w relationship is measured empirically under equilibrium conditions. It is then used in Darcy-based simulations for all dynamic conditions. This is only valid if there is a guarantee that this relationship is unique for a given flow process (drainage or imbibition) independent of dynamic conditions; (b) It is also known that P ^c?CS ^w relationship is flow process dependent. Depending on drainage and imbibition, different curves can be achieved, which are referred to as ??hysteresis??. A thermodynamically derived theory (Hassanizadeh and Gray, Water Resour Res 29: 3389?C3904, 1993a) suggests that, by introducing a new state variable, called the specific interfacial area (a ^nw, defined as the ratio of fluid?Cfluid interfacial area to the total volume of the domain), it is possible to define a unique relation between capillary pressure, saturation, and interfacial area. This study investigates these two aspects of capillary pressure?Csaturation relationship using a dynamic pore-network model. The simulation results imply that P ^c?CS ^w relation not only depends on flow process (drainage and imbibition) but also on dynamic conditions for a given flow process. Moreover, this study attempts to obtain the first preliminary insights into the global functionality of capillary pressure?Csaturation?Cinterfacial area relationship under equilibrium and non-equilibrium conditions and the uniqueness of P ^c?CS ^w?Ca ^nw relationship.     

Correction to: Darcy–Brinkman–Forchheimer Model for Film Boiling in Porous Media

Transport in Porous Media - A correction to this paper has been published: https://doi.org/10.1007/s11242-021-01631-0     

Pore-Scale Imaging and Modelling of Reactive Flow in Evolving Porous Media: Tracking the Dynamics of the Fluid–Rock Interface

Transport in Porous Media - Fluid–mineral and fluid–rock interfaces are key parameters controlling the reactivity and fate of fluids in reservoir rocks and aquifers. The interface...     

Influence of Viscous Fingering on Dynamic Saturation–Pressure Curves in Porous Media

We report on results from primary drainage experiments on quasi-two-dimensional porous models. The models are transparent, allowing the displacement process and structure to be monitored in space and time during primary drainage experiments carried out at various speeds. By combining detailed information on the displacement structure with global measurements of pressure, saturation and the capillary number Ca, we obtain a scaling relation relating pressure, saturation, system size and capillary number. This scaling relation allows pressure–saturation curves for a wide range of capillary numbers to be collapsed on the same master curve. We also show that in the case of primary drainage, the dynamic effect in the capillary pressure–saturation relationship observed on partially water saturated soil samples might be explained by the combined effect of capillary pressure along the invasion front of the gaseous phase, and pressure changes caused by viscous effects in the wetting fluid phase.     

Infiltration of Salt Solute in Homogeneous and Saturated Porous Media – An Analytical Solution Evaluated by Numerical Simulations

In many cases various land disposal activities (e.g. infiltration, injection wells) constitute an important potential source of groundwater contamination. Using a 2D physical model, the behaviour of the infiltration of a salt solute, locally injected in a homogeneous and saturated porous medium, has been analysed. Under various experimental conditions (density effects, injection flow rate) the salt solute penetrates the porous media and leads to a steady-state regime inside the mixing zone. By using experimental observations, the basic equations describing the flow and transport phenomena can be simplified and an analytical solution obtained. Its validity is subject to numerical verification. The numerical model, based on the development of the mass balance equation expressed by its conservative form, uses a combination of the mixed hybrid finite element (MHFE) and discontinuous finite element (DFE) methods. The efficiency of this numerical model was previously verified on standard benchmarks, for example Elder's problem and Henry's problem. In the first step, the qualitative good agreement between the experimental and numerical results enabled us to use the numerical model in order to verify some hypotheses resulting from visual observations. Thus, the numerical results reveal the existence of a steady-state regime inside the mixing zones. Nevertheless, both its vertical and longitudinal extensions are less than those observed in the physical model. In the second step, the numerical results enable to establish the validity domain as well as the accuracy of the proposed analytical solution.     

Instability of Hadley–Prats Flow with Viscous Heating in a Horizontal Porous Layer

The onset of convective rolls instability in a horizontal porous layer subject to a basic temperature gradient inclined with respect to gravity is investigated. The basic velocity has a linear profile with a non-vanishing mass flow rate, i.e., it is the superposition of a Hadley-type flow and a uniform flow. The influence of the viscous heating contribution on the critical conditions for the onset of the instability is assessed. There are four governing parameters: a horizontal Rayleigh number and a vertical Rayleigh number defining the intensity of the inclined temperature gradient, a Péclet number associated with the basic horizontal flow rate, and a Gebhart number associated with the viscous dissipation effect. The critical wave number and the critical vertical Rayleigh number are evaluated for assigned values of the horizontal Rayleigh number, of the Péclet number, and of the Gebhart number. The linear stability analysis is performed with reference either to transverse or to longitudinal roll disturbances. It is shown that generally the longitudinal rolls represent the preferred mode of instability.     

Beyond Kozeny–Carman: Predicting the Permeability in Porous Media

Transport in Porous Media - Various processes such as heterogeneous reactions or biofilm growth alter a porous medium’s underlying geometric structure. This significantly affects its...     

Modeling Variable Density Flow and Solute Transport in Porous Medium: 2. Re‐Evaluation of the Salt Dome Flow Problem

Case 5, Level 1 of the international HYDROCOIN groundwater flow modeling project is an example of idealized flow over a salt dome. The groundwater flow is strongly coupled to solute transport since density variations in this example are large (20%).Several independent teams simulated this problem using different models. Results obtained by different codes can be contradictory. We develop a new numerical model based on the mixed hybrid finite elements approximation for flow, which provides a good approximation of the velocity, and the discontinuous finite elements approximation to solve the advection equation, which gives a good approximation of concentration even when the dispersion tensor is very small. We use the new numerical model to simulate the salt dome flow problem.In this paper we study the effect of molecular diffusion and we compare linear and nonlinear dispersion equations. We show the importance of the discretization of the boundary condition on the extent of recirculation and the final salt distribution. We study also the salt dome flow problem with a more realistic dispersion (very small dispersion tensor). Our results are different to prior works with regard to the magnitude of recirculation and the final concentration distribution. In all cases, we obtain recirculation in the lower part of the domain, even for only dispersive fluxes at the boundary. When the dispersion tensor becomes very small, the magnitude of recirculation is small. Swept forward displacement could be reproduced by using finite difference method to compute the dispersive fluxes instead of mixed hybrid finite elements.     

Darcy–Carreau Model and Nonlinear Natural Convection for Pseudoplastic and Dilatant Fluids in Porous Media

Transport in Porous Media - The linear and weakly nonlinear stability analyses are carried out to study instabilities in Darcy–Bénard convection for non-Newtonian inelastic fluids. The...