Modeling of Rock Permeability Damage and Repairing Dynamics Due to Invasion and Removal of Particulate from Drilling Fluids

The investigation discussed in this paper was motivated by the need for model which is able to simulate both permeability reduction of hydrocarbon formations due to the mud-component invasion during over-balance drilling, casing/cementing, workover operations, and dynamics of permeability repairing during well cleanup. The paper focuses on development and validation of model to describe internal mud cake (IMC) dynamics, placing special emphasis on dynamics of the IMC removing during well cleanup procedure. Set of laboratory experiments with clay slurry injection and subsequent brine water backflow in samples of Bentheimer sandstone is discussed. The specific of these experiments is that backflow was carried out with alternating rates (“multirate” backflow). It is shown that the conventional deep-bed filtration model is not able to reproduce the dynamics of multirate backflow. The stochastic model we suggest takes into account pore size distribution and describes the mobilization of trapped particles within individual groups of pores within a “pore ensemble.” We provide simulation results to show that the suggested model reasonably reproduces permeability dynamics during both clay slurry injection and brine water backflow stages.     

Effective Solution through the Streamline Technique and HT-Splitting for the 3D Dynamic Analysis of the Compositional Flows in Oil Reservoirs

In this article we suggest a new phenomenological mathematical model for the groundwater transport of colloid particles through porous media which is able to describe some significant effects experimentally observed but not captured within the framework of the classic approach. Our basic idea is to consider both the pure water and the colloid suspension as two thermodynamic phases. Using the network models of porous media, we simulated numerically the transport process at the pore-scale. By averaging the result derived, we have obtained the relative permeabilities for both phases, the percolation threshold for suspension flow, and the effective suspension viscosity. Due to specific laws of colloid particles repartition between various classes of pores, the relative permeability of suspension happens to be a highly nonlinear function of saturation, very far from the diagonal straight line. This determines a difference between the macroscale phase velocities. The suspension velocity is shown to be higher than that of water in major cases, only if the colloid particles are not too large. The suggested model predicts and describes in a closed form the effect of colloid transport facilitation observed experimentally.     

Two-Phase Flow Properties Prediction from Small-Scale Data Using Pore-Network Modeling

The objective of this work is to evaluate the prediction accuracy of network modeling to calculate transport properties of porous media based on the interpretation of mercury invasion capillary pressure curves only. A pore-scale modeling approach is used to model the multi-phase flow and calculate gas/oil relative permeability curves. The characteristics of the 3-D pore-network are defined with the requirement that the network model satisfactorily reproduces the capillary pressure curve (P_c curve), the porosity and the permeability. A sensitivity study on the effect of the input parameters on the prediction of capillary pressure and gas/oil relative permeability curves is presented. The simulations show that different input parameters can lead to similarly good reproductions of the experimental P_c, although the predicted relative permeabilities K_r are somewhat widespread. This means that the information derived from a mercury invasion P_c curve is not sufficient to characterize transport properties of a porous medium. The simulations indicate that more quantitative information on the wall roughness and the node/bond aspect ratio would be necessary to better constrain the problem. There is also evidence that in narrow pore size distributions pore body volume and pore throat radius are correlated while in broad pore size distributions they would be uncorrelated.     

Permeability anisotropy due to consolidation of compressible porous media

The characterisation of flow through porous media is important for all solid–liquid separation and fluid transport realms. The permeability of porous media can be anisotropic and furthermore, the extent of anisotropy can be increased as a result of an applied compressive force. However, the understanding of how anisotropy develops is incomplete. An overview of research on permeability anisotropy is given and an expression for predicting anisotropy as a function of void ratio is offered. The two underlying assumptions of the proposed model are: flow in different directions occurs within the same network of pores and deformation is primarily due to the compression of the particles in the direction of the applied force rather than due to particle rearrangement. The assumption of network connectivity allows permeability anisotropy to be described as a function of flow path tortuosity only. Results are presented for hydraulic anisotropy measured in lignite that has been upgraded by a compression dewatering method known as mechanical thermal expression. The lignite permeability is shown to be up to eight times greater in the direction perpendicular to compression, suggesting that the rate of dewatering could be significantly increased by choosing the drainage to also be perpendicular to the direction of the applied compressive force. It is illustrated that the proposed anisotropy model can be used to accurately predict the experimentally determined permeability anisotropy ratios for lignite, as well as for other materials including sand, clay and kaolin.     

The structure of instantaneous reversals in highly turbulent flows

Structure of instantaneous flow reversals has been measured in a highly turbulent axisymmetric diffuser flow using pulsed-wire anemometry. In this 8° nominal included angle conical diffuser, the adverse pressure gradient (APG) is strong enough to cause appreciable instantaneous flow reversals (instantaneous backflow up to 30% of the time), but the time-averaged flow is non-separated. The results are compared with the other severe APG separating flows reported in literature. An increase in entry Reynolds number indicated a decrease in the size of near-wall instantaneous reversals region as well as a decrease in the magnitude of instantaneous backflow. Also, the region of instantaneous reversals moves slightly downstream at appreciably higher Reynolds numbers. The initiation and growth of instantaneous reversals in a conical diffuser was found to strongly influence the wall-layer and the central region. Present results also suggest that the instantaneous backflow should be considered for modelling of instantaneously-separating diffuser flows. In the final stages of a conical diffuser, the magnitudes of cross-stream pressure gradient were found to be appreciably larger than that of the longitudinal pressure gradient, indicating that accurate representation of a conical diffuser flow can not be achieved without considering V-momentum equation. A comparison of various separating flows revealed remarkable similarity of instantaneous reversals regions and distributions even in different flow configurations.     

Experimental displacement patterns in a 2 × 2 quadrant block with permeability and wettability heterogeneities—problems for numerical modelling

The effect of heterogeneities on miscible and immiscible flood displacements in 2D bead packs in quadrant form, 2 × 2 block heterogeneity, with either a permeability or a wettability contrast is the subject of this paper. The physical processes occurring during miscible and immiscible flow and displacement within permeability and wettability quadrant bead pack models have been studied experimentally. This geometry occurs in a number of situations relevant to hydrocarbon production: particularly faults where adjacent rocks have large permeability contrasts with rapid changes, in the laboratory with core butting, in reservoir simulation where grid blocks have different permeability and in reservoirs having near-wellbore damage problems. The model quadrants 1–4, had 1 and 4 and 2 and 3 with identical properties, either in permeability or wettability. Reported are complete unit mobility miscible displacements, then the effects of viscosity differences (mobility modifiers) and finally immiscible displacements on displacement patterns for initial linear injection. The experiments demonstrate that nodal flow occurs for both miscible and immiscible flow, but for immiscible flow there are boundary effects due to capillary pressure differences created by water saturation changes or wettability contrasts which can leave patches of isolated fluid within a quadrant. The displacement patterns for the different models and fluids change significantly with the viscosity and wettability changes, particularly for the immiscible displacements. This is due to the changing capillary pressure between the quadrant blocks as the water saturation change. These are difficult to address in numerical modelling but should be accounted for. Other effects include coupling of all physical processes governing the flow through the node and creations of microzones of trapped residual oil. Our displacement patterns can therefore be a valuable verification benchmark tool for numerical modelling and a calibration data source for those wishing to simulate the effects of capillary pressure under differing wettability conditions and for those investigating upscaling modelling procedures. However, the possible loss of physical reality when averaging must always be considered.     

Characterization of the crossover from capillary invasion to viscous fingering to fracturing during drainage in a vertical 2D porous medium

We experimentally studied the displacement of a viscous wetting fluid (water) by an inviscid non-wetting fluid (air) injected at the bottom of a vertical Hele-Shaw cell filled with glass microbeads. In order to cover a wide parameter space, the permeability of the porous medium was varied by using different bead size ranges and diverse air flow rates were generated by means of a syringe pump. A LED light table was used to back illuminate the experimental cell, allowing a high speed camera to capture images of the drainage process at equal time intervals. The invasion occurred in intermittent bursts. Image processing of the bursts and fractal analysis showed successive transitions from capillary invasion to viscous fingering to fracturing during the same experiment, dependent on the medium permeability, the air injection flow rate, and the vertical position in the cell. The interplay between the capillary, viscous and gravity forces determines the nature of the invasion pattern and the transitions, from capillary invasion to viscous fingering with decreasing fluid pressure on one hand and from viscous fingering to fracturing with decreasing effective overburden pressure on the other hand.     

Pulsatile flows of Leslie–Ericksen liquid crystals

Capillary pulsatile flows of calamitic (rod-like) and discotic nematic liquid crystals are analyzed using the Leslie–Ericksen equations for low-molar mass liquid crystals, using computational, analytical, and scaling methods. The dependence of flow-enhancement and power requirement on frequency, amplitude, pressure drop wave-form, molecular geometry is characterized. The unique roles of orientation-dependent local viscosity and backflow (orientation-driven flow) on flow-enhancement and power requirement are elucidated. The local viscosity effect is shown to be a significant factor in flow-enhancement at all pressure drops, but only affects power requirement at higher pressure drops. Backflow has weak effects on flow-enhancement and large effects on power requirements at low average pressure drops. Amplitude, frequency, and molecular geometry effects are clearly manifested through viscosity and backflow. A detailed comparison with predictions for power law fluids shows a clear correspondence between these non-Newtonian fluids and nematic liquid crystals. The unique distinguishing feature of pulsatile flows of liquid crystals is found to be backflow, such that power increases with increasing frequency, a featured that does not exist in other non-Newtonian fluids due to lack of a strong flow driven by restructuring/re-orientation processes. Future use of these new results may include measurements of viscoelastic parameters that control backflow.     

Aerodynamic analysis and validation by using three turbulence models for narrow trench configuration

Three turbulence models SST Gamma Theta, k-ω and k-ε which all found in ANSYS CFX were used. Velocity contours, pressure coefficient profiles and turbulence levels contours were discussed. Results indicate that three models calculated the cross flow boundary layer with different thicknesses. This leads to a difference in calculation each the momentum of the cross flow fluid closer to the jet exit and cooling performance. At M = 0.5 from range X/D = 5:40, the average centerline effectiveness values in case SST Gamma Theta model and k-ω model were 21 and 9.2 % larger than k-ε model, respectively. While at high blowing ratios 1 and 1.4, the effectiveness values in case SST Gamma Theta model and k-ω model were 16, 24.4 % and 32, 46 % less than k-ε model, respectively. k-ω model shows a larger backflow regions than others which a maximum negative value of u/u_∞ = ?0.198 is reached in zone 4.     

Measurement of permeability of microfluidic porous media with finite-sized colloidal tracers

We present two methods how the permeability in porous microstructures can be experimentally obtained from particle tracking velocimetry of finite-sized colloidal particles suspended in a liquid. The first method employs additional unpatterned reference channels where the liquid flow can be calculated theoretically and a relationship between the velocity of the particles and the liquid is obtained. The second method takes advantage of a time-dependent pressure drop that leads to an exponential decrease in the particle velocity inside a porous structure. From the corresponding decay time, the permeability can be calculated independently of the particle size. Both methods lead to results comparable with permeabilities derived from numerical simulations.     

An Experimental and Numerical Study of Relative Permeability Estimates Using Spatially Resolved $$T_1$$- z NMR

Relative permeability is a key characteristic describing flow properties of petroleum reservoirs, aquifers and water retention of soils. Various laboratory methods, typically categorised as steady-state, unsteady-state and centrifuge are used to measure relative permeability and may lead to different results. In recent years, 1D MRI, NMR \(T_2\) and \(T_1\) profiling have been applied for the characterisation of rock cores. It has been shown that spatially resolved NMR in conjunction with centrifuge technique may provide high-quality capillary pressure curves. Combining Burdine and Brooks–Corey models enables estimation of relative permeability from capillary pressure curves. This approach assumes a strong relationship between capillary pressure and relative permeability known to be complex. Here we compare a generalised approach of Green, which relies on saturation profiles set by various capillary drainage techniques, to a NMR relaxation approach. Comparisons are performed experimentally and numerically using three sandstone rocks to test the influence of rock morphology. The numerical part includes simulation of a centrifuge capillary drainage by applying morphological drainage transforms on high-resolution 3D tomograms. \(T_1\) responses along the sample are simulated using a random walk technique. The NMR relaxation-based approach is then compared to LBM simulated relative permeability and to experiment. The study confirms the applicability of NMR relaxation methods for relative permeability estimation of water-wet rocks and validates a numerical approach against experiment.     

Flow of a two-phase medium in a laval nozzle

The back reaction of particles on a gas flow in Laval nozzles was investigated experimentally. Experimental data were obtained that characterize the change produced by the particles of a solid phase in the shape of the sonic line, the pressure distribution on the nozzle profile, and the configuration of the shock waves in the jet. Flow rate coefficients are given for different nozzle profiles and mass fraction and sizes of the particles in the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 107–111, January–February, 1981.     

Influence of phase behavior on chemical flood transport phenomena

We propose that there are two classes of temporal development in the degradation of permeability of porous media due to deposition of fines: (1) Deposition, and therefore permeability degradation, is localized to bands growing orthogonally to the average local flow direction, and (2) permeability degradation occurs in stripes parallel to the local flow direction. These latter stripes do not influence total permeability much as they develop. When these stripes are allowed to develop, they coalesce and worm holes form. We discuss how imposing different flow conditions such as constant flow and constant pressure influence the deposition process. Our conclusion is that constant pressure conditions typically lead to a slower permeability degradation compared to constant flow conditions as a direct consequence of the formation of low-permeability bands. We test our ideas by numerical simulations on a simple model for fines migration and deposition in porous media.     

Co-history Matching: A Way Forward for Estimating Representative Saturation Functions

Core-scale experiments and analyses would often lead to estimation of saturation functions (relative permeability and capillary pressure). However, despite previous attempts on developing analytical and numerical methods, the estimated flow functions may not be representative of coreflood experiments when it comes to predicting similar experiments due to non-uniqueness issues of inverse problems. In this work, a novel approach was developed for estimation of relative permeability and capillary pressure simultaneously using the results of “multiple” corefloods together, which is called “co-history matching.” To examine this methodology, a synthetic (numerical) model was considered using core properties obtained from pore network model. The outcome was satisfactorily similar to original saturation functions. Also, two real coreflood experiments were performed where water at high and low rates were injected under reservoir conditions (live fluid systems) using a carbonate reservoir core. The results indicated that the profiles of oil recovery and differential pressure (dP) would be significantly affected by injection rate scenarios in non-water wet systems. The outcome of co-history matching could indicate that, one set of relative permeability and capillary pressure curves can reproduce the experimental data for all corefloods.     

Spontaneous Switching between Permeability Enhancement and Degradation in Fractures in Carbonate: Lumped Parameter Representation of Mechanically- and Chemically-Mediated Dissolution

Principal mechanical and chemical processes contributing to the observed spontaneous switching from net decrease in permeability to net increase in a fracture in carbonate are examined. The evolution of permeability, and related fracture aperture, is represented through a lumped parameter model. The significant processes of pressure solution beneath bridging asperities, transport of dissolved mass to the fracture void, and subsequent precipitation or dissolution within the fracture void enable the principal characteristics of observed behavior to be followed. The evolution of dissolved mass concentration in the pore fluid is followed for arbitrary applied stress, temperature, and pH conditions, with appropriate feedback to the evolution of fracture permeability. Comparisons with experimental measurements in limestone (Polak et al., 2004, Water Resour. Res. Vol. 40, W03502, doi: 10.1029/2003GL017575) show satisfactory agreement for the evolution of fracture aperture and to a lesser degree in calcium concentrations in the effluent pore fluid. Importantly, the spontaneous switching in permeability change, from aperture reducing to aperture increasing, with no change in environmental conditions, is replicated without the need for an ad hoc trigger. Although this switch is accurately replicated, the lumped parameter model is incapable of replicating the rapid observed growth in permeability that directly follows. This inability results from the assumed form of the lumped asperity model, that is incapable of representing the spatially distributed change in aperture that is seen to occur within the fracture. Despite this inconsistency, the model is shown capable of representing the principal behaviors evident in the response.     

Long-term Petrophysical Investigations on Geothermal Reservoir Rocks at Simulated In Situ Conditions

In the course of stimulation and fluid production, the chemical fluid–rock equilibrium of a geothermal reservoir may become disturbed by either temperature changes and/or an alteration of the fluid chemistry. Consequently, dissolution and precipitation reactions might be induced that result in permeability damage. In connection with the field investigations at a deep geothermal doublet, complementary laboratory-based research is performed to address these effects. The reservoir is located at a depth of 4100 to 4200 m near Groß Schönebeck within the Northeast German Basin, 50 km north of Berlin, Germany. Within the reservoir horizon, an effective pressure of approximately 45 MPa and a temperature of 150°C are encountered. Furthermore, the Lower Permian (Rotliegend) reservoir rock is saturated with a highly saline Ca–Na–Cl type formation fluid (TDS ≈ 255 g/l). Under these conditions we performed two sets of long-term flow-through experiments. The pore fluid used during the first and the second experiment was a 0.1 molar NaCl-solution and a synthetic Ca–Na–Cl type fluid with the specifications as above, respectively. The maximum run duration was 186 days. In detail, we experimentally addressed: (1) the effect of long-term flow on rock permeability in connection with possible changes in fluid chemistry and saturation; (2) the occurrence and consequences of baryte precipitation; and (3) potential precipitations related to oxygen-rich well water invasion during water-frac stimulation. In all substudies petrophysical experiments related to the evolution of rock permeability and electrical conductivity were complemented with microstructural investigations and a chemical fluid analysis. We also report the technical challenges encountered when corrosive fluids are used in long-term in situ petrophysical experiments. After it was assured that experimental artifacts can be excluded, it is demonstrated that the sample permeability remained approximately constant within margins of  ±50 % for nearly six months. Furthermore, an effect of baryte precipitation on the rock permeability was not observed. Finally, the fluid exchange procedure did not alter the rock transport properties. The results of the chemical fluid analysis are in support of these observations. In both experiments the electrical conductivity of the samples remained unchanged for a given fluid composition and constant p-T conditions. This emphasizes its valuable complementary character in determining changes in rock transport properties during long-term flow-through experiments when the risk of experimental artifacts is high.     

Computational Modeling of Fluid Flow through a Fracture in Permeable Rock

Laminar, single-phase, finite-volume solutions to the Navier–Stokes equations of fluid flow through a fracture within permeable media have been obtained. The fracture geometry was acquired from computed tomography scans of a fracture in Berea sandstone, capturing the small-scale roughness of these natural fluid conduits. First, the roughness of the two-dimensional fracture profiles was analyzed and shown to be similar to Brownian fractal structures. The permeability and tortuosity of each fracture profile was determined from simulations of fluid flow through these geometries with impermeable fracture walls. A surrounding permeable medium, assumed to obey Darcy’s Law with permeabilities from 0.2 to 2,000 millidarcies, was then included in the analysis. A series of simulations for flows in fractured permeable rocks was performed, and the results were used to develop a relationship between the flow rate and pressure loss for fractures in porous rocks. The resulting friction-factor, which accounts for the fracture geometric properties, is similar to the cubic law; it has the potential to be of use in discrete fracture reservoir-scale simulations of fluid flow through highly fractured geologic formations with appreciable matrix permeability. The observed fluid flow from the surrounding permeable medium to the fracture was significant when the resistance within the fracture and the medium were of the same order. An increase in the volumetric flow rate within the fracture profile increased by more than 5% was observed for flows within high permeability-fractured porous media.     

Finite Element Modelling of Flow Through a Porous Medium Between Two Parallel Plates Using The Brinkman Equation

Despite the widespread use of the Darcy equation to model porous flow, it is well known that this equation is inconsistent with commonly prescribed no slip conditions at flow domain walls or interfaces between different sections. Therefore, in cases where the wall effects on the flow regime are expected to be significant, the Darcy equation which is only consistent with perfect slip at solid boundaries, cannot predict velocity and pressure profiles properly and alternative models such as the Brinkman equation need to be considered. This paper is devoted to the study of the flow of a Newtonian fluid in a porous medium between two impermeable parallel walls at different Darcy parameters (Da). The flow regime is considered to be isothermal and steady. Three different flow regimes can be considered using the Brinkman equation: free flow (Da > 1), porous flow (high permeability, 1 > Da > 10⁻⁶) and porous flow (low permeability Da < 10⁻⁶). In the present work the described bench mark problem is used to study the effects of solid walls for a range of low to high Darcy parameters. Both no-slip and slip conditions are considered and the results of these two cases are compared. The range of the applicability of the Brinkman equation and simulated results for different cases are shown.