The work presents a reinterpreted and generalized version of Bernoulli’s equation (BE), which gives consistent results for entire pipe flow system varying geometry and design, in line with empirical data and fundamental principles and laws of physics. Reinterpreted since the role played by the internal water pressure is crucially changed compared to BE and generalized since viscous pressure is included. The framework developed uses thermodynamics, i.e. chemical potential, which consequently implies that BE should be formulated as a pressure balance between external supply and internal demands. External supply is delivered e.g. by a pump whereas internal demands are represented by changes in dynamic-, hydrostatic-, and viscous pressure. The external-internal distinction together with the requirement that entire systems must be considered, changes the way BE should be interpreted and applied since it conventionally only considers local subsections of the system. Entire systems must be considered since external pressure supplied depends on internal flow resistance terms, which further depends on the entire system. The pressure terms should additionally be applied according to respective physical content. High dynamic pressure value (high speed) is e.g. not the cause of decrease in internal water pressure as conventionally assumed. The generalized BE will be especially useful for design and engineering purposes but could also impact on how the Euler- and Navier–Stokes equations should be interpreted and applied. A condition for the cavitation phenomenon is additionally discussed.

A diffusive equilibrium thermodynamic framework for two-phase capillary drainage and imbibition pressure in thin horizontal cylindrical tubes partially wetted by water is presented. The approach is based on the sum of external- and internal chemical potential for each fluid phase, which equals zero when diffusive equilibrium occurs. The difference between non-wetting and wetting phase chemical potential either externally- or internally therefore defines and generalizes the capillary pressure concept. Difference in external chemical potential is generated using e.g. pumps whereas internal difference is due to changes in interfacial areas and phase entropies. The latter is included under the assumption of non-interacting phases, which therefore give a logarithmic expression for varying fluid saturations. The assumption requires, however, fit parameters since the reference state cannot be calculated ab initio on the macroscopic level. Simple tube geometry allows for development of analytical capillary pressure expressions for drainage and imbibition. The drainage expression is compared with experimental data for homogeneous porous media and excellent match is observed considering the rough assumption related to non-interacting phases. Heterogeneities induced by pore geometry and wettability effects may induce more uncertainties and deviations in more realistic porous media. The framework is general and consistent and supported by several additional observations. Jurin’s law can e.g. be derived therefrom. It could therefore have important consequences for engineering and science disciplines where simplified tube-like geometry is a relevant representation of more complex systems.

Summary An analytical expression for the bottomhole pressure (BHP) difference between a vertical injection and a production well is derived based on a balance between the external and internal chemical potential and the principle of least flow resistance. This expression is used to explain and interpret the impact of vertical grid refinement and varying permeability distributions of the fine gridblocks on simulated BHP differences and injectivity indices. The two potentials are manifested as the BHP difference and the viscous pressure loss given by Darcy’s law, respectively. The expression is, furthermore, developed by dividing the total water injection rate into flow paths, which involves the sum of the relationship between distance traveled, effective permeability, and volumetric flow rate for each path. It therefore enables physical justifications for the BHP and injectivity index (II) variations, as it is based on fundamental laws and principles. That is, it explains why and, in principle, how a total constant volume rate injected is distributed into a set of paths based on grid properties and nature’s tendency to minimize the external work density supplied, i.e., the BHP difference. The expression is first used to explain the decrease in BHP difference and corresponding increase in II upon refining one of the layers in the original grid in the vertical direction. The decrease is explained as an effect of reduced crossflow distances in the vertical direction when water is flowing in the finer compared to the coarser grid. Shorter crossflow distances require less supply of external work density and thereby lower BHP differences. It is, furthermore, used to explain BHP differences when populating refined gridblocks with different permeability distributions when one layer is divided into two and 20 layers. All the distributions have equal flow capacities. The BHP difference between the wells increased, corresponding to a decrease in the II for the two-layer case when the permeability contrast between the layers increased. It is explained as an effect of the reduced total flow capacity of the two composite refined layers as one of them becomes almost impermeable. The reduced total flow capacity consequently forces more water to crossflow out of the refined composite region, which requires a higher BHP difference. The same trend is likewise observed for the refined 20-layer cases, where the BHP difference increases, with a corresponding decrease in the II, as the contrast between the maximum and minimum permeability layers grows. The analytical expression has only been used for a limited set of refinement cases in the vertical direction herein, so further testing on other flow systems should be performed. Grid refinements laterally are also expected to contribute significantly to BHP variations, as the decrease in crossflow distance upon implementing grid refinement conventionally is even larger than in the vertical direction highlighted here. It should, from a more practical engineering point of view, additionally also be tested in reservoirs containing more than one fluid phase and under transient flow conditions, although accuracy will decrease the more the fluid properties of the second phase deviate from water and the flow departs from stationary conditions. The expression highlights nature’s holistic approach as the complete flow distribution changes upon altering permeability values locally. It may, therefore, have implications for how upscaling should be performed and support procedures conventionally used when determining flow fields through heterogeneous geological formations.

Water constitutes an increasing proportion of the production stream from mature oil field. To reduce discharges to the sea, produced water is often reinjected into the reservoir. Produced water typically contains traces of oil and particles, which may reduce injectivity upon reinjection. A common assumption for understanding particle filtration and deposition in these systems is that the flow remains constant. However, due to operational constraints, maintenance, and unforeseen events, the flow frequently varies. As new power sources, including offshore wind, are utilized to power injection facilities, more dynamic flow conditions are expected. This paper investigates the combined impact of reinjection of produced water and cyclic injection on particle filtration and formation damage. The study utilized dynamic core displacement on Bentheimer sandstone core samples, with brine containing suspended quartz particles used as a simplified analogue for produced water. The core sample was placed in a core holder designed with multiple pressure ports, allowing for the monitoring of differential pressures along the core during injection. Three different injection cycles were applied during the core flooding process and compared to a continuous injection baseline. The results demonstrated that both internal and external filter cakes were formed during core flooding for all injection cycles. The results further indicated that when the sample is subjected to injection cycles of increasing frequency, a more compressed and stable external filter cake is formed. The internal filter cake penetrated the core to an estimated depth of 5 mm to 12 mm, whereas the external filter cakes exhibited permeabilities below 10 mD. Our findings indicated that samples with lower permeability are more susceptible to formation damage caused by invaded particles while seeing shallower invasion zone.

This work presents a governing equation (GE) for two-phase flow in porous media connecting capillary pressure to frictional pressure loss and external chemical potential supplied to a system in either stationary or diffusive equilibrium. It is based on the difference between non-wetting and wetting phase chemical potential (physically, pressure or energy density), which leads to a generalization of the capillary pressure concept. The difference in phase internal chemical potentials is characterized by changes in both interfacial areas and entropy densities due to variation in fluid saturations and is balanced by the system external chemical potential supplied. A definition of the capillary pressure concept is formulated based on the diffusive equilibrium criterion. The GE can explain the origin of the dynamic capillary pressure term, hysteresis, and connect the shift in the capillary pressure curve upon injecting water phases with varying salinities to all the other chemical potentials acting. It can connect, constrain, and potentially quantify all effects which can be formulated in terms of chemical potentials since it is based on a balance equation all two-phase flow systems must obey when either in stationary or diffusive equilibrium.

Abstract Produced water re-injection in hydrocarbon extraction is nowadays common practice mainly enforced by environmental concerns and regulations. Produced water contains, however, significantly higher levels of solids and oil emulsions than regular water used for injection purposes. The negative impact these constituents have on water injectivity has been known for a long time and documented in several publications as far back as the 1980s. Injectivity decline has in general also obtained increased focus over the last decades due to increased interest in CO2 injection for storage purposes as well as increased focus on reduced power consumption and emissions, all through the value chain. The topic is therefore closely linked to important issues related to environment, storage, power consumption, emissions, and economy. Decline in injectivity vs. time is hence expected when produced water re-injection is implemented but improved predictions for how it will evolve is important for reservoir management purposes. Since such predictions conventionally are generated using simulators, robust and reliable experimental input data is crucial. There are many publications focusing on performing controlled laboratory experiments and develop theoretical models to interpret the data. Generating experimental data in cases where the aqueous phase also contains solids and oil emulsions is, however, associated with significantly more challenges and higher uncertainties than conventional tests, which involve homogeneous fluid phases. They originate from the tendency both solids and oil emulsions have to segregate in opposite directions all through the test period. Such additional challenges may in worst case induce uncertainty levels preventing conclusions to be drawn. The aim of the current work focuses on the improvement and standardization of laboratory procedures and practices for mitigating the two major additional challenges related to segregation. They were established by performing numerous tests in several different laboratories using synthetic produced water with well-defined particles and oil emulsions added together with outcrop sandstone cores. The consequence could be increased reproducibility and reduced uncertainty in the final experimental results reported from such tests. The result is therefore a recommended practice list where each bullet point is described in detail sequentially as the test evolves. Additionally, some recommendations related to microbial issues are included if tests planned also include such challenges. Some experimental results are also included obtained using the practices and standards described. It is believed that the recommended practices contributed to increased reproducibility and reduced the uncertainty in all results including these. Increased emphasize on practices and standards for reducing the negative impact from lack thereof could perhaps also facilitate more inter-laboratory result comparisons in the future as they likely impact the results reported significantly.

Uptake of oxygen by hemoglobin (Hb), described by the oxygen-Hb dissociation curve, is obviously important for the existence of all vertebrates. Its sigmoidal curve shape indicates that oxygen binds more tightly if sites already are occupied, commonly referred to as the cooperative effect. The effect has been challenging to understand and quantify ever since its experimental demonstration in 1904. Here, we derive an ab initio analytical expression for the dissociation curve based on the fundamental principle of uniform oxygen chemical potential and absolute activity throughout the system at equilibrium using the grand partition function. The resulting analytical dissociation expression therefore only has four molecular oxygen-Hb binding energies as free variables, which are determined by fitting the analytical expression to measured data. The corresponding resulting negative reaction enthalpies identified in increasing magnitude are, ΔH1=−41.6, ΔH2=−48.8, ΔH3=−51.2, andΔH4=−51.8kJ/mol, in the range observed experimentally. The difference between ΔH1 and ΔH4 is ∼10kJ/mol, smaller than the maximum enthalpy difference measured experimentally, ∼16.7kJ/mol. Hence, the cooperative effect can therefore be explained, from an energy point of view, as caused by the reaction enthalpy difference between ΔH1 and the three subsequent enthalpy values. No impact of Hb’s spatial and structural properties is assumed. The finding highlights the importance of identifying the ligand-receptor molecular binding energies, and thereby the reaction enthalpies, under different conditions as a way for calculating not only the oxygen-Hb but ligand-receptor dissociation curves in general under various conditions, a priori, since the procedure for determining these curves ab initio has been established.

This work addresses the explanation for the threshold pressure sometimes observed in porous media flow leading to so-called non-Darcian flow where a certain external pressure is required to initiate macroscopic flow. Using the Langevin equation, it is first shown that the rapid fluctuating random pressure spikes on each side of the grains in porous media caused by molecular collisions between fluid molecules and the mineral surfaces generate a root mean square (RMS) pressure value even in the absence of macroscopic flow. The RMS pressure can be interpreted as a zero-rate resistance pressure term, which will act as a threshold resistance pressure over which the externally supplied pressure must exceed to initiate macroscopic flow. Using the fluctuation dissipation theorem, the RMS threshold pressure value is quantified, and its magnitude is mainly governed by the specific surface area of the medium and the thermal kinetic energy of the fluid molecules, i.e., the system temperature. Measurable threshold pressures are therefore expected to occur in media with high specific surface areas which is in line with empirical observations. Plotting Darcy velocity vs. pressure gradient for different threshold gradients shows good agreement with measured data. The presence of the zero-rate pressure resistance term further supports the existence of a thermal resistance term previously introduced under dynamic flow conditions. In the latter case, the re-active viscous resistance term must also be included to obtain the total flow resistance. A generalized form of Darcy’s law is therefore proposed, which accounts for the small pressure fluctuations observed in the absence of macroscopic flow, and therefore contains two resistance terms under dynamic conditions, i.e., viscous- and thermal resistances. Another implication of the nature and presence two resistance terms is also discussed with respect to flow resistance during aqueous vs. gaseous flow, essentially the Klinkenberg effect. This work connects porous media flow description, more specifically the mechanism underlying the thermal resistance mode, to well-known phenomena which require considerations about the molecular nature of fluids, e.g., the fluid-particle phenomenon called Brownian motion. It is shown that the fluid-porous medium system in the static case essentially is Brownian motion in the limit of large size porous media. The effect of the molecular collisions is, however, averaged out in this case and therefore insufficient to generate “Brownian motion” but can instead be measured and used to characterize the system. Such an interpretation is supported experimentally by the ever-present small pressure fluctuations of thermal origin, which show red or Brownian noise behavior.

This work presents a method for calculating the difference in temperature jumps observed at the solid–water and water–solid interfaces when heat is flowing under steady state conditions from a hot to a cold solid separated by an intermediate water phase. The method is based on a hypothesis stating that the entropy flux is maximized where the heat flux is constrained, i.e., at the solid–water interfaces. By focusing on the entropy rather than the heat flux and by maximizing its value vs the magnitude of the temperature jump over the interfaces where the latter is constrained, simple analytical expressions for the jump differences independent of the actual heat flux are established only depending on the absolute temperature of the hot and cold solid. The results show that the temperature jump at the hotter interface, therefore, must be higher than the jump at the colder because of the differences in absolute temperature between the two interfaces, supported by many observations. The results, furthermore, show that the temperature jump asymmetry between the two interfaces should increase with decreasing absolute temperature of the system. The work, therefore, finally indicates that there are two quantities contributing to the magnitude of any temperature jump, the heat and entropy flux. More investigations about their relationship under different conditions are encouraged since the topic is not systematically acknowledged and, therefore, investigated in the literature.

Abstract Laboratory core flooding experiments performed are used to predict fluid flow behaviour in the reservoir. Most reservoirs are in a reduced state. However, iron minerals in the extracted cores may become oxidized going from the reservoir to the laboratory. Oxidation of the core can affect wettability and thereby relative permeability curves used for reservoir simulation studies. The aim of this work was to study the potential effect of core oxidation on relative permeability. Steady state relative permeability experiments with in-situ saturation measurements and use of live oil have been performed on one composite core at 10 different saturations. The core was tested under three different conditions: oxidized (Exp. 1), reduced (Exp. 2) and re-oxidised (Exp. 3). The core was cleaned, fluid saturated and aged before each test. In Exp. 2, the core plug was chemically reduced. All fluids used in this experiment were oxygen free. In Exp. 3 the core plug was treated using fluids to oxidise the core. Fluids injected and extracted, and core samples were analysed using a range of methods. Results showed that the measured relative permeability curves from Exp. 2 and 3 were similar and significantly different from the results from Exp. 1. The attempt to restore the core material to the initial state (oxidized) after Exp. 2 by exposing the core material to fluids with oxygen was seemingly not successful. The observation was also supported by unsteady state flooding measurements which indicated that as long as the core remained fluid saturated, it behaved as a reduced core, even after extensive exposure to oxygen containing liquids. The crossing point of the oil and water relative permeabilities indicate that the core was more water wet in reduced state compared to before the oxidized state. Differences in chemical composition were also detected between extracts from Exp. 1 - 3. The conclusion is that significant differences in steady state relative permeabilities of oil and water in oxidized and reduced states were observed for the iron containing core investigated. The results also indicate that if the core is kept fluid saturated, effects of oxidation may be significantly delayed.

The analogy between electrical and fluid flow resistance has been applied to describe the fluctuating pressure of thermal origin observed over single-phase fluid-saturated porous rock samples in equilibrium, i.e., in the absence of any macroscopic fluid flow. It is shown that fluctuating voltages in the electrical case, known as Johnson–Nyquist noise, and fluctuating pressures in the fluid case can be described by the fluctuation dissipation theorem (FDT). The average square of the fluctuating fluid pressure is therefore proportional to the generalized driving force, i.e., the thermal kinetic energy of the molecules (kBT), the remaining resistance in the absence of any macroscopic flow called the thermal resistance RT, and the bandwidth of the pressure fluctuations, Δf. The theoretical power spectral density (PSD) curves differ for the electrical and fluid cases being white in the former and red in the latter, i.e., it falls off as Constant/f2 for increasing frequency. The calculated theoretical predictions are in general in good agreement with observed root mean square pressure values measured over a brine-saturated sandstone rock sample at 50 and 80 ℃ and air-saturated chalk and sandstone samples at 80 ℃, using reasonable values for the specific surface areas and bandwidths, although experiments indicate a stronger temperature dependency than predicted. Furthermore, the observed PSD curves are falling off as Constant/f2 for frequencies above unity according to the prediction, i.e., they represent red noise. Pressure fluctuations or noise over single-phase fluid-saturated rock samples can be adequately characterized by the “fluid version” of the Johnson–Nyquist noise expression. The demonstration of red noise adds additional support to the existence and influence of the thermal resistance term previously introduced under dynamic conditions, because fluid behavior descriptions should be consistent at all levels. The results connect observations of macroscopic fluid behavior in porous media to actions occurring on the molecular level in a consistent way relying on the well-established explanation for Brownian motion and diffusion phenomena introduced by Einstein in 1905 and subsequent generalizations, i.e., the FDT. They may also serve as the basis for explaining non-thermal fluctuating pressures observed in two-phase flow through porous media under dynamic conditions. It is hypothesized that such fluctuations fundamentally are caused by interfacial capillary waves, which again are macroscopic manifestation of molecular thermal fluctuations, always present at the interface between two immiscible fluids. The consequences of the work in a more general sense are that new ways and methods for describing porous media flow in both single- and multiphase flow modes must be further developed since two rather than one resisting force mode is acting on every fluid phase present.

A new approach to determine the transient period towards steady state pore flow velocity for fluids propagating through porous media under constant pressure condition is presented. The transient expression relates to the mean pore velocity rather than the fluid pressure conventional considered when characterizing transient behavior in porous media. It is based on the general, resistance force-velocity relationship, and is therefore analogous to the approach used when calculating transient periods for objects falling through resisting liquid fluids and for the increase in electric currents towards respective steady state values. The transient is caused by inertia forces and characterized by a relaxation time comprising fluid density and viscosity together with porous medium properties as porosity and absolute permeability. Results show that the transient period increases with decreasing medium porosity and fluid viscosity and with increasing fluid density and absolute permeability of the medium. The transient period is negligibly small for typical fluid/medium property values characterizing typical subterrain sandstone reservoirs. Significant transient periods, occasionally observed during laboratory fluid injection tests, are therefore caused by other time-dependent processes not captured by the transient expression presented herein, e.g., fines migration or electrokinetic phenomena.

A thermodynamic analysis of the cause for contact angle (CA) hysteresis has been performed for a system comprising water, air, and a solid matrix. The results show that the entropic contribution to the hysteresis work expression always induces a positive difference between cosine to the receding and advancing CAs, respectively, so the inequality, (cos theta R -cos theta A) > 0, always holds. The inequality therefore shows that the receding CA necessarily always must be lower than the advancing CA, theta R < theta A. The fundamental cause for CA hysteresis, according to the present analysis, is therefore the entropic contribution to the Helmholtz free energy and thereby given by the system temperature, the absolute value for the reaction energy of formation and the entropies of the fluids involved, respectively.

This work demonstrates that an additional resistance term should be included in the Navier-Stokes equation when fluids and objects are in relative motion. This is based on an observation that the effect of the microscopic molecular random velocity component parallel to the macroscopic flow direction is neglected. The two components of the random velocity perpendicular to the local mean flow direction are accounted for by the viscous resistance, e.g., by Stokes' law for spherical objects. The relationship between the mean- and the random velocity in the longitudinal direction induces differences in molecular collision velocities and collision frequency rates on the up- and downstream surface areas of the object. This asymmetry therefore induces flow resistance and energy dissipation. The flow resistance resulting from the longitudinal momentum transfer mode is referred to as thermal resistance and is quantified by calculating the net difference in pressure up- and downstream the surface areas of a sphere using a particle velocity distribution that obeys Boltzmann's transport equation. It depends on the relative velocity between the fluid and the object, the number density and the molecular fluctuation statistics of the fluid, and the area of the object and the square root of the absolute temperature. Results show that thermal resistance is dominant compared to viscous resistance considering water and air in slow relative motion to spherical objects larger than nanometer-size at ambient temperature and pressure conditions. Including the thermal resistance term in the conventional expression for the terminal velocity of spherical objects falling through liquids, the Stokes-Einstein relationship and Darcy's law, corroborates its presence, as modified versions of these equations fit observed data much more closely than the conventional expressions. The thermal resistance term can alternatively resolve d'Alembert's paradox as a finite flow resistance is predicted at both low and high relative fluid-object velocities in the limit of vanishing fluid viscosity.

The conventional and a generalized form of Darcy’s law for the absolute permeability of porous media at low flow rates has been derived using the Langevin equation. It can account for the conventional viscous resistance and additionally a term referred to as thermal resistance. The latter is hypothesized to stem from the relationship between the mean and molecular random velocity along the mean flow direction. The key to note when using the Langevin equation as the starting point for the derivation is that the frictional resistance occurring are the same whether the fluid or the object is moving relative to each other. Hence, a spherical particle moving through a stagnant fluid will experience the same friction as a fluid flowing past a stagnant spherical particle. Experimental data for absolute permeability at different temperatures indicate a temperature dependency, which can be accounted for by the thermal resistance term, which so far has been neglected. A procedure for matching and predicting pressure variation and absolute permeability vs. temperature is described. The generalized form of Darcy’s law presented should be of interest in all sciences where fluid transport in porous media occurs at varying temperatures, e.g., in soil science, hydrology, geothermal, chemical and petroleum engineering. It has also been discussed in relation to the Kozeny–Carman equation.

Microbial generation of H2S in-situ due to injection of seawater, so-called reservoir souring, can induce severe negative impact in terms of health, safety, maintenance and economics in mature oil fields. Historical H2S data have been collected for decades at the Norwegian Continental Shelf fields. The Gullfaks field in the North Sea has been negatively impacted by severe H2S production, whereas another comparable North Sea field, called Field B, only has minor H2S challenges. The difference in cumulative H2S development in Gullfaks wells are described as Type I, II and III by an empirical classification. Type I wells show rapid increase in the H2S concentration produced (<1 year or <1 mill. Sm3produced water), whereas Type III show a significant delay and slowly rising H2S production (>10years or >10 mill. Sm3produced water). Type II can be considered as a mixture of Type I and III. To understand the difference between these types, a microbial reservoir souring model called the Extended Growth Zone (EGZ) model has previously been introduced. Differences in H2S development described by the three different well types are supported by the consequences of the EGZ model for both fields on field, platform/installation and well level. The results support that the overall drainage water injection strategy implemented is the single-most important cause governing the development of produced H2S over the lifespan of a field: •High H2S production is expected when cold seawater is injected directly into oil bearing zones (Type I behavior). •Relatively lower H2S production is expected when cold seawater is injected in the water zone below oil bearing zones (Type III behavior). The EGZ model describes the development of reservoir souring based on two main hypotheses (1) Degradation of crude oil as carbon source (2) The importance of temperature on the rate of H2S generation as summarized: •The expected H2S development according to the EGZ model agrees with observed data from Gullfaks both on platform level (GFA>GFC>GFB) and for individual wells. •The observed slow development of H2S for Field B observed for platform and well bore level is explained by the presence of Type III wells only. •Type III wells in Gullfaks and Field B behave similarly within the uncertainty range for H2S production.

A thermodynamic analysis of the capillary drainage, imbibition and hysteresis pressure in two‐phase porous media systems at different temperatures has been performed. Expressions for the work required or gained when the fluid saturations change, proportional to the capillary pressures, are presented. The expressions are determined using the criterion that changes in Helmholtz free energy equals zero at equilibrium. From these expressions, the variation of capillary drainage, imbibition and hysteresis pressures for increasing temperature are determined without calculating the actual capillary pressure values. The temperature dependency results show that the capillary drainage pressure declines with a fractional rate for increasing temperature with contribution from two terms, the air‐water interfacial tension and the entropy term, which give a total fractional reduction of −0.0082 K−1, close to the value −0.0084 K−1 found experimentally. The capillary imbibition pressure increases with a total fractional rate versus temperature equal +0.004 K−1 in line with the observation that water tables in soils are lifted upon day‐time heating. The total fractional temperature variation of the hysteresis capillary pressure term declines with a rate of −0.0143 K−1 for increasing temperature in qualitative agreement with experimental data. The results support the hypothesis that the thermodynamic approach applied under given assumptions can account for the major effects determining the temperature dependency of two‐phase capillary pressure in porous media. Furthermore, the entropy term adds an additional hysteresis term to the conventional term, even under isothermal conditions. The capillary pressure hysteresis phenomenon is therefore caused by two effects: Differences in interfacial areas including contact angle hysteresis plus the entropic contribution.

Fluid-fluid momentum transfer can cause higher flow resistance when fluids flow in opposite directions as compared to the same direction. Conventional modelling of flow in porous media using simple, saturation dependent relative permeabilities does not account for such variations. We consider a generalized theory for multiphase flow in porous media based on mixture theory, where fluid mobilities follow from water-rock, oil-rock and water-oil interaction terms defined in momentum equations. Under strictly co- or counter-current flow modes, the generalized model produces explicit relative permeability expressions dependent on the flow mode, saturations, viscosities and interaction parameters. New expressions for counter-current relative permeabilities are derived assuming zero net flux, representative of counter-current spontaneous imbibition. These functions are compared to previously derived co-current relative permeabilities (assuming equal phase pressure gradients). The functions are incorporated into analytical solutions for forced and spontaneous imbibition (FI and SI) using the theory by Buckley and Leverett (1942) and McWhorter and Sunada (1990), respectively. Our results show that when accounting for viscous coupling; Counter-current relative permeabilities are always lower than co-current ones, including the end points. Both phase curves are reduced by the same saturation dependent coefficient. Increased viscous coupling in the FI case led to a more effective displacement, seen as an increased front saturation and average water saturation behind the front. For counter-current SI, increased viscous coupling resulted in lower imbibition rate. Increased viscosities reduces both oil and water counter-current relative permeabilities, and predict greater reduction in imbibition rate than only modifying the viscosities. The analytical solutions for SI were in agreement with numerical solutions of both a conventional and generalized model. The solutions for SI could be scaled exactly to a square root of time curve for arbitrary input parameters in the generalized model, especially including the strength of viscous coupling.

This paper presents a new convenient and easy-to-use method to analyze and calculate measurable quantities in 1-D counter-current (COUC) spontaneous imbibition (SI) processes. Cumulative water imbibed vs time can be calculated both up to and after the water has contacted the no-flow boundary as well as the time required to contact the no-flow boundary. The model's applicability for the whole process is a big advantage compared to other models which only are valid before this event. The method is developed based on a hypothesis that a frontal advance equation (FAE) also can be established for 1-D COUC SI processes in line with the Buckley - Leverett method for forced water imbibition. A relationship between distance traveled, x, by a diffusing water saturation, S-w and time of the form, x similar to root D(S-w)t, is identified from the microscopic theory for diffusion. The proportionality factor between distance and time is the square root of the capillary diffusion coefficient, assumed to be constant for a given water saturation, S-w. The corresponding cumulative water volume diffusing into the medium for the same water saturation, S-w is calculated from the diffusion equation using the same constant capillary diffusion coefficient D(S-w). The final step to establish the FAE is to recalculate the cumulative volume of water imbibed to an equivalent distance (ED) waterfront, i.e., the distance traveled by the imbibing water phase having the same cumulative volume water imbibed as calculated from the diffusion equation. The FAE is similar to the water saturation characteristics in the BL method. The results for cumulative water imbibed vs time is giving a highly accurate approximation when compared to analytical solutions (i.e., the McWhorter and Sunada solution), deviating slightly from the correct value due to the use of constant diffusion coefficients. The difference can be corrected by a factor in the range 1.0-1.24 for the four data sets considered. The conclusion obtained by comparing the FAE method result to the only experiment data set in the literature where all input data are available, GVB-3 (Bourbiaux and Kalaydjian, 1990), is that the new FAE cannot be falsified within the input parameter uncertainty range. The FAE method is qualitatively corroborated by the results from four 1-D COUC SI cases using synthetic input data as well comparison with many other test results reported in the literature.

Summary This paper presents a numerical study of water displacing oil using combined cocurrent/countercurrent spontaneous imbibition (SI) of water displacing oil from a water-wet matrix block exposed to water on one side and oil on the other. Countercurrent flows can induce a stronger viscous coupling than during cocurrent flows, leading to deceleration of the phases. Even as water displaces oil cocurrently, the saturation gradient in the block induces countercurrent capillary diffusion. The extent of countercurrent flow may dominate the domain of the matrix block near the water-exposed surfaces while cocurrent imbibition may dominate the domain near the oil-exposed surfaces, implying that one unique effective relative permeability curve for each phase does not adequately represent the system. Because relative permeabilities are routinely measured cocurrently, it is an open question whether the imbibition rates in the reservoir (depending on a variety of flow regimes and parameters) will in fact be correctly predicted. We present a generalized model of two-phase flow dependent on momentum equations from mixture theory that can account dynamically for viscous coupling between the phases and the porous media because of fluid/rock interaction (friction) and fluid/fluid interaction (drag). These momentum equations effectively replace and generalize Darcy's law. The model is parameterized using experimental data from the literature. We consider a water-wet matrix block in one dimension that is exposed to oil on one side and water on the other side. This setup favors cocurrent SI. We also account for the fact that oil produced countercurrently into water must overcome the so-called capillary backpressure, which represents a resistance for oil to be produced as droplets. This parameter can thus influence the extent of countercurrent production and hence viscous coupling. This complex mixture of flow regimes implies that it is not straightforward to model the system by a single set of relative permeabilities, but rather relies on a generalized momentum-equation model that couples the two phases. In particular, directly applying cocurrently measured relative permeability curves gives significantly different predictions than the generalized model. It is seen that at high water/oil-mobility ratios, viscous coupling can lower the imbibition rate and shift the production from less countercurrent to more cocurrent compared with conventional modeling. Although the viscous-coupling effects are triggered by countercurrent flow, reducing or eliminating countercurrent production by means of the capillary backpressure does not eliminate the effects of viscous coupling that take place inside the core, which effectively lower the mobility of the system. It was further seen that viscous coupling can increase the remaining oil saturation in standard cocurrent-imbibition setups.